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NORTHEASTERN UNIVERSITY
Graduate School of Engineering
Thesis Title: Design, Fabrication and Modeling of Microfabricated Inductively Coupled
Plasma Sources
Author: Felipe Iza
Department: Electrical and Computer Engineering
Approved for Thesis Requirement of the Master of Science Degree
______________________________________________________ _________________
Thesis Advisor: Jeffrey A. Hopwood
Date
______________________________________________________ _________________
Thesis Reader: Nicol E. McGruer
Date
______________________________________________________ _________________
Thesis Reader: Carey M. Rappaport
Date
______________________________________________________ _________________
Thesis Reader:
Date
______________________________________________________ _________________
Department Chair: Fabrizio Lombardi
Date
Graduate School Notified of Acceptance:
______________________________________________________ _________________
Director of the Graduate School: Yaman Yener
Date
NORTHEASTERN UNIVERSITY
Graduate School of Engineering
Thesis Title: Design, Fabrication and Modeling of Microfabricated Inductively Coupled
Plasma Sources
Author: Felipe Iza
Department: Electrical and Computer Engineering
Approved for Thesis Requirement of the Master of Science Degree
______________________________________________________ _________________
Thesis Advisor: J. A. Hopwood
Date
______________________________________________________ _________________
Thesis Reader: N. E. McGruer
Date
______________________________________________________ _________________
Thesis Reader: C. M. Rappaport
Date
______________________________________________________ _________________
Thesis Reader
Date
______________________________________________________ _________________
Department Chair: F. Lombardi
Date
Graduate School Notified of Acceptance:
______________________________________________________ _________________
Director of the Graduate School: Yaman Yener
Date
Copy Deposited in Library:
______________________________________________________ _________________
Reference Librarian
Date
DESIGN, FABRICATION AND MODELING OF MICROFABRICATED
INDUCTIVELY COUPLED PLASMA SOURCES
A Thesis Presented
by
Felipe Iza
to
The Department of Electrical and Computer Engineering
in partial fulfillment of the requirements
for the degree of
Master of Science
in
Electrical Engineering
in the field of
Electronic Circuits and Semiconductor Devices
Northeastern University
Boston, Massachusetts
July 2001
Design, Fabrication and Modeling of mICP Sources
Abstract
ABSTRACT
Microsystems that integrate mechanical and optical structures have been
fabricated for a wide range of applications in recent years. This thesis focuses on the
design, fabrication and performance of microfabricated inductively coupled plasma
(mICP) sources with the ultimate goal of integrating them in plasma-based microsystems.
Large ICP sources are extensively used in the semiconductor industry because of
their high efficiency, high ion density and plasma controllability. mICP sources,
however, present a poorer performance than their large system counterparts. It is the aim
of this thesis to investigate the factors that limit the performance of mICP sources in
order to obtain design guidelines for more efficient designs.
A typical mICP source consists of a planar spiral-like coil microfabricated on a
glass substrate. Using single-turn coils instead of spirals allow us to flip over the devices
such that the coil is adjacent to the plasma. This increases the coupling between the coil
and the plasma while keeping the fabrication process requirements down to one mask.
Previous mICP source experiments and models suggested that increasing the
frequency of operation would lead to better performance of mICP sources. Although this
is true at low frequencies, no efficiency improvement is observed at frequencies much
larger than the electron collision frequency (>3). A new model that incorporates the
effect of the electron inertia on the conductivity of the plasma seems to agree with the
experimental results.
ii
Design, Fabrication and Modeling of mICP Sources
Abstract
For mICP sources operating at ~1GHz the new model predicts maximum
efficiency at pressures of a few torr. The main factor limiting the efficiency of the device
at high frequency is the coil resistance, which is increased to ~20 times the DC value by
the proximity effect.
iii
To my fiancée, Myung Hee Kim,
and to my parents, Felipe Iza and Amparo Pérez,
for their love and support
iv
Design, Fabrication and Modeling of mICP Sources
Acknowledgements
ACKNOWLEDGMENTS
There are many people who in one way or another have helped me in coming to
Northeastern University and in completing this thesis. I would like to show them my
gratitude and thank all of them.
I want to express my deepest and sincere gratitude to my academic and thesis
advisor, Dr. J. A. Hopwood, for his guidance and support during these last two years. His
many helpful insights, suggestions and detail discussions have made the completion of
this work possible. Without his direction and motivation I would not have been able to
pursue this thesis in a field in which I had absolutely no background.
I would also like to express my gratitude to the rest of faculty, staff, and
colleagues at the Plasma Engineering Laboratory and the Microfabrication Laboratory
(MFL) Group at Northeastern University for their friendship, support and help. In
particular, I would like to thank Michael Miller for his not always appreciated work in
keeping the labs up and running, Weilin Hu for his practical suggestions working in the
lab, and Patricia Nieva, Xiaoqing (Vivian) Lu, Xiaoji Yang, and Juan Carlos Aceros for
their friendship and always interesting discussions.
I would also like to thank Claudia Costanzo, Program Director of the Commission
for Cultural, Educational and Scientific Exchange between the United States of America
and Spain, for giving me the opportunity to come to Boston under a Fulbright
scholarship, and her help and advice with the bureaucracy that is always hidden behind
v
Design, Fabrication and Modeling of mICP Sources
Acknowledgements
these scholarships. I would also want to thank ENDESA for sponsoring my studies
through the Fulbright Program.
Finally, I would like to express my deepest gratitude to Dr. Fernando Arizti for his
help and support when coming to the USA seemed almost impossible.
This project was also supported by the National Science Foundation under Grant
No. DMI-0078406.
vi
Design, Fabrication and Modeling of mICP Sources
Table of Contents
TABLE OF CONTENTS
ABSTRACT
ii
ACKNOWLEDGEMENTS
Error! Bookmark not defined.
TABLE OF CONTENTS
vii
1 .- INTRODUCTION
1
1.1 .- TYPES OF MICROPLASMA SOURCES
2
1.2 .- MICP SOURCES: PREVIOUS DESIGNS
4
1.3 .- WHAT IS PLASMA
7
1.3.1 .- Sheaths
8
1.3.2 .- Plasma Conductivity
11
2 .- NEW MICP SOURCE DESIGN
13
2.1 .- MICP SOURCE MODEL
13
2.2 .- FREQUENCY SELECTION
16
2.3 .- COUPLING COEFFICIENT IMPROVEMENTS
17
2.4 .- COIL PARAMETERS
21
2.4.1 .- Coil Resistance
21
2.4.2 .- Coil Inductance
22
2.5 .- PLASMA PARAMETERS
24
2.5.1 .- Plasma Resistance
24
2.5.2 .- Plasma Inductance
26
2.6 .- COIL WIDTH SELECTION
26
2.7 .- MATCHING NETWORK
27
3 .- FABRICATION
31
3.1 .- FABRICATION ISSUES
32
3.1.1 .- Photolithography
33
3.1.2 .- Gold Electroplating
34
vii
Design, Fabrication and Modeling of mICP Sources
Table of Contents
4 .- EXPERIMENT DESCRIPTION
36
4.1 .- SET UP
37
4.2 .- PROBES
39
4.2.1 .- Probe Design
42
5 .- PERFORMANCE OF THE NEW MICP SOURCE
5.1 .- ION DENSITY AND ELECTRON TEMPERATURE CALCULATION
44
44
5.1.1 .- Step 1: Plasma Potential
47
5.1.2 .- Step 2: Probes Potential
48
5.1.3 .- Step 3: Area Ratio
49
5.1.4 .- Step 4: Ion Current
49
5.1.5 .- Step 5: Electron temperature
51
5.1.6 .- Step 6: Ion density
52
5.2 .- FREQUENCY OF OPERATION AND MATCHING
52
5.3 .- ELECTRON TEMPERATURE
54
5.4 .- ION DENSITY
54
6 .- NEW MICP SOURCE MODEL
57
6.1 .- NEW PLASMA MODEL
57
6.2 .- NEW EFFICIENCY EXPRESSION
62
6.2.1 .- Efficiency As A Function Of The Frequency Of Operation
63
6.2.2 .- Efficiency As A Function Of The Power Absorbed By The Plasma
66
6.2.3 .- Efficiency As A Function Of Pressure
68
6.3 .- APPROXIMATION FOR LARGE AND MICROFABRICATED ICP SOURCES
71
6.3.1 .- Frequency Of Operation
73
6.3.2 .- Pressure And Power Absorbed By The Plasma
74
6.4 .- MODEL AND EXPERIMENTAL RESULTS AGREEMENT
76
7 .- LOSSES IN MICP SOURCES
77
7.1 .- SKIN EFFECT
77
7.2 .- PROXIMITY EFFECT
79
viii
Design, Fabrication and Modeling of mICP Sources
Table of Contents
7.3 .- CAPACITIVE COUPLING
85
7.4 .- EXPERIMENT RESULTS AND MODEL PREDICTIONS WITH LOSSES
86
8 .- CONCLUSIONS AND FUTURE WORK
87
9 .- REFERENCES
89
APPENDICES
91
APPENDIX I: MINIMUM GLASS THICKNESS
92
APPENDIX II: COUPLING COEFFICIENT
94
APPENDIX III: PROGRAM USED TO DESIGN THE NEW MICP SOURCES
95
APPENDIX IV: MATCHING NETWORK DESIGN
104
APPENDIX V: 5-MM SINGLE-TURN MICP SOURCE PARAMETERS
106
APPENDIX VI: FABRICATION PROCESS TRAVELER
107
APPENDIX VII: PROBE MEASUREMENT CURVE FITTING
109
APPENDIX VIII: PROXIMITY EFFECT IN A SINGLE TURN COIL
118
INDEX OF FIGURES
Figure 1.1 Voltage distribution in the plasma
9
Figure 1.2 Induced electric field in the plasma region
4
Figure 1.3 Ion density of three mICP sources made on copper clad epoxy board in
argon at 370mtorr, 1.3W [7]
5
Figure 1.4 Ion density created by different mICP sources @ 350mtorr, 1W
6
Figure 2.1 ICP source model
13
Figure 2.2 Equivalent ICP source model
14
Figure 2.3- Magnetic field in a mICP source
17
Figure 2.4 a) Multi-turn coil with cavity etched at the back of the wafer b) Single
turn coil flipped over
18
Figure 2.5 Multi-turn mICP source
19
Figure 2.6 Coupling coefficient as function of the separation between the coil and
the plasma
20
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Design, Fabrication and Modeling of mICP Sources
Table of Contents
Figure 2.7 Cross section of the coil
22
Figure 2.8 Wire loop
22
Figure 2.9 Wire loop approximation
23
Figure 2.10 Electric field and electron density distribution in the plasma region
25
Figure 2.11 Predicted power efficiency of a 5mm single loop mICP source
27
Figure 2.12 Matching network schematics
28
Figure 2.13 Single turn mICP source
29
Figure 3.1 mICP source on plastic substrate
32
Figure 3.2 Cracks in the photoresist
34
Figure 3.3 mICP sources fabricated a) DI water wet before electroplating b) Soapy
solution wet before electroplating
35
Figure 4.1 mICP source mounted on package and bonded to the glass tube
37
Figure 4.2 Experiment set up
38
Figure 4.3 Typical voltage-current characteristic for a single Langmuir
40
Figure 4.4 Typical voltage-current characteristic for a double probe measurement
41
Figure 4.5 Probes a) double probe b) coaxial probe
42
Figure 5.1 Typical voltage-current characteristic for a coaxial probe in the mICP
44
Figure 5.2 Coaxial probe schematic
46
Figure 5.3 Iterative process for calculating the electron temperature and the ion
density
47
Figure 5.4 Inner and outer conductor potential
49
Figure 5.5 Regions in the voltage-current characteristic of a coaxial probe
50
Figure 5.6 Ion current fitting
51
Figure 5.7 Ion density and the power reflection coefficient as function of frequency
for a constant amplitude input signal of –8dBm (~150mW) a) with the
25% additional tuning capacitor added b) without the additional tuning
capacitor. (Device in flipped over configuration)
53
Figure 5.8 Electron temperature a) Device I b) Device II (Flipped over)
54
Figure 5.9 Ion density generated by the new mICP sources
55
Figure 6.1 New ICP source model
58
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Design, Fabrication and Modeling of mICP Sources
Table of Contents
Figure 6.2 Characteristic curves of a resistance inversely proportional to the power
it dissipates
59
Figure 6.3 Voltage across the plasma impedance
60
Figure 6.4 Equivalent circuit for the new ICP source model
60
Figure 6.5 ICP Source efficiency as function of the frequency of operation
64
Figure 6.6 Ion density vs. frequency of operation for 3 different mICP sources
operating in Argon at 300mtorr, 1.3W. From Hopwood et al. [7]
65
Figure 6.7 ICP Source efficiency as function of the power absorbed by the plasma
67
Figure 6.8 Efficiency as function of the power absorbed by the plasma and the
frequency of operation for a constant pressure
68
Figure 6.9 Equivalent Plasma Resistance
74
Figure 7.1 Non-uniform current distribution due to the skin effect
77
Figure 7.2 a) Current distribution in the coil b) Equivalent current distribution using
the skin depth
78
Figure 7.3 Eddy currents in the coil
80
Figure 7.4 Non-uniform current distribution due to the proximity effect
80
Figure 7.5 Coil Effective Resistance Decomposition
82
INDEX OF TABLES
Table 1.1 Comparison of different microplasma sources
4
Table 4.1 Test conditions
36
Table 6.1 Large and microfabricated ICP source comparison
72
Table 7.1 Coil resistance increment as function of frequency
79
Table 7.2 Efficiency loss due to the proximity effect when the frequency is
increased from 690 MHz to 818 MHz
84
xi
Design, Fabrication and Modeling of mICP Sources
Introduction
1.- INTRODUCTION
It was in 1965 when Gordon Moore, then Fairchild Semiconductor's R&D
director, made the famous observation that chip capacity doubles every 18 moths. Since
then the number of transistors on a chip has increased more than 3,200 times leaving
behind a knowledge and technology that in the last decades has been used to fabricate not
only transistors but also other structures in a micro-scale (micromachines).
The integration of microelectronic circuitry into micromachined structures brings
up the possibility of fabricating completely integrated systems (microsystems) that have
the same advantages of low cost, reliability and small size as traditional integrated
circuits. Microsystems that integrate mechanical and optical structures have been
fabricated for a wide range of applications including automotive, telecommunications,
biochemistry, bioengineering and consumer electronics.
This thesis focuses on the design, fabrication and performance of microfabricated
inductively coupled plasma (mICP) sources with the ultimate goal of integrating them in
a plasma-based microsystem. Although many large scale ICP sources use helical coils, 3dimensional structures are costly and hard to fabricate with conventional microfabrication
processes. Therefore planar structures are desirable for fabricating a cost effective
microplasma source. Plasma-based microsystems would find application in gas analyzers,
ion thrusters, sterilizers, plasma displays and pixel-addressable plasma processing.
Page - 1 -
Design, Fabrication and Modeling of mICP Sources
Introduction
ICP sources are typically modeled as air-core transformers. In this thesis we show
that the model used for large ICP systems is not appropriate to describe the performance
of microfabricated ICP sources operating at high frequencies. A new model that
incorporates the effects of the electron inertia and the power and pressure dependence of
the plasma impedance is presented in chapter 6. This new model agrees with the
experimental results obtained in this thesis as well as with the results from previous
generations of mICP sources.
1.1.- TYPES OF MICROPLASMA SOURCES
Several microplasma sources operating by different principles have been reported
in recent years. In this section we look at the advantages and disadvantages of each
approach, and thereby justify our interest in mICP sources.
Direct current (DC) plasma sources have been fabricated for optical emission
detectors [2],[3]. Although the voltage applied between the electrodes can be reduced as the
dimensions get smaller, DC microplasma sources still require high voltages (~800V)
which for certain applications can be inconvenient and dangerous. However these devices
are easy to fabricate, compact and require simple electronics to operate them. The main
drawback of these devices is the electrode erosion due to the constant bombardment of
ions driven by the perpendicular electric field. This erosion limits the usage of the
devices to few hours of operation.
Capacitive coupled plasma sources,[4],[5] present a longer life than DC sources
Page - 2 -
Design, Fabrication and Modeling of mICP Sources
Introduction
because the electrodes can be protected with low sputtering yield materials even if these
are insulating. Capacitive coupled plasma sources are simple to fabricate and compact,
but they require more complicated electronics than DC sources to drive them. The fact
that the electric field is perpendicular to the electrodes limits the ion density achievable
with these devices. As the power applied to the device increases, so does the energy lost
by the ions accelerated in the sheath regions resulting in little ion density gain. Typically
capacitive coupled plasma sources, although more efficient than DC sources, produce ion
densities 10 times smaller than inductively coupled or microwave plasma sources.
Large microwave plasma sources are popular due to their high efficiency.
However the dimensions of the device are strongly related to the frequency of operation.
A microfabricated plasma source of few millimeters would require frequencies of
operation of the order of 10 to 100 GHz, or by the same token, microwave sources
operating at ~1G would be of the order of several centimeters in size.[6]
Finally inductively coupled microplasma sources have been reported recently.[7],[8]
The schematic of a microfabricated inductively coupled plasma source is shown in Figure
1.1. A planar spiral like coil generates a magnetic field that induces an electric field in the
azimuthal direction. Since the electric field is parallel to the wall, increasing the power in
the device does not translate into a higher energy loss due to ions being accelerated in the
sheath region. Therefore, higher ion densities can be achieved than in DC and capacitive
coupled plasmas.
Page - 3 -
Design, Fabrication and Modeling of mICP Sources
Introduction
Coil
H
Glass wafer
Sea
l
Plasma
E
Glass tube
VACUUM REGION
Figure 1.1 Induced electric field in the plasma region
Table 1.1 summarizes qualitatively the pros and cons of different microplasma
sources in terms of their size, electronic complexity, plasma intensity and life of the
device.
Microplasma type
Size
Electronic
complexity
Plasma
density
Life
DC
Small
Simple
Low
Short
Capacitively coupled
Small
Medium
Medium
Medium
Inductively coupled
Medium
Medium
High
Long
Large
Complex
High
Long
Microwave
Table 1.1 Comparison of different microplasma sources
1.2.- MICP SOURCES: PREVIOUS DESIGNS
Several mICP sources have been developed at Northeastern University. This
section presents a review of these designs that help us to put the new design in
perspective and see the evolution in the performance of the mICP sources.
Page - 4 -
Design, Fabrication and Modeling of mICP Sources
Introduction
The first attempt to fabricate a miniaturized inductively plasma source was a
20-turn coil wound around a 6mm Pyrex tube. The performance of this device was
strongly limited by the losses in the coil, which is a concern in all mICP sources.
The next mICP sources were fabricated as planar spiral-like coils on a 1-oz copper
clad epoxy board. 3-turn 5-mm coils, 5-turn 10-mm coils and 6-turn 15-mm coils were
fabricated and tested. The planar structure of these devices makes them compatible with
microfabrication techniques. It was noticed that the efficiency of these devices increased
with the frequency of operation (See Figure 1.2), although the ion density obtained was
an order of magnitude lower than in large ICP sources.
1
.
2
e
+
1
0
1
.
2
e
+
1
0
1
.
0
e
+
1
0
IonDesity(cmIonDesity(cm -3 ) -3 )
1
.
0
e
+
1
0
8
.
0
e
+
9
8
.
0
e
+
9
6
.
0
e
+
9
6
.
0
e
+
9
4
.
0
e
+
9
4
.
0
e
+
9
2
.
0
e
+
9
2
.
0
e
+
9
0
.
0
0
0
.
0
0
1
0
0
1
0
0
2
0
0
3
0
0
2
0
0
3
0
0
F
r
e
q
u
e
n
c
y
(
M
H
z
)
1
5
m
m
c
o
i
l
1
0
m
m
c
o
i
l
1
5
m
m
c
o
i
l
5
m
m
c
o
i
l
1
0
m
m
c
o
i
l
5
m
m
c
o
i
l
4
0
0
5
0
0
4
0
0
5
0
0
F
r
e
q
u
e
n
c
y
(
M
H
z
)
Figure 1.2 Ion density of three mICP sources made on copper clad epoxy board in argon at
370mtorr, 1.3W [7]
Page - 5 -
Design, Fabrication and Modeling of mICP Sources
Introduction
This lower ion density is due to an increase in the surface to volume ratio of mICP
sources that leads to higher wall recombination. A low efficiency is also due to the losses
in the coils and perhaps due to a higher capacitive coupling between the coil and the
plasma. As mICP sources shrink down, these effects become more pronounced and lead
to worse efficiencies (lower ion densities for the same RF power).
The last generation of mICP sources is a 3-turn 5-mm spiral-like coil fabricated of
gold on a 700m thick glass wafer. It operates at ~450MHz and it can generate ion
densities of 1011cm-3 while consuming 3W.
Figure 1.3 shows the evolution in the efficiency of mICP sources including the
new design that is the subject of this thesis.
New mICP
source
IonDesity(cm -3 )
1
e
+
1
1
1
e
+
1
0
mICP source on
glass wafer
mICP source on
wound copper clad epoxy
on a Pyrex
board
tube
1
e
+
9Coil
1
e
+
8
01
0
02
0
03
0
04
0
05
0
06
0
07
0
08
0
09
0
0
F
r
e
q
u
e
n
c
y
(
M
H
z
)
Figure 1.3 Ion density created by different mICP sources @ 350mtorr, 1W
Page - 6 -
Design, Fabrication and Modeling of mICP Sources
Introduction
1.3.- WHAT IS PLASMA
Plasma, in the microfabrication context, is a weakly ionized gas in which free
electrons and ions move randomly in every direction. The term weakly ionized means
that the density of electrons and ions is much smaller than the density of neutral
molecules in the gas (typically less than 1%). Since free electrons are generated by
stripping electrons from neutral atoms, electron-ion pairs are generated and the number of
electrons and ions in the plasma is essentially the same. Therefore on average the plasma
can be considered neutral.
Energy is needed to strip electrons from neutrals in order to start and maintain the
plasma. This energy can be of various origins and in the case of mICP sources electrical
energy is used. If the energy flowing into the plasma is insufficient, the plasma
recombines into a neutral gas.
In a low pressure, electrically-driven plasma the electrons are much hotter than
the ions, and the plasma is said to be in non-thermal equilibrium. The high reactivity of
the ions generated in the plasma and the low temperature of the gas are used in the
semiconductor industry in many fabrication processes.
The next sections present basic relationships in plasma theory that will be needed
in future chapters. The expressions are not derived and the reader is referred to
reference [1] for a detailed explanation and derivation of these formulae.
Page - 7 -
Design, Fabrication and Modeling of mICP Sources
Introduction
1.3.1.- Sheaths
Although the plasma is essentially neutral, this is not true at plasma chamber walls
where the plasma forms a region called the sheath. In this region the ion density is larger
than the electron density. The formation of the sheath is due to the higher mobility of the
electrons that diffuse faster than the ions from the bulk of the plasma to the walls. As
electrons pile up at the walls, a potential is formed that repels additional electrons and
accelerates the ions toward the walls.
The potential difference between the walls (floating potential Vf) and the plasma
(plasma potential Vp) is such that the number of electrons reaching the wall balances out
with the number of ions so no net current flows. Between the sheath and the bulk of the
plasma there exists a region called the presheath where a small electric field exists to
match the boundary conditions in between the sheath and the bulk of the plasma (See
Figure 1.4).
Assuming that electrons follow the Boltzmann distribution, the electron flux to the
wall is given by:

