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Investigation of plasma parameters characteristic of radio frequency and magnetized plasmas using Langmuir probes and microwave interferometry

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INVESTIGATION OF PLASMA PARAMETERS CHARACTERISTIC OF
RADIO FREQUENCY AND MAGNETIZED PLASMAS USING LANGMUIR
PROBES AND MICROWAVE INTERFEROMETRY
by
Pamela Semrad-Doolittle
A dissertation submitted in partial fulfillment of the
requirements for the degree of
Doctor o f Philosophy
(Analytical Chemistry)
at the
University of Wisconsin-Madison
2000
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R eaders' Page, This page is not to he hand-written except lor
A dissertation entitled
Investigation of Plasma Parameters Characteristic
of Radio Frequency and Magnetized Plasmas Using Langmuir
Probes and Microwave Interferometry
submitted to the Graduate School of the
University of Wisconsin-Madison
in partial fulfillment of the requirements for the
degree of Doctor of Philosophy
by
Date of Final Ora! Examination: O ctober 30, 2000
Month & Year Degree to be awarded:
December 2000
August
May
Signature, Dean of Graduate School
1?. [)nw?Lu<f'/&fl
Q
'V'QjyJ- O
. Z T y y rftP s\/)
I
R e a d e rs’ Page. This page is not to be hand-written except for the signatures
Pamela S. Semrad-Doolittle
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INVESTIGATION OF PLASMA PARAMETERS CHARACTERISTIC OF RADIO
FREQUENCY AND MAGNETIZED PLASMAS USING LANGMUIR PROBES AND
MICROWAVE INTERFEROMETRY
Pamela Semrad-Doolittle
Under the supervision o f Professor R. Claude Woods
at the University o f Wisconsin —Madison
Plasma parameters o f a 27.12 MHz full wave helical resonator discharge source
and a magnetically enhanced DC hollow cathode discharge source were obtained using
Langmuir probes and microwave interferometry. The RF discharge tube diameter was
10 cm and the approximate length of the antenna structure was 1 m. A mathematical
model was employed to predict the antenna length which would produce the desired fullwave resonance on the antenna at a given plasma density, and this coil length varied
varied from 8 - 10 m for densities from 109 - 10n cm '3. Column densities for nitrogen
and argon plasmas in the 10 —100 mTorr pressure range were measured using microwave
interferometry, yielding 1 x 1013 cm'2 for a 70 mTorr, 500 W argon plasma, and
2.5 x 1012 cm'2 for a 70 mTorr, 1000 W nitrogen plasma. Special Langmuir probes with
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filtering networks were constructed and used to obtain partial axial spatial distributions o f
plasma parameters for this source. For a 100 W, 13 mTorr nitrogen plasma there is a
0
moderate density region, with volume densities in the low 10
approximately the geometric center o f the antenna structure.
cm'
l
range, in
The volume density
decreases by a factor o f 10 or more near the ends and outside o f the antenna.
The axial and radial spatial distribution o f ion densities and electron temperatures
were also measured in a 3 m long by 15 cm diameter magnetically enhanced, electron
beam DC glow plasma.
Measurements taken by Langmuir probes and microwave
interferometry confirmed that the ion column density is increased by approximately a
factor o f 2 in the presence of a 264 G magnetic field. The column density o f a 13 mTorr
nitrogen plasma with an applied axial magnetic field o f 264 G was measured to be
2.3 x 1012 cm'2 using microwave interferometry. For the same experimental conditions
Langmuir probe experiments reveal that the magnetic field extends the high density
region, which starts at the cathode, down the length of the tube to approximately 150 cm
from the front o f the hollow cathode. In this extended region the volume densities are
approximately 3 x 1010 cm'3, but they decrease substantially between 100 and 150 cm
before reaching a fairly steady value o f 9 x 109 cm'3. The spatial distribution o f bulk
electron temperatures in the electron beam plasma is fairly constant, with measured
values o f
0.9 eV. The plasma potential is also fairly constant under these experimental
conditions with values near -25 V along almost the entire length of the 3 m long tube
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(with the anode grounded and the cathode at 1050 V) although a gradual, measurable rise
o f about 4 V was recorded between 50 cm and 275 cm (near the anode).
Approved:
Date
R. Claude Woods
Professor o f Chemistry
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Acknowledgements
I would like to express my gratitude to my advisor, Professor R. Claude Woods,
for his guidance, friendship, and encouragement in learning about a detailed and difficult
research topic. He is the smartest, kindest, and most patient and honest person I have
ever met. If a scientist can be thought o f as an artist, then Claude would be one in the
same. I can only hope to have half of the physical intuition he possesses about this very
complicated science. I will always strive to give explanations as clearly and concisely
(and without notes) as he always does.
I have made many friends here during the course o f my graduate studies. Dave
Olszewski and Charley Langley, my fellow group members, have been a particular
source o f strength and laughter. I will carry with me fond memories o f time spent and
conversations with Yansheng Men, Mary Jezl, Ion Abraham, Yansheng Pak, and Ken
Lai. I am also deeply appreciative to Dan Sykes, who helped cover things related to my
job as I worked to finish the research and writing necessary to graduate.
I would like to say a special thank you to the faculty o f the analytical sciences
division, who provided me with the opportunity to work as their lab director in
conjunction with finishing up my research. The completion o f this work took longer than
anyone expected. Despite this fact, they continued to show good faith in my promise to
finish. I would especially like to thank Professor Bob Hamers for providing me with the
initial opportunity.
I am also deeply grateful to Professor Lloyd Smith, my current
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V
supervisor, for helping me to prioritize the many commitments I have both personally and
professionally, in order to be the most effective in fulfilling all o f them.
The chemistry department at the UW has an am azing network o f professional
support staff, and their efforts can be found throughout the lab I did my research in.
Vince Fitzgerald was my best electronics teacher and family councilor. Mike Green was
always willing to stop everything he was doing to help troubleshoot a signal problem. I
would also like to thank Jerry Lancaster for putting his hands into HV power supplies,
when I was too chicken to do it. The experts from the machine shop, in particular Ed
Vasiukevicius, Kendall Schneider, Rick Pfeifer, and Jerry Stamn, are the primary artists
o f the apparatus in my lab. For the most part, I only drew the pictures, and they weren't
as polished as the results would suggest.
I would like to thank my parents for everything they've done to help me realize
my dreams. My parents raised me to think for myself, and taught me I can do anything
with hard work and planning. When I was younger, my independence was a source of
stress for them. But I think they realize now that a strong, independent character has
some advantages. My Dad called me this morning to say that he is thankful to have me
as his daughter. Mom and Dad, I thank God every day for your presence in my life.
Finally, completing this work has been the hardest on my own family, and I am
the most grateful to my husband and children for doing they best they could to help me
finish. I know I have not been the easiest person to live with over the past few months.
My husband, Dan, drew many o f the graphics in this thesis during early morning hours,
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while everyone else was sleeping. My girls, Cassy and Nichole, have patiently waited
weeks for a trip to the zoo and a game o f goldfish. My son, Kevin, has done more baby­
sitting o f little girls than any 12 year-old boy should ever be expected to do. I love you
all very much, and please know that you all are the most important people o f my life.
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Table o f Contents
Abstract.................................................................................................................... i
Acknowledgements................................................................................................v
Chapter 1
Overview..................................................................................................................1
1.1 In t r o d u c t i o n .................................................................................................................................................. 1
1 .2 M o t i v a t i o n ......................................................................................................................................................3
1.3 A
SUMMARY OF THE WORK PRESENTED IN THIS THESIS............................................................... 7
R e f e r e n c e s ...............................................................................................................................................................11
Chapter 2
Theoretical and Empirical Description o f the Full Wave Helical Resonator
Discharge................................................................................................................12
2 .1
In t r o d u c t i o n ................................................................................................................................................12
2 .2 T h e o r y
o f o p e r a t i o n ............................................................................................................................... 14
2 .3 M a t h e m a t i c a l
2 .4 R e su l t s
m o d e l o f t h e s o u r c e ..........................................................................................17
o f t h e m o d e l a n a l y s i s ....................................................................................................... 3 0
2 .5 C o n s t r u c t i o n
d e t a il s o f t h e f u l l - w a v e h e l ic a l r e s o n a t o r s o u r c e ...............33
R e f e r e n c e s .............................................................................................................................................................. 4 3
Chapter 3
Design, Construction, and Practical Application o f Langmuir Probe and
Microwave Interferometry Diagnostic T o o ls...................................................44
3 .1
In t r o d u c t i o n .............................................................................................................................................. 4 4
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3 .2 C o n v e n t io n a l
3.3
L a n g m u ir
a n a l y s i s o f t h e p r o b e t r a c e .......................................................................4 6
p r o b e m e a s u r e m e n t s in a n
RF
p l a s m a ............................................................. 61
3.3.1 R F probe design and construction.........................................................................64
3.3.2 RF passive filtering options...................................................................................75
3 .4 T h e
m ic r o w a v e in t e r f e r o m e t e r ................................................................................................... 8 7
R e f e r e n c e s ..............................................................................................................................................................9 7
Chapter 4
Investigation o f Plasma Parameters in the Helical Resonator Discharge
99
4 .1 I n t r o d u c t i o n .............................................................................................................................................. 9 9
4 .2 E x p e r im e n t a l
d e t a i l s o f t h e a r g o n p l a s m a .....................................................................1 0 2
4 .3 E x p e r im e n t a l
d e t a i l s o f t h e n i t r o g e n p l a s m a .............................................................. 1 07
R e f e r e n c e s ........................................................................................................................................................... 1 26
Chapter 5
' Characterization o f a Magnetically Enhanced Electron Beam Discharge
Source.................................................................................................................. 127
5 .1 In t r o d u c t i o n ..............................................................................................................................................1 2 7
5 .2 T h e
h o l l o w c a t h o d e g l o w d i s c h a r g e a p p a r a t u s .......................................................... 129
5 .3 M o d e s
5 .4 U
s in g
o b s e r v e d in t h e h o l l o w c a t h o d e s o u r c e .......................................................... 133
L a n g m u ir
p r o b e s f o r t h e a n is o t r o p ic c a s e .........................................................13 7
5 .5 E x p e r im e n t a l R e s u l t s ....................................................................................................................... 14 6
R e f e r e n c e s ........................................................................................................................................................... 160
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ix
Epilogue.............................................................................................................161
Appendix A Detailed Description o f the Pressure Control System........... 166
Appendix B Mathcal Worksheet Helical....................................................... 176
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1
Chapter 1
Overview
1.1 Introduction
The purpose o f experiments conducted in our laboratories is to enhance the
understanding o f the chemistry and dynamics exhibited by electrical discharges.
Of
particular interest are low pressure, high density plasmas, where the electron densities are
in excess of 1010 cm'3, and electron temperatures range from 2 —6 eV. The widespread
applications of these discharges span such technical fields as thin film deposition,
semiconductor processing, and production o f transient species for fundamental
spectroscopic studies. Microwave spectroscopy is the primary experimental technique
utilized in our laboratories for the study of molecular ions. Producing sufficient densities
o f ions o f interest is o f such critical importance that it has become a study all its own.
Since the production o f ions depends on the chemistry o f the discharge, how this
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2
chemistry is affected by such parameters as pressure, supply current and voltage,
magnetic fields, and how the plasma is actually coupled to the exciting source are
crucially important. Techniques that can reliably measure plasma parameters, however,
are problematical in a variety of ways.
Currently our labs have the capability o f generating plasmas using a DC hollow
cathode discharge source, with the option o f magnetically enhancing plasma in the
discharge tube using a solenoid magnet, as well as a radio frequency source, which was
modified for part o f the work described in this thesis. The goal o f the work presented in
this thesis was to develop experimental strategies that would lead to better understanding
the chemistry and physics o f the discharges produced in these sources. The diagnostic
tools used in these investigations were primarily Langmuir probes o f planar geometries
and microwave interferometry. We used the probes to measure plasma parameters as
they evolved radially and axially in the chamber. As we describe further in the next
section, Langmuir probes provide a powerful and flexible method that can measure local
plasma parameters. While the experimental apparatus is straightforward and economical
to construct, analysis of the probe curve is a non-trivial problem, and is not fully
understood for cases where the plasma potential greatly fluctuates, or in situations where
the plasma is anisotropic because of applied magnetic fieids. Microwave interferometry
is essentially a refractive index measuring experiment, which measures the density o f
electrons in a plasma by integrating a phase shift along the line-of-sight o f a
monochromatic beam o f microwave radiation. While this experiment fails in providing
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3
information about local structure inside our sources, the line integrated electron densities
are very accurate. In order to determine how well a probe was working, given the variety
o f conditions we imposed on it, we integrated over the axial profile o f densities as
determined by the probe and compared the result with data collected using microwave
interferometry.
1.2 M otivation
Many areas of technology have benefited from advances in plasma processing.
Continued progress in such technical fields, e. g., in the semiconductor industry, depends
on furthering the fundamental understanding o f the physical and chemical mechanisms at
work in order to optimize plasma processes.
Electrons bear the responsibility o f
sustaining the discharge, which occurs mainly by electron impact ionization.
The
available energy that electrons have that they can provide to the plasma through collision
processes is usually described in terms o f the electron temperature, which is only strictly
meaningful if the local electron velocities or kinetic energies are described by a MaxwellBoltzmann distribution. In the more general case, the distribution o f energies present in a
particular environment must be described by the complete electron energy distribution
function (EEDF).
Accurate determination o f these quantities is necessary in the
characterization and optimization o f plasma processes.
The Langmuir probe is used
specifically to collect the data from which these quantities are determined.
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4
A s an electron travels through a gas, it can drive several other processes besides
ionization. Other such kinetic processes include electron impact excitation, electron-ion
recombination, electron impact dissociation or rearrangement, attachment, detachment,
and elastic scattering. For a particular collision pair, e. g., an electron colliding with a
nitrogen molecule, there exists a certain probability for each of these events to occur.
The probability that a particular event will occur is given by the cross-section for the
defined process, which is a function o f the collision energy, primarily the electron energy.
The sum o f the individual cross-sections for that collision pair is called the total collision
cross-section.
Electron impact ionization ranks as by far the most important kind o f collision
sustaining a discharge. The likelihood o f ionization occurring by this event is represented
by the ionization cross-section. If the energy provided by an electron colliding with an
atom is below a minimum energy required to facilitate ionization, z. e., the ionization
threshold, the reaction occurs with zero probability and has zero cross-section. Thus the
other factor important to this process is the electron energy. In order for ionization to
occur, the electron must possess energy in excess o f the ionization potential o f the atom
or molecule.
Since the process o f ionization can be treated as a chemical reaction, one can
consider a reaction rate for the process.
If z is the rate of ionization per colliding
electron/sec, then
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5
zne = —^ = C nena ={qive)nena
at
(1)
where C is a rate constant and equals the average o f the product o f the ionization crosssection qt and the electon velocity ve. In Eq. (1), the number o f electrons is ne and the
concentration o f neutrals is n0.
The average (q y e} is found by integrating over all velocities o f the known energy
distribution function f(e) de. In a Maxwellian, isotropic plasma, electrons will follow a
Maxwell-Boltzmann distribution with the average energy, <s> = lkTe/2, where k is the
Boltzmann constant and Te the electron temperature. Let's assume the Maxwellian case
for the moment. Then the final expression for the ionization rate constant looks like
(2)
where the Maxwell-Boltzmann distribution o f electron energies is given by
(3)
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6
t
ffe)
ffs)
tail
e
Figure 1.1.1 Illustrating the ionization rate integral
where me is the mass of the electron and e is the energy.
This integral is represented pictorially in Figure 1.1.1. The EEDF is represented
by fE(e). The dash-dotted curve corresponds to the integrand o f Eq. (2) and the shaded
area underneath it is the rate constant for ionization, (q,ve) . The number o f electrons in
the tail o f the distribution that will participate in the ionization process is doubly shaded.
This treatment has been found to hold in all cases where the distribution is known and
multi-stage processes are absent [1]. Other processes such as excitation, recombination
and scattering all have corresponding cross-sections. Determining the rate constants for
these events follows the same procedure that was illustrated above.
In all cases, the
EEDF is a fundamental quantity used in determining the kinetics o f processes in a
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7
discharge. This remains true regardless o f the nature o f the excitation source, the gases
used in generating the plasma, or the size o f the chamber. If the distribution o f electron
kinetic energies is not Maxwellian, which in glow discharge or non-equilibrium plasmas
it frequently is not, then the shape o f the function / e (s ) can change dramatically, but the
formula in Eq. (2) is still valid. By obtaining the EEDF, a fundamental understanding o f
the dynamics of a discharge can be realized.
1.3 A summary of the work presented in this thesis
A radio frequency (RF) discharge source operating at 27.12 MHz, originally
constructed by W. T. Conner [2], was previously used in this laboratory as an RF
discharge source. Use of this source proved problematic, since the plasma routinely
traveled to the pumping port, where it could potentially eat away o-ring and teflon seals,
as well as degrade the pump oil. Glass beads and glass wool were placed in the pump
port to discourage this event, however the addition o f these deterrents also decreased the
conductance o f the gas flow path to the pump, and as a result, decreased the maximum
flew rate at which one could operate a plasma for any desired pressure. This source was
modified as part of the current work to become a full-wave helical resonator plasma
source, according to a paper by Vinogradov and Yoneyama [3].
We hoped this
modification would help improve our measurements in two ways.
First, inductive
sources are reported as confined within the resonator volume at pressures above 10
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8
mTorr [4]. This would minimize the problem o f plasma extending down through the
pumping port, decreasing the opportunity for degradation o f seals and oil, and improving
the purity o f the plasma.
Second, inductive plasmas are reported as producing
significantly higher density, low temperature plasmas [5, 6] than the plasma sources we
currently employ. Increasing the densities of molecular ions present would improve the
signal quality o f the microwave spectra of interest. Some theoretical consideration o f this
source and a description o f its construction are given in Chapter 2.
The diagnostic tools utilized in the characterization o f the full-wave helical
resonator, as well as the magnetically enhanced (ME) DC glow discharge, are described
in Chapter 3.
Langmuir probes provide information regarding ion densities, electron
temperatures, plasma and floating potentials, and EEDFs, while capacitive probes
measure the RF fluctuations o f the plasma potential. A probe is to some extent always an
intrusive diagnostic technique. Probes used for these experiments were constructed with
minimal perturbation to the plasma in mind. Careful analysis o f the I-V curves obtained
by Langmuir probes must be based on valid assumptions for reliable results. Analyses o f
the I-V trace are based on the Druyvesteyn second derivative method for obtaining the
electron energy distribution function (EEDF).
The effects of RF fluctuations that
potentially distort the I-V trace are of particular concern when using Langmuir probes as
a diagnostic technique.
A filter formed by cascading connections o f inverted L-type
networks o f transmission lines [7], as well as several configurations o f passive, lumped
element, filter network, were employed to minimize this distortion [8, 9]. We provide a
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9
general discussion o f the effectiveness, as well as the limitations, o f this kind o f filter
apparatus. Microwave interferometry was the benchmark technique that allowed us to
draw conclusions about how accurately the probes performed in providing us plasma
densities. The theories used to process the characteristic trace fail under the conditions
imposed on the probes in our RF and ME sources, but neither magnetic fields nor RF
oscillations critically affect the interferometry measurement or the mathematical
treatment o f the microwave interferometric data, at least for our experimental conditions.
We discuss this technique in detail at the end o f Chapter 3.
A study o f the full-wave helical resonator is presented in Chapter 4.
The
fluctuations in the plasma potential for the helical resonator source measured hundreds o f
volts. We discuss what we believe were the strengths and flaws of probe constructions
and filtering networks we attempted to use. While we were not able to provide rigorous
quantitative information regarding the plasma parameters in the source, we can show
evidence o f inductive character in certain regions of the plasma. We also present the line
integrated electron densities determined by microwave interferometry for a variety o f
pressure and power conditions.
A DC glow discharge with an axial magnetic field has previously been used to study
molecular ions [2]. In that study, it was shown that the magnetic enhancement o f this
source results in a very different kinetic environment for ions and electrons.
In this
environment, a beam o f very energetic electrons, propagating down the center o f the
chamber, is responsible for sustaining the discharge.
Microwave spectroscopy
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10
experiments confirmed that, in the presence o f an axial magnetic field, the signal strength
for molecular ions was greatly enhanced. While a qualitative model was developed in
previous work to describe the physical properties o f this type o f plasma, this source had
yet to be characterized analytically. A problem charactizing this plasma using Langmuir
probes is predicting the effect o f the magnetic field on the current collected by the probe.
The electrons, in particular, no longer move through the plasma in a random path.
Rather, they gyrate down the length o f the cylinder about in a helical path about their
own guiding center. For this reason, the theories used to extract plasma parameters from
the characteristic trace fail. The ion current, on the other hand, is less affected by the
magnetic field enhancement. It is possible, at least approximately, to analyze the ion
current part o f the trace to extract local density information. In Chapter 5 this qualitative
model o f the magnetically enhanced negative glow, electron beam discharge is discussed,
and the corresponding experimental results are presented.
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II
References
[1] A. Von Engel, Electrical Plasmas: Their Nature & Uses, _Taylor & Francis LTD.,
London 1983, p. 42.
[2] W. T. Conner, Ph. D. Thesis (Physics), University o f Wisconsin-Madison (1989).
[3] G. K. Vinogradov and S. Yoneyama, Jpn. J. Appl. Phys. Vol. 35Pt. 2, No. 9A, (1996).
[4] P. Bletzinger, Rev. Sci. Instrum. 65 (9), (1994).
[5] J. Keller, J. C. Forster, and M. S. Barnes, J. Vac. Sci. Technoi. A 11(5) (1993).
[6] J. Hopwood, Plasma Sorce Sci. Technoi. 1, 109 (1992).
[7] K. Shimizu, A. Hallil, and H. Amemiya, Rev. Sci. Instrum. 68 (4), (1997).
[8] A. P. Paranjpe, J. P. McVitie, and S. A. Self, J. Appl. Phys. 67(11), p. 6718-6727,
(1990.
[9] V. A. Godyak, R. B. Piejak, and B. M. Alexandrovich, Plasma Sources Sci. Technoi.,
1, p. 36-58, (1992).
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12
Chapter 2
Theoretical and Empirical Description of the Full Wave Helical
Resonator Discharge
2.1 Introduction
Microwave spectroscopy is used in our laboratory experiments to obtain
qualitative and quantitative information regarding transient species present in a plasma.
Typically, molecules o f interest are unstable species made up of two to four atoms.
Theoretical work, performed by other group members involved in this research thrust,
model the molecular and electronic structures and the rotational and vibrational modes o f
excitation, which gives our experimental work a good estimate o f the expected
spectroscopic frequencies. The standard source generating molecules for these studies
has been a DC glow discharge operating in a "normal" mode, meaning 50 - 1000 mA
current and up to 1500 V between the hollow cathode electrode and the grounded anode.
Plasmas generated by this source are typical o f laboratory generated plasmas, with
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13
electron temperatures between 1 and 5 eV and ion temperatures o f 100 to 500 K, and
fairly dense, with electron densities ranging from 109 to 1011 cm'J.
Excitation
mechanisms in this type o f plasma are described elsewhere [1].
The transient ions o f interest often exist in very small populations, which makes
our work in detecting their presence difficult, and makes analysis o f these species by
microwave spectroscopy even more challenging.
Thus exploring sources that would
produce larger densities o f these molecules by different mechanisms is o f particular
interest in the experimental thrust o f our research group.
A RF discharge source
operating at 27.12 MHz, originally constructed by W. T. Conner [2], was previously used
in this laboratory to study the reaction mechanisms characteristic of this type o f source.
This type o f plasma was reported to produce a higher density o f radicals, with more
highly excited vibrational states than the DC glow discharge [3]. Many problems were
encountered in trying to implement this source as a useful and practical tool.
problem arose due to the internal construction o f the tube.
One
The cylinder containing the
plasma has no grounded surface inside o f it. It is made o f pyrex, and capped o ff with
teflon or polycarbonate windows.
The intended purpose of this construction was to
m inim ize contamination due to the sputtering o f metal inside the source. An unintended
result, however, was that the plasma current had no path to ground inside o f the source.
The nearest grounded surface was the pumping port.
So upon ignition, the plasma
current sought out the grounded surface o f the pumping port, where it could potentially
eat away o-ring and teflon seals, as well as degrade the pump oil. Glass beads and glass
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14
wool were placed in the pump port to discourage this event, by providing additional
surface area for ions and electrons to recombine.
The addition o f these deterrents,
however, also decreased the conductance o f the gas flow path to the pump. Even after
upgrading the old pumps to a system that utilized a Roots blower and large mechanical
pump, the base pressure o f the system was about 5 mTorr.
The helical resonator plasma source has been reported to efficiently generate
dense, cold plasmas at low pressure [4, 5]. It utilizes simple hardware, and it does not
require a DC magnetic field. It can, in principle, operate without a matching network,
and has been reported to exhibit a high O. For these reasons we have pursued improving
upon the source originally constructed by Conner by modifying it to be a full wave
helical resonator. This chapter describes empirically how the source is supposed to work,
then discusses a model predicting the density and coupling conditions of the plasma with
the antenna, and finally gives details o f the physical construction o f the source.
2.2 Theory o f operation
Helical resonator discharges are inductive sources with a resonance structure.