1
e  n es v e e
4
q(Vp  Vf )
kTe
1
8 k Te 
 n es
e
4
π me
q(Vp  Vf )
kTe
Eq. 1.1
where nes is the electron density at the edge of the sheath, v e the average electron
velocity, Te the electron temperature, me the electron mass, q the electron charge and k
Page - 8 -
Design, Fabrication and Modeling of mICP Sources
Introduction
the Boltzmann’s constant.
On the other hand, ions are accelerated by the field in the presheath to the Bohm
velocity and do not follow a Boltzmann distribution. The ion flux to the wall in terms of
the Bohm velocity (uB) is given by:
i  nis u B  nis
kTe
Mi
Eq. 1.2
where nis is the ion density at the edge of the sheath, and Mi the ion mass.
Sheath
Pre
sheath
Plasma Bulk
(ne  ni)
Pre
sheath
Sheath
nes = nis
ni
ni
ne
ne
Wall
Wall
Sheath
Plasma
Potential
(Vp)
Floating
Potential
(Vf)
Pre
sheath
Plasma Bulk
(Vp)
Pre
sheath
Sheath
e
e
i
i
Wall
Wall
Figure 1.4 Voltage distribution in the plasma
An expression for the difference between the plasma potential and the floating
Page - 9 -
Design, Fabrication and Modeling of mICP Sources
Introduction
potential in terms of the electron temperature can be found by setting the electron flux
equal to the ion flux:
(Vp  Vf ) 
1 k Te  M i 

ln 
2 q
 2 π me 
Eq. 1.3
The ion and electron density at the edge of the sheath is related to the ion and
electron density in the bulk of the plasma:
n es  n e e

1
2
 n is  n i e

1
2
Eq. 1.4
The Debye length is a characteristic length scale in a plasma. It is a measure of the
distance that the potential of a charged object penetrates into the plasma and is
proportional to the sheath thickness.
λ De 
ε o Te
q ne
Eq. 1.5
where o is the vacuum permittivity.
For a biased body the sheath thickness (s) can be calculated using Child’s law:
 2Vp  V   4
2

s
λ De 
3
Te


3
where V is the potential of the biased body.
Page - 10 -
Eq. 1.6
Design, Fabrication and Modeling of mICP Sources
Introduction
1.3.2.- Plasma Conductivity
Another important characteristic of the plasma that will be needed in future
chapters is the plasma conductivity. The plasma conductivity is given by:
σ
ε oω2pe
jω  ν
Eq. 1.7
where pe is the electron plasma frequency,  is the frequency of operation and  the
electron-neutral collisional frequency.
The electron plasma frequency is the fundamental characteristic frequency of the
plasma and represents the frequency at which the electron cloud oscillates with respect to
the ion cloud. It is given by:
ω 2pe 
q2ne
ε o me
Eq. 1.8
Combining Equation 1.6 and 1.7 we find another expression for the plasma
conductivity:
σ
q2ne 1
jω  ν me
Eq. 1.9
This expression shows the dependence of the plasma conductivity on the electron
density. At low frequencies the plasma can be considered a poor conductor with
Page - 11 -
Design, Fabrication and Modeling of mICP Sources
Introduction
conductivity proportional to the electron density. However at high frequencies the
conductivity becomes complex and the plasma behaves inductively.
Page - 12 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
2.- NEW MICP SOURCE DESIGN
In this chapter we describe the motivation and the procedure followed to design
the new mICP source.
2.1.- MICP SOURCE MODEL
Modeling is a useful technique to understand the behavior of a system and
identify the parameters that affect its performance. Inductively couple plasma sources are
typically modeled as an air-core transformer with the coil (source) acting as the primary
of the transformer and the plasma (single current loop) as the secondary (Figure 2.1).
This model is a direct representation of the physical phenomena occurring in an
ICP source. Rc represents the coil resistance, Lc the coil inductance, Lp the inductance of
the plasma due to the single loop of current induced by the coil, Rp the plasma resistance
due to the collisions of the electrons within the current loop, and k the coupling
coefficient between the coil and the plasma.
Rc
I
1
V
Coil
Mk L c L p
Lc
I
Rp
Lp
2
Plasma
Figure 2.1 ICP source model
Page - 13 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
Simple manipulations of the circuit in Figure 2.1 lead to the following equivalent
circuit:
Rc
I
1
R eq  R p
Req
R 2p  ω2 L2p
L’ = Lc + Leq
R’
V
k 2ω 2 L p Lc
Leq   L p
k 2ω 2 L p Lc
R 2p  ω2 L2p
Figure 2.2 Equivalent ICP source model
where Req and Leq are the equivalent plasma resistance and inductance referred to the
primary of the transformer.
The power efficiency of an ICP source can then be calculated as the ratio of the
equivalent plasma resistance divided by the total resistance of the circuit in Figure 2.2:
η
R eq
R c  R eq

k 2ω2 L p Lc R p
Eq. 2.1
R c R 2p  ω 2 L2p R c  k 2 ω 2 L p L c R p
This expression can be simplified for the case of a mICP source and a large ICP
system as follows:
η
R eq
R c  R eq

k 2ω 2 L p Lc R p
R c R 2p  k 2ω2 L p Lc R p
For R p  ω L p
(mICP)
Eq. 2.2-a) b)
η
R eq
R c  R eq
Page - 14 -
2

k Lc R p
R c L p  k 2 Lc R p
For R p  ω L p
(Large ICP)
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
It is clear that although the efficiency of a large ICP source does not depend on
the frequency of operation (a well-known experimental observation), for a mICP source
the efficiency depends on the square of the frequency. This fact had been reported in
previous mICP sources and will be further investigated in this work.
It should also be noted that the efficiency of both large and micromachined ICP
sources depends on the square of the coupling coefficient. Therefore the model predicts
that a relatively small improvement on the coupling coefficient and a small increase in
the frequency of operation can lead to much more efficient mICP sources since the
efficiency depends on the square of these two parameters.
The rest of the parameters that affect the efficiency do have a smaller impact than
the coupling coefficient and the frequency of operation although they should also be
taken into account during the design. Equation 2.2-a can be rewritten as :
η
R eq
R c  R eq

1
For R p  ω Lp
 1   Rc   Rp 
1  2 2 


 k ω   Lp   Lp 
(mICP)
Eq. 2.3
As we might expect, the efficiency gets better as the ratio between the resistance
and the inductance decreases. Therefore the coil should be designed to maximize the
inductance while minimizing the resistance. The same is true for the plasma, although its
parameters depend on the type of gas and the power supplied to the mICP source and
therefore they are not design variables.
Page - 15 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
The new mICP source is designed and tested to corroborate these predictions and
obtained more intense plasmas with a more efficient design.
2.2.- FREQUENCY SELECTION
Increasing the frequency of operation of mICP sources leads to better
performance. However, the frequency of operation is limited by three factors:

Physical: The operating frequency should be lower than the self-resonance
frequency of the coil.

Economical: High frequency electronics for a power supply might get too
expensive as the frequency increases and eventually limit the viability of a
plasma-based microdevice.

Practical: The signal generator used for the testing of the new mICP source
(HP8656A) and the power amplifier (EIN603L) do not go beyond 1GHz.
With this three factors in mind the frequency of operation for the new mICP
source was chosen to be 900MHz. This frequency is below the self-resonance frequency
of previous multi-turn coils and therefore should be well below the self-resonance of the
new mICP source which is based on a single loop coil. Moreover ~1GHz is in the
frequency range at which mobile phones and other telecommunication equipment work,
and therefore a final power supply design could take advantage of the electronics already
available at these frequencies. And finally, 900MHz is a frequency at which we can test
the devices, having some margin of error for mismatches than can occur during the
Page - 16 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
fabrication process.
2.3.- COUPLING COEFFICIENT IMPROVEMENTS
The coupling coefficient (k) of equation 2.1 is a measure of how much of the
magnetic field generated by the coil actually intersects the plasma region. Figure 2.3
shows a schematic of the plasma source and the plasma region in which magnetic field
(H) lines have been sketched. It is easy to see that as the coil is separated from the plasma
region fewer and fewer lines reach the plasma, and therefore the coupling coefficient
tends to zero. On the other hand if the coil comes in intimate contact with the plasma, all
the lines generated by the coil will intersect the plasma and the coupling coefficient
becomes 1.
H
Coil
Glass wafer
Seal
Plasma
VACUUM REGION
Glass tube
Figure 2.3- Magnetic field in a mICP source
In a mICP source, the coupling coefficient is limited by two factors: the thickness
of the glass substrate (wafer) where the device is fabricated and the sheath width of the
Page - 17 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
plasma. The sheath of the plasma can not be ignored as the glass substrate gets thin and
there is not much to do to minimize it other than making the plasma more intense (See
equation 1.6). It is possible to use thinner substrates to increase the coupling coefficient.
However, the thickness will eventually be limited by the mechanical strength needed to
withstand the pressure difference between the plasma region (in vacuum) and the outer
world (atmospheric pressure). The maximum coupling coefficient can be achieved by
etching a cavity at the back of the wafer only under the coil (Figure 2.4-a). However,
even in this case the thickness cannot be reduced beyond ~200m. The minimum
thickness calculation can be found in Appendix I.
Wire Bond
One-turn coil
Coil
Glass
wafer
Plasma
Cavity
Glass
Tube
Thin
protective film
Glass
Tube
Plasma
Figure 2.4 a) Multi-turn coil with cavity etched at the back of the wafer b) Single turn coil flipped
over
A way to get rid of the separation due to the glass wafer is flipping over the
device (Figure 2.4-b). Since no substrate separates the plasma region from the coil, it is
Page - 18 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
necessary to electrically isolate the coil and the plasma as well as protect the coil from
being sputtered by the plasma.
This protective/insulating layer can be very thin because the mechanical strength
required to withstand the pressure difference between the plasma region (in vacuum) and
the outer world (atmospheric pressure) is provided by the substrate which can be as thick
as needed. Since the new device is to be tested in argon, a layer of photoresist can be used
as a protective/insulating layer. However a protective layer more resistant to oxygen
containing gases can be obtained by using other substances such as spin-on-glass.
Wire bond
Figure 2.5 Multi-turn mICP source
Multi-turn coils (spirals) need a wire bond to connect the center of the coil with
one of the pads (Figure 2.5). This wire bond does not allow us to flip over the device
because it is not insulated and makes the sealing of the plasma chamber difficult. For this
reason the new mICP source is designed as a single-loop coil which keeps the fabrication
process simple and allows us to flip over the device to investigate the effect of the
coupling coefficient in the overall performance of the device.
Page - 19 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
By approximating the coil and the plasma by two coaxial circular conductors, it is
possible to calculate the coupling coefficient between these two. The coupling coefficient
between two circular loop conductors can be calculated using the Neumann’s formula
[9]
for the mutual inductance of two circular loop conductors. The calculation involves
elliptical integrals and the details can be found in Appendix II. Figure 2.6 shows the
variation of the coupling coefficient as function of the separation between the two coils,
or in our case, the coupling coefficient as function of the separation between the coil and
the plasma.
By flipping over the device we can reduce the separation from ~850m (glass
wafer + plasma sheath) down to ~400m (plasma sheath only) and therefore improve the
coupling coefficient by a factor of ~1.7.
1
.
0
0
.
8
0
.
6
CouplingCoeficnt(k)
0
.
4
0
.
2
0
.
0
0
2
0
0
4
0
0
6
0
0
8
0
0
1
0
0
0
S
e
p
a
r
a
t
i
o
n
(
u
m
)
Figure 2.6 Coupling coefficient as function of the separation between the coil and the plasma
Page - 20 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
2.4.- COIL PARAMETERS
The new mICP source uses a 5-mm single turn coil to create a plasma. The reason
for the single turn (instead of a spiral as in previous designs) is explained in the previous
section. The diameter was chosen to be the same as in the previous design so designs can
be compared under similar conditions.
The new mICP source is made of gold. Although copper and silver have lower
resistivity, experience shows that they oxidize quickly while gold circuits do not present
any degradation in performance even after several months. Silver or copper devices may
be reconsidered for the flipped over devices since these devices do require a protective
layer. A new fabrication process would need to be developed however.
The thickness of the coil is chosen to be ~10m as in previous designs. Thicker
films would not reduce the resistance of the coil significantly due to the skin effect but
would be substantially more difficult to fabricate.
Given the number of turns, the outer diameter of the coil, its thickness and the
electrical properties of the gold, the coil should be designed to maximized the power
efficiency of the source. The only parameter left to be determined is the width of the coil.
We therefore need to express all the parameters in the mICP model in terms of this
parameter so the optimum width can be found.
2.4.1.- Coil Resistance
The coil resistance can be calculated as function of the coil width using:
Page - 21 -
Design, Fabrication and Modeling of mICP Sources
Rc  ρ
New mICP Source Design
2 π rave
A effec
Eq. 2.4
where  is the resistivity of gold at the operating temperature, rave is the average radius of
the coil, and Aeffec is the effective cross section area of the coil that incorporates the skin
effect. Figure 2.7 shows the effective area in a cross section schematic of the coil.
A effec  w  h  (w  2 δ)(h  2 δ)
rout
rave
h
...where δ the skin depth and is given by :

δ
Aeffec
w
1
f πσμ
Figure 2.7 Cross section of the coil
2.4.2.- Coil Inductance
It is not trivial to find an analytical closed form expression for the inductance of a
rectangular cross section loop. For this reason we need to use an approximation in order
to estimate the actual value of the inductance. Two methods have been used to predict the
inductance, both leading to similar values:

Wire approximation:
A wire loop self-inductance is typically calculated as: [9]
r
d
  8r  
L c  μ r  ln    2 
 d 
Figure 2.8 Wire loop
Page - 22 -
Eq. 2.5
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
where  is the permittivity of the gold, r is the average radius and d is the
diameter of the wire.
In order to use this expression we need to approximate the coil by a wire using
the average radius (rave) and a wire diameter equal to the height of the coil (h)
as shown in Figure 2.9.
rave
h
Figure 2.9 Wire loop approximation

Semiempirical formula by Sunderarajan et al.: [10]
Several semiempirical equations have been developed for the calculation of
different geometry spirals. One of them is the equation developed by
Sunderarajan et al. which can be particularized for a single turn coil:
Lc 
di
n
w