The antenna length is a multiple of /i/4 , where X is the wavelength of the excitation
frequency. The antenna, cut at the prescribed length, is coiled around the outside o f a
cylindrical chamber. Quarter wave structures are grounded at one end, while half-wave
structures are grounded at both ends o f the antenna. When an integral number o f quarter
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15
waves just fit along the length of the antenna, the structure becomes resonant and
generates more intense electromagnetic fields. The walls o f the tube are a dielectric.
Power from the antenna inductively couples to the plasma much like it does from the
primary to the secondary o f a transformer.
The plasma acts as a single turn, lossy
secondary conductor coupled to the resonating structure o f the antenna [6].
Quarter-wave and half-wave structures are widely employed in industry for
semiconductor processing and thin film deposition. While these sources are labeled as
inductive, inevitably some capacitive coupling also occurs. The RF voltages along the
coil can induce capacitive currents, resulting in large fluctuations o f the plasma potential.
Large plasma potentials cause more plasma to form outside o f the coil, particularity near
grounded surfaces. The source built by Conner was supposed to operate as a quarter
wave helical resonator. In this case, the end o f the antenna closest to the pumping port
was grounded. The other end, which was near the gas entrance, was left open.
A full-wave structure, constructed by Vinogradov and Yoneyama, was reported to
internally compensate capacitive currents, thus minimizing the possibility o f external
capactive currents that generate plasma outside the source [7]. In the cited paper, a
qualitative description on the operation of this source is provided. Figure 2.2.1 details the
RF current and voltage amplitudes along a cylindrical inductor comprising this full-wave
structure. Both ends o f the antenna are grounded. Voltage and current occur as standing
waves along the length o f the antenna structure, which is coiled around the outside of the
discharge tube and cut to approximately the length of the exciting source. Where the
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16
current magnitude equals 0 nodes develop and minimums in the plasma density are
expected. In this case, three lobes o f plasma are formed inside the source. These lobes
ideally capacitively couple to each other so strongly that the capacitive coupling to
external grounds is eliminated. Capacitive coupling to the antenna, on the other hand,
will tend to be strongest and to create the most ionization just where the above mentioned
nodes occur (see Fig. 2.2.1).
We have constructed a full-wave antenna o f this type with the hope o f improving
on the source originally built by Conner, and utilizing it as a source for microwave
spectroscopy. The details o f the construction are described later in this chapter. We
observed that the plasma was not contained inside the source for powers greater than
100 W, no matter what the pressure. This would imply that the plasma was still at least
partially capacitively coupled to the source, and the capacitive coupling between the
plasma and ground was not sufficiently compensated by the internal special structure o f
the plasma.
This could happen in particular if the antenna proved non-resonant.
Experiments, which we describe later on, were performed that led us to believe the
resonance condition was at least partially satisfied. Based on those results, we sought out
a model that would help us understand the complicated interactions that lead to favoring a
particular mode o f coupling.
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17
Voltage j
Current i
L ength A long the C y lin d er
F ig u r e
2 .2 .1
Id e a l v o lta g e a n d c u r r e n t d is trib u tio n s o f a fu ll w a v e h e lic a l r e s o n a to r .
S h a d e d a rea s
r e p r e s e n t v o lu m e s o f p la s m a th a t fo rm in s id e th e s o u rc e .
2.3 Mathematical model o f the source
Seeking a more complete description o f how this source should work, a
mathematical model developed by M. A. Lieberman was adapted to the dimensions o f
our full-wave resonator [8]. The model uses a continuous current sheet to represent RF
current flowing through the antenna helix, and unfolds the cylindrical geometry o f the
source into a rectangular geometry. This "developed sheath model" yields the dispersion
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18
relation between /? and a>, where fi represents the axial wavenumber and co is the
excitation frequency o f the source.
The relationship is derived by modeling the RF
current in the helix structure o f the antenna as a continuous current sheet. The cylindrical
geometry o f the antenna is deformed into rectangular coordinates.
This dispersion
relation gives information about the relationship between the antenna length, plasma
density, and excitation frequency, for a given type o f resonance condition imposed on the
antenna (quarter-wave versus full-wave resonance, etc.) to be exactly satisfied.
This
model show two modes o f wave propagation are possible. Lieberman calls the first mode
a coax mode, but this is non-resonant, since the axial wave length is many times longer
than the antenna. The second mode is referred to as the helix mode, and is is this mode
that we seek to make exactly resonant. By adapting this model to the dimensions and
structure o f our particular source, we hoped to predict how to tune the antenna length to
achieve an exact resonance for any given plasma density.
A simple picture of the developed sheath model is shown in Figure 2.3.1. The
cylindrical coordinates r, 0, and z are represented by rectangular coordinates x, y,and z,
respectively. The variable a represents the radius of the discharge tube, b is radius o f the
coil, and c is the radius o f the outer conductor o f the source. The current in the coil is
represented by a sheet in the y, z plane at x=b with conductivity infinity in the I direction
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19
F ig u re 2 .3 .1 R e p r e s e n ta tio n o f a slic e o f th e d is c h a r g e tu b e , w h e r e a is th e ra d iu s o f th e p la s m a re g io n , b is
th e ra d iu s o f t h e c o il, a n d c is th e r a d iu s o f th e o u te r c o n d u c tin g s h ie ld . T h e c e n te r o f th e t u b e lie s a lo n g
th e z a x is . A ls o , tf/ is th e a n g le m a d e b e tw e e n t, a v e c to r p e r p e n d ic u la r to th e h e lix w ir e s a n d th e z a x is , o r
b e tw e e n I, th e v e c t o r a lo n g th e h e lix w ire s , a n d t h e y a x is . A s th e c o il a p p r o a c h e s th e lim it o f m a n y c lo s e ly
s p a c e d tu rn s , ^ .a p p r o a c h e s 0 , w h ile a t v e r y fe w tu r n s ( < l ) , ^ a p p r o a c h e s kJ2.
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20
and conductivity zero in the t direction. The directionality o f I and t with respect to y and
z, respectively, is measured by the angle iff. W ave solutions for all electric and magnetic
field components vary as e
(Later on one must superimpose two counter-
propagating traveling waves o f this type in order to create a standing wave that will
satisfy the boundary conditions at the axial ends o f the antenna structure.) The axial
electric and magnetic fields obey the two dimensional Helmholz equation:
( 1)
(2)
where k=a>/c0 is the free space wavenumber and (3 is the axial wavenumber, or 2ttI'kaxiaiSimilarly,
G F = Q .7if.
The tangential electric fields m ust become zero at the surface o f the
outer conductor, and they must be even with respect to changing the sign o f x near x=0.
Using the above stated conditions and integrating Eqs. (1) and (2), the axial electric and
magnetic fields can be described by the following equations, where the subscripts a, b,
and c refer to the regions associated with the plasma, that between the plasma and the
coil, and that between the coil and the outer conductor surface, respectively:
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E za = A * c o sh (p ax)
(3)
H za = B * coshQvc)
(4)
E zb = C * cosh(/?D(x —a)) + F * sinh(j30(x —a)
(5)
/ / , A = E * cosh(j?0(x - a)) + F * sinh(/?0(x - a))
(6)
E zc = G * sinh(p„ (x - c))
(7)
H :c = H * cosh( p 0(x - c )).
(8)
In the above equations, p a is the transverse wavenumber in the plasma, while p a is
transverse wavenumber in the vacuum region:
p . = ^ / 3 1 - k 1s p
.
(9)
( 10)
Here sp is the plasma permittivity for the case o f a collisionless plasma:
sp =1
a)2
Y
P ,
CO
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(11)
22
where cop is the electron plasma frequency
®p = J —
(1 2 )
V
and e is the absolute value o f the charge on the electron, m is the mass o f an electron, n is
the plasma density, and s0 is the permittivity of free space.
The electric fields along the x and y directions can be obtained from the axial
fields using Maxwell's equations. The results are:
p
E
ax
z J IS
p
p
H
p~
LVL
ax
ax
ax
03 )
(K )
os)
06)
The boundary conditions for the problem require continuity in Ey, Ea Hy>and H- at the
plasma edge, where x=a. Imposing these conditions for Ea for example, implies that
E:a=E:b. Using equations Eqs. (3) and (5) and evaluating at x=a gives
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23
C = A *cosh(paa) .
(17)
For Hz we use Eqs. (4) and (6) and impose the boundary conditions at x=a to obtain
E = B* cosh (,paa) .
(18)
Using Eq. (14) to obtain Ey from Eqs. (4) and (6) and evaluating at x=a yields
Fpa = Bpa svah.{paa) ,
(19)
while using Eq. (16) to obtain Hy from Eqs. (3) and (5) and evaluating at x - a leads to
DPa£o
=A
£ p£ oPo
sin(p ad) .
(20)
Now we must consider the boundary conditions for the fields at the surface o f the
coil, where x=b. We have already stated that a sheet of infiniteconductivity in the I
direction and zero conductivity in the t direction representsthe current in the coil. Thus
Elb = E lc =Q
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(21)
24
and
Elb= E lc.
(22)
The magnetic field is continuous along the I direction, so that
H lb= H lc.
(23)
The electric and magnetic fields in the I and t directions can be derived from the
respective components in the y and z directions by a rotation by angle yr.
E = E y cos y/ + E , sin y/
(24)
Et = —E y sin y/ + E. cos y/
(25)
H, = H y cos y/ + H . sin y/
(26)
H t = —H y sin y/ + H . cos y/
(27)
By substituting Eqs. (24) - (27) into Eqs. (21)-(23), which define the boundary conditions
at the coil, the following four equations result:
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25
\e sinhp 0( b - a ) + F coshp a(b - <2 )]cos^
(28)
Po
+
[C cosh p a(b —a) + Dsinh p a(b —a )]sin ^ = 0
G s ia y /= = i^ - H c o s iff
jap
p^ f J _ a^jr p
(29)
p^ Qj _
^ ^ s in
ip +
Po
[Ccosh p 0(b - a) + Dsinh. p a(b - a)]cos iff =
(30)
H sinhp 0(b - c ) s i n ^ + G sinhp a(b —c)cosiff
Po
J (O S a
[C sinhp 0( b - a ) + D coshp a(b - n)]cos\ff +
Po
[Ecoshp a(b - a) + F s in h p a(b - a)]sin y/ =
(31)
^ <oe° G coshp 0(]b—c)cosyr + /T coshp a( b - c ) sin ^ .
The result o f the above derivation is four equations (Eqs. (17) - (20)) describing
the boundary conditions o f the electric and magnetic fields in the region o f the plasma,
and four more equations (Eqs. (28) - (31)), describing the boundary conditions o f these 2
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M(P, <*>)=
c o sh [p a(P , a ) (b - a ) ]
0
-1
0
0
e p(o ))sinh[p a(p , 0))a]
0
-sin h [p „(P , co)a]
-c o sh [p a(p , a))a]
0
0
0
0
0
/?sin(y/)
0
0
0
-s(P , to)
S sin ( yJ)
0
0
0
0
0
0
0
0
S c o s( y/)
t(p , (o)S cos(y/)
S c o s( y/)
t(P , a)Scos(y/)
0
0
1
0
0
0
0
s(P , co)
0
0
0
0
0
0
-t(P , © )S co s(y /)
-t(P , t&)Rcos(y/)
0
0
s in ( ^ )
0
0
-t(p , c o )co s(^ )
-t(P , co)/,s in ( ^ )
<2sin(y/)
0
t(P , co)Ssin({//)
-Rs\n(y/)
t(P , co )/?sin (^)
- S s in ( ^ )
-Pcas(y/)
-t(p , co )(2 sin (^ )
F ig u re 2 .3 .2 T h is sh o w s th e 8 x 8 m a trix d e riv e d from th e c o e ffic ie n ts o f e ig h t e q u a tio n s a n d e ig h t u n k n o w n s. T h e u n its o f th e c o e ffic ie n ts a re
d im e n sio n le ss. V a ria b le /?= co sh [p 0(p , co)(b-a)], 5 = sin h [p 0(p , co)(b-a)], P= sin h [p 0(P , co)(b-c)], a n d Q= c o sh [p 0(p , co)(b-c)], V a ria b le s / a n d s
a re fu n c tio n s o f p a n d a , a n d w e re u se d to m ak e th e u n its o f th e c o e ffic ie n ts d im e n sio n le ss.
27
fields for the coil. This set o f eight linear, homogeneous equations must be solved for
the eight unknowns A - K. W e can generate an 8x8 matrix, shown in Fig. 2.3.2, from the
coefficients o f the unknowns. The imaginary part of the original equations was divided
out algebraically. The coefficients o f the matrix were converted to dimensionless units
by multiplication with appropriate terms. We also define variables s=
, and t=
Po
.
CPo
This 8x8 matrix must be singular, i. e., its the determinant must equal 0, in order for these
homogeneous equations to have a non-trivial solution. Values o f J3 and a>that make this
condition true are found, and a dispersion relation J3(co) results.
In the case where there is no plasma present in the discharge chamber, the
boundary value problem simplifies. The electric and magnetic fields o f the plasma region
are no longer important to the problem, since this region is indistinguishable from the
surrounding one. Once again, integrating the two dimensional Helmholz equations and
setting a=0, the rest o f the fields are described as
Ezb = C * cosh p 0x
(32)
H .b = £ *cosh p ax
(33)
E.c = G * s in h p 0( x - c )
(34)
H .c = H * cosh p 0(x —c).
(35)
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28
To restrict the fields at the surface o f the coil, we impose the boundary conditions
described in Eqs. (21) - (23). The following four equations result:
£ sin h p 0&•c o s ^ + C coshp06 •s in ^ = 0
(36)
Po
G sin if/ = ^ C°^L
- H cos yr
Po
(37)
E sinh p 0b - sin y/ + C cosh p a(,b —a ) •cos yr =
Po
(38)
H sinh p a {b - c) sin y/ + G sinh p a(b - c) cos if/
Po
J - 6-'? C sinh p ab ■co sy/ + E coshp ab -sm y / =
P°
(3 9 )
G cosh p a(ib - c) cos y/ + H cosh p a(b - c) sin y/.
Po
Now the problem consists o f a set o f four linear, homogeneous equations with four
unknowns. We can generate a singular matrix using the coefficients o f the unknowns, the
determinant of which must be 0. The matrix is shown in Fig. 2.3.3. In this case, we
found the determinant symbolically:
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29
D et\A\ = —t cosh p ab •sinhp ac - sin2 y/ +
M(p,
(40)
}=0-
t3 sinh p ab ■cosh p ac ■cos2 y/
cd)=
cosh[p0(P, co)6]sin(^)
- s(P, co)sinh[p0(P, co)6]cos(^)
0
0
cosh[p0(P, co)6]cos(^)
s(p, co)cosh[p0(P, co)6]cos(^)
s(P ,
co)cosh[p0(P, co)6]sin(^)
cosh[p0(P, co)6]sin(^)
0
0
sin(^)
-s(p, co)cos(^)
-Pcos( y/)
-s(P, co)jPsin(^)
-s(P, ®)Qcos(y/)
O sin(y/) —
F ig u re 2 .3 .3 S im p lif ie d fo rm o f F ig . 2 .3 .2 , in th e c a s e w h e r e n o p la s m a is p re s e n t in th e tu b e . V a r ia b le P=
s in h [ p 0(P , c o )(b -c )], a n d Q= c o s h [ p 0(P , co)(b -c)]
Comparison o f the results from this equation to those from the larger set for <3=0 provide
a convenient check of their correctness.
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30
2.4 Results o f the model analysis
A Mathcad™ worksheet HELICAL was written to extract values o f /? and co that
satisfy the singular matrix M. The dispersion relation f5(a>) was determined numerically
for our source dimensions, namely
a = 5 cm
b= 10 cm
c = 40 cm
L = 100 cm,
where L is the axial distance between the grounded ends o f the antenna. Using the values
defined above for the variables related to the source geometry (depicted in Fig. 2.3.1), we
calculated the dispersion relation o f P(co).
The graphs o f P(oo), with density as a
parameter, are presented in Figure
The angular frequency variable co was
converted to
f
by
f= c o /2 7 r.
2.4.1
We considered plasma electron density parameters o f
moderate (109 cm'3) and high (1011 cm '3) values, and the no plasma condition.
The
'T .jtf
special case f3h = ----- — , the "helix wave" which corresponds to a wave propagating at
cQs in ^
the velocity o f light along the actual coil winding, is also plotted in Fig. 2.4.1. (Note that
Lieberman's paper has tan(if/) instead o f sin {if/) in the formula for /?/,.) The special case
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31
o f a wave propagating straight down the axis at the velocity of light, the "axial wave", is
also shown in the figure.
For the full wave resonator, we require fiL= 2k, i. e., L=Aaxiai,
since one full cycle o f RF can resonate along the axial antenna length.
New Dispersion Relation for the Full-Wave Helical Resonator
12
♦ no plasma
■ 10E9 coax
10E9 helix <
■
A
10
8
helix wave ji
axial wave j
6
4
2
Me
0
0
10
20
30
40
50
60
f(MHz)
F ig u r e 2 .4 .1 T h e d is p e r s io n r e la tio n p r e s e n te d h e re h e lp s to p r e d ic t th e c o u p lin g c o n d itio n o f t h e p la s m a
a n d d e n s itie s , g iv e n th e g e o m e trie s o f th e c h a m b e r a n d th e e x c itin g s o u r c e .
Lieberman shows that for f3 <
the plasma exists in a non-resonant "coax"
mode, where the plasma floats at a much higher voltage than the outer shield.
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The
32
plasma potential fluctuates several hundred volts, following the fluctuations in the
antenna, with respect to the outer conducting surface. For fi> J3f„ the plasma exists in the
potentially resonant "helix" mode, where the plasma and outer conducting surface remain
Resonance Frequency (MHz)
Predicted Density (cm'J)
26
10y
30
10“
Table 2.4.1 Resonant conditions for generation of an inductive plasma.
at nearly the same potential, while the antenna exists at a much higher potential compared
to the other two elements. We interpret this mode as an inductive resonance mode, where
the plasma acts as a single turn, lossy secondary conductor coupled to the resonant
structure o f the antenna, although the Lieberman paper does not mention inductive or
capacitive coupling. The calculated (3 for our source is 6.3 m '1, if L is 1 m. Looking at
the graphs o f (3(co) in Fig. 2.4.1, we can find the resonant frequencies for any given
electron density for the source. (Plots for other values of n can easily be added). In
particular, at 27 MHz, a resonant coupling is possible for densities o f around 1010 cm'3.
This relationship would be very useful if we could actually change the frequency
of our exciting source, since, according to the graph, we could potentially maximize the
apparent electron density by changing the excitation frequency. Also by changing the
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33
excitation frequency, we might be able to optimize the inductive condition, which would
simplify the Langmuir probe experiments, since the fluctuations in the plasma potential
would be much smaller. The excitation frequency o f the RF source is not tunable for our
system. The source puts out a fixed, 27.12 MHz. Based on the model, however, for any
given electron density, e. g., 1010 cm'3, one should be able to select an antenna length that
would lead to an approximate resonance.
2.5 Construction details of the full-wave helical resonator source
An illustration o f the plasma chamber apparatus is given in Figure 2.5.1. This
figure shows our final version, while the discussion here begins with the initial design.
The plasma is contained in a 4” ID, 60” long pyrex tube, with an additional 12" long
pyrex cross at one end. The long pyrex tube is wrapped with lA" teflon tubing for air,
water or liquid nitrogen cooling. The end o f the pyrex tube is capped with a teflon lens
that was customized to attach to the gas manifold providing a port for gas inlet. The end
o f the cross is covered with a second teflon lens for microwave experiments. This lens is
removed easily, and can be replaced with a custom-made polycarbonate window,
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IN
v
m
v
10
VACUUM
F ig u re 2.5 .1
PUMP
D ia g ra m o f th e H R d isc h a rg e c h a m b e r a p p a ra tu s: (1 ) te flo n len s/g a s in let, (2 ) F a ra d a y sh ie ld , (3 ) o u te r sh ie ld a n d su p p o rt
stru c tu re , (4 ) c o ile d a n te n n a , (5 ) re m o v a b le w in d o w , (6 ) g ro u n d p la n e , (7 ) c o n n e c tio n s to g ro u n d m ad e w ith strip s o f O F H C c o p p er.
35
allowing for the insertion o f Langmuir probes into the tube and for visual observation.
The bottom o f the cross exits to a new RUVAC model WSU500 Roots blower and
Leybold TRIVAC dual stage rotary vane pump, both filled with inert Fomblin fluid. An
improved base pressure o f 0.5 mTorr is achievable in this system. Gases are introduced
to the manifold by an MKS1159B flow control system. A pressure control system was
designed as part o f the current research effort, and it automatically adjusts the flow o f gas
to maintain a set point pressure chosen by the user. A detailed description o f the pressure
control system can be found in Appendix A.
An MKS 120 capacitance manometer
monitors the gas pressure. Note that the plasma sees no metal inside the tube, eliminating
noise due to electrode sparks and contamination generated by sputtering o f metal. The
Faraday shield and a second outer shield were added after initial experiments indicated
the plasma was capactively coupled to the source. These two shields and the ground
connections were made out o f sheets o f oxygen-free, high conductivity (OFHC) copper.
We discuss the construction details and the importance of their presence later on in this
section.
A block diagram of the excitation apparatus is shown in Figure 2.5.2.
Our
antenna was fashioned around a 40" long piece of PVC tubing that had an OD o f 11".
We cut a 36.5' piece o f 1/4" copper tubing as a first approximation to the wave length
corresponding to our exciting frequency of 27.12 MHz (assuming a pure helix wave).
We first wrapped the copper tubing around an 11" wood form, while a lathe turned the
form. The coiled antenna was then slid over the PVC pipe. The coils were manually
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36
spread apart to a measured spacing o f 2". The discharge tube, with cooling and insulation
in place, was inserted into the PVC pipe. The antenna coil was connected in series to the
water cooling o f the discharge tube. Both ends of the coil were initially grounded to a
stainless steel 40" bar. The RF is supplied to the antenna by a Henry Radio 100-B solid
state exciter operating at 27.12 MHz. This exciter drives a Henry Radio 2K Classic-X
1000 W linear amplifier. The 3000 V needed to power this amplifier
matching
box
amplifier
supply
exciter
F ig u re 2 .5 .2 B lo c k d ia g r a m o f th e e le c tric a l e x c ita tio n to th e a n te n n a . A ll c o n n e c tio n s a r e m a d e u s in g H V
c a b le a n d U H F c o n n e c to r s u n le s s s ta te d d if f e r e n tly in th e te x t. A ll c o n n e c tio n s t o g r o u n d a re c la m p e d ,
u s in g 1" s tr ip s o f O F H C c o p p e r .
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37
is supplied by a Hipotronics 805-2A DC power supply, connected to the amplifier using
HV cable and type C connectors. The connection from the amplifier to the antenna coil
is made through a brass slider, which clamps on to the copper tubing using a screw.
When loose, the slider can be adjusted easily to a new position on the antenna.
The antenna structure must resonate at the exciting frequency. For a full wave
resonator, one wavelength of the exciting frequency must resonate along the length o f the
antenna, which is wrapped around the outside o f the discharge chamber.
RF from Rhode-Schwartz
50 Q termination^
To antenna
To channel A of the
network analyzer
Figure 2.5.1 Block diagram of the 50 ohm impedance bridge used to determine the 50
ohm matching point on the antenna.
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38
This wavelength, however, deviates from the vacuum value in the presence o f the plasma
impedance, which depends on the plasma density, as described earlier.
Other factors
affect the resonance frequency, e. g., the spacing o f the coils as they are formed around
the chamber, and the point at which the source is connected to the coil. In principle, if
the length and spacing are correct, and if the connection from the source to the antenna is
made to match the output impedance o f the source—in our case 50 ohms—it should be
possible to operate the system without a matching network. Initially, we did not use an
antenna tuner to minimize the standing wave ratio (SWR) and maximize the forward
power to the antenna. We found the 50 ohm matching point by using a network analyzer,
a Rhode-Schwartz frequency generator, and a 50 ohm impedance bridge as illustrated in
Figure 2.5.1.
The 50 ohm termination and the antenna oppose each other, since the
reflected power is what is measured by channel A. This scheme for finding the resonance
and match conditions unfortunately can not be used when the plasma is on, since the high
power RF would destroy the sensitive instrumentation.
The matching point was found by adjusting the slider to a point where the
network analyzer showed the resonance to be 27 MHz. With the slider in place, we tried
to turn on the plasma. Without a match box, the best standing wave ratio (SWR) we
could achieve was 2, which means that half o f the power to the antenna was being
reflected back to the source. After trying many different points of attachment for the
slider, we decided to use a Nye Viking 3000 W antenna turner model MB-V-A as a
matching box in order to minimize the SW R and maximize the forward power to the
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39
antenna. With this addition, we were able to achieve safe operating parameters for the
equipment, using the full range o f 1000 W from the amplifier, as well as to
approximately establish the resonance condition required to generate the plasma.
Using the network analyzer was one way to find what frequency resonates along
the length o f the antenna. Another way to detect a resonance is to sample the electric
fields generated on the outside o f the coil. We took a small neon lamp and affixed it to a
teflon rod. This rod was passed along the outside o f the antenna surface. Provided that a
resonant structure exists, the neon lamp should light in the presence of the intense electric
fields generated by it. The intensity o f the lamp should be proportional to the magnitude
o f the electric fields. In particular, light should extinguish when V=0, which would be
the situation at both grounded ends and in the center of the structure. This was in fact
observed, indicating that the resonance condition was satisfied, at least approximately.