μ n 2 d avg c1   c 2 
2
 ln 
  c3 ρfill  c4 ρfill  Eq. 2.6
2
  ρfill 

...where for a single turn coil:
c1 ,c 2 ,c3 ,c 4 are constants
n 1
dout
ρfill 
d out  2  rout  w 
d out  2  rout  w 
d avg  d out  w
Page - 23 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
Both methods give similar values of inductance and provide us with an expression
for calculating the inductance as function of the coil width (w).
2.5.- PLASMA PARAMETERS
In the model presented in section 2.1, the plasma is modeled with an inductor (Lp)
and a resistor (Rp) as a single loop coil. The diameter of this imaginary coil is set by the
diameter of the mICP source coil that induces the electric field in the plasma region.
2.5.1.- Plasma Resistance
The plasma resistance is harder to predict than the coil resistance because the
plasma conductivity and the electric field vary across the plasma. The plasma
conductivity depends on the electron density which is not constant across the plasma. The
electron density is maximum at the center of the tube and zero at the walls as a result of
the diffusion process of the electrons within the plasma. Typically the radial electron
density distribution within a cylindrical chamber volume can be considered a type-1
Bessel function.
On the other hand, the electric field induced in the plasma is zero at the center of
the tube, reaches a maximum somewhere under the coil and decreases as we move
towards the glass tube. For these calculations the electric field has been approximated by
a sinusoidal. Figure 2.10 shows a qualitative graph of the ion density and the electric field
in the plasma as function of the radius.
Page - 24 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
Electron density (ne)
Electric field (E)
r=0
r
center of the tube
radius of the tube
Figure 2.10 Electric field and electron density distribution in the plasma region
The plasma resistance can then be calculated as a parallel connection of
differential cylinders, each having a constant electron density, and therefore constant
conductivity.
r
d
r
Glass
Tube
rtube
1
h σ(r)

dr
0
RP
2 π r E(r)
h
Eq. 2.7
...where  is the conductivity of the plasma
E is the electric field
Plasma
differential
element
h is the plasma length
Rp the plasma resistance
The conductivity as function of the electron density is given by Equation 1.8, and
therefore it is possible to perform the integral knowing the variation of the electric field
and the electron density as function of the radius r. The peak value of the electric field
and the ion density depend on the power absorbed by the plasma.
Page - 25 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
Sheaths have been neglected in this calculation and the exponential variation of
the electric field along the longitudinal axis of the tube has been approximated by a
constant field within an effective plasma length (h).
It might seem that the plasma resistance is not calculated accurately and actually it
is true. However, we can vary the power absorbed by the system to control the electron
density and, therefore, the resistance of the plasma. Notice that we are not trying to
estimate the plasma resistance accurately, but rather we are developing a model to
perform an analysis that leads us to the best possible coil design. And the best coil design
is independent of the plasma characteristics, although the actual efficiency will depend on
the plasma resistance.
2.5.2.- Plasma Inductance
Since the current flowing in the plasma also forms a single loop, the plasma
inductance is approximately the same as the coil inductance.
2.6.- COIL WIDTH SELECTION
The expressions developed in sections 2.4 and 2.5 can be used in equation 2.1 to
calculate the efficiency of the mICP as function of coil width. A MATLAB program that
performs this calculation can be found in Appendix III. The efficiency as function of the
width is presented in Figure 2.11.
The plot suggests that the thicker the coil the better the performance of the mICP
Page - 26 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
source gets. Although the result also suggests that the efficiency would be maximum
when the coil becomes a disk, this result is not accurate as the inductance formula used
for this calculation was derived only for coils (coil width << coil radius). Therefore that
part of the graph is ignored and the coil width is chosen to be half the radius (1.25 mm).
1
.
0
0
.
8
0
.
6
Eficeny(%)
0
.
4
0
.
2
0
.
0
0
5
0
0 1
0
0
01
5
0
02
0
0
02
5
0
03
0
0
0
C
o
i
l
W
i
d
t
h
(
u
m
)
Figure 2.11 Predicted power efficiency of a 5mm single loop mICP source
2.7.- MATCHING NETWORK
Once the coil width is selected the values of the coil resistance, coil inductance,
plasma resistance and plasma inductance in our model are determined. These values lead
to a coil-plasma equivalent circuit that presents an arbitrary input impedance. However, it
is desired that the input impedance of the mICP source equals the output impedance of
the power supply at the frequency of operation, so the power transfer from the power
Page - 27 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
supply into the mICP source is maximized.
To achieve this, a matching network consisting of two capacitors is used (Figure
2.12). The input impedance equals the power supply impedance (typically Rsupply = 50 )
for the following values of capacitance:
Ct
Input
impedance
Rc
Cm
Matching
network
Ct
Rp
M
Lc
Coil
Lp
Ct 
Cm
Matching
network
 L' ω 

R
supply
L' ω 
R’
L’
 R'  R'  ω

Plasma
Cm 
Input
impedance
1
1
Ct ω
2

1  
2 
  ω
 R'  L' ω 
C
ω

t

 
Coil
+
Plasma
Figure 2.12 Matching network schematics
The derivation of these expressions for the required capacitances can be found in
Appendix IV. Note that the matching can be achieved only if Rsupply > R’, which holds
true for a mICP source.
The matching network is implemented by two interdigital capacitors fabricated in
gold. The digits of the capacitors are 10m wide, 8m high and they are separated by
Page - 28 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
10m wide gaps. The product “length x number of digits” was calculated extrapolating
data from previous designs and assuming that in a first approximation the capacitance is
proportional to the number of digits and their length.
Once the product is calculated, small aspect ratio modifications were made to
guarantee that the ohmic loses in the capacitors are negligible. The results obtained for a
5mm single turn coil mICP source can be found in Appendix V.
Two additional capacitors were fabricated next to the device to shift the frequency
of operation and investigate the performance of the device at different frequencies. The
capacitance of these additional capacitors is 25% of the tuning and the matching
capacitor respectively (Figure 2.13).
Single turn coil
Matching Capacitor
Cm
Tuning Capacitor
100m
Ct
Digits of the Interdigital
Capacitor
Additional capacitors
Figure 2.13 Single turn mICP source
Page - 29 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Design
Finally, the single turn coil and the matching network are arranged such that
parasitic loops are minimized. The two capacitors share a base and the connections to the
coil are very compact. Figure 2.13 shows a picture of an actual device and a close-up
view of the interdigital capacitor.
Page - 30 -
Design, Fabrication and Modeling of mICP Sources
Fabrication
3.- FABRICATION
A traveler for the fabrication of the mICP source can be found in Appendix VI.
The single turn mICP source can be fabricated as follows:
Sputtering
TiW
Glass Wafer
Au
Cr
Beginning with a 700m glass wafer, a 1000 Å seed
layer of gold is deposited over a 300 Å chrome
adhesion layer. Then a 300 Å TiW layer is
deposited on the gold film to improve the
photoresist adhesion.
Photolithography:
Two coats of AZ®P4620 photoresist are spun to get
a ~15m layer. The photoresist is then exposed and
developed.
TiW plasma etch
The TiW layer is plasma etched in O2+SF6 where it
has been revealed after the photoresist developing.
Gold electroplating
The device is electroplated to a thickness of ~8m
using the photoresist as a mold.
Photoresist strip
Once the device is grown, the photoresist is
stripped.
TiW, Au & Cr etch
Finally, the metal layers are stripped, TiW and gold
in a wet etch and Cr in a O2+CF4 plasma.
Alternatively, the three metal layers can be
physically removed using an ion beam etcher.
Page - 31 -
Design, Fabrication and Modeling of mICP Sources
Fabrication
Protective film
If the device is to be flipped over, the next step is to
apply a protective layer. In the experiments
described in the next chapters a coat of AZ®P4620
has been used as a protective layer.
Finally the wafer is diced and each device placed in a plastic substrate that acts as
a package. A SMA connector is also mounted on the same plastic substrate and the
device is wire bonded to the connector. Figure 3.1 shows a mICP source mounted in the
plastic substrate with a photoresist protective layer on the coil region.
SMA connector
Plastic substrate
Photoresist protective layer
Figure 3.1 mICP source on plastic substrate
3.1.- FABRICATION ISSUES
The fabrication process of a single turn mICP source is fairly simple. However
problems might arise during the photolithography process and the gold electroplating.
Page - 32 -
Design, Fabrication and Modeling of mICP Sources
Fabrication
3.1.1.- Photolithography
The photolithography process is probably the most delicate part of the process
since the photoresist is going to act as a mold later during the electroplating step. Any
defect in the photoresist will translate later into a defect in the final device. Sometimes
these defects are tolerable, but some others are fatal. In general defects due to particles
sitting in the photoresist translate in broken digits in the capacitor since gold will not
grow in that particular spot during the electroplating process. A broken digit does not
have a significant effect on the overall performance of the device as long as the number
of broken digits is small.
The real problems are normally due to cracks and adhesion failures in the
photoresist layer. More often than not, the photoresist layer cracks during the baking
process after the developing step. Reducing the photoresist layer thickness seems to help,
but since the photoresist acts as a mold it cannot be made thinner. Gold seed layers of 600
Å and 1200 Å have been tried to see if they translated into underlying films of lower
stress, but in both cases the photoresist did crack.
Fortunately most of the cracks end in the outer part of the device and do not
propagate across the devices (See Figure 3.2). A more subtle defect in the photoresist
layer occurs when the photoresist does not adhere completely to the TiW layer and forms
a cavity underneath the photoresist digits. These cavities do get filled with gold during
the electroplating process, short-circuiting the capacitor and ruining the device.
Page - 33 -
Design, Fabrication and Modeling of mICP Sources
Fabrication
Figure 3.2 Cracks in the photoresist
3.1.2.- Gold Electroplating
During the electroplating process the defects in the photoresist layer translate into
defects in the actual device. Those defects need to be addressed during the
photolithography process. However, there is an additional issue during the gold
electroplating process which is gas bubbles being trapped in between photoresist digits.
This effect is more prevalent near the bases of the capacitors, and as a result, the gold
plating does not occur in these areas. Figure 3.3-a shows a device from a previous
generation of mICP sources in which most of the digits are damaged due to bubbles
trapped near the bases of the capacitors during the electroplating process.
It was found that dipping the wafer in a soapy solution before starting the
electroplating process wets the photoresist, changes the surface tension and significantly
reduces the number of bubbles that get trapped in between the digits of photoresist
(Figure 3.3-b).
Page - 34 -
Design, Fabrication and Modeling of mICP Sources
a)
Fabrication
b)
Figure 3.3 mICP sources fabricated a) DI water wet before electroplating b) Soapy solution wet
before electroplating
Page - 35 -
Design, Fabrication and Modeling of mICP Sources
Experiment Description
4.- EXPERIMENT DESCRIPTION
Two identical 5-mm single-turn mICP sources have been tested using four
different conditions to investigate the effect of frequency of operation and coupling
coefficient on the source efficiency. These conditions are summarized in the Table 4.1:
Device
Coupling
coefficient
Frequency
Condition 1
I
Low
High
Condition 2
I
Low
Low
Condition 3
II
High
High
Condition 4
II
High
Low
Table 4.1 Test conditions
The first device (Device I) was tested facing outwards from the vacuum chamber
(tube) with the glass substrate in between the coil and the plasma region (in similar
fashion to previous designs). On the other hand, Device II was flipped over to bring the
coil as close as possible to the plasma region to improve the coupling coefficient. Each
device was tested at two different frequencies of operation as the resonant frequency was
varied by adding an additional capacitor to Ct as described in section 2.7.
For each of these conditions the mICP source maintained argon plasmas at 100,
200, 300 and 400 mtorr and the RF power was swept from 200 to 1000 mW for each
pressure. Although the device is capable of sustaining the plasma at pressures of at least
Page - 36 -
Design, Fabrication and Modeling of mICP Sources
Experiment Description
12 torr, the probe theory used to diagnose the plasma assumes non-collisional sheaths
which limit the validity of the calculations to pressures under 400 mtorr.
4.1.- SET UP
Once the device has been fabricated, mounted in the plastic package and wire
bonded to the SMA connector, it is attached to a glass tube that acts as vacuum chamber
(Figure 4.1). The glass tube is terminated in a metal flange that attaches the tube to a
body where the gas inlet, the mechanical pump and pressure gauges are connected.
Figure 4.1 mICP source mounted on package and bonded to the glass tube
A signal generator and a RF amplifier are used to supply power to the mICP
source. A dual directional coupler, a RF switch and a RF power meter are used to
measure the forward and reflected power. Figure 4.2 shows a schematic and actual
pictures of the experiment set up.
The gas used for the experiment is argon, which is fed through the gas inlet, and a
mechanical pump is used to create vacuum in the chamber. A capacitive pressure gauge
Page - 37 -
Design, Fabrication and Modeling of mICP Sources
Experiment Description
1. Single turn mICP source
2. 5/16” Glass tube (vacuum
chamber)
7
1
6
8
2
3. Signal generator
HP8656A (.1 - 990MHz)
4. RF Power amplifier
EIN603L (+30dB)
5
4
5. RF coupler
Pasternack PE2217-20
30dB
3
818.000
-
7.3
6. RF Switch
919C70200
11
13
M
M
K
MK
KSSS
0.53
7. RF Power Sensor
HP8482H
8. RF Power meter HP435A
12
9. Gas inlet (Argon)
10. Needle Valve (SS4)
10
11. Pressure gauge
MKS390HA (10 torr)
9
12. Signal Conditioner
MKS270B
13. Gas outlet (to pump)
14
14. Electrostatic plasma probe
Controller HIDEN ESP004
8
12
3
4
11
9
10
9
2
5
1
7
8
10
7
Figure 4.2 Experiment set up
Page - 38 -
6
13
Design, Fabrication and Modeling of mICP Sources
Experiment Description
measures the pressure in the chamber and a needle valve at the inlet allows us to vary the
pressure by altering the gas flow.
The plasma is diagnosed using a thin Langmuir probe that is discussed in
following sections. An electrostatic plasma probe controller drives the probe and
measures the voltage-current characteristic of the plasma. A computer is used to store the
data for future analysis.
4.2.- PROBES
One of the most commonly used methods for measuring the ion density in a
plasma is by means of Langmuir probes. This plasma diagnosis technique consists of
introducing a metallic probe in the plasma. When the probe is biased to different
voltages, current flows through it and a voltage-current characteristic curve that depends
on the ion density and the electron temperature in the plasma can be obtained.
As the probe is introduced in the plasma, a sheath forms around the probe and the
probe is driven to the floating potential. The probe size needs to be small so the plasma is
not significantly perturbed by the probe. When the probe is at the floating potential the
flux of electrons reaching the probe equals the flux of ions being collected and the net
result is a zero current flowing through the probe.
However if the probe is externally driven to a voltage different than the floating
potential, the electron and ion fluxes are no longer in equilibrium and a net current does
Page - 39 -
Design, Fabrication and Modeling of mICP Sources
Experiment Description
flow through the probe. Figure 4.3 shows a typical voltage-current characteristic for a
single Langmuir probe.
I
I
Plasma
V
V
Figure 4.3 Typical voltage-current characteristic for a single Langmuir
When the probe is driven negatively with respect to the floating potential,
electrons see a higher barrier to reach the probe and the electron flux decreases. On the
other hand, the ion flux does not change with the applied voltage (it is limited by the
Bohm velocity) and the overall result is an ion current being collected by the probe.
Following the sign convention of Figure 4.3 this corresponds to a negative current. When
the voltage applied is negative enough (V >> Te / q) the electron flux is negligible and the
current flowing through the probe is approximately constant. This corresponds to the ion
saturation current and the slight increase as the voltage becomes more negative is due to
an increase in the ion collecting area due to larger sheaths.
Similarly, when the probe is driven positively with respect to the floating
potential, more electrons have enough energy to reach the probe and the electron flux
increases. Since the ion flux is independent of the applied voltage as long as the applied
voltage is smaller than the plasma potential (V<Vp), the net result is electrons being
Page - 40 -
Design, Fabrication and Modeling of mICP Sources
Experiment Description
collected by the probe. Following the sign convention of Figure 4.3 this corresponds to a
positive current. As the voltage continues increasing the ion current becomes negligible
and the current increases exponentially until electron saturation current is reached.
In the new mICP source the plasma is generated in a glass chamber where no
voltage reference is available. This makes it impossible to use a single probe to diagnose
the plasma as the applied voltage will simply drive the plasma potential which is
otherwise floating.
Therefore, a double probe scheme as shown in Figure 4.4 is needed.
V
I
I
A1
A2
Plasma
V
Figure 4.4 Typical voltage-current characteristic for a double probe measurement
In a double probe scheme each probe behaves as a single probe. When no voltage
is applied between the two probes, both probes sit at the floating potential and no current
flows through the probes. When a voltage V is applied between the probes, one probe is
driven positively and the other one negatively with respect to the floating potential. The
voltage of each probe will be such that their difference is the applied voltage V and the
Page - 41 -
Design, Fabrication and Modeling of mICP Sources
Experiment Description
net ion current collected by the negative driven probe equals the net electron current
collected by the positive driven probe. If both probes are identical, a symmetric voltagecurrent characteristic is obtained as shown in Figure 4.4. The current at large positive and
negative voltages is limited by the ion saturation current of the probes.
4.2.1.- Probe Design
Due to the reduced dimensions of the plasma (5-mm in diameter) it is very
important to use very small probes to perturb the plasma as little as possible. Two probes
(See Figure 4.5) have been used to test the devices, both of them following the double
probe scheme.
Probe 1
Probe 2
a)
b)
Figure 4.5 Probes a) double probe b) coaxial probe
The first design is a symmetrical double probe consisting of two silver-coated
wires 0.008” in diameter, 3mm long. The two wires are separated forming a U shape to
Page - 42 -
Design, Fabrication and Modeling of mICP Sources
Experiment Description
prevent their sheaths from overlapping. The second design consists of a coaxial cable in
which the inner conductor (silver-coated 0.008” in diameter) acts a one of the probes and
the outer conductor (copper 0.034” in diameter) as the other probe. The first probe is
2mm long whereas the length of the second probe depends on the plasma length.
The symmetrical double probe has the advantage of symmetry, however, it
perturbs the plasma more significantly than the coaxial probe. The main limitation of the
symmetrical double probe comes from the fact that the ion density in the plasma changes
along the radius of the chamber (See Figure 2.10). Since the two probes need to be
separated ~1.5mm to guarantee that their sheaths do not overlap when a voltage is
applied, the ion density around each probe is quite different for this small plasma and the
probe theory that assumes that both probes see the same plasma cannot longer be used.
On the other hand the coaxial probe is asymmetric, but it perturbs the plasma less
than the symmetrical double probe and it is not affected by the ion density changes along
the radius of the chamber.
The results presented in the next sections were obtained with a coaxial probe
described above.
Page - 43 -
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
5.- PERFORMANCE OF THE NEW MICP SOURCE
Probe measurements where recorded for the conditions described in section 4.
Each voltage-current curve is obtained by sweeping the voltage across the probes from
-50 to +50V in intervals of one half volt and measuring the current flowing through the
probes at each voltage.
5.1.- ION DENSITY AND ELECTRON TEMPERATURE CALCULATION
A typical voltage-current characteristic of the coaxial probe is shown in Figure
5.1. It can be seen that it is clearly asymmetric compared to Figure 4.4 and this is due to
the fact that in the coaxial cable the probes have different areas.
4
0
0
3
0
0
2
0
0
Probecurnt(A)
1
0
0
0
1
0
0
2
0
0
6
0
4
0
2
0
0
2
0
4
0
6
0
A
p
p
l
i
e
d
V
o
l
t
a
g
e
(
V
)
Figure 5.1 Typical voltage-current characteristic for a coaxial probe in the mICP
Page - 44 -
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
The current flowing through each probe is determined by the probe potential with
respect to the plasma potential and it is the net result of two process: electrons reaching
the probe and ions being collected by the probe. Therefore the current in each probe can
be separated in two components, namely an ion current and an electron current.
Assuming that electrons follow the Boltzmann distribution, the electron current in
the probe is given by:

1
I e   q A n es v e e
4
q(Vp  V)
kTe
Eq. 5.1
where A is the probe area including the sheath around the probe and the rest of
parameters are as described for equation 1.1.
On the other hand since ions are accelerated to the Bohm velocity (uB) by the field
in the presheath, the ion current is given by:
Ii  q A nis u B
Eq. 5.2
When there is no voltage applied in between the two probes, both of them are
driven to the floating potential and no net current flows through the probes.
I  Ii  Ie
0   q A n is
n is
Page - 45 -
q(Vp  Vf )



1
u B     q A n es v e e kTe 


4



1
u B  n es v e e
4
q(Vp  Vf )
kTe
Eq. 5.3
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
So the final expression for the net current in each probe as function of the probe
potential is:
I
qVp

1
I  q A n es v e e kTe
4
V
Plasma
Vf
qV
 kT

kTe
e
e  e 




Eq. 5.4
From Figure 5.2 it is clear that the current that flows through the probes (inner and
outer conductors of the coaxial probe) and that in the power supply need to be the same
as the three are connected in series. Therefore it is possible to find a relationship between
the voltage applied in between the probes (V) and the current flowing through the
probes (I) as function of the area ratios of the probes and the ion currents of each probe.
I  I 2   I1
A1
A2
I1
Plasma
I1  I1i  I1e
I 2  I 2i  I 2e
I2
Coaxial
Probe
V
I
 V1  Vp 
1

I1e   q A1 n e v e exp 
4
 Te 
 V2  Vp 
1

I 2e   q A 2 n e v e exp 
4
 Te 
V  V1  V2
Figure 5.2 Coaxial probe schematic
Page - 46 -
V
I  I1i A1 Te

e
I 2i  I A 2
Eq. 5.5
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
The experimental data (voltage-current characteristic) can be fitted with this
expression and thereby obtain the electron temperature and the ion density of the plasma.
A MATLAB program that solves the fitting problem can be found in Appendix VII.
1
2
3
4
5
6
Initial
guess
Plasma
potential
Guess
2 Area
ratio
Probes’
potential
Area
ratio
Ion
4
current
Electron
temperature
Te=3eV
Vp
A1/A2
V 1, V 2
A1/A2
Ii1, Ii2
Te
5
Ion
density
ni
Ion
curre
A1/ temperature and the ion density
Figure 5.3 Iterative process for calculating the electron
nt
A2
Ii1, Ii2
Figure 5.3 shows a flow chart of the iterative fitting process followed to calculate
Ion
the electron temperature and the ion density from the experimental
data. It starts with an
curre
nt
initial guess for the electron temperature and iterates until the value of the electron
A1/
A2
Ii1, Ii2
temperature converges. Normally no more than three iterations are required. In each
Ion
iteration an inner iteration is performed to calculate the areacurre
ratio and the potential of the
nt
probes at each applied voltage. The next sections describe each
Ii1, Ii2of the steps greater detail.
5.1.1.- Step 1: Plasma Potential
A1/
A2
Taking the floating potential (Vf) as the voltage
Ion
curre
reference,
nt
Ii1, Ii2
(Vp) is given by equation 1.3:
A1/
M2 i 
1 kTe  A

VP 
ln 
2 q  2 π m e 
Ion
curre
nt
Ii1, Ii2
where Mi is the ion mass and me the electron mass.
A1/
A2
Ion
curre
nt
Ii1, Ii2
Page - 47 Ion
curre
the plasma potential
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
5.1.2.- Step 2: Probes Potential
We set the floating potential as the voltage reference (Vf = 0). Since the electron
density is approximately equal to the ion density, it is possible to rewrite equation 5.4 for
each probe as follows:
qVp
qV1


1
kTe 
I1  q A1 n i v e e
1  e kTe

4

qV2

1
kTe 
I2  q A2 ni v e e
1  e kTe

4


qVp








Eq. 5.6
Dividing both expressions and taking into account that both probes have the same
current (I1 = –I2):
qV1

A1  1  e kTe
1 

qV2
A2 
kTe
1 e





Eq. 5.7
Finally, since the applied voltage V=V2-V1, the potential of each probe can be
calculated as function of the applied voltage, the electron temperature and the area ratio:

A
 1 1
kT
A2
V2  e ln 
qV

q
A1 kTe
e
 1 
A2

Page - 48 -






;
V1  V  V2
Eq. 5.8
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
4
0
0
Figure 5.4 shows the voltage at
3
0
0
the inner and outer conductor of the
2
0
0
Probecurnt(A)
1
0
0
coaxial probe as function of the applied
0
voltage.
1
0
0
2
0
0
6
0
4
0
2
0
0
2
0
4
0
6
0
A
p
p
l
i
e
d
V
o
l
t
a
g
e
(
V
)
5.1.3.- Step 3: Area Ratio
4
0
When each of the probes is
Plasma potential
2
0
Floating
potential
0
driven to a large negative potential, the
current in the probe reaches the ion
Probevltage(V1,2)inVolts
2
0
saturation current. The ion saturation
4
0
6
0
6
0
Outer conductor
Inner conductor
4
0
2
0
0
2
0
current is proportional to the area of the
4
0
6
0
A
p
p
l
i
e
d
v
o
l
t
a
g
e
i
n
b
e
t
w
e
e
n
p
r
o
b
e
s
(
V
)
i
n
V
o
l
t
s
probe and therefore the area ratio is
Figure 5.4 Inner and outer conductor potential
given by the current ratio of the probes
when they are driven to the same potential. The dashed arrows in Figure 5.4 show how to
find the current ratio for the calculation of the area ratio.
Steps 2 and 3 are repeated until convergence in the value of the area ratio is
found.
5.1.4.- Step 4: Ion Current
The voltage current characteristic has three clearly differentiated regions (Figure
5.5). In region 1 the applied voltage drives the inner conductor into large negative
Page - 49 -
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
voltages collecting ions and having a negligible electron current. Similarly in region 3 the
applied voltage the outer conductor is driven into large negative voltages collecting ions
and having a negligible electron current. The region 2 is a transition region in which both
probes have significant electron current.
4
0
0
3
0
0
2
0
0
Region
Region
Region
1
2
3
Probecurnt(A)
1
0
0
0
1
0
0
2
0
0
6
0
4
0
2
0
0
2
0
4
0
6
0
A
p
p
l
i
e
d
V
o
l
t
a
g
e
(
V
)
Figure 5.5 Regions in the voltage-current characteristic of a coaxial probe
Let r be the radius of the probe (inner or outer) and s the thickness of the sheath
around the probe, then the ion current is given by:
Ii  q A nis u B  q 2π  r  s  L nis u B
Eq. 5.9
As the applied voltage drives the probe into higher negative voltages the sheath
around the probe gets wider and therefore the effective area of the probe increases. The
sheath can be calculated using equation 1.5 and substituting back in equation 5.9:
Page - 50 -
Design, Fabrication and Modeling of mICP Sources
I i  q A n is

2
u B  q 2π  r 
743

3

Performance Of The New mICP Source
Te  2  Vp  V  


n es 
Te

3
4

 L n is u B


Eq. 5.10
I i  I o  α  Vp  V 
3
4
Io and  are parameters that depend on the electron temperature and the ion
density and therefore can not yet be calculated directly.
4
0
0
Region 1
Region 2
Region 3
3
0
0
2
0
0
Fitted curve
Ii outer
Curent(A)
1
0
0
0
Fitted curve
1
0
0
2
0
0
6
0
Ii inner
4
0
2
0
0
2
0
4
0
6
0
A
p
p
l
i
e
d
v
o
l
t
a
g
e
(
V
)
Figure 5.6 Ion current fitting
However these parameters can be obtained by fitting this expression in region 1
and 3 of the voltage-curve characteristics (where the electron current is negligible).
Figure 5.6 shows the voltage-current characteristic of plasma and the fitted ion currents
for each probe.
5.1.5.- Step 5: Electron temperature
The last step in the iterative process is to calculate the new electron temperature.
Taking natural logarithms in equation 5.5:
Page - 51 -
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
 I  I1i A 2  V
 
ln 
 I 2i  I A1  Te
Eq. 5.11
The left hand term can be calculated using the measured current, the ion current,
and the area ratio calculated in previous steps. Then the curve is fitted into a line and the
electron temperature is obtained from the slope of the line.
5.1.6.- Step 6: Ion density
The ion density at the edge of the sheath is calculated from the ion current
expression in equation 5.10. Then from equation 1.4 the ion density in the bulk of the
plasma is calculated.
5.2.- FREQUENCY OF OPERATION AND MATCHING
The first thing to do once the device is set up to be tested is to determine the
frequency of operation of the device. This frequency is close to the parallel resonance
frequency of the device and ideally corresponds to the 50 input impedance.
The optimum frequency of operation can be found by sweeping the frequency and
measuring the plasma intensity for a constant input signal. Figure 5.7 shows the ion
density as a function of the frequency of operation around the resonant frequency for
device II (flipped over configuration) with and without the additional capacitor for a
constant amplitude input signal. The lower plots in Figure 5.7 correspond to the power
reflection coefficient which is defined as the ratio of the reflected power to the forward
Page - 52 -
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
power.
1
.
2
e
+
1
1
6.0e+10
5.5e+10
1
.
0
e
+
1
1
5.0e+10
IonDesity(cm -3 )
4.5e+10
Ion Density (cm-3)
8
.
0
e
+
1
0
Plasma cannot
be maintained
6
.
0
e
+
1
0
4
.
0
e
+
1
0
4.0e+10
3.5e+10
3.0e+10
2.5e+10
2.0e+10
1.5e+10
2
.
0
e
+
1
0
1.0e+10
5.0e+9
0
.
0
6
6
5
6
7
0
6
7
5
6
8
0
6
8
5
6
9
0
6
9
5
7
0
0
7
0
5
7
1
0
7
1
5
7
2
0
7
2
5
7
3
0
7
3
5
0.0
770
F
r
e
q
u
e
n
c
y
(
M
H
z
)
790
800
810
820
830
840
Frequency (MHz)
a)
b)
0
.
7
2=P reflctd /P forwad
0
.
5
2=P reflctd /P forwad
780
0
.
4
0
.
3
0
.
6
0
.
5
0
.
4
0
.
3
0
.
2
0
.
2
0
.
1
0
.
1
PowerRflctionCeficnt
PowerRflctionCeficnt
0
.
0
0
.
0
7
7
0 7
8
0 7
9
0 8
0
0 8
1
0 8
2
0 8
3
0 8
4
0
6
6
5
6
7
0
6
7
5
6
8
0
6
8
5
6
9
0
6
9
5
7
0
0
7
0
5
7
1
0
7
1
5
7
2
0
7
2
5
7
3
0
7
3
5
F
r
e
q
u
e
n
c
y
(
M
H
z
)
F
r
e
q
u
e
n
c
y
(
M
H
z
)
Figure 5.7 Ion density and the power reflection coefficient as function of frequency for a constant
amplitude input signal of –8dBm (~150mW) a) with the 25% additional tuning capacitor added b)
without the additional tuning capacitor. (Device in flipped over configuration)
We notice that the resonant frequency of the device is well below the design
frequency (900 MHz or 805MHz when the additional tuning capacitor is added). This
frequency shift seems to be due to better quality capacitors. Since the number of digits
were designed extrapolating the results from previous designs but the number of working
digits has increased by minimizing the bubble-forming during the electroplating, the
Page - 53 -
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
capacitors obtained have a larger capacitance than the design value and the resonant
frequency shifts down.
5.3.- ELECTRON TEMPERATURE
The electron temperature can be obtained from the probe measurements as
described in section 5.1. Figure 5.8 shows the electron temperature obtained as a function
of pressure under different power conditions.
5
400mW
600 mW
800 mW
1W
4
3
2
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Electron Temperature Te (eV)
Electron Temperature Te (eV)
5
200 mW
400 mW
600 mW
800 mW
1W
4
3
2
0.05
0.10
0.15
0.20
Pressure (torr)
a)
0.25
0.30
0.35
0.40
0.45
Pressure (torr)
b)
Figure 5.8 Electron temperature a) Device I b) Device II (Flipped over)
As expected the electron temperature decreases as the pressure increases, and it is
independent of the power consumed by the source. Since the electron temperature is
essentially determined by the gas type, chamber dimensions and pressure of operation, it
is also independent of whether the device is flipped over or not.
5.4.- ION DENSITY
The more interesting information comes from the probe measurements of the ion
Page - 54 -
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
density. The ion density per watt absorbed by the plasma gives the efficiency of the mICP
source. Figure 5.9 shows the ion density as function of pressure and power for two
devices tested at two different frequencies. Points with the same symbol at any given
power correspond to different pressures and points of the same color to the same device.
As the power increases so does the ion density for all pressures and higher
pressures lead to higher ion densities for the same power. The new mICP sources perform
better than the previous generation and ion densities of ~1011 can be achieved with only
1W.
1e+11
400 mtorr
300 mtorr
200 mtorr
Device II (Flipped over) at 690 MHz
Device II (Flipped over) at 818 MHz
Device I at 690 MHz
Device I at 800 MHz
8e+10
Ion Density (cm-3)
100 mtorr
6e+10
4e+10
2e+10
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Power (W)
Figure 5.9 Ion density generated by the new mICP sources
More surprising is the fact that the efficiency not only did not improve as the
Page - 55 -
Design, Fabrication and Modeling of mICP Sources
Performance Of The New mICP Source
frequency increased but in fact it dropped about a ~50% for a ~17% increase in the
operating frequency. The model presented in section 2.1 does not explain this loss of
efficiency and a more accurate model is presented in the next section to explain this
behavior.
Flipping the device over clearly improves the efficiency of the mICP source. A
gain of ~40% is achieved by improving the coupling coefficient. Although it was clear
from the model in section 2.1 that the efficiency would improve, the model does not
consider the capacitive coupling between the coil and the plasma which also increases as
the device is brought closer to the plasma. This data, however, shows that the gain in
inductively coupled power into the plasma is larger than any increase in the capacitive
losses.
Page - 56 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
6.- NEW MICP SOURCE MODEL
We saw in the previous chapter that the model used to design the mICP source
cannot explain all of the experimental results. This model considered the plasma
conductivity to be a real constant, and therefore modeled the plasma simply as a resistor.
The conductivity of the plasma, however, is given by equation 1.7. Although for
frequencies much lower than the electron-neutral collisional frequency the conductivity
can be approximated as a real, it is in general a complex quantity.
6.1.- NEW PLASMA MODEL
Let’s reconsider the plasma model we had (an air-core transformer) and
incorporate the complex component as well as the power dependence of the plasma
conductance. From equation 1.7 we can obtain the resistivity of the plasma as:
ρ
1
ν
ω

j
2
σ ε o ω pe
ε o ω 2pe
Eq. 6.1
Substituting the plasma electron frequency (pe) from equation 1.8 and recalling
that the ion and electron density are approximately the same in the bulk of the plasma:
ρ
me ν
m ω
 j 2e
2
q ni
q ni
Eq. 6.2
Finally since the ion density (ni) is proportional to the power absorbed by the
Page - 57 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
plasma (Pp), we can write the impedance of the plasma as:
α2 ν
α 2ω
zp 
j
 R p  j ω Li
Pp
Pp
Eq. 6.3
where  is a proportionality constant.
From equation 6.3, the new ICP source model needs an additional inductance to
incorporate the effect of the inertia of the electrons (Figure 6.1). At frequencies lower
than the collisional frequency, electrons in the plasma undergo many collisions during a
period of the excitation field and therefore the plasma impedance is dominated by the
resistance due to the collisions. On the other hand when the frequency of operation is
larger than the electron-neutral collisional frequency, the electrons undergo very few
collisions during a period of the excitation field. Therefore the resistance due to the
collisions become negligible and the electron behavior becomes dominated by inertia.
Rc
I
1
V
Coil
Mk L c L p
Lc
Li
Lp
I
Rp
2
Rp 
α2 ν
Pp
α2
Li 
Pp
Eq. 6.4
Plasma
Figure 6.1 New ICP source model
We should notice that the inductance due to the electrons’ inertia is not coupled to
the inductance of the coil and therefore it must remain a separate circuit element in the
Page - 58 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
model.
The plasma resistance (Rp) and inductance (Li) are not constant as they depend on
the power absorbed by the plasma. Moreover the plasma resistance also depends on the
electron-neutral collision frequency.
Let us now consider a circuit element that has a resistivity inversely proportional
to the power it dissipates. Such an element has an infinite resistance when there is no
power applied to it, and as the power is increased the resistance decreases in such a way
that the voltage across the terminals of the element remains constant. Figure 6.2 shows
the characteristic curves of such a component that behaves in the same way as the plasma
resistance.
I
β2
β2
β2 R
R

 2
P I2 R
V
β
R
Eq. 6.5
Vβ
where 2 is a proportionality constant.
I
V
R
I
R
Power

V
Figure 6.2 Characteristic curves of a resistance inversely proportional to the power it dissipates
Therefore, going back to the new ICP source model, the voltage across the plasma
Page - 59 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
resistance in the model is constant with respect to the power absorbed by the plasma and
equal to α ν in equation 6.3.
If we now add the inductance due to the inertia of the electrons (Figure 6.3), it is
possible to show that the voltage across the inductance is also independent of the power
dissipated in the plasma. As more power is absorbed by the plasma, the plasma becomes
more dense and the resistance decreases, thereby maintaining constant the voltage across
the plasma (Vp).
Li 
I
+
+
α2
Pp
VLi
I
I
-
α ν
Rp
VRp  α ν
+
Vp
2VRp
-
-
α2ν
Rp 
Pp
VLi  jω
α
ν
ω

Vp  α ν  1  j 
ν

Figure 6.3 Voltage across the plasma impedance
The model in Figure 6.1 is not easy to work with and it is preferable to refer the
plasma to the primary as shown in Figure 6.4:
Rc
I
1
V
Req
Lc
Leq i
Leq p
R eq  R p
k 2ω 2 L p L c
R 2p  ω 2 (L p  L i ) 2
Leq i   Li
Leq p   L p
k 2ω 2 L p L c
R 2p  ω 2 (L p  Li ) 2
k 2ω 2 L p L c
R 2p  ω 2 (L p  Li ) 2
Figure 6.4 Equivalent circuit for the new ICP source model
Page - 60 -
Eq. 6.6
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
Substituting equation 6.4 in the expression for the equivalent resistance of the
plasma, we obtain:
R eq 
α 2 ν k 2ω2 L p Lc Pp
α 4  ν 2  2   2ω2 L pα 2 Pp  ω2 L2p Pp2
Eq. 6.7
The current flowing through the coil (primary of the transformer) can be easily
expressed in terms of the power dissipated in the plasma.
2
I =
Pp
R eq
=
α 4  ν 2 +ω2  +2ω 2 L pα 2 Pp +ω 2 L2p Pp2
α 2 ν k 2ω 2 L p L c
Eq. 6.8
When the plasma extinguishes, no power is dissipated in the plasma (Pp=0) but
there is current flowing through the coil. This current is the minimum current necessary
to induce the voltage at which the plasma can be sustained inductively (Vp).
Summarizing, we can conclude that:
1. In addition to the inductive behavior due to the single loop current induced by the
coil in the plasma, the plasma presents an inductive behavior due to electron
inertia that becomes dominant when running the plasma at frequencies larger than
the electron-neutral collisional frequency. This inductive behavior is modeled by
an inductor that is not coupled to the coil inductance.
2. The voltage across the plasma impedance is independent of the power dissipated
in the plasma. This agrees with the fact that the electric field strength in a plasma
Page - 61 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
is independent of the power [11].
3. An ICP source can maintain plasma only when a minimum current in the coil has
been reached. This agrees with the well-known fact that there exists a threshold
power bellow which a plasma cannot be inductively maintained [12].
6.2.- NEW EFFICIENCY EXPRESSION
The efficiency of the ICP source can be easily calculated from Figure 6.4 as the
ratio of the power dissipated in the plasma equivalent resistance to the total power
dissipated in the circuit.