Still, plasma found its way down the pumping port. We began to seek out ways to
reduce the amount of capacitive coupling occurring between the plasma and the coil.
According to Lieberman [9], a Faraday shield slotted in a direction parallel to the electric
field will allow the inductive field to penetrate while shorting out the capacitive fields in
the r and z directions. We constructed a shield to fit over the PVC pipe and under the
antenna out o f a sheet of OFHC copper. Using a welding torch, we slotted the sheet
along the length o f the tube with 1" spacings. We wove sheets of mylar between the slots
to ensure each piece was electrically isolated. After wrapping the pipe with the shield, we
generously wrapped the entire shield with mylar sheets in order to electrically insulate it
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40
from the antenna, which slid over it. Above 100 watts, however, the mylar proved to be
an inadequate insulator between the coil and the shield, as hot spots developed where
arcing occurred and the mylar was melted. We replaced the outer wrapping with a high
voltage paper insulator, which proved to be much more effective. We also added another
shield that externally covered the entire antenna. The purpose o f this external shield was
to localize the return field that develops around the outside o f the antenna. The necessity
o f this shield was also emphasized by the Lieberman model discussed in Sect. 3.3 and by
private conversations with M. Lieberman. In the previous configuration, the return field
was delocalized as it took up the entire room. We also had a big problem with radiating
RF, which not only interfered with the electronics implemented in this experiment, but
also could be detected in experiments going on elsewhere in the building. It was hoped
that adding this shield would minimize this RF noise. The shield was made by wrapping
a sheet o f OFHC copper around a cylindrical support skeleton that fit over the antenna
apparatus. The ends o f the shield were then clamped together using two 40" stainless
steel bars and six screws. Both ends of the antenna were clamped to this shield, and then
connected to ground. Finally, we worked to improve all of the connections to ground in
order to minimize ground loops, and the added stray capacitance and inductance intrinsic
to the braid shield we had used previously. All connections were made with 1" strips o f
OFHC copper that were either soldered, clamped, or bolted to their termination points.
With the above modifications, we were able to generate plasmas using the full
range o f power available from the source, with the SWR measuring no more than 1.3.
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41
We still had problems with radiated RF interfering with some o f the electronics in the
room above 300 W power. At powers near 1000 W, it sometimes happened that plasma
would form in the glass gas manifold that was built along side the discharge chamber. We
were able to m inim ize this possibility somewhat by adding a small piece o f glass wool
just before the point at which gas entered the chamber. Experimental specifics o f these
plasmas are detailed in Chapter 4; however, some qualitative comments are given here.
In several sources [7, 10], it was reported that an inductive plasma was contained inside
the antenna structure. Bletzinger reports that in low pressure cases (in his case, pressures
less than 10 mTorr), an inductive plasma will extend downstream in some cases because
the longitudinal electric field contributes strongly to the drift o f charged particles. We
observed that only plasmas at pressures o f 10 mTorr or more, and using powers o f 150
watts or less, were confined to the antenna structure.
W e found, however, plasma
extending out of both ends o f the antenna at higher powers, even at pressures as high as 1
Torr. The nitrogen plasmas that appeared confined were a rose pink color. As the power
was increased above 150 W, the plasma would abruptly pop out o f the end o f the tube.
The SWR would change significantly at this point, starting at 1.3 when it was confined,
and changing to 2.0 when it extended downstream. At that point the match could be
adjusted back to a reasonable value using the matching box. As the power was increased
to 1000 W, the nitrogen plasma maintained pink hues, but brightened (or whitened)
considerably.
We also experimented with argon plasmas.
extended out of the tube no matter the pressure or power.
These plasmas always
The argon plasma was a
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42
bluish-purple color, and also brightened (whitened) as the power increased to 1000 W.
The problems of radiated RF interference were also much worse in argon than in
nitrogen.
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43
References
1. W. T. Conner, Ph. D. Thesis (Physics), University o f Wisconsin-Madison, p. 25-72
(1989).
2. W. T. Conner, Chapter 2.
3. J. L. Destombes, C. Demuynck, and M. Bogey, Phil. Trans. R. Soc. Lond. A, 324, p.
347 (1988)
4. EC. Niazi, A. J. Lichtenberg, M. A. Lieberman, EEEE Transactions on Plasma Science,
23:(5), p. 833-843 OCT (1995).
5. J. H. Keller, J. C. Forster, M S. Barnes, J. Vac. Sci. Technol. A, 11(5) p. 2487, (1993).
6. M. A. Lieberman and A. J. Lichtenberg, Principles o f Plasma Discharges and
Materials Processing, John Wiley & Sons, NY, (1994).
7. G. K. Vinogradov and S. Yoneyama, Jpn. J. Appl. Phys. Vol 35 (1996)
8. M. A. Lieberman, A. J. Lichtenberg, and D. L. Flamm, "Theory o f a Helical Resonator
Plasma Source", Memorandum No. UCB/ERL M90/10, (1990).
9. M. A. Lieberman and A. J. Lichtenberg, Principles o f Plasma Discharges and
Materials Processing, John Wiley & Sons, NY, (1994), p. 405.
10. P. Bletzinger, Rev. Sci. Instrum.: 65(9) p. 2975-9 (1994).
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44
Chapter 3
Design, Construction, and Practical Application o f Langmuir Probe and
Microwave Interferometry Diagnostic Tools
3.1 Introduction
A Langmuir probe is a powerful and very widely applied tool that can measure local
plasma parameters inside o f a discharge. In certain cases, where the plasma potential is
constant, the plasma is isotropic, and the electron energy distribution is Maxwellian, the
analysis o f the characteristic trace is relatively straightforward.
The plasma sources
investigated in this work do not conform to these simple conditions, and as result, we
needed to work hard to understand ways to extract meaningful information from the
trace. In Sect. 3.2 we discuss a general interpretation o f Langmuir probe measurements;
we consider when the theories work, and when they begin to fail. In that section we also
discuss the general experimental set up, and describe the probes used in analyzing the
magnetized plasma in Chapter 5. Specific conditions imposed on the probe current and
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45
how we approached the analysis o f curves obtained in the magnetized plasma, are
discussed in more detail in Chapter 5.
The full wave helical resonator described in Chapter 2 was reported to form an
inductive plasma.
This inductive plasma, in principle, had the unique property of
internally compensating for capacitive currents.
This compensation supposedly
minimized the external capacitive currents that normally develop between the plasma and
other conducting, exterior features o f the chamber, such as, the antenna and other
conducting surfaces in and around the vacuum system. Our initial impressions were that
our newly constructed source did not exhibit these unique desirable features. The plasma
was not confined to the antenna region, for example, and initial experiments seemed to
indicate that plasma formed by our source was more capacitive than inductive in nature.
The only way we had to determine if the local structure resembled, in any way, the
structure illustrated in Figure 2.2.1, was to use Langmuir probes.
Using a probe to
sample an RF plasma, however, is not a trivial problem, since it is imperative that only
the DC component o f the plasma current be collected by the probe. This problem is well
known and has been investigated using a variety of capacitive and inductively coupled
plasmas. Soon after our investigations began, it became clear that, even though many
papers have been written on the RF case, most o f those applications still suffered from
RF contamination.
Section 3.3 will discuss the special problems faced when using
Langmuir probes for the RF case, as well as the particular problems we encountered as
we worked to obtain meaningful data.
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46
Microwave interferometry measures the density of electrons present in the path o f
a monochromatic beam o f microwave radiation as it propagates through the plasma
volume. This technique does not give any information regarding the local structure, r. e.,
the spatial distrubution, inside the chamber.
The density data from it, however, is
probably the most quantitative obtained out o f all the experiments we performed on these
sources. The measurement is the least affected by the magnetic or RF fields imposed on
the plasma. The experiment has a simple design, and provides quick results that are easy
to process.
We compared line integrated plasma densities from this experiment with
those obtained by the probe, in order to gauge how well the probe experiment was really
working. A general description o f the interferometry experiment and the experimental
design are presented in Sect. 3.4.
3.2 Conventional analysis o f the probe trace
Since their inception over 70 years ago, Langmuir probes have become a crucially
important plasma diagnostic technique.
In principle, a probe can obtain detailed
information regarding parameters that fundamentally affect the chemistry in a discharge.
In spite o f their simplicity, however, a probe is to some extent always an intrusive
diagnostic technique. Careful analysis of the I-V curve, based on valid assumptions by
the user, is imperative for reliable results. Ion densities, electron temperatures, plasma
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47
potentials and the electron energy distribution functions (EEDFs) can all be determined in
favorable cases.
The probe acts as a small electrode in contact with and biased relative to the
plasma. The plasma in this simple case is quasineutral everywhere except around the
probe.
In the probe region, electric fields penetrate into the plasma and locally
breakdown this quasineutral state, and a space-charge sheath is created. The thickness of
the sheath depends on the density, the electron temperature o f the plasma, and the voltage
drop across the sheath. The characteristic shielding distance o f a plasma is defined by the
Debye length, Xd, which in MKS units is given by
(1)
Constants in the above expression are defined as follows:
G0 =
the permittivity of free space
k= the Boltzmann constant
Te—the electron temperature
«e=electron density
e=charge o f the electron.
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48
Conventional methods o f probe curve analysis depend on the collisionless motion
o f particles once inside the sheath. For this condition to be satisfied, the mean free paths
o f ions and electrons (A, and /Le, respectively) must be large compared to the sheath size.
A second condition for obtaining accurate results relates to the perturbation the probe
imposes on the plasma. As the probe bias Vb relative to the plasma potential Vp increases,
it is desired that the probe maintains its planar geometry as far as the plasma particles are
concerned. As the bias increases, the sheath thickness between the probe and the plasma
also increases, because the electric field penetrates further into the plasma. The shielding
distance ds related to the probe perturbation can be approximated using the ChildLangmuir Law [ 1]:
13/4
—
d =
(64k T
r* -v ,
■Ar
kT.
(2)
Here, the prefactor is approximately 1.25, and the expression in square brackets is the
voltage drop across the sheath as a multiple of the electron temperature.
When ds
becomes comparable to the radius o f the probe, the planar symmetry condition is broken.
The sheath thickness, as given by Eq. (3), is approximately a few (3-10) times Ap.
A presheath region exists beyond the sheath, where weak electric fields still penetrate
into the plasma. In this region ions are accelerated by the moderate electric fields which
develop there. The thickness o f the presheath depends strongly on the local molecule
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49
density and is approximately A,-. Bohm determined that in order for ions to enter the
sheath, they must possess a velocity, v0 defined
(3)
where M is the mass of the ion [2]. The voltage drop across the presheath adjusts itself to
accelerate thermal ions from the bulk plasma to the Bohm velocity as they reach the
sheath edge.
The probe, to some extent, is always an intrusive diagnostic technique. Charged
particles are lost not only to the probe surface, but also to the probe holder. If losses
occurring to the probe and holder are comparable to losses that occur at the walls plus
other recombination processes, the perturbation will be too large for one to extract
meaningful current traces. It must be true, therefore, that
(4)
where Ip and //, are the currents lost to the probe and probe holder, respectively, and Iw is
the plasma current lost to the walls o f the chamber.
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50
Once biased relative to Vp, the probe collects current from the plasma.
A
characteristic trace is presented in Fig. 3.2.1. Electrons are repelled when the probe is
Characteristic Trace
Q.
-30
-20
-10
40
F ig u r e 3 .2 .1 T h e c h a r a c te ris tic c u r r e n t v s . v o lta g e tra c e o f a L a n g m u ir p ro b e . T h e a b o v e tr a c e w a s ta k e n
in D C c a th o d e - g e n e ra te d n itro g e n p la s m a 4 c m f r o m th e c a th o d e , u s in g 2 0 m T o r r p r e s s u r e , 9 5 0 V , 2 0 0 m A .
biased very negative relative to Vp (Fi<~25 V in Fig. 3.2.1), and in this region of
saturated ion current, only positive ion current is collected. As its bias becomes more
positive, the probe begins to collect electrons. At the point where 7 = 0 , the ion and
electron currents collected by the probe are equal. This point (Vb~35 V in Fig. 3.2.1)
corresponds to the floating potential Vj for the plasma. As the bias is further increased,
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51
ions are increasing repelled, while more electrons are collected, although both species
still contribute to the current.
In this so-called electron retardation region, a nearly
exponential rise in the current occurs, until the probe bias reaches the plasma potential.
At that point (Vp = Vb ~ 38 V in Fig 3.2.1) and beyond, the rise levels o ff perceptively as
the probe is saturated with electron current. The electron saturation region develops in
this region where Vb>Vp. As long as the plasma potential Vp is well defined for the probe
trace, meaning Vp remains at a constant value or at a constant separation from the
instantaneous probe voltage Vb over the RF cycle in the RF case, the point Vp = Vb can be
easily observed as the "knee" of the trace.
As long as the probe maintains an effective planar geometry, the electron
temperatures show a Maxwellian distribution, and the inequalities mentioned above are
satisfied, the mathematical equations describing the current collection by the probe are
one dimensional. In this case, the equations can be greatly simplified. As reviewed in
Sudit's and Lai's thesis works, [3, 4], the results of these derivations show that the
electron current Ie collected by the probe is described by
(5)
where the electron saturation current is
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52
(6)
'e
Similarly, the positive ion saturation current, Ii0 is given by
/ to«0.6L 4 en
—
V m,
where Ap is the area o f the probe surface.
(7)
In the case where the distribution o f the
electron temperature is Maxwellian, by taking the natural logarithm o f both sides o f Eq.
(5) we obtain
( 8)
and Te can be obtained from the slope o f the resulting line. Additionally, as long as Vp is
known, a reliable measurement o f Ie allows one to calculate the electron density from
Eqs. (5) and (6).
If the electron velocity distribution is isotropic, but not Maxwellian, the equations
presented in Eqs. (5), (6), and (7) no longer apply. The electron current is then described
as
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53
I e = 2nApe \v.dv, J v /( v ) /v ,
where v z0 =
- 2 e (V b - V p)
(9)
. Taking the numerical first and second derivatives o f the
probe trace are the main steps in the Druyvesteyn second derivative method [5], which
provides the energy distribution o f electrons collected by the probe:
( 10)
Here, / e ( s ) is the EEDF, where
s=
obtain the electron density, ne.
e(Vb-Vp). By integrating over the EEDF we can easily
Note that in taking the first derivative, the plasma
potential can be found by locating the maximum o f the resulting curve. In taking the
second derivative, the plasma potential can be found be evaluating where the resulting
curve crosses the x-axis. Furthermore, the effective electron temperature Ts can be found
by
e
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(11)
54
The distribution function is sometimes presented as F(s), the electron energy probability
f(s)
function (EEPF), where F(e)= - W , and differs from the EEDF only by a constant factor
y] S
o f V J . Using this expression in the Druyvesteyn equation to describe the distribution o f
electron energies provides a somewhat simplified result, where
Sample EEDF for an Isotropic, Maxwellian Plasma
0.6
0.5
0.4
cu 0.3
5:
Q
DJ
w
0.2
0
2
4
6
8
10
12
eV
F ig u r e 3 .3 .2 S a m p le E E D F w a s ta k e n 4 c m f ro m th e c a th o d e in a D C g lo w n itr o g e n p la s m a u s in g 2 0 m T o r r
p r e s s u r e , 2 0 0 m A , a n d 9 5 0 V fro m th e s u p p ly . T h is E E D F w a s d e r iv e d fro m th e c u r v e p r e s e n te d in F ig .
3 .2 .1 .
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55
d2/T
dVb
=W
\ ^ F&)4 " \ Mm.
(12)
The EEPF is normally plotted on a semi-logarithmic scale, for reasons described below.
The EEDF for an isotropic, approximately Maxwellian distribution o f velocities is
presented in Fig. 3.3.2. The units o f the x-axis are given in eV, but are related to the
temperature by 1 V=T 1,600 K. The EEDF begins at e = 0, which is also the point where
the probe bias equals Vp. Electrons possessing no kinetic energy in the bulk plasma can
reach the probe surface at this bias voltage. As the probe bias becomes negative relative
to Vp, electrons must possess enough energy to overcome the repelling voltage o f the
probe.
A Maxwellian distribution has a well-defined electron temperature.
Cold
electrons can still penetrate to the surface for small, negative changes o f Vb\ they are soon
repelled, however, as the probe bias becomes more and more negative.
The EEPF from the same probe curve is illustrated in Fig. 3.3.3. This is plotted
on a logarithmic scale using Eq. (12). The slope of the resulting line for a Maxwellian
distribution should be constant, /. e„ the plot should be a straight line, with a value
inversely proportional to the effective Te. (The apparent dip near s = 0 is an artifact of
the probe perturbation of the plasma.) For the range of electron energies given between 04 eV, the EEPF appears somewhat linear. The value o f Te is the same as from the EEDF,
if we use only the straight part o f the EEPF in this energy range, and the value is 0.72 eV,
which tells us that the plasma at this axial position in the tube is very cold. Recall from
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56
Chapter 1 that only the tail of the EEDF contributes to the ionization o f molecules in the
plasma.
If the tail is as small as expected for a pure Maxwellian distribution at this
temperature, then the plasma electrons at this particular position in the plasma do not
have enough energy to cause any ionization or to sustain the plasma.
The EEPF,
however, shows with good resolution that, although present in m uch smaller quantities,
there do occur other populations o f more energetic electrons. In fact, Fig. 3.3.3 suggests
there are three distinct populations o f electrons, represented by the three maximums at
2 eV, 7 eV, and
Sample EEPF Plot for Nitrogen
0
2
4
6
8
10
12
14
16
-0.5
-1.5
fN
-2.5
-3.5
-4.5
eV
F ig u r e 3 .3 .3 S a m p le E E P F w a s ta k e n 4 c m f ro m th e c a t h o d e in a D C g lo w n i tr o g e n p la s m a u s in g 2 0 m T o r r
p r e s s u r e , 2 0 0 m A , a n d 9 5 0 V fro m th e s u p p ly ; it is d e r iv e d fr o m th e c u rv e p r e s e n te d in F ig . 3 .2 .1 .
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57
11 eV. Because o f the improved resolution in plotting the EEPF over the EEDF, and the
ability to recognize Maxwellian behavior as a straight line on the semi-log plot,
distributions are often presented in this format.
The theory presented in this section assumes that the sheath does not expand and
lose its planar geometry at voltages much more negative than the floating potential. In
practice, however, at voltages very negative relative to Vf the sheath does expand,
specifically outward at the edge o f the probe surface, and begins to change its shape from
planar to hemispherical. Because this increases the effective surface area o f the probe,
and would in turn affect the calculation for the ion densities, we also analyze the ion
saturation currents according to the equation presented by Johnson and Holmes [6]. In
their paper, the expanding sheath is modeled as a filled-in half torus, which expands as Vb
becomes more negative than Vf. Ion densities are obtained by the following expression
/, = / „ [ l + < / ( n - F , / ] .
(13)
In the above equation, Iio is the ion saturation current expressed in Eq. (7), d is an
empirically determined constant, and y is an index o f expansion for the sheath. Once the
plasma potential is known, a plot o f ln(dI/d(Vb - V p)) vs ln(Vb - V p) gives a straight line
with a slope o f y-l. Then plotting /, vs. (Vb —Vp) y gives a straight line with its intercept
equal to Ii0. As long as the electron temperature is known, this value for the current can
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58
be used in Eq. (7) to find the plasma density.
By studying low pressure, high density
plasmas Johnson and Holmes found 0.7<y<0.8. In some cases, we found y to be in this
reported range, while in other cases we found y to be very close to 1. Ion densities
determined using this method o f analysis are referred to as “corrected ion densities” in
subsequent chapters.
A block diagram o f the probe experiment is provided in Figure 3.2.2. Special
control and processing software was developed for a VAXStation 3200 computer running
the VMS operating system by Sudit [7], The computer controls an AAV11-D digital to
analog (D/A) converter, which produces a 12 bit, + 10 V triangle wave at step
frequencies up to 10 kHz. This output is sent to a probe interface circuit originally
D/A
VAX
workstation
Computer
interface box
A/D
F ig u r e 3 .2 .2 B lo c k d ia g ra m o f th e p ro b e e x p e rim e n ta l d e s ig n .
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+50V
+ 10V
From D/A
0
—
V
20K
:A02(p^N
20K
A V
Rs
[*vj
-H h
xlO
V
-0
to probe
_T\
-K
V
-50V
to A/D
©
= e a rth g ro u n d
V
= F lo a tin g g ro u n d (V p ro v id e d b y K e p c o 0 -1 0 0 0 V p o w e r su p p ly .)
V
F ig. 3 .2 .3 In terfa ce c irc u it p ro v id e s h ig h v o lta g e iso latio n b e tw e e n th e p la sm a & c o m p u te r.
VO
60
designed by Sudit [7] and improved in this research, which provides the necessary high
voltage isolation between the plasma and the computer system. The basic components o f
the measuring circuit are illustrated in Fig. 3.2.3. The measuring circuit first divides the
triangle wave output o f the D/A converter by two. The result is input to a AD202KN
isolation amplifier, which isolates the ground chassic from a floating circuit common
provided by a Kepco APH1000M HV DC power supply set to approximately the floating
potential o f the plasma. A floating amplifier multiplies the triangle wave by a factor o f
10. The floated, ±50 V triangle wave is the bias voltage o f the tip. The return current
collected by the probe is measured as a voltage drop across a sampling resistor. The
resulting signal is coupled back to the A/D converter using another AD202KN isolation
amplifier, which isolates the high voltage o f the floating common from the rest o f the
grounded A/D converter.
We needed to build special probes for measurements in the RF plasma, and the
details and designs are presented in the next section. For the magnetized plasma, we used
a planar probe design as specified in Figure 3.2.3. A 3.0 mm diameter, 0.1 mm thick
molybdenum disk was spot welded onto a 1/16" diameter tungsten rod. The opposite end
o f the rod was ground flat to allow for a good silver-braze connection to a 320 cm long
piece o f insulated 17 gauge copper wire. The vacuum seal at the probe tip was made
using Torr-Seal, an epoxy glue.
The holder is bent near the actual probe as shown, so
that most of it lies along the bottom o f the discharge tube to minimize the holder
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61
perturbation, and to support the weight o f the 10' long probe holder. The probe tip can
also be rotated to the tube center or away from it, as desired.
F ig u re 3 .2 .3 P la n a r p r o b e u s e d in th e m a g n e tiz e d p la s m a s .
3.3 Langmuir probe measurements in an RF plasma
In the simple case presented in the previous section, the plasma potential was
assumed well defined. For the RF plasma, the plasma potential with respect to ground
fluctuates tens and sometimes hundreds o f volts through a cycle o f the applied field, even
though the average electron temperature and density characteristics of the plasma remain
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62
Characteristic Traces in Nitrogen
20 mTorr, 40 seem, 100 W
RFI
Filtered I
•3.5
-2.5
CL
0.5
-40----- -30----- -20----- -10-0.5-0.
volts
F ig u r e 3 .3 .1
T h e R F -I tr a c e s h o w s in a d e q u a te f ilte r in g o f th e R F flu c tu a tio n s . T h e f ilte r e d tr a c e sh o w s
im p r o v e d c la r ity o f th e e le c tro n re ta r d a tio n r e g io n .
relatively stable. The fluctuations occur not only at the fundamental frequency of the
power supply, but also at harmonics o f the fundamental. The effects o f these fluctuations
on Langmuir probe measurements are well known [8, 9]. The time scale for obtaining a
probe trace in this environment is much larger than the RF cycle, and as a result, the
current measured for a particular bias voltage is really a time-averaged value. Without
proper compensation, the probe trace and any subsequent analyses are inevitably
corrupted. This smearing o f the characteristic trace is illustrated in Fig. 3.3.1. From the
figure, we observe that the electron retardation region is particularly affected.
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63
EEDFs in Nitrogen
20 mTorr, 40 seem, 100 W
0 .0 7
unfiltered |
0 .0 6
L
filtered
i
0 .0 5
>
0 .0 4
§
0 .0 3
Q
0.02
0.01
0.00
0
10
20
30
40
50
eV
F ig u r e 3 .3 .2 C o m p a r is o n o f E E D F s o b ta in e d fr o m filte re d a n d u n f ilte r e d p r o b e a s s e m b lie s . T h e u n f ilte r e d
a s s e m b ly y ie ld s a tr a c e th a t d is p la y s s e v e r e d is to r tio n o f th e E E D F . T h e f ilte r e d a s s e m b ly le a d s to a m u c h
im p r o v e d tr a c e , b u t it s till d is p la y s s ig n if ic a n t d is to r tio n , a s o b s e r v e d in th e b r o a d e n in g o v e r th e to p o f th e
d is tr ib u tio n .
The portion of the trace where high-energy electrons apparently contribute to the current
is extended with respect to the true population of these electrons. As a result, the electron
temperature is severely overestimated, as illustrated in EEDFs pictured in Fig. 3.3.2.
Note that while the filtered trace illustrates improved tracking o f the electron retardation
region, there still occurs a large amount of RF interference, as can be seen by the
broadening over the top of the distribution function.
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64
To preserve the integrity o f the I-V trace, the probe must faithfully follow the
fluctuations in the plasma potential. There are several ways to eliminate the effects of
RF fluctuations on the probe measurement. Braithwaite et al. implemented a method that
actively compensates for fluctuations by using a feedback circuit to bias the probe with
the same fundamental frequency and phase used to generate the plasma [10].
Other
authors have attempted this active compensation method with moderate success [11, 12].
In all o f these cases, however, only fluctuations associated with the fundamental
frequency are corrected for. Harmonics o f the fundamental frequency also contribute to
the distortion o f the probe curve [13].
Our approach to improving the measurement
involved optimizing the probe construction in conjunction with using a network o f filters
tuned to the fundamental frequency, and the first and second overtones to correct for the
fluctuations in the plasma potential [14-16]. Details o f the probe construction are
discussed in Sect. 3.3.1, while the filter applications are presented in Sect. 3.3.2.
3.3.1 R F probe design and construction
If the probe is forced to follow all frequency components o f the plasma potential
oscillations, then only DC current generated by the external bias voltage is collected. It is
theoretically possible to create this condition through the construction o f the probe. The
approach we have followed involves adding a second metallic surface o f larger area than
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65
Zi
auxiliary probe
Al 'V A - f i ________
cs
probe tip
Ccoupling
jyvvjl
V
Z2
: g stray
Zin
F ig u r e 3 .3 .1 A s c h e m a tic r e p r e s e n ta tio n o f th e m e a s u r in g c ir c u it e q u iv a le n t to a L a n g m u ir
p r o b e . T h e c u r r e n t s o u r c e is th e p la s m a . T h e v a r ia b le s Z / a n d Z2 a r e in d ic a te d b y d o tte d
r e c ta n g le s d ra w n a r o u n d th e r e s p e c tiv e c ir c u it c o m p o n e n ts th a t c o n tr ib u te to t h e ir v a lu e .
the main probe tip, which we call the auxiliary probe. A block diagram o f an equivalent
circuit is provided in Figure 3.3.1. The plasma potential, in this case fluctuates at a
fundamental frequency co, as well as at its overtones. The idea is that probe tip can be
forced to follow the fluctuations in real time if the auxiliary probe has a lower impedance
coupling to the plasma than the main probe. The coupling o f the auxiliary and main
probe to each other occurs through a capacitor. The path o f the current to the collecting
surfaces o f the probe tip and the auxiliary probe has both capacitive and resistive
components.
The variables Rp and Cp represent the sheath resistance and sheath
capacitance related to the probe tip, while Rs and Cs represent the corresponding auxiliary
probe components values. The diagram shows that the equivalent circuit o f the probe
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66
acts like a voltage divider with respect to the plasma potential.
The impedances
associated with thie various capacitive couplings are frequency dependent, and the
combined impedances important to the probe’s operation are labeled above as Z/ and Z?.
The capacitance values associated with the probe tip and auxiliary probe surfaces are
small and will be explicitly calculated below.
The sheath resistances are difficult to
predict and depend- upon how far the bias potential is from the plasma potential. Thus Rs
may or may not b « smaller than l/coCs. If we neglect Rs, then Cs and CCOuPimg are in
series, and their com bined impedance will be dominated by whichever is smaller. It is
clear that
Vm = V
Zl— ,
m PZ,+Z2
(14)
where Vm is the voltage value measured at the output end o f the probe. We conclude
from the above expression that maximum impedance to ground must be achieved through
the relative values o f |Z,| and |Z ,|. The goal is o f course to make Cs and Ccoupling as large
as possible and CStr&y as small as possible.
Based on E-q. (14) we pursued probe designs that would emulate the equivalent
circuit drawn in Figure 3.3.1. Our first design was adapted from a probe described by
Godyak [17]. The first version o f the probe we built had the same construction as the one
used in experiments soon to be described by Men [18]. The probe design is presented in
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67
Figure 3.3.3.
The first construction attempt used two concentric SS tubes, insulated
electrically, to couple RF fluctuations from the plasma to the bias voltage driving the
probe tip, with the outer surface o f the outer cylinder serving as an auxiliary probe. A
3.0 mm diameter, 0.1 mm thick molybdenum disk was spot welded to a thin gold wire to
form the two-sided planar probe tip. The wire was pulled through a hole drilled into a
0.055" ED by 0.125" OD by 2.2" long piece o f stainless steel tubing and then silver
soldered to it. This piece o f tubing was covered with teflon heat shrink tubing, and then
inserted into a 0.180" ID by 0.250" OD by 2.2" long piece o f stainless steel tubing. The
teflon heat shrink served as the dielectric for the capacitance existing between the outer
conductor in contact with the plasma, and the inner conductor, which was in contact with
the probe. The probe tip sat about 2 mm above the pieces o f assembled tubing. One end
o f a length o f RG319U coaxial cable was stripped to expose 2.5 cm o f the center
conductor. The center conductor was then silver soldered to the inner piece o f tubing. A
second piece o f teflon heat shrink was placed over the solder connection and exposed
wire. The coaxial cable was then inserted into a 150 cm long lA" OD piece of pyrex
tubing. An epoxy glue was used to form the vacuum seal between the pyrex and stainless
steel tubing. Ceramic paint was used around the probe surface area to form the vacuum
seal.
The values o f capacitances shown in Figure 3.3.1 can be approximated by
assuming values for the plasma density and electron temperatures.
Lieberman [19]
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According to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o-
7
F ig u re 3.3 .3 D ia g ra m o f th e M a rk 1 p la n a r p ro b e : (1 ) p la n a r p ro b e tip , (2 ) o u te r sta in le ss ste el tu b in g , (3 ) in n er sta in le ss ste el tu b in g , (4 ) T e flo n
h e a t sh rin k in su latio n , (5 ) T o rr-s e a l p lu g , (6 ) in su late d c a b le , (7 ) P y re x Vi" tu b in g .
On
oo
69
the sheath capacitance developed between a floating metal surface and a plasma can be
approximated by
The thickness of the sheath region that develops between a floating conducting surface
and the plasma, ds, can be approximated using the Child-Langmuir Law, given in Eq. (2).
Based on microwave interferometry measurements, which will be discussed in more
detail later in this chapter as well as in Chapter 4, we measured the volume density to be
approximately lx l0 9 cm'3. The electron temperatures in RF plasmas are known to be
moderately cold. We assume Te in this case to be around 3 eV. Given these assumptions,
and using Eq. (1) to calculate Xp, we calculate ds to equal 1 mm. Using Eq. (15) we
calculate Cs and Cp to equal 4.4 pF and 0.04 pF, respectively. The impedance at 27.12
MHz is very high through these two capacitors, so unless the sheath resistance Rs is
small, the overall impedance |Z,| may be much larger than desired. The capacitance
value for Ccoupling can be calculated using Gauss’s Law:
(16)
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70
where K is a dielectric constant. The geometry o f the capacitor is that o f two concentric
cylinders o f radii r/ and r2, with r 2 corresponding to the radius o f the outer cylinder.
From Eq. (16) we can obtain an expression for the electric field in terms o f r:
E(r) = — £ — .
2nrs„L
(17)
Capacitance is defined as the total charge accumulated on a surface divided by the
potential difference between the two surfaces. We can find the potential difference, V2Viyby integrating over Eq. (17) using the limits o f the radius values for the cylinders. The
result is
2 k s „KL
v2 -r,
In V
(18)
\ rx J
which is the equation used to calculate the capacitance between the two cylinders. Using
the geometries o f the cylinders in the probe construction, and 2.0 for the dielectric
constant o f teflon, CCOupUng is calculated to be 16 pF.
The original probe construction used an RG316U coaxial cable as the connecting
wire from the probe assembly to the connector outside the vacuum chamber. About 150
cm o f cable is used inside the probe holder, which is sometimes heated to temperatures in
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71
excess o f 100 C by the plasma. The shield braid is well grounded, and serves to isolate
the inner conductor from the fluctuations in the plasma. It is important that the cable
insulation does not melt, since doing so would short the center conductor to ground. Our
cable choice is suitable for temperatures up to 200 °C, with the insulation and outer
coating both made out of teflon. The nominal capacitance rating o f the cable, which
determines Cslray, is 95.8 pF/m. We obtain a value of 144 pF for the stray capacitance.
Given the above discussion, it is no surprise that the Mark 1 probe proved to do
an inadequate job o f following the plasma fluctuations, since the stray capacitance was 9
times larger than the coupling capacitance.
We attempted to increase the value of
Ccoupling by soldering an additional 350 pF capacitor between the inner and outer pieces of
tubing. The capacitance adds in parallel with the assembly coupling capacitance. We
observed no real benefit from this change in design. We also tried to increase the surface
area o f the auxiliary probe in contact with the plasma. We added this surface area in the
form o f four wings, as illustrated in Figure 3.3.4. With this added surface area, however,
we still observed no reproducible improvement in the RF smearing o f the probe trace.
We would pursue alternate probe designs, some of which will be described in more detail
below, but ultimately we would continue to reach the limit set by the large value o f stray
capacitance dominating the voltage divider. We worked to decrease this
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72
1.25"
F ig u r e 3 .3 .4
A d d e d s u r fa c e a r e a u s in g w in g s f o r th e a u x ilia ry p r o b e . C y lin d r ic a l a s s e m b ly d im e n s io n s
re m a in e x a c tly a s d e s c rib e d p re v io u s ly .
value by choosing alternate cable and probe holder materials. Other high-temperature co­
axial cables have capacitance ratings similar to the one we originally chose. We decided
to try an ordinary coaxial cable that reported much lower capacitance ratings. We settled
on using Belden RG62U cable, which was rated at 44.3 pF/m and reported to have a
maximum temperature o f use o f 80 C. We also shortened the glass probe holder size to
1 m. With these changes we were able to reduce Cstray significantly, although using a
capacitance meter, we still measured a value of 87 pF.
We also tried stripping the
RG62U outer and ground shielding, and then inserting the center conductor and
insulation into a i m long V*" OD SS piece of tubing. This construction proved the most
effective in minimizing Cstray, reducing the value to 65 pF, although it did not improve
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
plasma sheath
probe tip
■coupling
stray
F ig u re 3 .3 .5 D ia g ram o f M a rk 2 p ro b e a lo n g w ith e q u iv a le n t c iru it (1 ) tu n g ste n ro d , (2 ) u ra n iu m g la ss o r c e ra m ic p a in t in su latio n ,
(3 ) T o rr S eal v a cu u m se a l, (4 ) p y re x V4" tu b in g p ro b e h o ld e r, (5 ) R G 6 2 U c o -a x ia l c a b le in sid e p ro b e h o ld e r.
-j)
U
74
the overall quality o f the probe trace when used in conjunction with the above described
probe assembly.
We sought alternative probe designs that would provide a different coupling
mechanism between Vp and Vb. Based on conversations with Professor A. Wendt and M.
Patterson, we built a probe o f the design illustrated in Figure 3.3.5 a. A 3.0 mm diameter
14.2 cm long 2% thoriated tungsten rod was coated with an approximately 0.7 mm thick
layer o f uranium glass.
Uranium glass is reported to have a dielectric constant o f 4
(conversations with Kimball staff scientist). In this configuration, the capacitive coupling
provided by the auxiliary probe occurs through a capacitor with two dielectrics, the first
dielectric material being the plasma sheath, the second being the insulating material
covering the tungsten rod. The non-conducting surface o f the uranium glass eliminates
the resistive coupling of the plasma to the auxiliary probe. The effective value o f the
capacitor will be governed by the less efficient dielectric, in this case the sheath. The
probe impedance | Z p | develops from the parallel combination o f Cs and Cp, while the
frequency dependent impedance o f Cs!ray equals 1ljcoCstray In this case
K = - - - Pl C s)„ : VPCstray + (P o + Q )
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(19)
75
Once again, if we could make Cstray smaller than the sum of Cs and Cp, the probe would
faithfully follow the plasma potential.
For the dimensions given above, CSheath was
calculated to equal 16 pF. We attached this assembly to the SS holder containing the
stripped RG62U cable. According to Eq. (19) our probe assembly was still not in the
desired limit. Paranjpe et al. [20] describe a similar probe assembly in their work, where
the stray and probe capacitance values are approximately equal. That work describes a
passive tuning circuit that compensates for the effect o f Cstray on the fundamental RF
frequency. We employed a more complicated version o f a filter circuit to tune the probe.
The details o f the passive circuit we used are presented in the next section.
3.3.2 R F passive filtering options
In the best o f circumstances, the probe construction alone could provide the
necessary components to minimize the effects o f RF fluctuations on the probe
measurement. The basic premise o f that would be the RF voltage across the probe sheath
can be minimized by ensuring that the impedance between the probe and plasma, [ Z\ | ,
is small in comparison to the impedance between the probe and electrical ground, \Z i \ .
In practice, it is necessary to implement additional tuning or filtering networks to fulfill
the high impedance condition which forces the probe to follow the plasma potential. We
sought out resonant circuit designs that would impose high impedances for \ Z i \ at the
fundamental and overtone frequencies o f the excitation source. Successfully eliminating
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76
these RF components and tracking of the RF fluctuations of the plasma by the probe tip
would allow for the measurement o f the DC component of current as the probe was
biased relative to the plasma potential.
The initial filter design, which we used in conjunction with the Mark 1 probe,
used several L-shaped stages made from coaxial cable to form cascading resonant
networks. The individual networks theoretically imposed a high impedance condition for
a particular RF harmonic. A transmission line, or a coaxial cable in particular, has a
distributed inductance along the line and a distributed capacitance, which develops
between the two conductors. Our design was based on the work o f Shimizu et al. [21]
1.8 m
il
i
D.95
m
ii
ii
iI
2
4f
f
/
F ig u r e 3 .3 .6 C o n f ig u r a tio n o f th e tra n s m is s io n lin e c a b le f ilte r . T h e v a r i a b l e / i s 2 7 .1 2 M H z . E n d s o f th e
c a b le c o n n e c tin g th e p r o b e to th e m e a s u rin g c ir c u it u tiliz e B N C ty p e c o n n e c to rs . A ll o th e r e n d s a r e le ft
o p e n , c o v e r e d o n ly w ith e le c tr ic a l ta p e to p r o te c t fro m a c c id e n ta l p h y sic a l c o n ta c t.
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77
They claimed that the input impedance o f a coaxial cable can be made either infinite or
zero at (2 n + iy b y adjusting the lengths o f the cable used to construct the filter. High
impedances can be achieved by choosing the length o f the cable so that / =
, where u
is the nominal velocity o f propagation through the cable, which is a parameter for the
particular cable type. It can be easily located in manufacturers' catalogs or calculated
directly.
By connecting the cables as a cascade o f L-shaped networks, as illustrated in Fig.
3.3.6, the RF input voltage is cancelled by a reflected voltage returning from the output
with opposite phase. We implemented several versions o f this cable filter, using a variety
o f cable types with varying capacitive and inductive characteristics. The RF noise was
not effectively filtered. In fact, the more stages o f the filter we added, the worse the RF
contamination o f the probe trace seemed to be.
The electron retardation region was
frequently so corrupted that at no point did it cross zero to show a floating potential for
the plasma. After much time and work, it was concluded that there were some problems
with the theory behind the cable filter.
We still believe that from end-to-end, the
attenuation factors in the stop bands are indeed high for the specified frequencies. The
impedance to ground, however, seems limited by the cable capacitance, which is not
negligibly small. As a result the overall performance o f the filter seems to be limited by
the cable capacitance, which increases as more stages o f the filter are added. For the
three stage filter depicted in the figure, we had a total o f 13.3 m of coaxial cable. The
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78
calculated capacitance using the rating value from the catalog equals 590 pF, which
corresponds to an impedance o f 62 Q. at 27 MHz. To verify that the impedance to ground
was very small, we measured it using a network analyzer.
These measurements
confirmed our suspicions, with resulting impedance values equaling 47 Q. at 27 MHz, 20
O at 54 MHz, and 8 Q at 108 MHz. While the cable filter proved to work poorly as a
filter, due to its extraordinary length and flawed design concept, it worked very well as an
antenna. As a result, we believe it efficiently picked up on the RF radiation in the room,
which added to, rather than solved, the overall RF problem.
We attempted to use a series combination o f individual band-stop filters, each
constructed out o f a parallel combination o f an inductor and capacitor. We used this filter
design with both the Mark 1 and Mark 2 probes. The passive components o f each filter
were initially chosen such that
/ = „
2x ( L C y
(20)
w h e re /is the fundamental excitation frequency or one o f its overtones. A block diagram
o f the series combination o f filters is provided in Fig. 3.3.7. The filter components for
each frequency were enclosed in an aluminum box. The inductor components were fixed
in value, while the capacitive components could be tuned to achieve the maximum
impedance with respect to the RF frequency. A small hole was drilled through the cover
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79
to allow for the tuning o f the capacitor using a small plastic screwdriver. Connections to
the other filter components, the probe, and the measuring circuit were made with BNC
connectors mounted onto the filter boxes.
While this filter scheme proved the most effective for our experiments, we
encountered several problems. O f course the resonance conditions of the three circuits
together are not the same as those for the three separate one stage filters. The equation
for the impedance of the three stage network is more complicated, and the use o f Eq. (20)
can only provide a rough guess for the component values. Tuning the networks proved to
be extremely difficult.
Our procedure was to monitor RF fluctuations immediately
preceding the measuring circuit, with the plasma on and the filters in place, using a
Li
L2
L3
nrn ri nmm
■stray
m e a s u rin g
c irc u it
F ig u r e 3 .3 .7 B lo c k d ia g r a m o f th e f ilte r in g c o m p o n e n ts u s e d f o r L a n g m u ir p r o b e e x p e rim e n ts . V a lu e s o f
th e c ir c u it c o m p o n e n ts fo llo w : L i= 3 .5 p H , C t= 0 .9 — 5 0 p F , L 2= 0 .7 5 p H , C 2= 0 .9 — 5 0 p F , L3= 0 .2 2 p H ,
C 3= 0 . 9 - 5 0 p F .
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80
spectrum analyzer set to display the fundamental frequency o f 27.12 MHz, as well as the
first and second harmonics.
With the plasma on, we tuned the capacitors o f the
individual networks in order to minimize the residual RF measured just prior to the
measuring circuit of the probe.
With the filters connected in series, there occurs a
significant amount of cross talk between all o f the filter components. Trying to tune for
54.24 MHz, for example, severely affected the attenuation o f the fundamental. In fact,
tuning for one frequency usually meant that one or both of the other frequencies would be
detuned. The process was time consuming, and at best, we could eliminate the third
harmonic fluctuations, and reduce the fundamental and second harmonic to —29 dBm and
—54 dBm, respectively. The tuned condition was also very sensitive to the probe position
in the plasma.
It is probable that the stray capacitance (mostly due to the long coaxial cable
inside the probe holder), which impairs the filtering ability of the probe, also impairs the
effectiveness o f this particular configuration o f the filtering circuit. Using this filter in no
way improves the problem caused by the stray capacitance, which is the reason why the
probe can't effectively follow the fluctuations in Vp in the first place. As illustrated in
Fig. 3.3.7, the condition is not improved by adding the filters, since Cstray maintains its
effective value and location. The probe holder capacitance cannot be combined with any
other component values to remove its effect, so it keeps the value o f | Z%1, the
impedance to ground, at too small a value.
This is a fundamental flaw in this
arrangement.
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81
We also implemented a lumped element resonant circuit design developed by A.
Wendt, which was a modified version o f a circuit described in a paper by Paranjpe et al.
[22], for a tuned Langmuir probe. The circuit design is presented in Fig. 3.3.8. The
original circuit only filtered out the fundamental frequency and omitted £?, C 2, L 3 , and
Cj. In the design o f Fig. 3.3.8, Cstray is connected directly in parallel with C/, which is the
tuning capacitor for the fundamental frequency.
Therefore (Cstray+Cj) becomes
effectively a single element o f the filter design, which imposes the high impedance
condition on the probe tip. The series legs of the filter tune for the overtones o f the
fundamental RF frequency. It should be noted that the 1000 pF capacitor serves as an RF
short to ground, and occurs after the passive components o f the filter. Normally it would
stray
to computer
1000 pF
F ig u r e 3 .3 .8 L u m p e d f ilte r c ir c u it f o r th e tu n e d R F p r o b e . P a s s iv e c o m p o n e n t v a lu e s : L /= 0 .I 2 p H , C / = 0 .9
— 100 p F , L 2= 0 .2 0 p H , C 2= 0 .9 — 5 0 p F , L 3=0 .2 4 p H , C 3= 0 .9 - 1 0 0 p F . P o in t (A ) m a r k s a c o n n e c tio n p o in t
to a n o s c illo s c o p e a n d a w a v e g e n e r a to r .
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82
be present as the first component o f a low-pass filter. This RF shunt is essential to the
design o f this filter since it is what pulls C/ into a parallel combination with Cstray and
makes the high resonant impedance o f the filter look like the impedance o f the probe tip
to ground. In this design it is at least theoretically possible to completely tune out the
effect o f the stray capacitance at the resonance condition.
Initial values of the passive components were determined for us by Wendt using a
Mathematica program, which plots the impedance vs. frequency for the illustrated circuit.
Component values were adjusted by trial and error to shift the resonances observed on the
plot to the desired frequencies. We verified the choices o f passive component values
using a MathCad worksheet. In this worksheet, the frequency dependent impedance Z for
the lumped circuit was described mathematically using
terml ( f ) + term-, ( / ) + termz ( f ) ’
(21)
where the individual terms in the denominator are given by
(22)
term
(23 )
oa^
2
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83
term3 { f ) = —— — --------
.
(24)
^ r3 k A Q - l]
coC
Characteristic Impedances for the Lumped Circuit Filter
2000
1800 1600
1400 1200
e
JS
o
-
1000
800 600 400 200
-
0
25
50
75
100
MHz
F ig u r e 3.3.9
T h e o re tic a l im p e d a n c e m a g n itu d e
|z |
f o r th e lu m p e d c ir c u it filte r, u s in g in d u c to r
c o m p o n e n t v a lu e s g iv e n in F ig . 3.3.7, a n d th e fo llo w in g c a p a c ito r c o m p o n e n t v a lu e s : C/=50 p F , Caray — 70
p F , C 2= 2 2 p F , C j = 4 6 p F .
A theoretical plot o f | Z | vs. frequency for the lumped circuit is illustrated in Fig.
3.3.9. The inductive components o f the filter are fixed; the tuning occurs through small
changes in the capacitive values o f C/, C?, and C3 . The resonance conditions were
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84
optimized for 27.12 MHz, the fundamental excitation frequency o f the plasma source, as
well as its first and second overtones. The peaks in | Z | in the figure actually occur at
28 MHz, 53 MHz, and 84 MHz, although these values change dramatically with small
changes in any C. We should point out two issues related to the theoretical plot. First,
the theoretical impedances from this model actually go to infinity for the resonance
frequencies, because the model assumes ideal inductors and capacitors, and no dissipative
losses. The actual impedances we measured using a network analyzer were rather low,
on the order o f 1 KX2 for 27.12 MHz, with decreasing values for the successive harmonic
frequencies. This reduction in Z results from significant damping due to the resistive
components o f the filter, which include the resistance in the wires and connections, as
well as the capacitive impedance between the turns o f the inductor. These losses get
worse at higher frequencies, which is why we observe a successive decrease in the
effective impedances responsible for filtering the first and second overtones. We tried to
maximize the O o f the filter by hand-winding the inductors to insure wide spacing
between the turns, and using thick, high-conductivity copper wire to minimize the
intrinsic impedance o f the wire. We also used ceramic capacitors.
The second issue has to do with the choice o f passive component values used in
the lumped circuit.
The method used to obtain the initial values, in part, sometimes
involved guesswork on our part, which ultimately may have led to choosing values that
were not necessarily optimum values for our purposes. We were in fact able to calculate
the precise values o f the passive components needed in a simpler two frequency lumped
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85
circuit, which could resonate at the fundamental and first overtone frequencies.
The
circuit used to develop the mathematical model was similar to the one depicted in Fig.
3.3.8, with the difference being the omission o f L 3 and Cj. The impedance for the two
frequency circuit can be obtained as the inverse o f the admittance Y, where
Y
( l - a ) 2 LlCl)(l —G>2 L 2 C2) —a>2 LxC 2
jco (1 -(d 2 L 2 C2')
For the purpose o f this discussion, we will assume the stray capacitance is incorporated
into the value o f C/. The above expression can be simplified algebraically to a quadratic
equation. By letting Lz=2Li, one o f the solutions to the quadratic equation results from
choosing the capacitance values such that C? = 1/2 C/ and that L jC / = L 2 C2 = (42cox)~2.
The resonant character o f the circuit for the admittance Y can be obtained by rearranging
Eq. (25), and making the appropriate substitutions for the passive component values
(cof -
co2)(4 co{
a
1
-o r)C ,
.
(26)
jco( 2 co2 -co )
The impedance is the reciprocal o f the admittance, so the zeros o f Eq. (26), namely
co1 and C0 2 , are the poles o f Z. If we define coi=2n x 27.12 x 106 s '1, then as the signal
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86
frequency reaches this limit, the impedance approaches infinity. The second resonance
happens when the signal frequency approaches 2/=54.24 MHz. It can also be seen that
this particular choice o f inductors and capacitors leads to a zero of the impedance at
V2f s 38 M H z.