1
1


2
2
2
Rc


R
R

ω
(L

L
)
R
1
c  p
p
i

1 2 c
R eq 1 
k Lc
R p k 2ω 2 L p Lc
1
 R p Lp  L
 L 

  i  2 i 


 Rp
 Lp R p  L p


Eq. 6.9
If we compare this expression with the one developed earlier in section 2.1, we
realize that both expressions are the same except for a new term due to the inductive
component of the plasma impedance.
Rc
I
V
Mk L c L p
Lc
I
1

R
L p 
R
1 2 c  p 

k Lc  L p R p 

1

 L 
R
Lp  Li
R
1 2 c  p 

 2 i 
 Rp
k Lc  Lp R p  L p


Rp
Lp
1

2
Rc
I
V
Mk L c L p
Lc
1
Li
Lp
Rp
2
Coil
Page - 62 -
I
Plasma
Section 2.1
model
New ICP
model
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
6.2.1.- Efficiency As A Function Of The Frequency Of Operation
In this section we are going to analyze how the frequency of operation influences
the efficiency of an ICP source. In the previous model the efficiency would increase with
the square of the frequency of operation, and as the frequency tended to infinity the
efficiency converged to 1 (Equation 2.2-a). However, this is not the behavior observed in
real devices (Figure 5.10).
Let’s consider now the new model presented in section 6.1 and manipulate the
expression of the efficiency so we get an insight of the changes of efficiency due to the
frequency of operation.
It is clear from equation 6.3 that:
Li 
Rp
Eq. 6.10

And therefore the efficiency expression of equation 6.9 can be rewritten as:

1

Rc
Rc
1
R eq 1  k 2L
c
1
 R p L p  R p
 


 2 



 Lp R p  L p 
 
At frequencies lower than the collisional frequency
Eq. 6.11
(<<) the new model
converges to the simpler model used to design the mICP source. As the frequency of
operation becomes comparable to or larger than the collisional frequency (>), the new
term due to the electron inertia needs to be taken into account.
Page - 63 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
Equation 6.11 can be reorganized as:

1
RL
RR  1
2R
1 
1 2 c  2 c p  2 c p  2  2 
k Lc k Lc R p k Lc L p    
Eq. 6.12
It is clear from equation 6.12 that the maximum efficiency is obtained when the
frequency of operation tends to infinity. Unfortunately, little gain is obtained once the
frequency is larger than the collisional frequency ( > 3). And not only does the
efficiency not increase significantly, but it no longer tends to 1 as the frequency tends to
infinity. Figure 6.5 shows the typical variation of an ICP source efficiency as function of
the frequency of operation.

max
3

Figure 6.5 ICP Source efficiency as function of the frequency of operation
Let us consider a practical case in which argon plasma is generated at 300 mtorr.
The collisional frequency can be calculated as the product of the rate constant (km) times
the neutral gas density (ng):
  km ng
Page - 64 -
Eq. 6.13
Design, Fabrication and Modeling of mICP Sources
where n g 
New mICP Source Model
P
, P is the gas pressure, k the Boltzmann constant and T the gas
kT
temperature.
For T=400K and P=300mtorr, the neutral gas density is 7.24x1021 m-3. The rate
constant for argon plasma with an electron temperature of ~3eV is 10-13 m3/sec
[1]
.
Therefore the electron-neutral collisional frequency under this conditions is ~725 MHz.
If we now calculate the frequency at which the term 1/2 can be neglected in
equation 6.12:
1
1
1
1
1

 2  2 
 0.1 2    10  f 
2
2
 



2
Eq. 6.14
Therefore for argon at 300mtorr the model predicts that the efficiency
improvement due to an increase in the frequency of operation is negligible for
frequencies above

 360MHz.
2
1
.
2
e
+
1
0
IonDesity(cm -3 )
1
.
0
e
+
1
0
8
.
0
e
+
9
6
.
0
e
+
9
4
.
0
e
+
9
1
5
m
m
c
o
i
l
1
0
m
m
c
o
i
l
5
m
m
c
o
i
l
2
.
0
e
+
9
0
.
0
0
1
0
0
2
0
0
3
0
0
4
0
0
5
0
0
F
r
e
q
u
e
n
c
y
(
M
H
z
)
Figure 6.6 Ion density vs. frequency of operation for 3 different mICP sources operating in Argon at
300mtorr, 1.3W. From Hopwood et al. [7]
Page - 65 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
This prediction agrees with the experimental results obtained by Hopwood et al.
when measuring the argon ion density versus the frequency of operation with previous
mICP sources (Figure 6.6).
6.2.2.- Efficiency As A Function Of The Power Absorbed By The Plasma
In section 6.1 we showed that the plasma impedance does depend on the power
absorbed by the plasma. In this section we are interested in finding the variation of the
ICP source efficiency as a function of the power absorbed by the plasma.
The efficiency expression for an ICP source is given by equation 6.9. Substituting
equation 6.4 in the efficiency expression we can make explicit the power dependence of
the ICP source efficiency:
1

R
1 2 c
k Lc
 2 
 
Lp Pp  2


 2 

2
 L p Pp


 Lp Pp

 
Eq. 6.15
It is straightforward to see from equation 6.15 that when the power absorbed by
the plasma is zero or infinite the efficiency of the ICP source tends to zero. This behavior
is the similar to that of a transformer when the secondary is open-circuited or shortcircuited. When the power absorbed by the plasma goes to zero, the plasma behaves as an
open circuit. On the other hand, when the power absorbed tends to infinity, the resistivity
decreases and the plasma behaves as a short circuit.
For a given pressure and frequency of operation, the efficiency of the ICP source
Page - 66 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
is maximum for a certain power. The power absorbed that maximizes the efficiency can
be found by taking the first derivative of the efficiency (Equation 6.15) with respect to
the power and making it equal to zero. Figure 6.7 shows the variation of the source
efficiency as function of the power absorbed by the plasma.
d
2  1
1
  0  Pp max 
 2
2
dPp
Lp  
Eq. 6.16
1
max 
1
2R c  1
1
1 
 2
 
2
2
k Lc  
  

max
Pp max
Pp
Figure 6.7 ICP Source efficiency as function of the power absorbed by the plasma
Since the plasma efficiency is determined by the ratio between the coil resistance
(constant in our analysis) and the equivalent plasma resistance, it is expected that the
maximum efficiency occurs when the equivalent plasma resistance is maximum. It can be
easily checked that the maximum equivalent plasma resistance occurs when the plasma
Page - 67 -
Design, Fabrication and Modeling of mICP Sources
resistance equals L p
1
2
1 2

New mICP Source Model
, which is another way of expressing that the optimal
power absorbed by the plasma Ppmax is
2  1
1
 2 .
2
Lp  
Figure 6.8 shows a graph of the ICP source efficiency as function of power
absorbed by the plasma and frequency of operation for a constant pressure.
Figure 6.8 Efficiency as function of the power absorbed by the plasma and the frequency of operation
for a constant pressure
6.2.3.- Efficiency As A Function Of Pressure
We have seen that the collisional frequency plays an important role in determining
Page - 68 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
the efficiency of an ICP source, especially when the frequency of operation is comparable
to the collisional frequency. The collisional frequency is proportional to the neutral gas
density (equation 6.13) and therefore it is directly proportional to the pressure at which
the plasma is operated.
The efficiency expression for an ICP source is given by equation 6.9. Substituting
equation 6.4 into the efficiency expression, we can make explicit the collisional
frequency (pressure) dependence of the ICP source efficiency:
1

R
1 2 c
k Lc
 2 
 
L P  2
 2p p  
 2 

 L p Pp


 Lp Pp

 
Eq. 6.17
Similarly to what happens with the power absorbed by the plasma, when the
pressure tends to zero or infinity the ICP source efficiency goes to zero. As the pressure
decreases, so does the neutral gas density, and therefore the electron-neutral collisional
frequency decreases as well. Since the resistance is a measure of the collisions electrons
undergo as they move in the plasma, as the number of collisions tends to zero the plasma
behaves as a short-circuit. On the other hand if the pressure increases so does the
collisional frequency and the plasma resistance increases and ultimately behaves as an
open circuit.
There exists an optimum pressure at which the efficiency of the ICP source is
maximum. The collisional frequency that maximizes the source efficiency can be
Page - 69 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
calculated by differentiating the efficiency expression with respect to the collisional
frequency and making it equal to zero.
PL 

d
  0    max    1  p 2 p 
d
 

Eq. 6.18
max 
1
2R c 
2 
1 2
1 

k Lc  
Pp L p 
Since the plasma efficiency is determined by the ratio between the coil resistance
(constant in our analysis) and the equivalent plasma resistance, it is expected that the
maximum efficiency occurs when the equivalent plasma resistance is maximum. It can be
easily checked that the maximum equivalent plasma resistance occurs when the plasma

2 
resistance equals  L p 
 , which is another way of expressing that the optimal

Pp 

 PL 
collisional frequency max is   1  p 2 p  .
 

One should notice that the maximum efficiency does not correspond to the
maximum of the real part of the plasma conductivity. The maximum of the real part of
the plasma conductivity corresponds to the minimum plasma resistance which occurs
when =. However in ICP sources the maximum efficiency is obtained when the
plasma resistance reflected in the primary is maximum. Nevertheless for microfabricated
ICP sources (Rp>>Lp) operating at frequencies relatively close to the collisional
Page - 70 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
frequency, the optimum pressure is that for which the collisional frequency is
approximately equal to the frequency of operation (  ).
 L 
 L p 
PL 

 max    1  p 2 p     1  p     1 



 
R p 
R p 



For L p  R p
(mICP)
Therefore the maximum energy transfer into the plasma in a mICP source happens
to be when the frequency of operation and the collisional frequency are approximately
equal.
Plasmas are easiest to start at pressures where the collisional frequency equals the
frequency of excitation (=) as it was observed in previous mICP sources
[8]
. This
condition is independent of the plasma conductivity and the reflected plasma impedance
in the primary of the source (since there is no plasma before we start it). However for a
mICP source the collisional frequency at which the plasma is most readily started and the
collisional frequency that maximizes the source efficiency happen to be the same: =.
6.3.- APPROXIMATION FOR LARGE AND MICROFABRICATED ICP SOURCES
In previous sections we have analyzed the efficiency as function of the power
absorbed by the plasma, the pressure, and the frequency of operation. Now we would like
to particularize those expressions for large and microfabricated ICP sources so the factors
that limit the efficiency of each ICP source can be compared. Table 6.1 presents typical
values for a large ICP source and a mICP source and their efficiency as predicted by the
Page - 71 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
new model. The efficiency predicted by the model seems reasonable and in agreement
with values of actual devices.
Coupling coef. (k)
Frequency (f)
Plasma inductance due to
the single current loop (Lp)
Plasma resistance (Rp)
Coil resistance (Rc)
Collisional Frequency ()
Efficiency ()
Large ICP
mICP
.3
.5
13.56 MHz
800 MHz
600 nH
10 nH
4
100
0.5
0.5
200 MHz
700 MHz
75 %
20 %
Table 6.1 Large and microfabricated ICP source comparison
In terms of the circuit elements used to model the ICP source, the main difference
between a large ICP system and a microfabricated ICP source is the ratio between the
plasma resistance (Rp) and the inductance due to the single current loop induced by the
coil (Lp). In a large ICP source the plasma resistance is much smaller than the impedance
Lp. However as the dimensions of the ICP source shrink the cross section area of the
plasma single loop reduces and therefore the plasma resistance increases. Moreover, the
plasma inductance due to the single loop current is approximately proportional to the
square of the radius of the coil. As the radius is reduced, the inductance decreases
dramatically. Therefore in a mICP source the plasma resistance (Rp) is much larger than
Page - 72 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
the impedance Lp.
Another difference between large and microfabricated ICP sources is their
frequency and pressure range of operation. Large ICP systems operate at 13.56MHz and
typically with pressures on the order of several mtorr. Therefore large systems in general
satisfy the condition (<3) and thus the inductance due to the inertia of the electrons
can be neglected. On the other hand, the new mICP sources operate at frequencies of
hundreds of MHz and at pressures of a few hundred mtorr. Therefore mICP sources do
not always satisfy that < and thus the inductance due to the inertia of the electrons
cannot be neglected.
6.3.1.- Frequency Of Operation
Particularizing the efficiency of an ICP source given by equation 6.11 for large
and microfabricated ICP sources we obtain:
Large ICP source
(Rp<<Lp <3)

1
R L
1 2 c p
k Lc R p
mICP source
(Rp>>Lp )

1
R Rp  1
1
1 2 c

2
k Lc L p   2 
Eq. 6.19
Eq. 6.20
We observe that for large ICP systems the efficiency can be considered
independent of the frequency of operation. However for mICP sources the efficiency
Page - 73 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
improves as the frequency of operation increases as long as the collisional frequency is
larger than the frequency of operation.
6.3.2.- Pressure And Power Absorbed By The Plasma
We know from previous sections that there exist an optimum pressure and
optimum power for which the efficiency is maximum. And this maximum is obtained
when the equivalent plasma resistance is maximized for a given frequency of operation.
Figure 6.9 shows the equivalent plasma resistance as function of the plasma resistance.
Large ICP
Source
mICP
Source
Req
R eq  R p
k 2ω 2 L p Lc
R 2p  ω 2  L p  Li 
2
1
for a given pressure
ω2
1+ 2
ν
ωα 2
R p max =ωL p +
for a given power
Pp
R p max =ωL p
Rp max
Rp 
2 
Pp
Figure 6.9 Equivalent Plasma Resistance
While an increase in the plasma resistance in the case of a mICP source lowers the
equivalent plasma resistance, in a large ICP source it increases the equivalent plasma
Page - 74 -
Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
resistance.
In terms of power, increasing the power in a mICP source increases the efficiency
while in a large system decreases it. We can see in Figure 6.8 that once the maximum
efficiency point has been reached the efficiency decreases very slowly as the power is
increased. Large ICP systems operate in this rather flat region of the efficiency surface
(Figure 6.8) and therefore the efficiency can be considered constant for a relatively wide
range of power for large ICP systems.
Large ICP source
(Rp<<Lp <3)

mICP source
(Rp>>Lp )

1
R LP
1  2 c p2 p
k Lc  
1
R  1
1
1 2 c
 2
2

k Lc Lp Pp    
2
Eq. 6.21
Eq. 6.22
Similarly in terms of pressure, increasing the pressure (collisional frequency) in a
large ICP source leads to better performance. In the case of mICP sources increasing the
pressure increases the efficiency as long as the collisional frequency is smaller than the
frequency of operation (). If the pressure continues to increase, the efficiency of the
mICP source starts decreasing. Therefore for an mICP the best performance is obtained
when the collisional frequency equals the frequency of operation (=). For the new
mICP sources operating at ~750MHz, the best performance would be then achieved at
~2 torr.
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Design, Fabrication and Modeling of mICP Sources
New mICP Source Model
6.4.- MODEL AND EXPERIMENTAL RESULTS AGREEMENT
The new model presented in this chapter was intended to explain the experimental
results presented in chapter 5. The new model predicts that the efficiency will not
improve with the frequency if the frequency of operation is much larger than the
collisional frequency (>3). We have seen that for argon at 300 mtorr no further
improvement is obtained for frequencies above ~360MHz which agrees with published
data. Therefore, for the new mICP sources, the gain in efficiency as we increase the
frequency from 690MHz to 818MHz based in the new model should be completely
negligible. This agrees with the fact that no improvement is observed in the experimental
data.
However we have observed that the ion density actually drops when the frequency
of operation is increased from 690MHz to 818MHz (Figure 5.9), and therefore some
effects not yet considered in the model must account for this efficiency loss. These
effects will be described in the following chapter.
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Design, Fabrication and Modeling of mICP Sources
Losses in mICP Sources
7.- LOSSES IN MICP SOURCES
In chapter 6 we introduced a new ICP model that incorporates the effect of the
electron inertia. This effect becomes relevant as the frequency of operation reaches the
electron-neutral collisional frequency. Since the frequencies of operation of the new
mICP sources (690 MHz and 818 MHz) satisfy  >> 3, the new model predicts no
efficiency improvement when the frequency of operation is increased from 690MHz to
818MHz. Some effects not considered in the model must account for the efficiency loss
observed experimentally. In this chapter we introduce these effects that, for the sake of
clarity, had been left out of the discussion in chapter 6.
7.1.- SKIN EFFECT
As the frequency of operation increases, the current flowing through the coil tends
to crowd around the extremes of the coil presenting an effective resistance larger than the
DC value. This increase in the effective resistance of the coil is the so-called skin effect.
The magnetic field generated by the current flowing through the coil pushes the electrons
toward the surface of the coil, reducing the effective cross section area (Figure 7.1).
1
0
Figure 7.1 Non-uniform current distribution due to the skin effect
Page - 77 -
Design, Fabrication and Modeling of mICP Sources
Losses in mICP Sources
At high frequencies the current distribution within the conductor approximates an
exponential decay and can be mathematically treated by determining the skin depth (),
which is the distance from the surface of the conductor where the current is 37% the
current at the surface (Figure 7.2).
δ
I
1
f π σ μ

a)
I
b)
Figure 7.2 a) Current distribution in the coil b) Equivalent current distribution using the skin depth
The skin depth is inversely proportional to the square root of the frequency of the
current flowing through the coil. Therefore, by increasing the frequency of operation we
decrease the effective area the current flows through and the resistance of the coil
increases. Since the efficiency of the source is determined by the ratio between the coil
resistance and the equivalent plasma resistance (the equivalent plasma resistance does not
depend on  for frequencies of operation greater than the collisional frequency),
increasing the coil resistance worsens the overall performance of the device.
Table 7.1 shows the skin depth, effective cross section area and the normalized
resistance of the 1-turn coil used in the new mICP sources.
Page - 78 -
Design, Fabrication and Modeling of mICP Sources
Losses in mICP Sources
Frequency
Skin depth ()
Effective Cross
Section Area (Aeff)
Normalized Coil
Resistance
DC

12500 m2
1
690 MHz
3.874 m
9704 m2
1.288
818 MHz
3.558 m
8917 m2
1.402
Table 7.1 Coil resistance increment as function of frequency
Since both the skin effect and the proximity effect affect the current distribution
across the coil section, their effect is similar and will be discussed together in the next
section.
7.2.- PROXIMITY EFFECT
Similarly to the skin effect, the proximity effect affects the current distribution in
the coil, increasing the coil resistance at high frequencies. The proximity effect consists
in the non-uniform redistribution of the current due to the effect of an external magnetic
field generated by nearby currents. The non-uniform current distribution can be seen as
the superposition of two currents: the primary current flowing through the conductor and
the eddy currents induced by the external magnetic field.
The proximity effect is in general difficult to treat analytically. It is not the
purpose of the following calculations to determine exactly the losses due to the proximity
effect, but rather obtain an order of magnitude of these losses that can help us in the
understanding of the performance of the mICP source.
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Design, Fabrication and Modeling of mICP Sources
Losses in mICP Sources
If we think of the coil as two independent halves, the current flowing through one
of the halves generates a magnetic field that induces an electric field in the other half.
This induced electric field will drive eddy currents as shown in Figure 7.3.
B
B
I
B
I
Induced
Eddy
currents
Induced
Eddy
currents
Figure 7.3 Eddy currents in the coil
The superposition of the eddy currents and the primary coil current result in a
non-uniform distribution as the net current tends to crowd near the inner part of the coil
(Figure 7.4).
1
0
Figure 7.4 Non-uniform current distribution due to the proximity effect
Page - 80 -
Design, Fabrication and Modeling of mICP Sources
Losses in mICP Sources
This current distribution results in a smaller effective cross-section area which, as
in the case of the skin effect, increases the coil resistance and lowers the efficiency of the
device.
In order to quantify the efficiency loss due to the proximity effect we need to
calculate the power dissipated by the eddy currents. The power loss due to the eddy
currents as function of the coil current is derived in Appendix VIII and the result is
presented here:
PEddy
2
2 0.292 2 ˆI 2 rout


128 
Eq. 7.1
where Î is the peak primary current flowing through the coil, rout is the outer radius of the
coil,  the skin depth and  the resistivity of the gold. An equivalent resistance that
accounts for these losses is given by:
R Eddy 
PEddy
1
2
Î 2

2
2 0.292 2 rout

64 
Eq. 7.2
Therefore we are able to divide the effective coil resistance into three terms,
namely, the DC resistance of the coil, an effective resistance that accounts for the skin
effect and an effective resistance for the proximity effect (Figure 7.5).
If we now compare the magnitude of each component we realize that the
resistance due to the proximity effect is at least an order of magnitude larger than both the
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Design, Fabrication and Modeling of mICP Sources
Losses in mICP Sources
DC resistance and the skin effect resistance. Therefore we can conclude that in the new
mICP sources the skin effect can be neglected and that the coil resistance is dominated by
the proximity effect.
The new mICP sources have a Q factor of 37 at ~800MHz (experimental
measurement) which corresponds to an effective coil resistance of ~1. Notice that this
value is more than an order of magnitude larger than the DC and skin effect resistance
and therefore it is mainly due to the proximity effect. The equation 7.2 was derived using
an over-estimated external magnetic field since the magnetic field due to the currents
outside the coil has been ignored.
RDC
RDC ~ 50m
Rc
RSkin
REddy
RSkin ~ 20m
REddy ~ 9
Figure 7.5 Coil Effective Resistance Decomposition
Since the resistance due to the proximity effect depends on the square of the
frequency of operation, increasing the frequency from 690MHz to 818MHz results in a
coil resistance increase of ~ 40 %. This increment in the coil resistance lowers the
efficiency of the mICP source at the higher frequency.
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Design, Fabrication and Modeling of mICP Sources
Losses in mICP Sources
The efficiencies of the mICP source at 690MHz and 818MHz are respectively:

1
R
1 c
R eq
' 
1
R '
1 c
R eq '
Eq. 7.3
where the apostrophe corresponds to values at 818MHz.
At these frequencies the equivalent plasma resistance is independent of the
frequency (since >3) and proportional to the power absorbed by the plasma. Therefore
for a constant power dissipated in the device (coil + plasma), the relation between the
equivalent plasma resistance at 690 MHz and at 818 MHz is given by the efficiency at
those frequencies:
R eq ~
1
~ Pp   P
Rp

R eq
R eq '


'
Eq. 7.4
where Pp is the power absorbed by the plasma and P the total power absorbed by the
mICP source (coil + plasma).
Since the coil resistance is mainly due to the proximity effect, the variation of the
coil resistance as the frequency increases from 690 MHz to 818 MHz is known.
Therefore, using equation 7.3 and 7.4 we can rewrite the efficiency at 818MHz as:
Page - 83 -
Design, Fabrication and Modeling of mICP Sources
' 
1

Rc '
1
R eq '
1
R '
Rc c
Rc
1
'
R eq

Losses in mICP Sources

1
R ' 1 
1 c
R c '
Eq. 7.5
Solving equation 7.5 for ’, the efficiency loss as function of the initial efficiency
and the coil resistance increment is given by:
'


1
Rc '
1  
Rc

Eq. 7.6
Table 7.2 shows the efficiency drop as the frequency is increased from 619 MHz
to 818 MHz based on equation 7.2 assuming a 40 % increase in the coil resistance.
Assumed efficiency at
690 MHz: 
Efficiency drop at 818MHz due to
proximity effect: (1-’/)100%
40 %
60 %
45 %
50 %
50 %
40 %
Table 7.2 Efficiency loss due to the proximity effect when the frequency is increased from 690 MHz
to 818 MHz
In Figure 5.9 it was shown that the ion density decreased approximately 45% as
the frequency was increased to 818 MHz. Therefore the efficiency loss observed as the
frequency is increased from 619 MHz to 818 MHz can be justified by an increase in the
coil resistance due to the proximity effect.
Page - 84 -
Design, Fabrication and Modeling of mICP Sources
Losses in mICP Sources
7.3.- CAPACITIVE COUPLING
ICP sources are always accompanied by some capacitive coupling due to the
alternating voltage difference between the coil and the plasma. This voltage difference
accelerates the ions in the sheath and therefore influences the energy per ion needed to
sustain the plasma. Therefore, as the voltage in the coil increases, so does the average
voltage difference between the coil and the plasma and for the same ion density, more
energy is required to sustain the plasma, i.e. the efficiency decreases.
Moreover the sheath width is proportional to the voltage difference across the
sheath to the three fourths, so as the voltage increases the sheath region widens. In the
case of large ICP sources where the coupling coefficient between the coil and the plasma
is mainly limited by the chamber wall, this increase in the sheath width does not have any
further significant influence. However in the new mICP sources the coupling coefficient
is determined mainly by the sheath width. Thus as the voltage in the coil increases not
only does the energy per ion required to sustain the plasma increase, but the coupling
coefficient between the coil and the plasma becomes worse.
Thus, we can conclude that the capacitive coupling always limits the performance
of ICP sources because it increases the energy lost by the ions in the sheath. And in the
case of mICP sources, this effect is aggravated because as the voltage across the sheath
increases, the inductive coupling between the source and the coil decreases.
Page - 85 -
Design, Fabrication and Modeling of mICP Sources
Losses in mICP Sources
7.4.- EXPERIMENT RESULTS AND MODEL PREDICTIONS WITH LOSSES
In chapter 5 we presented experimental results and observed that the ion density
decreases ~50% as the frequency of operation increases from 690MHz to 818MHz. The
new mICP model introduced in chapter 6, however does not predict this decrease because
the coil resistance was treated as a constant. In this chapter we have analyzed how the
coil resistance increases with the frequency of operation. Since the proximity effect
dominates the coil resistance, the coil resistance increases approximately with the square
of the frequency of operation. Therefore increasing the frequency of operation from
690MHz to 818MHz translates into an increase of the coil resistance by ~40%. And as
we showed in section 7.2, this increase in the coil resistance accounts for an efficiency
drop of ~50% in the overall performance of the device, in agreement with the
experimental data.
In chapter 5 we also observed that the ion density increased ~40% when
increasing the coupling coefficient by flipping over the device. This increase is relatively
small because the coupling coefficient is limited by the plasma sheath width(Figure 2.6).
The increase in the sheath voltage increases the sheath width and the energy required per
electron-ion generated in the plasma. This explains the relatively low ion-density increase
obtained by flipping over the device.
Page - 86 -
Design, Fabrication and Modeling of mICP Sources
Conclusions and Future Work
8.- CONCLUSIONS AND FUTURE WORK
As we mentioned in the introduction, the ultimate goal is to design an efficient
mICP source that can be integrated in a MEMS device. The performance of the mICP
sources described in this thesis and the analytical model developed can help us in
understanding the limitations of these devices and point us towards the key issues that
need to be addressed in order to improve the efficiency of future generations of mICP
sources.
The conclusions and trends that can be drawn from the results presented in this
thesis are:

No further efficiency improvement can be achieved by increasing the
frequency of operation above the electron-neutral collisional frequency.
Another way of looking at it is that better performance is expected for mICP
sources at higher pressure (higher collisional frequency). Therefore it might be
interesting to develop some collisional probe theory or use other methods to
measure the ion density and thereby the efficiency of future mICP sources at
higher pressures of operation.

Bringing the ICP source in contact with the plasma (flipping the device over)
leads to higher ion densities, although the efficiency improvement is not as
high as initially expected. There are two reasons for this limitation in the
Page - 87 -
Design, Fabrication and Modeling of mICP Sources
Conclusions and Future Work
efficiency improvement. On one hand, the sheath limits the coupling between
the source and the plasma to less than 0.5 even for sheaths less than 200m
wide. On the other hand, since the voltage difference between the coil and the
plasma is dropped completely in the sheath when the device is flipped over,
more energy is lost in the sheath as the ions get accelerated by a larger
potential (in non-flipped over devices most of the voltage drops across the
glass substrate).

Coil losses are primarily due to the proximity effect. A new coil design that
minimizes the losses due to eddy currents in the coil (e.g. a coil that is divided
into threads) would lead to more efficient devices. The difficulty of treating
proximity effect analytically leads us to suggest the use of 3D electromagnetic
solvers to design future mICP sources.
Page - 88 -
Design, Fabrication and Modeling of mICP Sources
References
9.- REFERENCES
[1] Michael A. Lieberman, and Allan J. Lichtenberg, “Principles of Plasma Discharges
and Materials Processing”, Wiley Interscience 1994
[2] Jan C.T.Eijkel, Herbert Stoeri, and Andreas Manz, “A Molecular Emission Detector
on a Chip Employing a Direct Current Microplasma”, Analytical Chemistry, Vol.
71, No. 14, July 15, 1999
[3] Jan C.T.Eijkel, Herbert Stoeri, and Andreas Manz, “A DC Microplasma on a chip
Employed as an Optical Emission Detector for Gas Chromatography”, Analytical
Chemistry, Vol. 72, No. 11, June 1, 2000
[4] M.W. Blades Group-University of British Columbia, “Atmospheric pressure plasma
on a chip”, http://a103.chem.ubc.ca/micro.html
[5] Yoshiki, H.; Horiike, Y., “An Atmospheric Pressure Microplasma Source on a Chip
Using 13.56 MHz Capacitively Coupled Discharge”, Proceedings of Symposium on
Dry Process, 2000, vol. 22ND, pp. 13-18. Institute of Electrical Engineers of Japan
[6] A.M. Bilgic, U. Engel, E. Voges, M. Kuckelheim and J.A.C. Broekaer, “A new lowpower microwave plasma source using microstrip technology for atomic emission
spectrometry”, Plasma Sources Science and Technology, Vol. 9, No. 1, February
2000
[7] Yu Yin, Jason Messier, and Jeffrey A. Hopwood, “Miniaturization of Inductively
Page - 89 -
Design, Fabrication and Modeling of mICP Sources
References
Coupled Plasma Sources”, IEEE Transactions on Plasma Science, Vol. 27, No. 5,
October 1999
[8]
J. Hopwood, “A Microfabricated Inductively Coupled Plasma Generator”, Journal
of Microelectromechanical Systems, Vol. 9, No. 3, September 2000
[9] P. Silvester, “Modern Electromagnetic Fields”, Prentice-Hall 1968
[10] Sunderarajan S. Mohan, Maria del Mar Hershenson, Stephen P. Boyd, and Thomas
H. Lee, “Simple Accurate Expressions for Planar Spiral Inductances”, IEEE
Journal of Solid-State Circuits, Vol. 34, No 10, October 1999.
[11] J. Hopwood, C. R. Guarnieri, S. J. Whitehair, and J.J. Cuomo, “Electromagnetic
fields in a radio-frequency induction plasma”, J. Vac. Sci. Technol. A 11(1),
Jan/Feb 1993
[12] N. Forgotson, V. Khemka, and J. Hopwood, “Inductively coupled plasma for
polymer etching of 200 mm wafers”, J. Vac. Sci. Technol. B 14(2), 732 (1996).
[13] Walter D. Pilkey, “Formulas for Stress, Strain, and Structural Matrices”, J. Wiley
1994
[14] J.A. Hopwood, “Plasma Assisted Deposition”, in The Handbook of Nanophase
Materials, A. Goldstein, Marcel-Dekker, New York 1997
[15] Constantine A. Balanis, “Advanced Engineering Electromagnetics”, John Wiley &
Sons 1989
Page - 90 -
APPENDICES
Page - 91 -
Design, Fabrication and Modeling of mICP Sources
Appendix I
APPENDIX I: MINIMUM GLASS THICKNESS
In order to calculate the minimum thickness of the substrate that can withstand a
pressure different of ~1 atm, we approximate the substrate by a circular plate fixed at the
edge:
P
a
From
[13]
the deflection, radial momentum (Mr) and tangential momentum (M)
are respectively given by:
1
P a 2 1     3    2 
16
1
M   P a 2 1    1  3  2 
16
Mr 

r
a
D
E h3
12 1   2 
where P is the pressure applied to the membrane, a the radius of the membrane,  the
Poisson’s ratio, E the Young’s modulus and h the thickness of the plate.
The maximum momenta occur at the edge of the plate (=1) and are given by:
1
M r max   P a 2
8
Page - 92 -
1
M  max   P a 2 
8
Design, Fabrication and Modeling of mICP Sources
Appendix I
Since ||1, the maximum momentum is the radial momentum at the edge of the
membrane.
For a homogenous isotropic material the stress in the membrane is given by:
r 
Mr z
h
 