We also attempted to solve the equation for the three resonant
frequency condition. The result, however, was a cubic equation, which we were not able
to simplify analytically. We were able to verify our analytical result for the two resonant
frequency condition, using the MathCad worksheet developed for Eqs. (21) —(24), and
observed sharp resonances at 27.12 MHz and 54.24 MHz as predicted.
Although we did not achieve as much success with the filter of Fig. 3.3.8 as we
had hoped in terms o f removing the RF contamination from the probe trace, we now
believe this was due to some problems with the procedure we used to tune the
capacitance values to achieve resonance. Of course the actual probe and holder must be
in place when the tuning is done, because Cs,ray is part o f the filter, and tuning is
preferably done with the probe installed in the vacuum system. Professor Wendt has
recently suggested to us a tuning procedure that seems better than what we used and that
should be used in the future. Point A in the schematic o f Fig. 3.3.8 is connected to an
oscilloscope input and through a 10 K£2 resistor to a signal generator. The capacitors are
tuned to give maximum signal amplitudes on the oscilloscope at each o f the three
frequencies provided in turn by the signal generator. The oscilloscope probe becomes
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87
part o f the circuit and should be left connected while Langmuir probe measurements are
carried out.
3.4 The microwave interferometer
Microwave interferometry measures the line averaged, or column density of
electrons existing in the path of a monochromatic beam of microwave radiation. While
this technique does not reveal local variations that occur radially and axially in the
plasma volume, it does have advantages as an experimental technique: the measurement
is easy to make, the data is easy to interpret, and the values obtained for the column
densities are accurate.
The technique is based on the principle that the refractive index o f the plasma is a
function o f the electron density. As a result, a beam propagating through a vacuum will
undergo a phase shift when a plasma is turned on in its path. The number o f degrees o f
L
phase shift is to first order proportional
For a monochromatic beam o f
o
microwave radiation at frequency f passing through a discharge tube under vacuum o f
length L , the wavelength o f the beam in the vacuum is X0. Let Xp be the wavelength o f the
beam propagating through a plasma. The phase shift, in radians, that occurs by turning
the plasma on is
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where 1IX0 =f/c and we define Xp in terms o f the index of refraction o f the plasma «p:
1
—
f i np
(28 )
c
(29)
In Eq. (29), co=2nf and cop is the electron plasma frequency, given by
cop =
4;m,e2
m..
(30)
Making the above substitutions in Eq. (27), assuming that ne is constant along the path,
and solving for ne gives
nm. ( c|A^p
2
f -
2 tzL
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(31)
89
This result is discussed further in the microwave plasma diagnostics book by Heald and
Wharton [23].
In the case o f the RF plasma or the normal DC plasma, the result given in Eq. (31)
applies. The expression can be simplified for modest densities, where the quadratic term
in the phase shift can be neglected, to an easy to use formula given by
L
fn edx = 2.066f (Hz) A^(degrees),
0
(32)
which we have used to calculate the column electron densities of plasmas under a variety
L
o f conditions. The units o f jn edx are cm'2. Note that in this linear approximation the
o
assumption that ne is constant along the path is no longer required. By dividing Eq. (32)
by the length o f the plasma volume in cm, one can obtain the average volume electron
density in the path o f the beam through the plasma region. Since we can measure the
spatial variations directly using the Langmuir probe, we have reported the column
densities determined by microwave interferometry and compared them to the axial
integral o f the ion densities we obtained from the probe data.
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90
In the magnetized plasma, the expression for
np
may no longer apply. Radiation
from the source frequency propagating along the z-axis, which is initially linearly
polarized in the horizontal direction, is separated into two circularly polarized waves with
different refractive indices by the magnetic field, i. e., the plasma in the magnetic field
becomes a birefringent medium. When these waves recombine, the direction of linear
polarization may be rotated with respect to that o f the initial beam. Heald and Wharton
give the expression for hp in a magnetized plasma as
it/2
C02{o}±COh)
1-----f - ! - ----2
CO^CO±COb Y + v 2
(33)
np± =
1
+—
2
co2 (a>±cob)
arv
a)[(<» ±cob ) 2 + v2]
a>^Q)±a>b ) 2 + v2]
4B
where cob——— is the electron cyclotron frequency and v is the collisional frequency o f
me
the electrons [24]. Here B is the magnetic field strength in units o f tesla. We assume in
our case o f cold, low-pressure plasmas, that v is small. This assumption simplifies Eq.
(33) to
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For a source frequency o f 10.8 GHz (<0=6.786xlO10 s '1), propagating through plasma
with a density o f lxlO 10 cm'3(<0p=5.641xlO9 s '1), at a magnetic field o f 250 G
(<06=4.396x109 s '1), we calculate np^ to equal 0.9967 and np to equal 0.9963, which
can be compared to the refractive index for the same conditions without the magnetic
field, which equals 0.9963.
These changes in the refractive index also change the
expression for the density given in Eq. (32) to
L
( ri \
\nedx = 2 .0 6 6 / (Hz) 1± — A^(degrees)
o
V &)
for the two circularily polarized waves.
(35)
For our experimental conditions we have
calculated the errors associated with this change in the refractive index to be + 6.5%.
Since the detector actually measures the power for both waves at the same time, to a first
approximation the errors cancel. We can calculate the angle o f Faraday rotation, T, of
the plane o f polarization for our conditions using [25]
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92
*F=(Aii) n L I X ,
where X is the wavelength o f the source frequency.
(36)
For the experimental conditions
described above, we calculated this rotation using Eq. (36) to be 7.2°. Since this rotation
angle is smaller than the resolution o f our ability to align the rotation angle o f the
microwave horns at the end o f the discharge tube, we can neglect the effect o f the
magnetic field when calculating the densities using this method. As a result, for the
magnetized plasmas studied in Chapter 5, we have based our analysis o f the phase shifts
using Eq. (32).
A block diagram o f the interferometer experimental apparatus is provided in Fig.
3.4.1.
A Hewlett-Packard 8690 B sweep oscillator generates microwave frequencies,
which for our purposes propagate through X-band waveguides.
The output is split
between two paths in a magic Tee, one proceeding through the plasma while the other
proceeds to the phase shifter and attenuator. The frequency o f the microwave output of
the HP8690 is precisely measured using a wave meter and is in the region o f 10 GHz. An
attenuator dial just before hom 1 adjusts the microwave power through the plasma
chamber, while another attenuator, positioned just after the phase shifter, adjusts the
power o f radiation through that leg o f the bridge. The discharge tube is capped off with
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
diode
detector
phase
shifter
wave meter
attenuator
term.
8690 B
sweep
oscillator
term.
magic
magic
tee
tee
plasma
F ig u re 3.4.1 B lo c k d ia g ra m o f th e m ic ro w a v e in te rfe ro m e try e x p erim e n t.
thermistor
power
meter
94
special teflon lenses, which help to focus the microwave beam through the tube. The two
beams are reunited in a second magic Tee and then routed to the detector. The detector is
a model P486A thermistor mount connected to a Hewlett-Packard model 432A power
meter. The detector measures the power resulting from the constructive or destructive
interference occurring between the two beams at the second magic Tee.
Once the power through the two legs o f the apparatus is comparable, a minimum
in the power measured at the detector, with the plasma off, is found by tuning the phase
shifter, which adjusts the phase o f the wave traveling outside the plasma region to match
the phase of the wave propagating through the evacuated plasma chamber.
The
attenuators must also be adjusted to equalize the power in the two legs o f the bridge.
Next, the plasma is turned on, and the phase shifter and attenuator are adjusted until the
minimum at the detector is found once again. The resulting phase shift becomes A(f> for
Eq. (32). The plasma attenuates the microwave signal in addition to shifting its phase. In
argon plasmas the densities are often so high that the plasma leg o f the bridge is almost
opaque to the microwaves and the method fails, at least at the frequency o f this system.
The power measured at the detector is a pW level signal. It was very difficult
sometimes to distinguish at what phase the minimum occurred at the detector, given that
the power meter did not deflect over a large range. To compound this problem, the phase
shifter was a manual dial o f low resolution, with tick marks only at 0.1 degree intervals.
Because o f these two issues, we developed an averaging technique to determine the phase
shift. We picked a level on the power meter greater than the
m i n im u m
and tuned the
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95
i
F ig u r e 3 .4 .2 P ic tu r e o f th e is o la tio n d e v ic e fo r th e w a v e g u id e a p p a r a tu s
phase shifter below the minimum until the power reached this level. We then measured
the phase past the minimum, where the meter would once again settle on the chosen
level. We determined the final phase shift by averaging these two values.
We used this experimental technique on both the RF and DC plasmas.
The RF
plasma posed additional complications, as radiating RF was able to propagate through the
waveguide material to the detector. This caused huge offsets in the power measured by
the meter, and often saturated the detector to a point where we could no longer measure
the phase shift. We constructed a device that isolated most o f the waveguide from the
detector.
The isolation device is shown in Fig. 3.4.2.
In this configuration, the
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96
conducting materials o f the waveguide are separated by an air gap, which can be adjusted
easily by separating the horn spacing. The horn mount is made o f wood, to prevent any
conduction between the two surfaces. Using this isolation device completely eliminated
the noise saturation o f the detector, and allowed us to make meaningful measurements in
the phase shift.
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97
References
1. F. F. Chen, Introduction to Plasma Physics and Controlled Fusion Vol. I: Plasma
Physics, (Plenum Press, New York and London 1990), p. 294.
2. D. Bohm, in The Characteristics o f Electrical Discharges in Magnetic Fields,
Edited by A. Guthrie and R. K. Wakeling (McGraw Hill, N ew York 1949) p. 77.
3. I. D. Sudit, Ph. D. Thesis (Physics), University o f Wisconsin, 1992, chpt. 2.
4. C. F. Lai, Ph. D. Thesis (Electrical and Computer Engineering), University o f
Wisconsin, (1995), p. 23.
5. M. J. Druyvesteyn, Z. Phys. 64, p. 781 (1930).
6. J. D. Johnson and A. J. T. Holmes, Rev. Sci. Instrum. 61, p. 2628 (1990).
7. I. Sudit, Ph. D. Thesis, (1994) p. 124.
8. V. A. Godyak and R. B. Piejak, J. Appl. Phys. 68, p. 3157 (1990).
9. K. R. Stalder, J. Appl. Phys. 72, p. 1290 (1992)
10. N. J. Braithwaite, N. P. Benjamin, and J. E. Allen, J. Phys. E.: Sci. Instrum. 20, p.
1046 (1987).
11. T. I. Cox et al., J. Phys. D: Appl. Phys. 20, (1987).
12.. A. Ohsawa, M. Ohuchi and T. Kubota, Meas. Sci. Technol., 2, p. 801, (1991)
13. C. Lai et al., J. Vac. Sci. Technol. A., 11, p. 1199 (1993).
14. V. A. Godyak, R. B. Piejak, and B. M. Alexandrovich, Plasma Sources Sci.
Technol., 1, p. 36-58, (1992).
15. A. P. Paranjpe, J. P. McVittie, and S. A. Self, J. Appl. Phys. 67, p. 6718 (1990).
16. Y. Ye and R. K. Marcus, Spectrochimica Acta B 50, p. 997-1010, (1995).
17. V. A. Godyak et al., Plasma Sources Sci. Technol. 1, p. 36 (1992).
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98
18. Y. Men, Ph.D. Thesis, pending.
19. M. A. Lieberman and A. J. Lichtenberg, Principles o f Plasma Discharges and
Materials Processing, (John Wiley & Sons, Inc., New York 1994), p. 344.
20. A. P. Paranjpe, J. P. McVittie, and S. A. Self, J. Appl. Phys. 67, p. 6718 (1990).
21. K. Shimizu, A. Hallil, and H. Amemiya, Rev. Sci.Instrum. 68, p. 1730 (1997).
22. A. P. Paranjpe, J. P. McVittie, S. A. Self, J. Appl. Phys., 67(11), p. 6721 (1991).
23. M. A. Heald and C. B. Wharton, Plasma Diagnostics with Microwaves, New
York: (John Wiley & Sons Inc.), 1965, p. 124.
24. Heald and Wharton, 1965, p. 15.
25. Heald and Wharton, 1965, p. 20.
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99
Chapter 4
Investigation of Plasma Parameters in the Helical Resonator Discharge
4.1 Introduction
The plasma physics o f the helical resonator has yet to be understood completely.
Initial work by Vinogradov and Yoneyama provides an empirical description of the
inductive and capacitive balance internally sustained in the discharge, as well as a
physical interpretation o f the unique plasma phenomenon observed inside the tube [1].
Further work by these authors attempts to characterize the source using optical emission
spectroscopy[2] and Langmuir probes [3]. We, o f course, were particularily interested in
the Langmuir probe experimental procedure and results.
Their experiments were performed on argon and oxygen plasmas with standard
discharge conditions o f 1.4 Torr pressure, 2 kW power, and source frequency o f 27 MHz.
In their Langmuir probe experiments the authors implemented an auxiliary electrode and
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100
RF filters similar to those described by Godyak and Popov [4]. While the pressures and
power levels investigated in their work were much higher than those in our operating
conditions, the probe data in particular gave indications o f the presence o f strong
fluctuations o f Vp in their source. The EEDFs shown for the oxygen plasma, for example,
were similar in shape to the unfiltered RF EEDF shown in Chapter 3 o f this work, even
though the presented EEDF was measured in an inductive lobe o f the resonator.
Specifically, the distribution o f energies was interpreted to be non-Maxwellian, with
electron temperature distributions calculated in the 2-18 eV range. These results conflict
with other Langmuir probe studies done on inductive oxygen plasmas, which report Te to
be in the range o f 4-6 eV [5, 6] and show the distribution o f energies to be very nearly
Maxwellian [7]. The value for Te reported by Vinogradov et al. is suspicious by itself,
since values above 10 eV for this parameter are very rare. But the shape o f the EEDF
was an especially strong clue that perhaps their resonator lacked the claimed inductive
character and that their Langmuir probe techniques were questionable.
We began our investigation using argon in the 2-100 mTorr pressure range, with
the hope o f obtaining results comparable to those reported by Vinogradov et al. [3].
Their EEDFs taken in argon were reported to have a Maxwellian distribution, with Te
measured to be around 2 eV, although the EEDFs themselves were not displayed. In
trying to implement our experimental design, we found that both the Langmuir probe and
interferometry measurements were problematical in the argon discharge. We discuss this
investigation and the particular problems we encountered in Sect. 4.2.
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101
The bulk of our investigation used nitrogen plasmas o f varied power and pressure.
In Sect. 4.3, we present experimental details regarding various Langmuir probe filter
schemes implemented for our experiments. The problem of-RF contamination o f the
probe trace was never completely solved.
We show, however, that in certain spatial
regions, the probe traces and their EEDFs do improve. For all o f the configurations we
used, the central region o f the antenna exhibited reproducible improvement to the probe
trace relative to the other parts o f the plasma.
We propose that the measured
improvement is due to a lesser amount o f capacitive coupling o f the plasma at this spatial
location, which results in a reduction o f the RF fluctuations.
Also in Sect. 4.3 the column densites obtained using microwave interferometry
data are presented for the full range o f power, and for a variety o f pressures. This data
was useful in predicting how well the probe was working, since it gave us a reliable
measurement of the electron density. The column densities increase linearly with respect
to power.
Paranjpe et al. [8] suggest that the ion current o f the probe trace is least
affected by the RF contamination.
We have used this idea to calculate the spatial
distribution o f ion densities using the ion current measured by a probe biased very
negative relative to Vp. As discussed in Chapter 3, the ion density depends on Te. Since
we could not obtain a reliable measurement for this parameter, we used 4 eV for nitrogen
as an approximate value [9, 10].
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102
4.2 Experimental details of the argon plasma
Initial experimental work was done using argon as the sample gas.
With
pressures in the 10-100 mTorr range, the plasma would ignite with a 100 W minimum
power to the antenna. Initially, the plasma would extend to the pumping port. Reducing
the power to 20 W would cause the plasma to contract completely to the antenna region.
With the plasma contained, after optimizing the standing wave ratio (SWR) to maximize
the forward power, we could increase the power to 100 W before the plasma would
extend again. The extended plasma was not different from the confined plasma in color
or luminosity. Above 100 W, the plasma would abruptly extend to the pumping port, and
a mismatch in the SWR was observed. These visual, as well as measured, phenomena
suggest the possibility o f two modes for the plasma.
Column densities obtained by microwave interferometry for extended and
confined argon plasmas at pressures of 50 mTorr are presented in Fig. 4.2.1. We started
increasing the power with the plasma confined in the antenna.
Once the plasma
extended, the SWR was optimized and the power adjusted to 90 W. The power was then
varied for the extended plasma. At levels o f around 50 W, the plasma begins to contract
to the antenna region once again. Column densities scale linearly with respect to the
power in both cases. The graph shows a clear difference between the two cases in the
rate at which the densities increase as the power is raised. The density for the confined
plasma is higher than that measured for the extended plasma at powers lower than 70 W.
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103
Column Densities for Argon Discharge Compared
50 mTorr, 65 seem, varied power
180
160
■♦ confined ’
140
]■
t
i
extended f
2 120
100
o
40
0
50
100
150
200
p o w er (W )
F ig u r e 4 .2 .1 T h is g r a p h c o m p a r e s io n c o lu m n d e n s itie s f o r a r g o n p la s m a s a p p e a r in g c o n f in e d v e rs u s
e x te n d e d . O p e ra tin g c o n d itio n s w e r e 6 5 s e e m flo w a n d 5 0 m T o r r p re s s u re .
These initial experiments seemed to suggest that there were physical differences between
the two cases. At low powers, besides the visual differences observed, we also could
measure a larger density for the confined plasma, even though the overall plasma volume
was smaller for the confined case. Since inductive plasmas are reported to exhibit higher
density/lower temperature conditions compared to capacitively coupled discharges [11],
we thought this data gave some initial evidence o f inductive behavior. The second piece
o f evidence related to possible physical differences was the abrupt change in the SWR
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104
when the confined plasma would extend to the pumping port. This event results from a
change in the characteristic impedance of the plasma, which could be related to a
difference in the way the plasma couples to the antenna.
Column Densities for Argon
1400
1200
O
♦ 20.6 m T o r r i
■ 30.06 m T o r r j
1000
*• 60.2 m T o r r
£
o
800
■ 70.4 m T o r r
^
600
x 80.9 m T o r r i
• 90.5 m T o r r i
400
+ 100.1 m T o r r i
*33
<u
T3
200
0
J - f
100
200
300
400
500
6C
p o w e r (W )
F ig u r e 4 .2 .3 T h e c o lu m n d e n s ity f o r a n a r g o n p la s m a a s a fu n c tio n o f p o w e r , w ith p r e s s u r e a s a p a ra m e te r.
The confined plasma is only observed between 20 and 100 W for the pressures we
investigated.
Above 100 W only the extended version is observed.
Figure 4.2.3
illustrates the linear scaling o f the densities as the power is increased for a range o f 20 —
100 mTorr. For operating conditions o f 20 mTorr and 300 W, the column density equals
2 .3 x l0 12 cm'2. Dividing this column density by the length o f the plasma volume, which
is 183 cm, gives the volume density. This parameter is the average over the entire spatial
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105
distribution inside the tube. The local volume density in the densest parts of the plasma
might be up to twice this much. For operating conditions o f 20 mTorr and 300 W, for
example, we calculate the average volume density to be 1.2xl010 cm'3, which is in good
agreement with values obtained by Lai using a Langmuir probe for a capacitively coupled
argon plasma operating under similar conditions [12].
The phase shift measured by the interferometer became difficult to interpret above
500 W in argon. Even with the isolation apparatus described in Chapter 3, which greatly
reduced the overwhelming amount of stray 27.12 MHz RF that traveled along the
waveguide and greatly changed the power meter signal, at higher powers the transmitted
power was barely detectable. Figure 4.2.4 is a plot of the signal power through the leg o f
the interferometry bridge containing a plasma at 50 mTorr versus antenna power. The
figure shows that for argon, the microwave signal is completely attenuated for powers
greater than 500 W. A nitrogen plasma, however, does not attenuate the signal for the
range o f powers available from our source. Heald and Wharton show that for microwave
interferometry measurements in an RF plasma, there is a critical density
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106
Power M easured for Microwaves Through Nitrogen and Argon
5 0 m T o r r p r e s s u r e , v a r ie d p o w e r
0 .0 2 5
0.02
♦ argon
■ nitrogen
0 .0 1 5
£
S
♦ ♦
0.01
0 .0 0 5
♦
0
200
400
♦
600
800
1000
1200
P o w e r (W )
F ig u r e 4 .2 .4 P o w e r th r o u g h t h e le g o f th e b r id g e c o n ta in in g th e p la s m a is m e a s u r e d a t th e d e te c t o r u s in g
a r g o n a n d n itr o g e n p la s m a s .
O p e r a tin g p r e s s u r e f o r th e fig u re is 5 0 m T o r r .
T h e m ic r o w a v e s ig n a l is
a tte n u a te d b y a rg o n .
Below this critical value, the plasma is transparent, while above this value the plasma is
opaque and highly reflective [13]. Using our approximate source frequency o f 10.8 GHz,
the critical volume density for our system is calculated to equal 4x10 10 cm'3. According
to the data shown in Fig. 4.2.3, it appears the density in argon is indeed around
5x1010 cm'3 at a few hundred watts power. (Note we must divide the column density by
the length of the tube to obtain the average volume density, but the microwave
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107
attenuation is determined by the local densities and is very non-linear in density). As a
result, we were limited to measuring how the densities for argon scaled with power levels
below 500 W by the critical density problem.
Nitrogen plasmas have much lower
densities that never reach the critical density in our experiments.
We attempted to obtain information regarding the local structure o f the plasma
underneath the antenna for argon plasmas using Langmuir probes.
The problem o f
radiated RF was particularly bad in argon. When reaching power levels above 300 W,
the digital readouts in the room, including the flow controller and capacitance manometer
readout, would oscillate horribly and the computer keyboard would freeze up. On the
rare occasion when we could actually obtain a probe trace, it had little resemblance to a
typical characteristic trace. The derivative analysis was impossible to perform, due to the
noise level o f the trace. Given all o f these complications, we used nitrogen instead as our
sample gas in most o f this research.
4.3 Experimental details o f the nitrogen plasma
An empirical description o f the expected behavior o f the nitrogen plasma is given in
Sect. 2.5. Like argon, at low powers (in this case 150 W) the plasma was confined to the
antenna region. Above 150 W the plasma would abruptly extend down to the pumping
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108
port. With the plasma extended, however, we could not reduce the power below about
100 W before the plasma would contract to the antenna region.
Fig. 4.3.1 showrs a plot o f the column densities for a nitrogen plasma for varied
power and pressure conditions. Below 100 W the plasma is confined to the antenna
region, while above this value the plasma extends to the pumping port. The graph shows
that there is no break in the line at 100 W, which might indicate a mode difference
between the two visually different conditions.
Densities simply scale linearly with
power. There does appear, however, to be a critical pressure at which the density sharply
increases, occurring somewhere in the range o f 50-60 mTorr. This is inferred by the
distinct grouping of densities measured in the 10-50 mTorr range versus the grouping
observed for the 60-100 mTorr pressure scans. Sometimes such a grouping can result
from the experimental procedure. Leaving the discharge o ff overnight can, for example,
cause differences in the wall conditions from day to day. These differences can affect the
values o f the plasma parameters measured for a given day relative to those from another
day. We have observed such artifacts in our data analysis previously. The experiment
reflected in Fig. 4.3.1, however, was performed on a single day, with all the apparatus
components left on for the entire experiment.
This pressure grouping was also
demonstrated to be reproducible.
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109
Column Densities for Nitrogen
v a r ie d p o w e r a n d p r e s s u r e c o n d it i o n s
10 m T o r r
300
* — 2 0 m T o rr
3 0 m T o rr
250
4 0 m T o rr
5 0 m T o rr
o 200
O
1
6 0 m T o rr
4— 7 0 m T o r r
150
80 m T o rr
•I
9 0 m T o rr
100
100 m T o rr
0
200
400
600
800
1000
RF power (W)
F ig u re 4 .3 .1
T h e c o lu m n d e n s ity f o r a n itro g e n p la s m a a s a fu n c tio n o f p o w e r, w ith p r e s s u r e a s a
p a r a m e te r.
The graph also shows that the density is fairly insensitive to pressure, other than
the marked change between 50 and 60 mTorr.
Experimentally, it was difficult to
optimize the forward power to the antenna below 50 mTorr. Even at high powers, the
SWR was at best 1.4, which also indicates that the resonant condition o f the antenna was
not optimized for these densities. At 60 mTorr and above, as we observe a marked jump
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110
in the density, the SWR was also improved, approaching a value o f 1 at powers o f 800 W
or more at the high pressures.
These differences are perhaps an indication that the
antenna was more closely approaching a resonant condition w ith the plasma.
We attempted to sample the local structure internal to the antenna using Langmuir
probes. Distances in the following discussion are measured relative to the grounded end
o f the antenna nearest to the pumping port. The antenna region is 1 m in length. Probe
data reported with negative position are those taken outside o f the antenna on the end
nearest to the pumping port. As we have already discussed extensively in Chapter 3, we
were unsuccessful in solving the problem of RF corruption o f the probe trace. Our first
attempts at this measurement used the cable filter in conjunction with the Mark 1 probe.
A sample probe trace is show in Fig. 4.3.2. In order to obtain a trace using this probefilter configuration, we had to float the measuring circuit at +73 V relative to ground.
This seemed unusual, since both inductive and capacitive plasmas for nitrogen are
reported to have effective floating potentials within 20 to 30 V o f zero [11]. While the
characteristic knee is clearly apparent at a bias voltage o f 31 V, the electron retardation
region is severely broadened. There also occurs no point were the probe trace crosses
zero, i. e., there is no apparent floating potential. The features o f this curve are typical o f
the curves we obtained using this combination o f filter and probe construction. Even
though the probe traces were obviously spoiled, sampling the axial distribution o f
parameters under the antenna provided us with some interesting information.
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I ll
Trace and Derivative Analysis for Nitrogen
cable filter, 40 mTorr, 200 W
trace
first der.
sec. der.
-30
-20
2.5
-10
V
F ig u r e 4 .3 .2 P r o b e tr a c e a n d d e r iv a tiv e a n a ly s is u s in g th e c a b le f ilte r a n d M a r k I p r o b e d e s c rib e d in d e ta il
in C h a p te r 3 . P r o b e a r e a is 5 .6 c m 2. O p e r a tin g c o n d itio n s in n itr o g e n w e re 4 0 m T o r r a n d 2 0 0 W . P r o b e is
in s e r te d 6 5 c m in to th e a n te n n a r e g io n .
Outside o f the antenna structure, and in the regions between 0-45 cm and 70-100
cm inside the antenna structure, the probe curves obtained could not be analyzed because
they were so spoiled. Visually we could confirm the presence o f plasma at both ends of
the antenna. Between 45 - 70 cm, however, which is approximately the central region o f
the antenna, we were able to extract EEDFs o f reasonable quality. The axial evolution of
these EEDFs is presented in Fig. 4.3.3. From the graph, we see that the region with the
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112
EEDFs for Nitrogen in the Central Region of the Antenna
40 mTorr pressure, 200 W
%
F ig u r e 4 .3 .3 A x ia l e v o lu tio n o f E E D F s in th e c e n tr a l r e g io n o f th e a n te n n a . M e a s u r in g c ir c u it o f th e p ro b e
w a s f lo a te d a t + 7 3 V . T h is d a ta w a s o b ta in e d u s in g th e c a b le filte r in c o n ju n c tio n w ith th e M a r k 1 p r o b e .
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113
best probe data occurs off center from the expected maximum in the density as predicted
by Vinogradov [14], and depicted in Fig 2.2.1.
This indicated that the resonance
condition was approximately present, but not exactly realized for this antenna length.
While we believe that the cable filter design is fundamentally flawed, we should point out
that the EEDFs are reasonable, and provided us with reasonable values for Te for
nitrogen, in the 2 —4 eV range [9, 10]. Also, as depicted in the EEPF presentation in
E E P F s for Nitrogen in the Center o f the Antenna
400 mTorr, 200 W
0
to
F ig u r e 4 .3 .4 E E P F e v o lu tio n in th e c e n tra l r e g io n o f th e a n te n n a f o r n itr o g e n . D a ta ta k e n w ith th e c a b le
f ilte r in c o n ju n c tio n w ith th e M a r k 1 p ro b e . P r o b e a r e a is 5 .6 c m 2.
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114
Fig. 4.3.4, the distribution of energies is near Maxwellian, which is inferred by the linear
slope o f the EEPFs. We believed at the time, however, that the lack o f readable traces
obtained from other regions o f the discharge was due to poor filtering. We were also
suspicious o f the rather large voltage needed to float the measuring circuit in order to
obtain the traces, since RF plasmas are reported to have effective plasma potentials
relatively close to zero [9, 10].
For these reasons, we sought out other methods to
improve the probe traces.
Our next attempt at sampling the local structure used the passive filters in series,
described in Chapter 3, along with the Mark 1 probe with added capacitance. Similar to
the attempts at a full axial scan using the cable filter, we were unsuccessful in obtaining
meaningful traces outside o f the central region of the antenna. A sample probe trace as
well as the first and second derivative analysis is presented in Fig. 4.3.5, for a probe
positioned at 50 cm, i. e., in the exact center of the antenna. The measuring circuit was
grounded when using this probe-filter configuration.
The probe trace shows a fairly
prominent knee, a steep electron retardation region, and a fairly long, level ion saturation
region. This trace also shows a floating potential o f about 12 V. The first and
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115
Trace and Derivative Analysis for Nitrogen
series filter, 2 0 m T o rr, 100 W
■2.5tra c e
'
I s td e r . |
2nd der. j
-50
-40
-30
-20
-10
40
bias voltage (V)
F ig u r e 4 .3 .5 P r o b e tr a c e a n d d e riv a tiv e a n a ly s is u s in g th e s e r ie s p a s s iv e f ilte r a n d M a r k 1 p r o b e d e s c rib e d
in d e ta il in C h a p te r 3 . O p e r a tin g c o n d itio n s in n itr o g e n w e r e 2 0 m T o r r a n d 100 W . P r o b e is in s e r te d 5 0
c m in to th e a n te n n a re g io n . P r o b e a r e a is 5 .6 c m 2.
second derivative reveals the knee at 31 V, which corresponds to the plasma potential.
This trace shows the difference between Vp and Vf to be approximately 18 eV. For a
plasma with a Maxwellian distribution o f electron energies, it is well known that the
difference between Vf and Vp should be approximately four times Te. In this case using
Eq. (11) from Chapter 3, we obtain an effective electron temperature of 5.2 eV, which is
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116
approximately four times less than the difference between Vf and Vp.
This result
indicates that the high energy tail o f the probe trace was virtually free from the RF
smearing.
Thus we believe this trace in particular provides accurate information
regarding the local plasma parameters in the central region o f the antenna.
Distribution of EEDFs in the Central Region o f the Antenna
2 0 m T orr, 100 W
F ig u re 4 .3 .6 E E P F e v o lu tio n in th e c e n tr a l r e g io n o f th e a n te n n a fo r n itr o g e n . D a ta ta k e n w ith th e s e r ie s
p a s s iv e f ilte r in c o n ju n c tio n w ith th e M a r k 1 p r o b e .
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117
The axial evolution o f EEDFs inside the central region o f the antenna is shown in
Fig. 4.3.6 for the series filter used with the Mark 1 probe. The operating conditions of
the discharge in this experiment were 20 mTorr and 100 W. Probe traces outside o f the
region represented in the figure did not have a clear knee, and showed very little
distinguishable ion current, although all traces did cross the x-axis. Comparing this axial
scan with the one shown using the cable filter in Fig. 4.3.2, it is apparent that the overall
RF smearing o f the trace is worse. The EEDF at 50 cm corresponds to the probe trace
shown in Fig. 4.3.5, and is the only one whose shape approaches what is expected for a
cold, Maxwellian distribution o f electrons. As the probe is moved away from the central
position, the RF contamination becomes increasingly worse, until the probe curves can no
longer be interpreted at all.
Despite the obvious distortions to the probe curves, we
considered the results to be an overall improvement over the cable filter, since the probe
trace possessed all the characteristic features pointed out earlier in Chapter 3.
Specifically, we consider the probe curve shown in Fig. 4.3.5 to be one o f the
most reasonable obtained during this research project. Thus we provide explicit pictures
o f the EEDF and corresponding EEPF for the curve in Fig. 4.3.7 (a) and (b), respectively.
The EEDF shows a nearly Maxwellian distribution o f electron energies, which is
confirmed by the linear shape o f the EEPF. The analysis using Eq. (11) from Chapter 3
yields a Te o f 5.2 eV. The electron density, determined by integrating over the EEDF, is
calculated to be 2.4 x 109 cm'3. Calculating the ion density using Eq. (7) from Chapter 3,
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118
F ig u r e 4 .3 .7
E x p li c it r e s u lts o f p r o b e c u r v e a n a ly s is f o r th e tr a c e illu s tr a te d in F ig . 4 .3 .5 . T h e E E D F is
p r e s e n te d in ( a ) . T h e E E P F is p r e s e n te d in (b ) . O p e r a tin g c o n d itio n s in n itr o g e n w e r e 2 0 m T o r r p r e s s u r e ,
EEDF Measured in the Antenna Center for Nitrogen
0.1
0.09
0.08
2L 0.07
a
0.05
0.04
0.03
0.02
0.01
5
£
-
0.01 0
10
20
30
40
50
60
e n e rg y (e V )
EEPF from Probe Measuring in the Central Antenna Region
0
10
20
30
40
-1.5
Q.
-2.5
-3.5
-4.5
e n e rg y (V )
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50
60
119
we obtain a value o f 3.2 x 109 cm"3, which is a reasonable value compared to the electron
density determination. These values are much higher than the average volume density
expected based on the microwave interferometric data for these conditions, which is
approximately 5.1 x 108 cm'3. The difference suggests that the density at this axial
position is much higher than the densities in the other regions o f the tube.
We should point out two issues related to the data presented in Figures 4.3.6 and
4.3.7. First, the Maxwellian probe curve was obtained in the geometric center o f the
antenna. This is unlike any other data we collected during this project, which suggests
that a higher density region exists where fluctuations in Vp decrease, but this region exists
off o f the geometric center of the antenna. Given that the filtering capabilities of our
Langmuir probe equipment was inadequate, we believe that the result presented in Fig.
4.3.7 is due to a minimum in the fluctuations in Vp in that region o f the discharge. Given
that this curve was measured in the geometric center o f the antenna, it is possible that the
resonance condition was present for that particular trace, and at the time the plasma was
inductive. The second issue relates to the confinement of the plasma. For this and other
experiments, the plasma visually appears to extend to the pumping port, suggesting that
the plasma region is not confined to the antenna. But given the large difference between
the average volume density of the interferometric data and the probe data, it is possible
that a higher density region does develop and is confined to the antenna region. Other
processes, which we discuss later on, may be responsible for our visual observation.
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120
Using the lumped filter in conjunction with the Mark 2 probe, we were able
analyze some o f the traces in other regions o f the antenna structure.
Figure 4.3.7
illustrates the evolution o f EEDFs for more o f the length o f the antenna structure.
Axial Distribution of EEDFs Inside the Antenna Structure
20 mTorr, 100 W
Qj
O
F ig u r e 4 .3 .7 A x ia l d is tr ib u tio n o f E E D F s in s id e th e a n te n n a f o r n itr o g e n . O p e r a tin g c o n d itio n s w e r e 2 0
m T o r r a n d 10 0 W .
T r a c e w a s ta k e n e v e ry 5 c m .
B la n k s w e r e le f t w h e re th e c u r v e s c o u ld n o t b e
in te rp re te d . D a ta o b ta in e d u s in g th e lu m p e d e le m e n t f ilte r a n d th e M a r k 2 p r o b e . P ro b e a r e a is 2 .8 c m 2.
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121
We see once again, that in approximately the middle o f the antenna structure, the EEDFs
improve, even though they are still severely distorted. The lumped filter seemed to be the
least effective in forcing the probe to follow Vp, at least as we were implementing it.
Even though a direct analysis o f the probe trace is impossible given the seriously
elevated values calculated for Te, the ion portion part o f the curve may be analyzed to
infer the ion densities o f the plasma.
Local Ion Densities Inside the Antenna
2 0 m T o rr, 100 W
1 .4
E
1 -2
Vi
0.8
o
O'
o
X
C
(L>
0.6
0 .4
0.2
-4 0
-20
0
20
40
60
80
100
120
cm
F ig u r e 4 .3 .8 A x ia l s p a tia l d is trib u tio n f o r io n d e n s itie s a lo n g th e le n g th o f th e a n te n n a s tr u c tu r e . A v a lu e
o f 4 e V w a s u s e d a s th e e ffe c tiv e e le c tr o n te m p e r a tu r e .
D a s h e d lin e s m a r k th e e n d s o f th e a n te n n a
s tr u c tu r e .
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122
We have measured the ion current for the probe biased very negative with respect to Vp
and assumed an effective electron temperature o f 4 eV, to approximate the local ion
densities in a nitrogen plasma using operating conditions o f 20 mTorr and 100 W. The
density profile is given in Fig.4.3.8. The ion current data was extracted using the lumped
element filter and the Mark 2 probe. The breaks in the line correspond to regions where
the ion current did not level off with respect to the bias voltage, so that no density could
be derived.
The desired spatial distribution for the presence of volumes o f plasma was
discussed previously in Chapter 2 and illustrated in Fig. 2.2.1. In this ideal case, where
the current magnitude equals 0 for the standing wave developed along the length o f the
antenna, nodes develop in the magnetic field and minima in the plasma density are
expected and were reported by Vinegrodov and Yoneyama [1]. These minima in the
density ideally occur at X
A and % o f the length o f the antenna. The data in Fig. 4.3.8
reveals a maximum in the density at 60 cm and several minima; most prominent are those
at 5, 40, 75, and 90 cm. The maximum in the data appears off center by about 10 cm
from the desired maximum location indicated in Fig. 2.2.1. The measured densities in the
apparent node regions approach zero only at 5 and 90 cm, which are both near the wellgrounded ends o f the antenna. If the other minima (at 40 and 75 cm) occur because of
minima in the magnetic field (which develops in proportion to the magnitude o f current
along the antenna length), then their high density values seem to suggest that the coupling
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123
is only partially inductive. The off center maximum also indicates a deficiency in tuning
the length o f the antenna, although the presence o f the maximum and minima suggest that
at least a partial resonance has developed for these experimental conditions.
Simpson's rule, we calculated the column density to be 4.2 x 10
in
Using
o
cm'-, which is about a
factor o f two o f the 9.4 x 1010 cm'2 density measured by the interferometer.
All o f the figures presented showing the spatial distribution o f EEDFs and
densities inside o f the antenna, suggest that the RF problem seems to lessen in the middle
o f the structure, and that a high density region may exist inside o f the antenna. Most o f
the data shows that this improvement does not happen exactly in the center, rather it
occurs at 55-65 cm from the pumping port end o f the structure. This is confirmed again
by the ion density profile shown in Fig.4.3.8, which shows the maximum in the density
o ff center o f the antenna structure. Our data indicates that the antenna for our system
r e m a in s
poorly tuned in some cases. At the time o f this work, we did not have a good
procedure to experimentally measure the matching point for the power. A method should
be developed that can allow for this tuning to occur when the plasma is on. The length o f
the antenna coil must be optimized to develop a resonant condition. Simply stretching
out the coil over the tube will not optimize the resonance condition, since this will change
the effective directionality of the current represented by y/, which was explained in detail
in Chapter 2. The actual length o f the coil must be changed so that the value o f L is
preserved. Choosing the antenna length must involve an approximate knowledge o f the
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124
Antenna L ength Approxim ation vs. D ensity
for a 27.12 MHz frequency exciter
2 0 -j—
♦
1 8 ----1 6 -----
♦
1 4 ------
1 2 ------
E 1 0 -----
♦
♦ _____
8 ------
6 ----4 ------
2 -----
0 ----1.E+00
l.E+02
l.E+04
l.E+06
l.E+08
density (cm*3)
l.E+10
l.E+12
l.E+14
F ig u r e 4 .3 .9 T h e a n te n n a le n g th is a p p r o x im a te d w i t h r e s p e c t to th e d e n s ity f o r a 2 7 .1 2 M H z e x c itin g R F
so u rc e.
density present in the plasma volume, according to the previous discussion in Chapter 2.
The variable /? for a lxlO 8 cm'3 plasma generated by a source frequency o f 27 MHz
equals 6.7 m '1. Based on the model, the antenna region should be 0.94 m long, rather
than 1 m long, to make favorable the resonant condition. Thus the calculation suggests
that the antenna coiled around the tube should be shorter for this expected density.
Figure 4.3.9 illustrates the sensitivity o f the antenna length to the expected density
produced by our source, using Lieberman's model described in Chapter 2.
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Points
125
corresponding to the helix mode converge to where /? is 11 m, which is also the
wavelength o f 27.12 MHz. The microwave interferometry measurements show that the
source produces densities in the 109 - 1012 cm'3 range. For this range of pressure, the
length o f the antenna can differ by nearly 2.75 m, which is approximately 2.75 turns
difference around the plasma volume. In order to achieve a resonance condition for this
range of densities, one must have the ability to tune the antenna length dynamically as the
plasma conditions change. Our current configuration does not allow for the dynamic
tuning o f the antenna. The detuned length o f the current antenna is possibly the cause o f
a non-resonant coupling between the plasma volume and the antenna, particularly at the
two ends o f the structure. Finally, it may be the case that the current shielding apparatus
is insufficient. In the current structure, the RF field lines are allowed to exit the ends o f
the shield, where they will loop around and penetrate into the plasma region beyond the
antenna ends. This interference may disrupt the desired resonant condition, particularly
at the ends o f the antenna, and be a reason why the confinement o f plasma to the region
inside the antenna is so difficult to achieve. It is possible that completely enclosing the
coil is necessary to prevent the penetration of these magnetic fields into the plasma.
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126
References
1.
G. K. Vinogradov and S. Yoneyama, Jpn. J. Appl. Phys. 35, p. 1130-1133
(1996).
2. G. K. Vinogradov, V. M. Menagarishvili, and S. Yoneyama, J. Vac. Sci. Technol.
A 16(6), p.3164 (1998).
3. G. K. Vinogradov, V. M. Menagarishvili, and S. Yoneyama, J. Vac. Sci. Technol.
A 16(3), p. 1444(1998).
4. V. A. Godayk and O. A. Popov, Sov. Phys. Tech. Phys. 22, p. 461, (1977).
5. N. C. M. Fuller, M. V. Malyshev, V. M. Donnelly and I. P. Herman, Plasma
Sources Sci. Technol. 9(2) p. 116-127 (2000).
6. A. Schwabedissen, E. C. Benck, and J. R. Roberts, Phys. Rev. E, 55, p. 3450,
(1997).
7. J. T. Gudmundsson, Takas hi Kimura, and M. Lieberman, Plasma Sources Sci.
Technol., 8(2), p. 22-30, (1999).
8. A. P. Paranjpe, J. P. McVittie, and S. A. Self, J. Appl. Phys. 67, p6718 (1990).
9. C. Lai, B. Brunmeier, R. C. Woods, J. Vac. Sci. Technol. A, 13(4), p. 2090,
(1995).
10. Rev. Sci. Instrum. 69(2), p. 1200 (1998).
11. C. F. Lai, Ph. D. Thesis (Electrical and Computer Engineering) University of
Wisconsin, 1995, chpt. 4.
12. C. F. Lai, Ph. D. Thesis, University o f Wisconsin, 1992, p. 76.
13. M. A. Heald and C. B. Wharton, Plasma Diagnostics with Microwaves, (1965)
(John Wiley & Sons Inc.,) p. 12.
14. G. K. Vinogradov and S. Yoneyama, Jpn. J. Appl. Phys. 35, p. 1130 (1996).
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127
Chapter 5
Characterization of a Magnetically Enhanced Electron Beam Discharge
Source
5.1 Introduction
The study of plasma parameters characteristic to specific sources is important to
the understanding of the physical phenomenon causing the formation of ions in the
discharge.
Local parameters o f the plasma, e. g., electron and ion densities, electron
temperatures, and plasma potentials change radially and axially in the discharge tube.
Such variations in the physical conditions o f the plasma can have a great effect on the
species distribution of the plasma under investigation. For this reason we have pursued
experiments that will determine local plasma parameters using Langmuir probes.
Most o f the molecular ions studied in our group have been generated by hollow
cathode glow discharge sources. This source can generate plasmas in two distinct modes.
The normal DC glow mode produces plasmas o f moderate density (1 x 1010 cm'3).
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The
128
plasma has three distinct regions, which we will define as the negative glow, a dark
space, and a positive column. The negative glow, or electron beam, mode has also been
used to study molecular ions.
The densities o f ions generated in this mode can be
increased using an axial magnetic field. This mode appears to involve an extension o f the
negative glow region, normally observed inside and near the cathode in the normal glow
discharge, all the way to the anode.
The easiest and most cost effective way to obtain local plasma parameters is to use
Langmuir probes. Previously in our group, Isaac Sudit studied the accuracy o f Langmuir
probe theories used to analyze the I-V trace.
He reported on plasma phenomenon
observed in the normal DC glow discharges o f nitrogen and helium, including,
confirmation o f the generation o f high plasma density, with a negative glow plasma in the
region o f the cathode, a dark space, and a positive column made up o f stationary
striations. His study o f these unmagnetized plasmas, which used cylindrical probes, was
done with unprecedented resolution.
Ion and electron densities determined by the
Druyvestyn and derivative methods o f analysis were all reasonably self-consistent, which
showed that this instrumental method was a viable means by which to obtain correct and
absolute local plasma parameters for the isotropic case, i. e,. the case when no magnetic
fields are present and the velocity distribution o f charged particles is the same in all
directions.
The introduction o f an external magnetic field disrupts the isotropic state o f the
plasma. In this case, the magnetic field restricts the motion o f the charged particles o f the
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129
plasma that have velocities in directions other than that o f the magnetic field. As a result,
the theories used to analyze the I-V trace fail, for they depend on the random direction of
motion o f particles inside the plasma. To date, no satisfactory quantitative theory exists
to describe the collection o f electrons by a probe in the presence o f a magnetic field. The
ion collection current by the probe surface is least affected by magnetic field
enhancement. The work presented in this chapter explores using the ion current collected
by a probe as a means to obtain local plasma parameters. Plasma densities determined by
this method were confirmed by comparison to line integrated densities measured using
microwave interferometry.
Using this methodology, we investigated how local
parameters inside the chamber are affected by the presence and strength o f a magnetic
field.
5.2 The hollow cathode glow discharge apparatus
The discharge chamber used in these experiments has been described previously
[1], although many improvements have been made since this initial description.
A
diagram o f the chamber is provided in Fig. 5.3.1. The plasma is generated inside a 3 m
long, 15.2 cm diameter, glass cylinder. The cylinder is wrapped first by 0.64 cm teflon
tubing, by which the cylinder walls can be cooled using forced air, water, or liquid N 2 .
This first layer of cooling line is wrapped with a 3 cm thick layer o f insulation.
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A
130
solenoid magnet fits over this layer o f insulation, and is described in detail below. The
end o f the cylinder housing the cathode is capped by a custom-machined teflon cylinder
allowing for electrical and water-cooling connections to the cathode, as well as gas
introduction to the chamber from a gas manifold. Gases are introduced first to the
manifold using a MKS 1159B mass flow controller. The flow controller is regulated by a
pressure control system, which is described in detail in Appendix A. The pressure control
system provides electronic feedback to the mass flow controller in order to maintain a
precise pressure measured by a MKS 120 capacitance manometer. The vacuum system is
maintained using a RUVAC Roots blower and TRTVAC dual-stage rotary vane pump, or
a Varian VHS-4 diffusion pump, all o f which utilize liquid nitrogen traps. Using the
diffusion pump, a base pressure o f 2 x 10'6 Torr can be achieved. The pressure control
system, Roots blower, and mechanical pump are improvements resulting from the current
research effort.
The plasma is generated inside the cylinder using internal electrodes.
The
electrode serving as the cathode is a 12.7 cm OD, 9.5 cm ID, 22.8 cm long polished
stainless steel hollow cylinder that is water cooled internally. The placement and care of
the electrode was described previously [2]. The anode end o f the cylinder is capped off
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To the gas manifold
F ig u re 5.2.1 H o llo w c a th o d e d ia g ra m : (1 ) in se rtio n p o in t fo r p ro b e s, (2 ) a n o d e , (3 ) ion g a u g e , (4 ) c a p a c ita n c e m a n o m e te r c o n n e c tio n p o in t, (5 )
so lo n o id m ag n e t w ith w a te r c o o lin g , (6 ) h o llo w c a th o d e a n d c o n n e c tio n p o in ts (7 ) liq u id n itro g e n tra p , (8 ) g a te v alv e.
132
by a stainless steel, water cooled "T" which serves both as a well-grounded anode and the
exit port to the pumping system. The ends o f the cylinder can be capped o ff by teflon
lenses, for the purpose o f microwave spectroscopy experiments, or by transparent
custom-machined polycarbonate windows that allow for visual observation and the
introduction o f Langmuir probes into the chamber.
The solenoid magnet was fashioned previously in our group following the work o f
DeLucia et al. [3,4]. The magnet is constructed around a 30.5 cm OD aluminum pipe that
slides over the discharge tube and its cooling system. The pipe is first wrapped with 0.95
cm copper tubing for cooling purposes. A layer o f thin copper sheeting follows, around
which 10 gauge insulated copper magnet wire is wrapped. The magnet is further cooled
using water flowing through 0.95 cm copper tubing, which is wrapped around the magnet
perimeter.
Water is circulated to the magnet using a Varian 920010 magnet coolant
control system heat exchanger. The magnet is powered by a Varian V-7808 10 kW power
supply, which is also water-cooled.
The maximum current output is 100 A, which
corresponds to a maximum magnetic field o f 440 G, measured with a Bell Hall effect
gauss meter. The response o f the magnet is linear with respect to current output, with an
efficiency o f 4.4 G/A, as illustrated in Fig. 5.2.2.