3
12
M z
h3
12
where z is the distance from the center plane of the membrane in the axial direction.
Clearly the maximum stress occurs at the surface of the plate (z=h/2). Thus, the
maximum stress occurs at the edge of the membrane (=1) and at the surface (z=h/2).
The maximum stress the membrane has to withstand is then given by:
r max 
M r max 6
h2
1
 P a2 6
 8 2
h
And solving for the thickness of the plate (h):
h
3 P a2
4  r max
Assuming that glass can withstand a maximum stress of ~20Mpa, the thickness of
the plate that can support a pressure difference of ~1atm is given by:
h  0.6a
Thus for a radius of 3mm, a minimum thickness of ~180m is required.
Page - 93 -
Design, Fabrication and Modeling of mICP Sources
Appendix II
APPENDIX II: COUPLING COEFFICIENT
The next MATLAB program performs the calculation of the coupling coefficient
between two co-axial coils as function of the separation between the coils using
Neumann’s formula [9].
%Constant definition
uo=4*pi*1e-7;
%Coil 1 (mICP source)
n1=1;%Number of turns
a=2.5e-3*3/4; %Radius
%Coil 2 (Plasma)
n2=1; %number of turns
b=2.5e-3*3/4; %Radius
D=linspace(10e-6,1e-3,1000);%Separation between coils
%Mutual inductance calculation
k=sqrt(4*a*b./(D.^2+(a+b)^2)); %Eq 5-49 [9]
[K E]=ellipke(k.^2); %Eq 5-52 [9]
d=2.5e-3/2; %mICP coil width
M=uo*n1*n2*sqrt(a*b)*((2./k-k).*K-2./k.*E); %Mutual inductance Eq. 5-59 [9]
%Self inductance calculation
k=(1-(10e-6)^2/8/a^2); %Eq 5-49 [9]
[K E]=ellipke(k.^2); %Eq 5-58 [9]
L=uo*a*((2./k-k).*K-2./k.*E); %Self inductance Eq. 5-61 [9]
coupling_coeff=M/L; %M=ksqrt(L1*L2)
hold off
plot(D*1e6,coupling_coeff)
xlabel('Coils separation (um)')
ylabel('Coupling coefficient (k)')
title('Coupling coefficient vs coil separation')
axis([min(D)*1e6 max(D)*1e6 0 1])
hold on
zoom on
grid on
Page - 94 -
Design, Fabrication and Modeling of mICP Sources
Appendix III
APPENDIX III: PROGRAM USED TO DESIGN THE NEW MICP
SOURCES
---------HIERARCHY
---------Top.m
|
|---- Coil.m
|
|---- Plasma.m
|
|---- j_slice.m
|
|---- Cap.m
-----FILES
-----Top.m : This is the "executable" file. It calls to the other functions and present the
results in text and graphically. It contains all the design parameters (coil, plasma and
chamber). We fix the parameters in the m file and call it from the matlab workspace.
Coil.m: It calculates the resistance and inductance of the coil
Plasma.m: It calculates the resistance and inductance of the plasma
J_slice.m: It calculates the current density in a differential cylinder in the plasma
Cap.m: It makes two designs for a capacitor (different #of fingers/length of fingers
ratios)
-----TOP.M
-----%Global variables
global r_out; %Outter radious of the coil
global width_coil; %Width of the coil
global r_chamber; % Radious of the chamber
global h_chamber; %height of the chamber
global sd_gold;
global f; %RF Source frequency
global power_const_1;%Depends on power put in
global power_const_2; %(m^-3) depends on the power put in
%Coil
r_out=2.5e-3; %Outter radious of the coil (m)
width_coil=1e-6:1e-6:r_out; %width of the coil (m)
%width_coil=1500e-6;
h_coil=10e-6; %height of the coil (m)
Page - 95 -
Design, Fabrication and Modeling of mICP Sources
%Source
f=900e6; %RF source frequency (Hz)
power_const_1=1; %Depends on power put in
power_const_2=1e17; %(m^-3) depends on the power put in
%Chamber
r_chamber=2.5e-3; %Radious of the chamber (m)
h_chamber=6e-3; %Height of the chamber (m)
P=1; %Chamber pressure (torr)
T=500; % Gas temperature (K)
%Coil
%---[Rc Lc]=coil(width_coil,h_coil,r_out,f);
%Plasma
%-----ne=1e17; %Electron density (m^-3)
ke=1e-13; %Electron collision rate (m^3/sec) (Function of Te)
[Rp Lp]=plasma(P,T,ne,ke);
%Equivalent impedance of the coil and the plasma
%----------------------------------------------k_coeff=.75; %Coupling coefficient (Estimation)
w=2*pi*f;
tmp=k_coeff^2*w^2*Lp.*Lc./(Rp.^2+Lp.^2*w^2);
Req=Rc+tmp.*Rp;
Leq=(Lc-tmp.*Lp);
%Source Efficiency
%----------------efficiency=(Req-Rc)./Req;
%Matching network
%---------------Rsource=50; %Source impedance (ohm)
%Capacitor in series with the load
C1=1./((Leq.*w-sqrt((Rsource-Req).*Req)).*w);
%Capacitor in parallel with the load
C2=(Leq*w-1./(C1.*w))./((Req.^2+(Leq*w-1./(C1*w)).^2).*w);
%Check
z1=Req+(Leq*w-1./(C1*w))*i;
z2=1./(C2*w*i);
z=z1.*z2./(z1+z2);
%Current in the coil
%------------------RF_Power=3; %RF Source power (W)
Vload=sqrt(RF_Power*Rsource); %Voltage in the load (V)
Icoil=Vload./abs(z1); %Current flowing through the coil (A)
Power_coil=Icoil.^2.*Rc; %Power dissipated in the coil (W)
Icap1=Icoil;
Icap2=Vload*C2*w;
Page - 96 -
Appendix III
Design, Fabrication and Modeling of mICP Sources
Appendix III
%Coil Selection (~Brooks coil criterion)
%--------------------------------------[aux index]=min(abs(width_coil-r_out/2));
%Capacitors design for the selected coil
[N1 length1 Rf1 N1_2 length1_2 Rf1_2]=cap(C1(index)); %Number and length of fingers
[N2 length2 Rf2 N2_2 length2_2 Rf2_2]=cap(C2(index)); %Number and length of fingers
%Results display
%--------------fprintf('\n\n\nDesign parameters\n');
fprintf('-----------------\n');
fprintf('Outter radious of the coil: r_out = %.3e m\n',r_out);
fprintf('Width of the coil: width_coil = %.3e m\n',width_coil(index));
fprintf('Frequency of operation: f = %.3e Hz\n',f);
fprintf('\nChamber dimensions\n');
fprintf('-----------------\n');
fprintf('Radious of the chamber: r_chamber = %.3e m\n',r_chamber);
fprintf('Height of the chamber: h_chamber = %.3e m\n',h_chamber);
fprintf('\nCoil\n');
fprintf('----\n');
fprintf('Coil Resistance Rc = %.3e ohm
fprintf('Coil Inductance Lc = %.3e H
(skin depth: %.3e m)\n',Rc(index),sd_gold);
(%.3e ohm)\n',Lc(index),Lc(index)*w);
fprintf('\nPlasma\n');
fprintf('------\n');
fprintf('Plasma Resistance Rp = %.3e ohm\n',Rp); %It has been assumed constant
fprintf('Plasma inductance Lp = %.3e H
(%.3e ohm)\n',Lp(index),Lp(index)*w);
fprintf('\nEquivalent circuit\n');
fprintf('------------------\n');
fprintf('Coil+Plasma equivalent resistance Req = %.3e ohm\n',Req(index));
fprintf('Coil+Plasma equivalent inductance Leq = %.3e H
(%.3e
ohm)\n',Leq(index),Leq(index)*w);
fprintf('\nMatching network\n');
fprintf('----------------\n');
fprintf('Matching capacitors (series): C1 = %.3e F
(%.3e
ohm)\n',C1(index),1/(C1(index)*w));
fprintf('Design 1: # fingers: %d
Length of finger: %.3e m\n',N1,length1);
fprintf('
Finger resistance= %.3e ohm
Power/finger= %.3e
W\n',Rf1,(Icap1(index)/N1)^2*Rf1);
fprintf('Design 2: # fingers: %d
Length of finger: %.3e m\n',N1_2,length1_2);
fprintf('
Finger resistance = %.3e ohm
Power per finger = %.3e
W\n',Rf1_2,(Icap1(index)/N1_2)^2*Rf1_2);
fprintf('\nMatching capacitors (parallel): C2 = %.3e F
(%.3e
ohm)\n',C2(index),1/(C2(index)*w));
fprintf('Design 1: # fingers: %d
Length of finger: %.3e m\n',N2,length2);
fprintf('
Finger resistance= %.3e ohm
Power/finger = %.3e
W\n',Rf2,(Icap2(index)/N2)^2*Rf2);
fprintf('Design 2: # fingers: %d
Length of finger: %.3e m\n',N2_2,length2_2);
fprintf('
Finger resistance = %.3e ohm
Power per finger = %.3e
W\n',Rf2_2,(Icap2(index)/N2_2)^2*Rf2_2);
Page - 97 -
Design, Fabrication and Modeling of mICP Sources
Appendix III
fprintf('\nPower considerations\n');
fprintf('--------------------\n');
fprintf('Power put in the system RF_Power = %.3e W\n',RF_Power);
fprintf('\nPower dissipated in the coil Power_coil = %.3e W
(%.3e A %.3e
V)\n',Power_coil(index),Icoil(index),Icoil(index)*sqrt(Rc(index)^2+(Lc(index)*w)^2));
fprintf('Power dissipated in the capacitor in series (Design 1) = %.3e W (%.3e A
%.3eV)\n',(Icap1(index)/N1)^2*Rf1*N1,Icap1(index),Icap1(index)/(C1(index)*w));
fprintf('Power dissipated in the capacitor in series (Design 2)= %.3e W (%.3e A
%.3eV)\n',(Icap1(index)/N1_2)^2*Rf1_2*N1_2,Icap1(index),Icap1(index)/(C1(index)*w));
fprintf('\nPower dissipated in the capacitor in parallel (Design 1)= %.3e W (%.3e A
%.3e V)\n',(Icap2(index)/N2)^2*Rf2*N2,Icap2(index),Icap2(index)/(C2(index)*w));
fprintf('Power dissipated in the capacitor in parallel (Design 2)= %.3e W (%.3e A %.3e
V)\n',(Icap2(index)/N2_2)^2*Rf2_2*N2_2,Icap2(index),Icap2(index)/(C2(index)*w));
fprintf('\nEfficiency (Width ideal capacitors) = %.1f%%\n\n\n',efficiency(index)*100);
%Plots
%----figure(1)
plot(width_coil,efficiency*100,width_coil(index),efficiency(index)*100,'o');
title('Efficiency =100*(Req-Rc)/Req');
xlabel('Coil Width (m)');
ylabel('Efficiency (%)')
grid on
figure(2)
hold off
plot(width_coil,Rc,width_coil(index),Rc(index),'o');
hold on
plot(width_coil,Req-Rc,'r',width_coil(index),Req(index)-Rc(index),'o');
plot(width_coil,Req,':',width_coil(index),Req(index),'o');
axis([0 max(width_coil) 0 1]);
title('Coil & Equivalent Plasma Resistance (ohm)')
xlabel('Coil Width (m)');
zoom on
grid on
figure(3)
plot(width_coil,Icoil,width_coil(index),Icoil(index),'o');
title('Current in the coil')
xlabel('Coil Width (m)');
ylabel('Current (A)');
grid on
figure(4)
hold off
plot(width_coil,Power_coil,width_coil(index),Power_coil(index),'o');
title('Coil, Plasma & Total power dissipation')
xlabel('Coil Width (m)');
hold on
plot(width_coil,Icoil.^2.*(Req-Rc),'r',width_coil(index),Icoil(index)^2*(Req(index)Rc(index)),'o'); %Power plasma
plot(width_coil,Icoil.^2.*(ReqRc)+Power_coil,width_coil(index),Icoil(index)^2*(Req(index)Rc(index))+Power_coil(index),'o'); %Total power
grid on
figure(5)
Page - 98 -
Design, Fabrication and Modeling of mICP Sources
Appendix III
hold off
plot(width_coil,C1*1e12,width_coil(index),C1(index)*1e12,'o')
hold on
plot(width_coil,C2*1e12,'r',width_coil(index),C2(index)*1e12,'o')
title('Matching Capacitors')
ylabel('Capacitance (pF)')
xlabel('Coil Width (m)');
grid on
figure(6)
plot(width_coil,Lc,width_coil(index),Lc(index),'o');
title('Coil inductance');
xlabel('Coil Width (m)');
ylabel('Inductance (H)');
grid on
-----COIL.M
-----function [Rc,Lc]=coil(width,h,r_out,f)
%function [Lc,Rc]=coil(width,h,r_out,f)
% width: width of the conductor %m
% h: height of the conductor %m
% r_out: external radius of the coil %m
% f: frequency at which the coil will operate %Hz
% Lc: inductance of the coil%H
% Rc: resistance of the coil%ohm
%Conditions
T_operation=500; %Operation of the temperature of the coil(K) (~ T of the plasma)
%Constants
uo=4e-7*pi; %air permeability (H/m)
%Gold
u_gold=uo; %permeability (H/m)
ro_gold_20C=2.4e-8; %Gold resistivity at 20C (ohm m)
alpha_gold=0.0034; %(1/K)
%Resistance
%---------%Conductivity
global ro_gold;
ro_gold=ro_gold_20C*(1+alpha_gold*(T_operation-293));
sigma_gold=1/ro_gold;
%Skin depth
global sd_gold;
sd_gold=1/sqrt(f*pi*sigma_gold*u_gold);
%Transverse Area
Area=width*h-(width-2*sd_gold)*(h-2*sd_gold);
%Resistance
Rc=ro_gold*2*pi*(r_out-width/2)./Area;
%Inductance
%---------%Simple accurate expressions for planar spiral inductances.- IEEE Journal of Solid-State
Circuits, Vol 34, No 10, Oct 99"
Page - 99 -
Design, Fabrication and Modeling of mICP Sources
n=1; %Number of turns
d_out=2*r_out; %Outter diameter (m)
d_in=2*(r_out-width); %Inner diameter (m)
d_avg=.5*(d_in+d_out); %Average diameter (m)
fill_factor=(d_out-d_in)/(d_out+d_in); %Fill factor
c1=1; c2=2.46; c3=0; c4=.2; %Constants of the method for a round spiral
Lc=uo*n^2*d_avg*c1/2*(log(c2/fill_factor)+c3*fill_factor+c4*fill_factor^2);
-----PLASMA
-----function [Rp,Lp]=plasma(P,T,ne,ke)
%function [Rp,Lp]=plasma(P,T,ne,ke)
%
% P: Chamber pressure (torr)
% T: Gas temperature (K)
% ne: Electron density (m^-3)
% ke: Electron colision rate (m^3/sec)
% r_out: Outter radius of the coil
% f: Source frequency
% Rp: Plasma resistance (ohm)
% Lp: Plasma inductance (H)
% Constants
me=9.11e-31; %Electron mass (Kg)
eo=8.8542e-12; %Vaccum Permittivity (F/m)
uo=4e-7*pi; %air permeability (H/m)
q=1.6e-19; %Electron charge (C)
k=1.3807e-23; %Boltzman constant (J/K)
%Global variables
global r_out; %Outter radious of the coil
global width_coil; %Width of the coil
global r_chamber; % Radious of the chamber
global h_chamber; %height of the chamber
global f; %RF Source frequency
global power_const_1;%Depends on power put in
global power_const_2; %(m^-3) depends on the power put in
%Resistance
%---------%Collision frequency
ng=P/k/T*133.322; %m^-3 (1torr=133.322Pa)
Vm = ng*ke;
%Calculate Rp
delta_r=r_chamber/1000;
Rp_inv=0; %Initialitation
for r = delta_r:delta_r:r_chamber-delta_r
length_slice=2*pi*r;
J_slice=j_slice(r);
ne_slice=power_const_2*besselj(0,((2.405*r)/r_chamber));
Page - 100 -
Appendix III
Design, Fabrication and Modeling of mICP Sources
Appendix III
%Plasma conductivity
w=2*pi*f;
sigma_slice=(q^2*ne_slice)/(me*Vm)*(Vm^2/(w^2+Vm^2));
E_slice=J_slice/sigma_slice;
V_slice=2*pi*r*E_slice;
% negative voltage
I_slice=delta_r*h_chamber*J_slice;
Rp_inv=Rp_inv+(1./(V_slice/I_slice));
end
Rp=1./Rp_inv;
%Inductance
%---------%Same as the coil (we have 1 turn coil!!)
n=1; %Number of turns
d_out=2*r_out; %Outter diameter (m)
d_in=2*(r_out-width_coil); %Inner diameter (m)
d_avg=.5*(d_in+d_out); %Average diameter (m)
fill_factor=(d_out-d_in)/(d_out+d_in); %Fill factor
c1=1; c2=2.46; c3=0; c4=.2; %Constants of the method for a round spiral
Lp=uo*n^2*d_avg*c1/2*(log(c2/fill_factor)+c3*fill_factor+c4*fill_factor^2);
--------JSLICE.M
--------function [j]=j_slice(r)
%function [j]=ne(r)
%r: radious at which the electron density is evaluated (m)
%j : current density at r (m^-3)
global
global
global
global
r_out;
r_chamber;
width_coil;
power_const_1;
j=power_const_1*sin(r*pi/r_chamber);
-----CAP.M
-----function [N_fingers,length,Rf,N_fingers2,length2,Rf2]=cap(C_desired)
%function [N_fingers,length]=cap(C_desired)
%C_desired: Desired value of the capacitance (F)
%N_fingers: Number of fingers
%length: Length of the fingers (m)
%Rf: Finger resistance (ohm)
%N_fingers2: Number of fingers (Modified to make the cap of less resistance)
%length2: Length of the fingers (m) (Modified to make the cap of less resistance)
%Rf2: Finger resistance (ohm) (Modified to make the cap of less resistance)
global sd_gold; %Skin depth. Calculated in coil.m
global ro_gold; %Gold resistivity. Calculated in coil.m
Page - 101 -
Design, Fabrication and Modeling of mICP Sources
Appendix III
%Data from previous designs from which we are going to interpolate
%Thickness and height of the fingers = 10e-6 m
%Separation between fingers = 10e-6 m
w=10e-6; %Width of the finger
h=10e-6; %Height of the finger
L_data=[4.7 3.3 1.3 1]*1e-3;
N_data=[390 273 107 83];
C_data=[65 32.6 5.3 2.8]*1e-12;
if(C_desired>max(C_data))
fprintf('\n\nWARNING: Extrapolation during the design of the capacitors!!\n')
fprintf('------------------------------------------------------------\n\n')
elseif (C_desired<min(C_data))
fprintf('\n\nWARNING: Extrapolation during the design of the capacitors!!\n')
fprintf('------------------------------------------------------------\n\n')
end
plot(C_data,C_data./L_data./N_data)
x=50:500; %Number of fingers
%The number of fingers and their lengths used in previous design follow a linear
relationship
p=polyfit(N_data,L_data,1);
%figure
%plot(N_data,L_data,'o',x,polyval(p,x));
%xlabel('# of fingers')
%ylabel('Length of finger (m)')
%The number of fingers and the capacitance can be approximated by a parabola
q=polyfit(N_data,C_data,2);
%figure
%plot(N_data,C_data,'o',x,polyval(q,x));
%xlabel('# of fingers')
%ylabel('Capacitance (F)')
[error n]=min(abs(polyval(q,x)-C_desired));
%Geometry
%-------N_fingers=x(n);
length=polyval(p,N_fingers);
%The design calculated is based on a given external ratio of the capacitors in previous
design
%Since in the 1-turn design the current is higher (2~3 times), the power dissipated in
the
%capacitor is a concern, and thus we are going to modify the design considering that the
%capacitance per unit length per number of finger is constant. Therefore multiplying the
number
%of finger and dividing their length by the same constant, the capacitance won't change
but the
%power dissipated can be decreased. (Less current per finger, and fingers of smaller
resistance
factor=1.5;
N_fingers2=N_fingers*factor;
length2=length/factor;
Page - 102 -
Design, Fabrication and Modeling of mICP Sources
%Resistance of one finger
%-----------------------Area=w*h-(w-2*sd_gold)*(h-2*sd_gold);
Rf=ro_gold*length/Area;
Rf2=ro_gold*length2/Area;
%fprintf('\n\n# fingers: %d \nLength of finger: %.3e m\nCapacitance: %.3e
F\n\n',N_fingers,length,polyval(q,N_fingers));
Page - 103 -
Appendix III
Design, Fabrication and Modeling of mICP Sources
Appendix IV
APPENDIX IV: MATCHING NETWORK DESIGN
Ct
Input
impedance
(zin)
R’
Cm
Matching
network
y in 
L’
1
 g jb
z in
Coil
+
Plasma
yin  jC m 
1

1 
R ' j  L '
C t 



1  



L
'


C t  
R'


yin 
 j  Cm 
2
2 




1
1
2
2

R '   L '
R '   L '
 

C t 

C
t


 

For the input impedance to be equal to the power source output impedance
(zsource= Rsource=50 ):
Re  y in  
R'

1 
R '2   L '
C t 

2

1
R source



1 


 L ' C 


t 

Im  y in    C m 
0
2 


1

R '2   L '
 


C
t  


Page - 104 -
Design, Fabrication and Modeling of mICP Sources
Appendix IV
And solving for Ct and Cm:
Ct 
1

 L' ω 
R
supply
L' ω 
Cm 
Page - 105 -

 R'  R'  ω

1
Ct ω
2
 2 
1  
 R'   L' ω 
 ω
C t ω  



Design, Fabrication and Modeling of mICP Sources
Appendix V
APPENDIX V: 5-MM SINGLE-TURN MICP SOURCE PARAMETERS
Output generated by the design program in Appendix 3 for a 5-mm single-turn
coil:
Design parameters
----------------Outter radious of the coil: r_out = 2.500e-003 m
Width of the coil: width_coil = 1.250e-003 m
Frequency of operation: f = 9.000e+008 Hz
Coil
---Coil Resistance Rc = 5.666e-002 ohm
Coil Inductance Lc = 5.110e-009 H
Plasma
-----Plasma Resistance Rp = 1.002e+003 ohm
Plasma inductance Lp = 5.110e-009 H
(skin depth: 3.392e-006 m)
(2.890e+001 ohm)
(2.890e+001 ohm)
Equivalent circuit
-----------------Coil+Plasma equivalent resistance Req = 5.252e-001 ohm
Coil+Plasma equivalent inductance Leq = 5.108e-009 H
(2.888e+001 ohm)
Matching network
---------------Matching capacitors (series): C1 = 7.435e-012 F
(2.378e+001 ohm)
Design 1: # digits: 128
Length of digit: 1.548e-003 m
Digit resistance= 7.061e-001 ohm
Power/digit= 2.462e-004 W
Design 2: # digits: 192
Length of digit: 1.032e-003 m
Digit resistance = 4.707e-001 ohm
Power per digit = 7.294e-005 W
Matching capacitors (parallel): C2 = 3.433e-011 F
(5.152e+000 ohm)
Design 1: # digits: 280
Length of digit: 3.379e-003 m
Digit resistance= 1.541e+000 ohm
Power/digit = 1.111e-004 W
Design 2: # digits: 420
Length of digit: 2.253e-003 m
Digit resistance = 1.027e+000 ohm
Power per digit = 3.292e-005 W
Power considerations
-------------------Power put in the system RF_Power = 3.000e+000 W
Power dissipated in the coil Power_coil = 3.236e-001 W
(2.390e+000 A 6.906e+001 V)
Power dissipated in the tuning cap (Design1)=3.151e-002 W (2.390e+000 A 5.685e+001V)
Power dissipated in the tuning cap (Design2)=1.400e-002 W (2.390e+000 A 5.685e+001V)
Power dissipated in the matching cap(Design1)=3.111e-002 W
Power dissipated in the matching cap(Design2)=1.382e-002 W
Efficiency (Width ideal capacitors) = 89.2%
Page - 106 -
(2.377e+000A 1.225e+001 V)
(2.377e+000A 1.225e+001 V)
Design, Fabrication and Modeling of mICP Sources
Appendix VI
APPENDIX VI: FABRICATION PROCESS TRAVELER
Wafer clean

Acetone, isopropanol wipe
Tri-X Ultrasonic Clean
10 min

Rinse
5 min

Pirana 2:1(H2SO4:H2O2)
10 min

Rinse
5 min

H2O2:NH4OH(3:1)
10 min

Rinse
10 min

250

Spin Dry @2000 rpm
3 drops in 800ml of DI water
Temperature: 75 C
Deposit Cr, Au and TiW
Run number
________
Base Pressure
________
Argon gas flow
0.045 sccm