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133
Field Strengths Obtained for Current Settings on
the Magnet
500
y = 4.4248x-0.266
400
cn 300
C/5
O 200
100
0
20
40
60
80
100
Amps
F ig u r e 5 .2 .2 M e a s u r e d a x ia l m a g n e tic f ie ld s tr e n g th f o r th e m a g n e t
5.3 Modes observed in the hollow cathode source
The hollow cathode source generates plasmas in two visually distinct modes. Modes
o f a 20 mTorr nitrogen plasma, for example, differ in color, as well as in the currents
drawn from the power supply for a particular voltage. We observe the normal DC glow
(NDC) mode to operate with a minimum current/voltage o f 150 mA/600 V to a maximum
o f 600 mA/3000 V. Our normal operating conditions in this mode are 200 mA and 1000
V. Three visually distinct regions of plasma exist in this mode. From the back o f the
cathode near the entrance o f gases from the manifold to about 35 cm in front o f the
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134
cathode, the plasma exists in a deep purple/lavender color. The color extends all the way
to a dark space, which has no color at all. The positive column then fills the rest o f the
cylinder from the dark space all the way to the anode. The positive column is rose-pink
in color and varies periodically in intensity along its length, from light to dark, due to
variations in local plasma parameters.
These alternating zones of luminescence and
darkness are called striations. The inside of the tube is not visible along most o f the
length due to the magnet and cooling apparatus positioned around it. We have observed
the striated intensity o f the positive column, however, in similar, visually accessible
systems.
Looking down either end o f the tube, the entire plasma brightens with an
increase in power and becomes deeper in purple hues with a decrease in power.
The mechanisms characteristic o f regions inside the NDC are well known and
have been discussed in detail previously [5, 6]. Secondary electrons emitted from the
cathode surface are accelerated radially inward by the electric field to energies equal to
about 3A that of the cathode voltage. These high-energy electrons bounce several times
o ff o f the cathode sheath (or cathode fall), colliding with neutral atoms and forming ions.
Ions and cold electrons diffuse out of the cathode region. These electrons are too cold to
cause any further ionization. The negative glow plasma is visually dim. The violet-blue
color is radiated from ions and occurs as a result o f the electronic emission in the
B2£ +—>X 2S+ bands o f N 2+, which emit at around 400 nm. The plasma potential remains
virtually constant throughout this region. The plasma diffuses out of the negative glow
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135
into the dark space, where most o f the ions neutralize. At this point the plasma potential
begins to increase. Some o f the cold electrons are accelerated through a double layer,
i. e., a local potential jump, immediately preceding the positive column. Electrons regain
enough energy to cause some ionizing collisions, reaching energies equal to or greater
than 15.5 eV, which is the ionization threshold for nitrogen. These electrons do not gain
nearly as much as the secondary electrons accelerated in the cathode fall. These electrons
are significantly hotter than the plasma electrons in the negative glow, and they are able
to promote neutral molecules to excited electronic states, which emit in the UV and red
wavelength regions.
As a result, the plasma in the region o f the positive column is
visually brighter and pink in color.
The negative glow electron beam (NGEB) mode, like the negative glow region o f
the NDC mode, is visually dim. The plasma seems to extend the entire length o f the
cylinder, although this is often not entirely visible. For nitrogen plasma using 20 mTorr
pressure and no magnet field, we observe this mode only at very low power settings and
often by chance alone do we see it. For 1000 V on the power supply, the plasma draws
about 2 mA current. Without a magnetic field, the plasma often flips to the normal mode
without warning. Using even small magnetic fields, however, increases the robustness o f
the mode and allows us to perform experiments on the NGEB mode in stable conditions.
The plasma in nitrogen appears purple/lavender in color from either end o f the cylinder.
The plasma visually brightens with an increase in the power. There seems to be an upper
limit to the voltages that sustain a stable NGEB mod. This limit is dependent on the gas
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136
sample as well as the pressure conditions. For example, using voltages higher than 1050
V from the supply often causes the plasma to flip back to the normal mode when using
nitrogen at pressures o f around 20 mTorr, even when using a strong magnetic field.
Since the bulk electrons o f the negative glow are too cold to produce ions through
collisions, there m ust exist a source o f hot electrons that can sustain the plasma. We have
visually observed an electron beam that originates from the center o f the hollow cathode
diameter, and extends about 3A the length of the cylinder. In his thesis work, Conner
provides a qualitative description o f this beam, and proposes that the beam forms from
ballistic electrons, or high-energy electrons, that originate from the cathode surface and
are accelerated and focused by the magnetic and electric fields. This beam is probably
the primary excitation mechanism responsible for generating and maintaining the plasma
in the NGEB mode [7]. We attempted to use Langmuir probes to measure the electron
temperature and density characteristics in this on-axis region o f the discharge. The plasa,
however, would extinguish or flip to the normal mode when the probe was placed in the
beam path. For this reason, the probe data was collected with the probe rotated 20° from
the central position o f the chamber. This phenomenon gives evidence for the fact that the
electron beam is essential in maintaining this method o f excitation.
We also explored the local plasma parameters o f the NDC mode with a small
magnetic field imposed on the plasma. We could produce the magnetically enhanced
normal glow (MEN) plasma using magnetic field strengths o f 54 G or less.
Fields
stronger than this would cause the plasma to abruptly flip to the NGEB mode. Visually,
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137
this plasma appeared to maintain both negative glow and positive column regions. This
could be observed by viewing the color of plasma at either end o f the chamber. We could
not, however, visually observe if the negative glow region was expanded by the magnetic
field, and if so, by how much.
5.4 Using Langmuir probes for the anisotropic case
A Langmuir probe is a conductor in contact with the plasma and biased relative to
the plasma potential.
In the experiments presented in this chapter, the conductor has the
shape of a planar disc. In the isotropic case, the plasma contains species o f both positive
and negative charges that possess an equal velocity distribution in all directions.
Electrons are more mobile, and typically possess more energy than ions in the plasma.
Depending on whether the probe bias is negative or positive relative to the plasma
potential, electrons are either repelled or attracted to the probe. Ions are also attracted or
repelled by the probe, but since they are not as mobile, they do not respond as quickly to
the perturbation. The perturbation o f the probe is confined to a volume defined as the
space-charge sheath.
Ions are accelerated in the pre-sheath region to a velocity vQ >
(,kT/M )112, so they can meet the Bohm sheath criterion and enter the sheath region. As
was discussed in Chapter 3, the theory used in interpreting the I - V trace depends on the
motion of collected particles inside the sheath and pre-sheath regions. The motion inside
the sheath region is assumed to be collisionless, and therefore is limited by the condition
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138
rp, rh l D « /U h .
(1)
If the above conditions are met, then the characteristic trace can be interpreted using the
methods described previously. Specifically, the isotropic EEDF is found by taking the
second derivative o f the characteristic trace
(2 )
Ine Ap dV
where e=e(F&-Vp). Integrating over the EEDF provides the local electron density, ne, as
well as the local, effective electron temperature Te where
n*
and
= f
(3)
d s
Te =
j^°f(e)e ds
(4)
Operating our source in the normal mode for a nitrogen plasma using conditions
o f 20 mTorr pressure, 200 mA and 950 V, typical local plasma densities in the positive
column are in the 109 cm’3 range. For these conditions, I d is about 0.4 mm, while the
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139
mean free paths o f ions and electrons are on the order o f cm [8]. By using a 3.0 mm
diameter planar probe, the conditions stated in (1) are satisfied.
Under the influence o f a magnetic field, the isotropic state o f the plasma is
compromised.
Rather than move randomly, charged particles move in helical orbits
along a guiding center path that is parallel to the electric and magnetic fields. The radius
o f this orbit is the Larmor radius, r^, where
eB
(5 )
Here, m is the mass o f the particle, v± is the velocity in the plane perpendicular to the
\2kT
magnetic field and usually takes the form of the thermal velocity, vx = ,/------, and B is
V m
the magnetic field strength. While vx is normally used in reference to neutral atoms and
molecules, in this case we are defining the thermal velocity for electrons and ions moving
in a plasma. Therefore T in this case is specifically the electron temperature, Te, or the
ion temperature Ti} respectively. We will show later on in this chapter that electron
temperatures are colder in magnetized plasmas, with an average value o f 0.5 eV. The
Larmor radius for a 0.5 eV electron in a 250 Gauss magnetic field is 0.1 mm. The
confinement o f the electron to this orbit means that the losses normally occurring at the
cylinder walls are diminished. On the other hand, the confinement seriously affects the
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140
way current flows to the probe. Predicting the path o f the electron as it approaches the
sheath area and probe surface becomes very complicated.
The path is influenced by
several variables, such as, the magnetic field, the orientation o f the probe with respect to
the magnetic field, the electric field o f the discharge, collisions with other particles, and
the electric field o f the probe as it penetrates the plasma.
Due to these complicated
interactions, the electrons no longer meet the conditions defined earlier in Chapter 3, and
therefore the theory used to analyze the portion o f the trace where the electrons contribute
is no longer adequate.
Ions propagating through the plasma are also affected by the axial magnetic field.
The effects, however, do not confine the ions as tightly as the electrons are confined. A
nitrogen ion, normally in the form o f N2+, with a thermal temperature o f 300 K gyrates
with a radius of 4 mm. While the motion o f the ion is confined, the diameter o f the
confinement is at least comparable to the diameter o f the probe. It is possible, therefore,
to analyze the portion o f the curve where ions primarily contribute to the probe current, at
least as a reasonable approximation. The equation for the ion current to a probe biased
very negative with respect to the plasma potential was given in Eq. (2) o f Chapter 3
(6)
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141
We found it necessary to use the expansion correction factor explained in Chapter 3 to
correct the ion densities calculated using this method.
The characteristic traces, as well as the first and second derivatives, o f a probe
measurement taken 16 cm from the front o f the cathode are shown in Fig. 5.4.1. The
probe curve obtained in the NDC plasma is a typical shape. Slopes o f the three regions of
the curve showing the ion saturation, electron retardation, and electron saturation currents
are typical given the experimental conditions [9]. Data presented in this chapter will
show that the magnetic field increases the density o f plasma produced in the chamber.
This can be observed qualitatively from the probe curves. The collected current at the
point o f the knee for the MEN plasma is about 3 times larger than the collected current
measured at the same point in the NDC probe curve. Indeed the local density o f the
MEN plasma determined by the theories presented above is approximately 3 times larger
than the NDC plasma. There occurs a decrease in the current collected for the NGEB
plasma, even though the calculated ion density is comparable to the local density
calculated for the MEN plasma.
The collection o f electron current in this case is
deficient, probably because the strength o f the magnetic field is strong enough to
diminish the collection of electrons by the probe.
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142
F ig u r e 5 .4 .1
T h e p r o b e c u r v e s a r e ta k e n 16 c m f r o m th e f r o n t o f th e c a t h o d e .
P r o b e a r e a is 2 .8 c m 2.
P r e s s u r e , f lo w r a te a n d th e s a m p le g a s a r e c o n s ta n t in a ll th r e e c a s e s : (a ) N D C g l o w , 2 0 0 m A , 9 5 0 V , n o
m a g n e tic f i e l d , ( b ) M E N g lo w , 5 4 G , 1 0 0 0 V , 2 5 0 m A , ( c ) N G E B g lo w , 2 6 4 G , 1 0 0 0 V , 2 m A . E a c h g r a p h
illu s tr a te s th e p r o b e c u r v e a n d r e s p e c tiv e d e r iv a tiv e s f o r th e th r e e m a g n e tic f i e l d e x p e r im e n ta l c o n d itio n s
d e s c r ib e d a b o v e .
(a)
Normal Glow Nitrogen Probe Curve Analysis
13.5 mTorr pressure, 60 seem, 0 G
-2.5
— -I
C/J
Q_
dl/dV
d2I/dV2
0 .5
-1 5
-10
0.5-
volts
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143
(b)
MEN Nitrogen Probe Curve Analysis
1 3 .5
m T o rr p re ssu re , 60 seem , 5 4 G
f
V ..... I--------- dl/dV
|........... d2I/dV2
1(
T/
7
J L
0
1
5
10
15
20
25
30
35
_
1 M0
45
1
U
v o lts
NGEB Nitrogen Probe Curve Analysis
13.5 m T o rr p re ssu re , 6 0 seem , 2 6 4 G
0 .2 0 -1
dl/dV
d2I/dV2
0 .1 5 -
0 . 10 a.
0 .0 5 -
i.00'
-5 0
-20
-30
v o l ts
-10
0 .0 5 -
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144
As a result, we observe a significant decrease in the electron current.
The EEDFs are obtained from the second derivative of the probe curve. The
distribution represents the energies of electrons that enter the sheath region, as the bias
voltage is swept negative with respect to Vp. The EEDFs o f the traces given in Fig. 5.4.1
are presented in Fig. 5.4.2. The units of the x axis are given in eV, but are related to the
temperature by 1 eV=l 1,600 K. A value of 0 eV corresponds to the point where the
Comparison of EEDFs in Nitrogen
13.5 mTorr pressure, 60 seem
1 .4
- - - NDC
|
M EN
|
/•\
NGEBI
="
0.8
§
0.6
u.
Q
0 .4
0.2
0
1
2
3
4
5
6
7
8
eV
F ig u r e 5 .4 .2 E E D F s in n itr o g e n 16 c m fro m th e c a th o d e . P r o b e a r e a is 2 .8 c m 2. P re s s u r e , f lo w r a te a n d
s a m p le g a s a re c o n s ta n t.
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145
second derivative equals zero, and therefore where Vb — Vp. At this point electrons need 0
energy to reach the probe surface. Increasing values o f eV in the EEDF correspond to the
increasing energies electrons must possess in order to reach the probe due to the negative
bias voltage relative to Vp. The EEDF for the NDC glow illustrates the Maxwellian
distribution and cold, effect electron temperature typical o f the negative glow region of
this type o f plasma. We believe this distribution to be an accurate representaion o f the
true electron energies present in this region of the plasma. The MEN glow shows a shift
in the effective Te to more energetic values. The EEDF is noisy, as are the first and
second derivatives shown in (b) of Fig. 5.4.1. The NGEB EEDF in the same position
shows a non-Maxwellian distribution, indicating that perhaps there occurs a population of
electrons possessing an effective distribution of around 3 eV. We believe the apparent
shifts in the effective Te in the presence of a magnetic field are an artifact o f the magnetic
field effects on the probe measurement, rather than a true representation o f the energies
present in the plasma. In the probe trace analysis, we have used the effective temperature
calculated by the EEDFs.
In the next section, we discuss how this error effects the
calculations we used to determine the ion densities.
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146
5.5 Experimental Results
The axial evolutions o f ion densities for nitrogen plasmas, using the three conditions
described above, are presented in Fig. 5.5.1. The pressure in each case is held at a
Comparison of Ion Densities for Nitrogen Plasma
13.5 mTorr pressure, 60 seem
70
-•-N D C
60
•-N G E B [
-•-M E N
i
50
40
30
20
10
0
0
50
100
150
200
250
300
cm from cathode
F ig u r e 5 .5 .1
C o m p a r is o n o f lo c a l io n d e n s itie s a lo n g th e z a x is o f th e d is c h a r g e tu b e . N o r m a l m o d e
p la s m a o p e r a te d a t 2 0 0 m A a n d 9 5 0 V . T h e N G E B m o d e p la s m a u s e d a 2 6 4 G m a g n e tic f ie ld , w ith 2 m A
a n d 1 0 1 0 V fro m th e s u p p ly . F ie ld s tr e n g th a p p lie d to th e M E N m o d e is 5 4 G , w ith 3 1 5 m A a n d 7 0 0 V
fr o m th e s u p p ly .
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147
constant 13.50 mTorr. Plasmas in the NDC mode o f operation utilize 200 mA current
and 950 V from the power supply. The MEN glow had an applied magnetic field o f 54
G, with 315 mA and 700 V from the power supply. The NGEB mode was sustained by a
264 G magnetic field, using 2 mA and 1050 V from the power supply. The probe data
show that the ion densities are significantly greater in both cases where a magnetic field
is applied. Comparing the normal mode data with and without the magnetic field, it
appears that the negative glow region is expanded. The sharp difference in densities
between the two regions, shown in the no-magnetic field case, seems to blur in the
presence o f the magnetic field. It also appears that the characteristic striations present in
the positive column o f the NDC case are not present when the magnetic field is applied.
The ion density profile of the MEN case appears to show features o f both the NDC and
NGEB profiles. In the region past about 50 cm, the density profile shows a similar shape
to the NGEB profile. Closer to the cathode, however, it seems that the production o f ions
follows the NDC case.
Evolution o f the electron temperatures for the three conditions described above,
are presented in Fig. 5.5.2. The axial evolution o f electron temperatures exhibited for the
NDC glow are typical for nitrogen have been accurately determined previously
[10].
Very cold electrons are characteristic o f the negative glow region. Some o f these cold
elections are accelerated through the double layer immediately preceding the positive
column to temperatures of around 4 eV. The process o f collisional cooling and then
reacceleration through subsequent double-layer regions is the reason for the striations
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148
Axial Evolution of Electron Temperatures for a Nitrogen Plasma
13.5 mTorr pressure, 60 seem
4 .5
:—
Te—neg. glow
4
;—
I e, norm-mag
3 .5
-------x -------------------------------------------------- 1 ♦
1
3
>
u 2 .5
4)
E-*
kJyW
r
^
t i l t A JU T *
O
1
~
4
Te-normal
. • • * * * * • ♦
1.5
I
1
•
•1
•/
•1
K
II
II
1
II
0 .5
0
0
50
100
150
200
250
300
cm from the cathode
F ig u r e 5 .5 .2 A x ia l e v o lu tio n o f th e e le c tr o n te m p e ra tu re s a re c o m p a r e d f o r th e th re e c o n d itio n s e x p la in e d
p re v io u s ly . E x p e rim e n ta l d e ta ils a r e th e s a m e a s th o s e d e s c r ib e d in F ig . 5 .5 .1
observed in both the density and temperature profiles. Note that the periodicity o f Te is
aliased past 150 cm. Variations in Te actually maintain the periodicity observed in the
regions between 38 and 120 cm.
Striations are not observed in the other two cases
investigated in these experiments. This is expected in the NGEB mode since there exists
no region, such as the positive column, where electrons are heated by a potential ramp.
Since this mode is simply an extension o f the negative glow present near the cathode of
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149
the normal mode, we expect the electron temperature to be very cold—approximately 0.2
or 0.3 eV. We measured the electron temperature to be near 1 eV, which is hotter than,
we expect. We do not present this temperature as a completely accurate value for reasons
discussed in section 5.2; the temperature is probably overestimated.
We believe,
however, the data supports the hypothesis that the temperatures are cold and do n o t
change significantly along the z axis. The electron temperatures measured for the M EN
case again show features present in both the normal and NGEB mode profiles. It appears
that no striations exist in the presence o f a magnetic field. This was also supported by the
ion density profile shown in Fig. 5.5.1. The temperatures from 0 - 150 cm are colder
than what was measured for the NGEB mode, but are elevated with respect to the
negative glow temperatures measured in the NDC glow.
Although it is probably true that the temperatures in both the NGEB and M EN
modes are approximately 0.2 —0.3 eV, in determining the ion densities, we have used Te
calculated from the EEDF. The apparent values calculated from the probe traces are
probably elevated by the magnetic field. Increasing the field strength seems to elevate
the apparent value even more. Since the ion density depends on, among other variables,
the electron temperature, this implies that the calculated ion density is probably small
with respect to the true value. We believe the error is relatively small, since the ion
density goes like the y ff^ , thus providing a value that is at least o f the correct order o f
magnitude. Also, since this error is systematic, we can still follow the spatial trends in
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150
Comparison of Plasma Potentials for Nitrogen
1 3 .5 m T o r r p r e s s u r e , 6 0 s e e m
0
50
100
150
200
250
300
-5 0
Vp NG EB
Vp M EN
Vp N D C
-100
;
1
iL
o -1 5 0
-200
-
-2 5 0
-3 0 0
cm
F ig u r e 5 .5 .3 A x ia l e v o lu tio n o f th e p la s m a p o te n tia ls f o r th e th r e e d if f e r e n t e x c ita tio n c o n d itio n s .
the temperature and density, even though the absolute densities may not be completely
accurate.
In the MEN glow, from the 150 cm m ark to the end o f the discharge chamber, Te
gradually increases by nearly 2.5 eV. This increase in Te can be explained by observing
the evolution o f local plasma potentials in Fig. 5.5.3.
In all three cases, we observe
relatively no change in Vp where the negative glow plasma is present. In the normal
mode, after the dark space region, there occurs a gradual increase in the plasma potential
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151
until the plasma terminates at the anode. This occurrence can also be observed in the
MEN case, although the increase in potential is somewhat blurred compared to the NDC
mode. The plasma potential remains a fairly stable -6 5 V until about the 150 cm mark,
where it starts its positive climb towards 0 V. After the 150 cm mark, the cold electrons
o f the negative glow are heated by the positive climb in the potential.
Another
consequence related to the increase in Te is that plasma electrons can now provide enough
energy for the collisional ionization and excitation o f neutrals in this region o f the
discharge. As a result, we visually observe a region appearing similar in color to the
positive column o f the NDC mode. A dark space is probably not present in the MEN
mode; rather the two observable regions blur together under the influence of the magnetic
field.
M ode of plasm a
nfp cm'2* 1011
nem cm'2* 1011
NGEB, 264 Gauss
58.13
23.35
MEN, 54 Gauss
50.14
55.76
NDC
28.21
18.60
T a b le 5 .5 .1
C o m p a r is o n o f th e d e n s itie s o b ta in e d u s in g m ic r o w a v e in te r f e ro m e tr y w ith th o s e d e te r m in e d
b y m e a s u r in g u n d e r th e a r e a o f th e c u r v e s sh o w n in F ig . 5 .5 .1 . D a ta is f o r a n itro g e n p la s m a , 1 3 .5 m T o r r ,
6 0 se e m .
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152
The ion column densities o f the plasmas described above can be calculated using
Simpson's Rule to approximate the area under the curves shown in Fig. 5.5.1. Column
densities were also measured directly using microwave interferometry. Recall that the
interferometry experiment measures the line integrated density o f electrons present in the
path o f the microwave beam propagating through the discharge tube. Table 5.5.1 shows
the correlation between the two measurements. For the case of the normal mode with an
applied magnetic field, the measurements agree to within 12%. The column densities
obtained for the other two cases using the probe data are significantly higher than what
was measured using microwave interferometry. The value is high by nearly a factor o f
two for the NGEB plasma, and high by 30% for the NDC case. Sudit [11] also observed
this correlation to be true for the normal case, and showed that the ion densities are
overestimated when using the ion current to extract values. We believe that significant
physical differences between the NGEB, MEN and NDC each affect the result o f the
expression that determines the ion density in unique ways. We expect, for example, that
Te is overestimated in the presence o f a magnetic field. If we use what we believe is a
more accurate value for the temperature, the resulting density is increased by
approximately a factor o f 2, which makes the correlation worse between the
interferometry and probe experiments.
We also know from our experiments that the
magnetic field increases the ion column density significantly, compared to the densities
produced in the normal mode.
Yet the measured ion current does not increase
proportionally with respect to the density, probably because Te decreases. While this
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153
tends to confirm that the electron temperature is indeed colder in the presence o f a
magnetic field, we cannot improve the accuracy o f the data until we know exactly what
happens to the temperature as the field strength is applied and its strength is increased.
Column Densities for Nitrogen Plasma in Normal Mode
13.0 mTorr, 60 seem, varied current and field strength
90
80
• 24 G
♦ 44 G
■ 53 G
70
_
60
* 50
* 40
° 30
20
I
10
0
50
100
150
200
250
300
350
mA current
F ig u r e 5 .5 .4 C o lu m n d e n s itie s d e te r m in e d b y m ic r o w a v e in te r f e r o m e tr y fo r a n itr o g e n p la s m a f o r v a r ie d
c o n d itio n s o f s u p p ly c u r r e n t a n d m a g n e tic f ie ld s tr e n g th .
W e investigated how the current and magnetic field strengths affected the
densities in both the normal and NGEB modes.
Fig. 5.5.4 shows how the densities
increase as the discharge current and magnetic field strengths are increased for the
normal m ode o f plasma. The relationship appears to be linear with respect to current.
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154
Column Densities for a Nitrogen Plasma Operating in the NGEB
Mode.
13.0 mTorr, 60 seem, varied current and field
25
88 Gauss
] ■ 176 Gauss
[ • 2 6 4 Gauss
20
15
I
10
0.5
1.5
2
2.5
3.5
m A m ps
F ig u r e 5 .5 .5
E le c tr o n c o lu m n d e n s itie s d e te r m in e d fay m ic r o w a v e in te r f e r o m e tr y fo r v a r ie d c u r r e n t a n d
m a g n e tic f i e l d c o n d itio n s
Interferometry data compiled for the NGEB discharge is presented in Fig. 5.5.5
and Fig. 5.5.6. As expected, increasing the supply current and magnetic field strengths
increases the column density for the plasma, as shown in Fig. 5.5.5. The relationship
between the current and the column density for a particular field strength and pressure is
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155
Saturation of Density for Increasing Field Strength for NGEB
Mode
13.56 mTorr, 60 seem, 1100 V, a few mAmps
30 ♦ density
25 -
20
_
♦
♦
-
o
*
15 -
E
°
10
-
5 -
0 0
100
200
300
400
500
Gauss
F ig u r e 5 .5 .6 E le c tro n c o lu m n d e n s itie s d e te rm in e d b y m ic r o w a v e in te rfe ro m e try . N itr o g e n p la s m a u n d e r
c o n s ta n t p re s s u r e , f lo w r a te a n d s u p p ly c u r re n t a n d v o lta g e c o n d itio n s .
T h e e f fe c t o f in c r e a s in g th e
m a g n e tic f ie ld s a tu r a te s a t a r o u n d 2 2 0 G .
linear. Irregularities in the curve are due to error in the readings o f the supply current off
o f the magnet, as well as the phase shift reading o ff o f the phase shifter. Both meters are
small, with low-resolution tick marks. The phase shifter reading also depends on a pW
signal, which is barely detectable using a fairly sensitive power meter. The effect o f the
magnetic field strength seems to have an upper limit to its ability to enhance the plasma
density. As observed in Fig. 5.5.6, this upper limit seems to occur at 220 G for 13.0
mTorr nitrogen in the NGEB mode.
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156
We did not observe a saturation in the density enhancement when increasing the
current and voltage supplied to the discharge. It is possible to determine if an increase in
the cathode voltage would increase the length o f the high density region further down the
tube or just increase the density at each point, leaving the profile the same, using a
Langmuir probe. Either o f these possibilities would increase the electron column density
(thus the ion column density) as measured by microwave interferometry.
Based on the probe data, we believe that the electron beam extends the full length
o f the chamber. The beam becomes diffuse due to collisions, and for this reason we lose
visual confirmation o f its presence. Ballistic electrons originate from the cathode surface,
obtaining energies near 1 keV after acceleration through the cathode fall in the first few
cm.
These electrons are primarily forward-scattered.
With collisions, these ballistic
electrons lose some o f their energy and scatter in other directions, causing the beam to
become more diffuse as it extends down the length o f the tube. Since we don't know the
cross-sections for the various collisional and scattering processes, we can't predict how
far it should extend given the initial conditions of the electrons. The data presented
describing the axial evolution of the ion densities, electron temperatures, and the plasma
potentials suggest that if the electron beam terminated, then a positive column should
follow due to the electric field gradient. The NDC and MEN modes show exactly this
phenomenon. Visually these modes are very different from the NGEB mode. Also, the
evolution o f measured plasma parameters show an increase in both the plasma potential
and the electron temperature values in the positive column.
This occurency is not
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157
observed in the axial evolution o f plasma parameters for the NGEB mode. The axial
evolution o f Vp in the NGEB mode suggests that any electric field is quite small (see
below), which is confirmed by the unchanging axial values for Te. Given this data, it is
reasonable to conclude that the beam sustains the ionization between the cathode and the
anode.
The beam electrons most likely do not carry the bulk o f the current flowing
between the cathode and the anode. Rather, cold electrons, which we detected with the
probe, carry the bulk o f the current. The current flowing through the cathode sheath is
entirely carried by ions falling out o f the bulk plasma. The secondary electrons emitted
from the sheath in the process, which form the beam, carry only a small fraction o f the
current. No other mechanisms are present that can create these ballistic electrons. While
these high-energy electrons are responsible for sustaining ionization processes, their
numbers are probably too few to sustain the 2 mA current drawn from the supply. The
only other electrons available to carry the current are the cold, bulk electrons. If this is
the case, then there must be a small electric field present, which drives the current.
In
fact, careful review o f the evolution of plasma potentials in the NGEB glow shows that
there occurs a gradual increase in VP, A V in magnitude, starting about 50 cm from the
cathode and continuing to the end of the tube.
We should point out that the density data for the NGEB mode shown in Fig. 5.5.1
is somewhat surprising given what is presented in the figures illustrating Vp and Te. Since
the plasma potential and electron temperatures are constant, we would expect the density
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158
profile to be uniform along the length o f the tube as well, rather than sharply decrease at
around 140 cm. We can provide several explanations for this observation, although they
are somewhat speculative. It's possible that the beam is indeed present along the length
o f the tube, but at around 140 cm, the ballistic electrons are sufficiently depleted or
reduced in energy to a point where they give less plasma density. This could explain the
decrease in ion density at the 140 cm mark, yet also justify why Te and Vp do not change
significantly. It is also possible the profile is at least partially an artifact o f the probe's
perturbation to the plasma.
Radial Ion Density Profile at 200 cm from Cathode
N 2, negative glow, 13 mTorr, 66 seem, 260 G
------------------------------------------------- 4 ! ♦ ni ( e 9 ) :
J .J
♦
Oon
X
rn
E
o
" ""
”
♦
♦
.. .........
.—
—■
♦
- —■ ■ -
A
£
1l . j
ii
—
♦
o
♦
c
L
1
A C
-6
-4
-2
0
2
4
6
cm from the center point
F ig u r e 5 .5 .7 P r o f ile o f th e r a d ia l s p a tia l d is tr ib u tio n 2 0 0 c m fro m th e c a th o d e in th e N G E B d is c h a r g e .
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159
There is other evidence that the beam is present at 200 cm from the cathode front.
We were able to radially sample the NGEB discharge at 200 cm from the cathode front,
without observing a flip in the mode. The radial density profile is presented in Fig. 5.5.7.
The dip in the density as the probe nears the region o f the beam is probably an artifact o f
the probe interference with the beam. This interference results in a decrease in the beam's
ability to provide ionizing collisions, which support the ion density. Further away from
this center, the probe does not interfere with this sustaining mechanism, which results in a
measurable increase in the density value.
Additional probe experiments that map the axial and radial evolution of plasma
parameters for other pressures, currents or magnetic fields can shed more light on the
behavior o f the beam electrons in the MEN and NGEB glow processes. Higher pressure
conditions, where the electron mean free path is smaller, may shorten the high density
region, while lower pressures may extend it as mentioned already.
Increasing the
discharge current or magnetic field strength might also extend the high-density region.
This kind o f information would be valuable for understanding the discharge mechanisms
and also for optimizing the ion production for microwave spectroscopy experiments.
Finally, a more quantitative analysis of the probe data would be very useful in
determining absolute plasma parameters evolving along the discharge length. For this to
happen, further study and development of reliable analysis methods for probe
measurements in the presence o f a magnetic field are necessary.
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160
References
1. R. C. Woods, Rev. Sci. Instrum., 44, 282 (1973).
2. I. D. Sudit, Ph. D. Thesis (Physics), University o f Wisconsin-Madison p. 29
(1992).
3. F. C. DeLucia, E. Herbst, G. M. Plummer, and G. A. Blake, J.. Chem. Phys. 78(5),
2312-2316(1983).
4. W. T. Conner, Ph. D. Thesis (Physics), University o f Wisconsin-Madison p. 1618(1989).
5. I. D. Sudit and R. C. Woods, Rev. Sci. Instr. 64, 2440-48 (1993).
6. I. D. Sudit, Ph. D. Thesis, p. 166-182 (1992).
7. W. T. Connor, Ph. D. Thesis, p.58 (1989).
8. I. D. Sudit, Ph. D. Thesis, p. 22 (1992).
9. I. D. Sudit and R. C. Woods, Rev. Sci. Instr. 64, 2440-48 (1993).
10. I. D. Sudit, Ph. D. Thesis, Chapter 5 (1992).
11. I. D. Sudit, Ph. D. Thesis, p. 289 (1992).
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161
Epilogue
The goals of the work presented in this thesis were (1) to investigate the
feasibility o f using a full wave helical resonator plasma source for molecular ions and
unstable molecules, which could then be detected using microwave spectroscopy, by
physically and electrically characterizing the plasma and (2) to measure the spatial
distribution o f densities, temperatures, and plasma potentials in a magnetized DC hollow
cathode source. Gaining this type o f deeper understanding of these discharges would be
very helpful in finding the experimental conditions that produce sufficient densities o f
unstable molecules.
We were partially successful in determining the feasibility o f the full wave
resonator for microwave spectroscopic studies. We still believe that this construction has
some potential advantages over the quarter wave structure originally implemented in our
laboratory and that it still may be a useful source for its intended purpose.
The
microwave spectroscopy experiment requires a long path length through the plasma. The
antenna length, which is wrapped around the tube, is only about 2.75 m long for the
quarter wave structure at 27.12 MHz. Because o f the large diameter o f the discharge tube
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162
plus insulation, this configuration allows for only about 2.75 turns around the system.
With fairly compressed turns the volume o f gas excited by the antenna is very short, and
probably not sufficient to produce measurable column densities o f unstable molecules.
Stretching out the turns to increase the length o f plasma excited by the antenna helps this
problem somewhat, but stretching it to an effective length o f 1 m would give such widely
spaced turns that it would spoil the current sheath approximation of the antenna coil
described in Chapter 2, and we would lose the ability to predict a properly tuned antenna
length. The full wave structure, on the other hand, provides us with 11 m o f antenna,
which wrap approximately 11 times around the plasma volume. These additional turns
provide us with a longer path length o f excited unstable molecules while preserving a
reasonable turn spacing.
In addition, if the capacitive currents are compensated for
internally by the plasma structure (as described in Chapter 2 and claimed by Vinegradov
and Yoneyama), then the radiating RF problem should be minimized since the plasma
should act more like a purely inductive source, but we have not confirmed this. The other
potential advantage o f this source is the reported confinement of the plasma inside the
antenna structure, which would yield several benefits, but we have not achieved this
reliably yet either.
While the RF helical resonator source would probably not replace
the magnetized DC plasma source, we believe exploring its feasibility is still worthwhile,
since it may produce higher concentrations o f certain species. Experiments exploring the
chemistry and physics o f this RF source may also be valuable to groups interested in
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163
other applications o f this type o f source, e. g., those researching plasma processing
techniques using RF sources.
We have found several reasons why the full wave helical resonator is not yet a
useful source for its intended purpose. Foremost is the practical problem o f radiated RF,
which will impair the function o f the sensitive electronics o f the microwave spectrometer,
and most other nearby instrumentation.
improve.
The shielding o f the source must therefore
Perhaps completely enclosing the antenna with a well-grounded shield (as
suggested in Chapter 4) will improve this problem. An equally critical issue is related to
choosing an appropriate antenna length that will satisfy the resonance condition along its
length. As described in Chapter 2, the length o f the antenna must be tuned with respect
to the density, and to obtain exact resonance it would need to be readjusted dynamically
when the density changed. A disadvantage o f the full wave resonator is that as the
antenna gets longer, it becomes somewhat more impractical to get it on resonance since
the change in length required for tuning is also proportionately greater. For our system,
the predicted antenna length for a 109 cm'3 is about 8 m long compared to a length o f
nearly 10 m long for a 1011 cm'3 density. The difference in length corresponds to 2
additional turns around the tube for the higher density.
We can generate densities
spanning this range easily with our source, and therefore tuning to the correct length,
which will fulfill the resonance condition, is a problem that must be solved. The easiest
solution is to tune the frequency o f the source rather than change the length o f the
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164
antenna. This would of course require a new RF power system, but even more important
would be greatly diminishing the radiated RF interference to avoid breaking FCC rules.
The Langmuir probe experiment is potentially a valuable tool in determining if
the full wave resonator is producing plasmas that will be useful for our purposes. We
were not able in this research to implement the probe experiment in such a manner as to
achieve its foil potential in an RF environment. We still gained, however, some useful
information from the characteristic traces, which we believe suggests that the source, as it
is currently implemented, has characteristics similar to those that were described by
Vinegradov and Yoneyama. As described in Chapter 4, there seems to be a high-density
region confined inside the antenna, the RF fluctuations reduce in the central region o f the
plasma, and there is some evidence for hints o f the expected nodal structure.
We have
explained design or procedural problems with the Langmuir probe experiments in
Chapter 3. We believe the lumped element RF filter, which folly compensates for the
stray capacitance in the Langmuir probe holder, is a promising solution to the RF spoiling
o f the trace. We have included what we believe is a sound procedure for tuning the filter
in Chapter 3. When this is properly implemented it should still be possible to obtain high
quality Langmuir probe data, although doing this has proved much more difficult than we
anticipated. This improvement in technique will be essential for the foil characterization
o f the helical resonator source.
The magnetized DC glow discharge is still the best source for generating unstable
molecular ions and radicals for our microwave experiments. The geometry provides us
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165
the desired long path length for the microwave beam, there are no fluctuating fields that
could impair the sensitive electronics o f the spectrometer, and the densities produced by
this source are in a useful range for the experiment.
This system has already been
successfully employed for microwave study o f many transient species. In Chapter 5, we
have provided a good method for measuring the spatial distribution o f plasma parameters
for this source using Langmuir probes. We have used it to map plasma temperature,
density, and potential in magnetized plasmas and have thus greatly increased our
understanding o f the internal structure o f the electron beam negative glow type discharge
that has proved so useful for microwave spectroscopy. This probe method can now be
applied with confidence to further explore experimental conditions that are optimum for
the production o f molecular ions. The effects o f the discharge voltage and the magnetic
field strength on the density and temperature profiles o f the source are issues that have
not been flilly addressed in this work, but more experiments exploring these relationships
could be very helpful in trying to understand the optimum conditions for producing
molecular ions.
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166
Appendix A Detailed Description of the Pressure Control System
I. Introduction
The production of ions in a plasma depends strongly on the pressure and the flow
o f gases in the system. Typically, the flow o f gases— and thus the initial pressure— is
established before the discharge is actually turned on. Once the discharge is started, the
pressure can vary several mTorr from the initial pressure. Since our experiments require
a stable flow rate of gases and steady-state pressure conditions, it's necessary to
continuously monitor the pressure, since the value changes with fluctuations in the
temperature o f the cooled walls, discharge current and voltage, as well as fluctuations in
the magnetic field. One can manually adjust the flow o f gases to re-establish the desired
pressure. This procedure, however, proves tedious, since the slightest adjustment can
cause large changes in pressure.
This is especially the case when operating with
pressures o f only a few mTorr.
This pressure control box operates as an interface between an MKS 510 power
supply read-out for a type 120 high accuracy pressure transducer and an MKS 247C 4
channel read-out for the 100 Series flow controllers. The pressure is set using a dial on
the control box, which in turn regulates the flow controller.
The control box then
monitors the output o f the pressure transducer and compares it to the pre-set voltage
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167
value.
Any difference will generate an error voltage that will drive the mass flow
controller to change the flow as needed to maintain the set pressure.
While this box was designed specifically as an interface for the MKS 51 OB and
the MKS 247C, minor changes in the circuit would allow this box to operate as an
interface between most electronic pressure transducer and mass flow control devices.
II. Theory o f Operation
The control box works as a simple feedback circuit to control the pressure. The
control o f pressure, however, involves more than just flow o f gas in and out o f a system.
Thus a general discussion o f pressure control is provided, followed by a detailed
description o f the pressure control circuitry.
The flow rate of a system at steady state is defined as the product o f the pumping
speed (1/s) and the pressure (Torr). Conductance, for example, in a narrow tube or a
valve adds in parallel to the pumping speed such that:
I - - U !
s sc c
(i)
where So is the initial pumping speed, C is the conductance of the system, and S is the
effective pumping speed. The smaller the value o f conductance, the more it will reduce
the effective pumping speed. Narrow tubing, elbows, valves, and rough surfaces are just
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168
a few examples o f features that reduce the conductance o f a system, and thus increase the
effect o f conductance on the pumping speed.
The system considered in this case consists o f two chambers, represented
schematically in Figure 1. The first chamber is a manifold where gases are initially
introduced into the system. The manifold is made o f pyrex tubing. This tubing adds
conductance to the system, since the dimensions are long and fairly narrow (about 1 inch
in diameter), as does a 3/8" diameter by 2' long glass connecting tube.
The second
chamber is where the plasma is formed. This chamber is also long, however it is 15 cm
in diameter. This results in negligible conductance added to the system.
In this lab we are interested specifically in the pressure o f the plasma chamber—
this is the pressure one would like to control. A problem arises here in trying to design a
circuit for this purpose. In steady state flow the mass flow must be equal all along the
Ft
F2
w-
\
m a n if o ld
f3
w
____________
W
S
p la s m a c h a m b e r
d if fu s io n p u m p
F ig u r e 1. S c h e m a tic o f a flo w s y s te m u s e d to in tr o d u c e g a s e s in to a p la s m a c h a m b e r .
F I is th e flo w
p r o d u c e d b y th e M K S 2 4 7 C F lo w C o n to lle r . F 2 a n d F 3 a re flo w s in to a n d o u t o f th e p la s m a c h a m b e r . S is
th e p u m p in g s p e e d o f th e d iffu s io n p u m p .
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169
the flow path.
Since the conductance of the discharge tube is much larger than the
manifold, the discharge pressure must be correspondingly lower. The conductance o f the
manifold itself is greater than that o f the smaller tube connecting it to the discharge tube.
Also, since the manifold is emptying into a m uch higher volume, lower pressure
chamber, it takes a considerably long time for the pressure to reach steady state inside the
second chamber.
If the circuit responds in haste, it will overdrive the mass flow
controller, overfilling the manifold. The manifold will slowly empty out; in the process,
however, the pressure in the plasma chamber will overshoot the pressure set by the
circuit.
The circuit will again over respond by turning the mass flow controller
completely off. By the time the transducer measures the effects of this, the pressure in
the manifold is too small to immediately provide flow to the second chamber.
The
pressure in the plasma chamber dives below the desired value. The circuit responds by
turning the mass flow controller completely on once again.
In short, unless a time
constant is incorporated into the circuit to slow down the response, the system pressure
will oscillate horribly.
One can predict the magnitude o f the time constant by characterizing the system
using linear, differentiation equations with constant coefficients to describe the steady
state pressures o f each chamber. Combining these equations and solving for the pressure
in the plasma chamber, one can predict what the pressure in the plasma chamber will be
given an initial flow out o f the controller. Also, one can graph the pressure with respect
to time to deduce how long it might take to reach a steady state. A FORTRAN program,
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170
PRESSURE, has been developed for this purpose. Given the initial conditions o f the
system, the volumes o f the first and second chambers, and the conductance o f the
manifold, a graph of pressure vs. time is displayed whereby the final pressure and the
time constant can be deduced. We have included a copy o f the program at the end o f this
appendix.
The time constant delay is incorporated into the circuit by means o f an integrator.
If the time constant matches roughly the time it takes for the second chamber to reach a
steady state, the controller works well. Damped oscillations may still occur.
These
oscillations can be minimized by small adjustments in the time constant by means o f a
trim-pot that adjusts the magnitude of the input resistance to the integrator.
The possibility remains that oscillations may occur, not because o f the controller,
but because of the diffusion pump. Overwhelming the pump with gas causes the oil in
the diffusion pump to be below the boiling temperature. This causes the pump to stop
working, i. e., it’s “choked”—until the oil heats up more. While the pump is choked,
pressure continues to increase in the chamber and, while the controller will turn the flow
off, it is impossible to maintain a constant pressure without a pump pulling gas out o f the
chamber. When the pump begins to work, pressure is reduced unless the pump is choked
again. If the effective pumping speed is known, one can calculate the maximum flow
into the second chamber using the equations given at the beginning of this chapter. If the
pump chokes because the user desires a higher pressure, closing the gate valve may solve
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171
the problem. This limits the flow o f gas by using conductance to decrease the effective
pumping speed. Thus a stable increase in pressure is allowed by the system.
III. Pressure Control Circuitry
The pressure control box operates as an interface between an MKS 510 power
supply read-out for a Model 120 high accuracy pressure transducer and MKS 247C 4
channel read-out for the 100 series flow controllers.
The circuit works as a simple
feedback circuit to control pressure. A schematic o f this circuit is included in Fig. 2 and
Fig. 3 at the end o f this appendix. A diagram o f the board is provided in Figure 4. The
readout for the pressure transducer is a 0-1 VDC linear output (with respect to pressure)
on pin four o f connector J3, with 1 VDC corresponding to 1 Torr. This voltage is input
into a LF353 operational amplifier. This voltage is multiplied by five and compared to
the 0-5 VDC reference voltage set by the operator. A differential amplifier with a gain o f
100 makes the comparison and outputs an on/off signal o f +•/- 12.5 VDC. This becomes
the input for the integrator, which provides the closed-loop time constant discussed in the
last section.
The time constant was approximated at 50 seconds using the program
PRESSURE. The integrator adds a 90 degree phase to the output o f the differential
amplifier. To correct for this phase shift, we use an inverting amplifier with a gain o f 1.
The 4 channel read-out accepts a 0-5 VDC linear input on pin 4 o f connector P6. This
input controls mass flow controller 1.
A voltage o f 0 VDC turns the controller
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172
completely off, while 5 VDC drives the controller full open at 200 seem (standard cubic
centimeter per minute1). The output o f the inverting amplifier has a range o f —12.5 to
+12.5 VDC. An absolute value and limiter circuit was added before the final output, in
order to protect the 4 Channel Read-out from voltages below zero or above 5 VDC.
The control box can be easily modified to work with different systems than the
one described here. The wire-wrap sockets eliminate the need to do a lot o f soldering.
First, as discussed in the last section, the closed-loop time constant must be adjusted. If
the value of the approximately estimated time constant differs largely from the one
employed here, the feedback capacitor as well as the input resistor can be easily changed.
Next, the magnitudes of the transducer output and the controller input need consideration.
Differences in the transducer output range from the one mentioned here can be scaled
accordingly, by changing the gain o f the X5 amplifier.
Differences in the input
characteristics of the controller can be accounted for, by changing the values o f the non­
inverting inputs o f the absolute value circuit.
1 1 seem corresponds to .0127 Torr* L/s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Schematic for Pressure Control Box Power Supply
Black
O n/O ff
Vi A m p, 120V fuse
W hite
Green
LB 1234
ADJ
LM3I7T
+ 12.5V
35V
1
_
“
2pF
Tant
IOOOu F
G round
35V
I
IOOOuF
IOOO, R l
- 12.5 V
ADJ
LM 337
Figure 2 Schem atic o f the pow er supply for the pressure control system .
o
u>
174
en
a
oo
■MGO
<N
op
+ 12.5 V
E
*/%o
4*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
175
o
O
c*-*
— o
o
— o o
OO
o
EO uT
GO
=L
oo
CG
oo
O
v>
•'T
cn
cn
—
O
oc
|
O
N
O
I
u ^ T
> I II
r m
r a
—
I
o
o
eo
oo
o
it.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3 Schematic of the circuit elem ents.
O
176
Appendix B
The Lieberman Helical Resonator Model
The following calculations are set up according to "Theory of a Helical Resonator Plasma Source".
The purpose is to model the resonance, loading, and heating mechanisms of a helical resonator
plasma of a given geometry. In the mentioned paper, eight equations with eight unknowns are
derived. For a non-zero solution set, the det [A] = 0, yielding a dispersion relation for the different
modes of plasma that can arise.
From the equations given in the paper, first lets construct [A]:
a := .05
eo := 8 .8 5 4 1 10~ 12
b := .10
p.
c := .40
:= 4-ti- 10” 7
i|/ := .1 0 0
L := 1.5
(
U .6 0 2 1 0
p o ( p ,c o ) := |p 2 - ffl2- 4-7t ~10
9 '6
J
-io
n2
( \
.
-I9l2
l l O 17
J ------------------------
8 .8 5 4 1 0 I2-9 .1 0 9 1 0 -31
p(co):=i-----------4- n-co —
p------10
8
tt
2 10l2-4-7t"-ep(co)
p a ( p ,c o ) := p —c o -----------------------9 -1 0 16
co := 27.12
The proceeding matrix was simplified. First, complex terms were divided out. Next, I used algebra
to make the coefficient expressions dimensionless. Finally, variables s and t were used to substitute
for the following expressions:
p olP .co
(a
\
2 -jc -c o IO 6
‘(p.®) := ------ :— :—
3.0-10 p o(p ,co
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
M(P, co)=
c o sh [p a(P , co)(b-a)]
0
0
0
-sin h [p #(P , co)a]
-c o sh [p a(P , co)a]
e p(co)sinh[pa(p , co)a]
0
0
0
0
0
0
0
0
0
-1
0
0
0
/?sin( I//)
0
S co s(y /)
t(p , co)Scos( y/)
0
0
0
-s (p , co)
S s in ( ^ )
0
S c o s( y)
t(P , co)Scos( i//)
0
0
0
s(P, w)
1
0
-t(p , co )S c o s(^)
0
t(P , co)Ssin(y/)
-/?sin(y/)
0
0
0
0
-t(P , co)/?cos( \j/)
0
t(P , co )/?sin (^)
-S sin (y /)
-Pcos(i//)
0
0
sin (y /)
-t(P , co)0 sin(yr)
0
0
0
0
0
-t(p , co)cos( I//)
-t((3, co)F’s in ( ^ )
< 3 sin (f)
F ig u re 2 .3 .2 T h is sh o w s th e 8 x 8 m a trix d e riv e d fro m th e c o e ffic ie n ts o f e ig h t e q u a tio n s a n d e ig h t u n k n o w n s. T h e u n its o f th e c o e ffic ie n ts a re
d im e n sio n le ss. V a ria b le fl= c o sh [p 0(P , co)(b-a)], S= sin h [p 0(P , co)(b-a)], P= s in h [p 0(P , co)(b-c)], a n d Q= c o sh [p 0(P , co)(b-c)]. V a ria b le s / a n d s
a re fu n c tio n s o f p a n d co, a n d w e re u se d to m ak e th e u n its o f th e c o e ffic ie n ts d im e n sio n le ss.
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