Pressure 12 mtorr
Chrome
5 min

DC-0.4A, Rotation speed 4
Gold
5 min

RF-300W, Rotation speed 4
TiW
5 min

DC-0.4A, Rotation speed 4
< 310-6torr

Visual inspection
PR Lithography
First PR Spin
S4620

Air Dry
5 min

Pre-Bake @ 90C
15 min

Air cooling
3 min

Second PR Spin
S4620

Air dry
5 min

Soft Bake @ 90C
1 hour

Hotplate

Light integral 450
Agitate
PR exposure - Canon
H2O:AZ400K (3:1)
2 min

Rinse
5 min

Page - 107 -
Hotplate
4000 rpm, 30 sec

Visual inspection
Hard bake @ 115C
4000 rpm, 30 sec
15 min

Hotplate
Design, Fabrication and Modeling of mICP Sources
Measure PR thickness
Appendix VI
________

DEKTAK
1 min

50 mtorr, 100 W
2min30sec

50 mtorr, 150 W
RIE Etch TIW
O2 plasma Descum
SF6:O2 (30:5) TiW etch

Visual inspection
Measure PR thickness
________

1h50min

10 min

Gold electro-plating
Au plate solution
Rinse
N2 Dry

Visual inspection

50mA, Duty cycle:1/4, Period:12msec
________

Strip PR (1165)
15 min

Temperature 85C
Strip PR (1165)
15 min

Temperature 75C
Rinse
10 min

O2 plasma Descum
3 min

________

H2O2 (TiW wet etch)
4 min

Rinse
5 min

Au etch
1 min

Rinse
5 min

Thickness measurement
PR Strip
Au thickness measurement
50 mtorr, 100 W
TiW/Au/Cr etch
RIE Chrome etch

ICP 240W, Bias 40W, O2 2.5mtorr, CF4 0.5mtorr

500 V DC, 0.5 A
Alternative TiW/Au/Cr etch
Ion Beam
Wafer Dicing
Page - 108 -
9 min

Design, Fabrication and Modeling of mICP Sources
Appendix VII
APPENDIX VII: PROBE MEASUREMENT CURVE FITTING
---------HIERARCHY
---------Top.m
|
|---|
|---|
|---|
|---|
|---|
|----
Vplasma.m
f_V1_V2.m
fit_isat.m
fit_te.m
fit_ni.m
ec.m
-----FILES
-----Top.m : This is the "executable" file. It calls to the other functions and present the
results in text and graphically.
F_V1_V2.m: Calculates in an iterative manner the voltage of each probe (inner and outer
conductor) and the area ratio.
Fit_isat.m: Calculates the error between the fitted ion saturation currents and the
measured current in the regions of interest.
Fit_te.m: Calculates the error between the fitted curve and the measured curve in the
region where the electron current is not neglegible.
Fit_ni.m: Calculates the error between the fitted curve and the measured ion saturation
current.
Ec.m: Calculates the collisional energy loss per ion-electron pair created [14]
-----TOP.M
-----clear;clc;
global
global
global
global
global
global
global
filename;
v_data;
i_data;
Io1;
Io2;
alpha1;
alpha2;
Page - 109 -
Design, Fabrication and Modeling of mICP Sources
global
global
global
global
global
global
global
global
global
global
global
global
global
global
global
global
global
Te;
l1;
r1;
Mi;
Area_ratio;
V1;
V2;
Vp;
indexni;
indexa1;
indexa2;
indexi11;
indexi12;
indexi21;
indexi22;
index1;
index2;
global indexte1;
global indexte2;
global indexl2;
l1=.2;%cm
r1=0.008/2*2.54;%cm
r2=0.034/2*2.54%cm
r_tube=7.94e-1;%cm
Mi=40;%amu
q=1.6022e-19;%C
dir
filename=input('File name: ','s');
fp1=fopen(filename,'r');
fp2=fopen('file.tmp','w');
[s n]=fscanf(fp1,'%s',1);
while(s(1)~='-')
[s n]=fscanf(fp1,'%s',1);
end
while n
fprintf(fp2,'%s ',s);
[s n]=fscanf(fp1,'%s',1);
fprintf(fp2,'%s\n ',s);
[s n]=fscanf(fp1,'%s',1);
end
fclose(fp1);
fclose(fp2);
load file.tmp
v_data=file(:,1);
i_data=file(:,2);
[i,index]=min(abs(i_data));
shift=v_data(index);
v_data=v_data-shift;
figure(1);hold off;zoom off;grid off
plot(v_data,i_data,'.g');hold on;
title('Choose voltage (>0) for Area ratio estimation');
Page - 110 -
Appendix VII
Design, Fabrication and Modeling of mICP Sources
Appendix VII
[v,i,button]=ginput(1);
[m indexa1]=min(abs(v_data+v));
[m indexa2]=min(abs(v_data-v));
if indexa1>indexa2
aux=indexa1;indexa1=indexa2;indexa2=aux;
end
plot(v_data(indexa1),i_data(indexa1),'*g',v_data(indexa2),i_data(indexa2),'*g');
title('Choose voltage (<0) for ni calculation');
[v,i,button]=ginput(1);
[m indexni]=min(abs(v_data-v));
plot(v_data(indexni),i_data(indexni),'og');
title('Choose voltage (>0) for plasma length calculation');
[v,i,button]=ginput(1);
[m indexl2]=min(abs(v_data-v));
plot(v_data(indexl2),i_data(indexl2),'og');
title('Set fitting region for Isat1: First point')
[v,i,button]=ginput(1);
[m indexi11]=min(abs(v_data-v));
plot([v v],[min(i_data) max(i_data)],'r');
title('Set fitting region for Isat1: Second point')
[v,i,button]=ginput(1);
[m indexi12]=min(abs(v_data-v));
plot([v v],[min(i_data) max(i_data)],'r');
title('Set fitting region for Isat2: First point')
[v,i,button]=ginput(1);
[m indexi21]=min(abs(v_data-v));
plot([v v],[min(i_data) max(i_data)],'r');
title('Set fitting region for Isat2: Second point')
[v,i,button]=ginput(1);
[m indexi22]=min(abs(v_data-v));
plot([v v],[min(i_data) max(i_data)],'r');
param_initial=[-50e-6 400e-6 -1e-7 1e-7];
param2_initial=[Te,0];
indexte1=0;
indexte2=0;
cont=0;
Te_old=0;%eV
Te=3;%eV Initial guess
while(abs(Te_old-Te)>5e-3)
cont=cont+1;
fprintf('\nIteration: %d
Vf=0;
Vp=Vplasma(Te);
Te:%.1f',cont,Te);
[V1 V2 Area_ratio]=f_V1_V2(Te);
param=fmins('fit_isat',param_initial,[0 1e-12 1e-12]);
Io1=param(1);
Io2=param(2);
alpha1=param(3);
alpha2=param(4);
param_initial=param;
Page - 111 -
Design, Fabrication and Modeling of mICP Sources
Appendix VII
i1_fitted=Io1+alpha1*(Vp-V1).^.75;
i2_fitted=Io2+alpha2*(Vp-V2).^.75;
i_fitted=(Area_ratio*exp(v_data/Te).*i2_fitted+i1_fitted)./(1+Area_ratio*exp(v_data/Te));
if(indexte1==0)
h=figure;hold off;zoom off;grid off
aux=log(abs((((i_data-i1_fitted)./(i2_fitted-i_data)))));
plot(v_data,aux,'.g');hold on;
title('Set fitting region: First point')
[v,i,button]=ginput(1);
[m indexte1]=min(abs(v_data-v));
plot([v v],[min(aux) max(aux)],'r');
title('Set fitting region: Second point')
[v,i,button]=ginput(1);
[m indexte2]=min(abs(v_data-v));
plot([v v],[min(aux) max(aux)],'r');
close(h);
end;
plot([v_data(indexte1) v_data(indexte1)],[min(i_data) max(i_data)],':g');
plot([v_data(indexte2) v_data(indexte2)],[min(i_data) max(i_data)],':g');
param2=fmins('fit_Te',param2_initial,[0 1e-3 1e-12]);
Te_old=Te;
Te=param2(1);
param2_initial=param2;
end
Vf=0;
Vp=Vplasma(Te);
[V1 V2 Area_ratio]=f_V1_V2(Te);
param=fmins('fit_isat',param_initial,[0 1e-12 1e-12]);
Io1=param(1);
Io2=param(2);
alpha1=param(3);
alpha2=param(4);
param_initial=param;
i1_fitted=Io1+alpha1*(Vp-V1).^.75;
i2_fitted=Io2+alpha2*(Vp-V2).^.75;
i_fitted=(Area_ratio*exp(v_data/Te).*i2_fitted+i1_fitted)./(1+Area_ratio*exp(v_data/Te));
%Graph for visual check
i1_fitted=Io1+alpha1*(Vp-V1).^.75;
i2_fitted=Io2+alpha2*(Vp-V2).^.75;
i_fitted=(Area_ratio*exp(v_data/Te).*i2_fitted+i1_fitted)./(1+Area_ratio*exp(v_data/Te));
figure(1);hold off
plot(v_data,i_data,'.g')
hold on
plot(v_data,i1_fitted,':m')
plot(v_data,i2_fitted,':m')
Page - 112 -
Design, Fabrication and Modeling of mICP Sources
Appendix VII
plot(v_data,i_fitted,'c')
plot(v_data(indexa1),i_data(indexa1),'*g',v_data(indexa2),i_data(indexa2),'*g');
plot(v_data(indexni),i_data(indexni),'or');
title('VI data & fitting curve')
plot([v_data(indexi11) v_data(indexi11)],[min(i_data) max(i_data)],':r');
plot([v_data(indexi12) v_data(indexi12)],[min(i_data) max(i_data)],':r');
plot([v_data(indexi21) v_data(indexi21)],[min(i_data) max(i_data)],':r');
plot([v_data(indexi22) v_data(indexi22)],[min(i_data) max(i_data)],':r');
%plot([v_data(indexte1) v_data(indexte1)],[min(i_data) max(i_data)],':r');
%plot([v_data(indexte2) v_data(indexte2)],[min(i_data) max(i_data)],':r');
xlabel('Voltage applied (V)')
ylabel('Current (A)');
%Ion density calculation
param=fmin('fit_ni',1,1e15,[0 1e-12 1e-12]);
ni=param(1);
%Plasma length calculation
Lde=743*sqrt(Te/ni)*1;%cm
sheath_1=sqrt(2)/3*Lde*(2*(Vp-V1)/Te).^.75;%cm
sheath_2=sqrt(2)/3*Lde*(2*(Vp-V2)/Te).^.75;%cm
l2=i_fitted(indexl2)/(2*pi*(r2+sheath_2(indexl2))*0.6*ni*1.6022e-19*sqrt(1.6022e19*Te/(1.6606e-27*Mi))*1e2)*1e-2 %cm
%Report
shift%V
Vf=0%V
Vp%V
Te%eV
ni%cm-3
Lde=Lde*1e-2;
sheath_1=sheath_1*1e-2;
sheath_2=sheath_2*1e-2;
[m index0]=min(abs(sheath_1-sheath_2));
floating_sheath=sheath_1(index0);
ion_flux=0.61*1.6022e-19*ni*sqrt(1.6022e-19*Te/(1.6606e-27*Mi))*1e5 %mA/cm2
%Power calculations
Area=2*pi*((r1+floating_sheath)*l1+(r2+floating_sheath).*l2+(r_tubefloating_sheath)*(l2+2e-3))+pi*(r_tube-floating_sheath).*2;%cm2
ub=sqrt(1.6022e-19*Te/(1.6606e-27*Mi))*1e2;%cm
Ei=Vp;
Pabsorbed=q*Area.*ni.*ub.*(Vp+2*Te+Ec(Te))
if(filename(3)=='x')
Pressure=10+str2num(filename(4)) %torr
else
Pressure=str2num(filename(3:4))/10 %torr
end
Power=str2num(filename(5:8))*1e-3; %W
P=Pressure/760*(1.013*1e5);%N/m2
T=300;%k Room temperature
k=1.3807e-23;%J/K
ng=P/(k*T)*1e-6 %cm-3
efficiency=Pabsorbed/Power;
figure(2)
hold off
plot([min(v_data),max(v_data)],[Vp Vp],':g');hold on
Page - 113 -
Design, Fabrication and Modeling of mICP Sources
Appendix VII
plot(v_data,V1,'y',v_data,V2,'r')
plot([min(v_data),max(v_data)],[Vf Vf],':b');
title('Voltages refered to the floating potential (Vf=0)');
xlabel('Applied voltage (V)');
ylabel('Potentials with respect to the floating potential(V)');
h=text((mean(v_data)+max(v_data))/2,max(V1)+4,'Inner Conductor');
set(h,'color','y');set(h,'FontSize',6)
h=text((mean(v_data)+min(v_data))/2,max(V2)+4,'Outer Conductor');
set(h,'color','r');set(h,'FontSize',6)
h=text((max(v_data)+min(v_data))/2,Vp+4,'Plasma');
set(h,'color','g');set(h,'FontSize',6)
h=text((mean(v_data)+min(v_data))*3/4,Vf-4,'V floating');
set(h,'color','b');set(h,'FontSize',6)
figure(3)
hold off
plot(v_data,sheath_1,'y')
hold on
plot(v_data,sheath_2,'r')
title('Sheath thickness')
xlabel('Applied Voltage (V)')
ylabel('Sheath (m)')
h=text((mean(v_data)+min(v_data))*3/4,min(sheath_2)+.2e-6,'Outer Conductor');
set(h,'color','r')
set(h,'FontSize',6)
h=text((mean(v_data)+max(v_data))/2,min(sheath_1)+.2e-6,'Inner Conductor');
set(h,'color','y')
set(h,'FontSize',6)
figure(2)
figure(1)
sigma_i=1e-14;%cm2 for Ar+
ion_mean_free_path=1/(ng*sigma_i)*1e-2 %m
fprintf('\nTe:%.1feV
ni:%.1ecm-3
Ion_flux:%.1fmA/cm2\n',Te,ni,ion_flux)
s=input('Save data (y/n): ','s');
if(s=='y')
fp=fopen('sum.txt','a');
fprintf(fp,'\n%.3e %.3e %.1f %.1f %.3f %.2e %.2e %.2f %.3e %.3e %.3e %.3e %.3e %.3e
%.3e
%.1f
',Pressure,Power,shift,Vp,Te,ni,ng,ion_flux,floating_sheath,max(sheath_1),max(sheath_2),i
on_mean_free_path,Lde,l2,Pabsorbed,efficiency*100);
fclose(fp);
disp('Data appended to sum.txt');
end
figure(3);zoom on;grid on
figure(2);zoom on;grid on
figure(1);zoom on;grid on
---------VPLASMA.m
---------function Vp=Vplasma(Te)
global Mi;%amu
%Assumed Vf=0
me=9.1095e-31;%Kg
Vp=.5*Te*log((Mi*1.6606e-27)*exp(1)/(2*pi*me));
Page - 114 -
Design, Fabrication and Modeling of mICP Sources
Appendix VII
---------F_V1_V2.M
---------function [V1,V2,Area_ratio]=f_V1_V2(Te)
global v_data;
global i_data;
global Vp;
global indexa1;
global indexa2;
Area_ratio=abs(i_data(indexa1)/i_data(indexa2));
V2=Te*log((1+Area_ratio)./(1+Area_ratio*exp(v_data/Te)));
V1=v_data+V2;
while (abs(V1(indexa1))-abs(V2(indexa2))>1)
if (abs(V1(indexa1))-abs(V2(indexa2))>abs(V1(indexa1-1))-abs(V2(indexa2)))
indexa1=indexa1-1;
else
indexa1=indexa1+1;
end
Area_ratio=abs(i_data(indexa1)/i_data(indexa2));
V2=Te*log((1+Area_ratio)./(1+Area_ratio*exp(v_data/Te)));
V1=v_data+V2;
figure(1);hold off;
plot(v_data,i_data,'.g');hold on
plot(v_data(indexa1),i_data(indexa1),'*g',v_data(indexa2),i_data(indexa2),'*g');
end
----------FIT_ISAT.M
----------function [t_error]=fit_isat(param)
%param----> Io1,Io2,alpha1,alpha2
global v_data;
global i_data;
global Vp;
global V1;
global V2;
global indexi11;
global indexi12;
global indexi21;
global indexi22;
global index1;
global index2;
Io1=param(1);
Io2=param(2);
alpha1=param(3);
alpha2=param(4);
i1_fitted=Io1+alpha1*(Vp-V1).^.75;
i2_fitted=Io2+alpha2*(Vp-V2).^.75;
error1=i1_fitted-i_data;
error2=i2_fitted-i_data;
Page - 115 -
Design, Fabrication and Modeling of mICP Sources
t_error=sum(error1(indexi11:indexi12).^2)+sum(error2(indexi21:indexi22).^2);
%figure(1);
%plot(v_data,i_data,'.g');
%hold on
%plot(v_data,i1_fitted,'m');
%plot(v_data,i2_fitted,'m');
------FIT_TE
------function [t_error]=fit_te(param)
global v_data;
global i_data;
global indexte1;
global indexte2;
global Area_ratio;
global Vp;
global V2;
global V1;
global Io1;
global Io2;
global alpha1;
global alpha2;
Te=param(1);
a=param(2);
fitted=a+v_data/Te;
i1_fitted=Io1+alpha1*(Vp-V1).^.75;
i2_fitted=Io2+alpha2*(Vp-V2).^.75;
aux1=abs((((i_data-i1_fitted)./(i2_fitted-i_data)))/Area_ratio);
error=log(aux1(indexte1:indexte2))-fitted(indexte1:indexte2);
t_error=sum(error.^2);
%figure(4);hold off
%plot(v_data(indexte1:indexte2),log(aux1(indexte1:indexte2)),'.g')
%hold on
%plot(v_data(indexte1:indexte2),fitted(indexte1:indexte2),'m')
------FIT_NI
------function [t_error]=fit_te(param)
global v_data;
global i_data;
global v_data3;
global i_data3;
global Io1;
global Io2;
global alpha1;
global alpha2;
global Te;
global l1;
global r1;
global Mi;
Page - 116 -
Appendix VII
Design, Fabrication and Modeling of mICP Sources
Appendix VII
global indexni;
global Vp;
global V1;
v=V1(indexni);
i=Io1+alpha1*(Vp-V1(indexni)).^.75;
ni=param(1);
Lde=743*sqrt(Te/ni);
s=sqrt(2)/3*Lde*(2*(Vp-v)/Te)^.75;
i_fitted=-2*pi*l1*(r1+s)*0.6*ni*1.6022e-19*sqrt(1.6022e-19*Te/(1.6606e-27*Mi))*1e2;
error=i_fitted-i;
t_error=sum(error.^2);
---Ec
---function [E]=Ec(Te)
x=[2 3 4 5];
y=[75 45 35 30];
E=interp1(x,y,Te);
Page - 117 -
Design, Fabrication and Modeling of mICP Sources
Appendix VIII
APPENDIX VIII: PROXIMITY EFFECT IN A SINGLE TURN COIL
B
The e.m.f. induced in a loop by the external
I
x
dx
magnetic field is given by Faraday’s law:
a = ¾ rout
E
rout
d
B  dS
dt 
where B is the magnetic field, dS is a surface
differential element and the integral is performed in
Induced
Eddy
currents
the area enclosed by the loop.
Assuming that the magnetic field is constant
in the region of integration and since the field is
perpendicular to the plane of the coil:
E
d
BS
dt
For a sinusoidal excitation of frequency , the e.m.f. induced in a loop is then
given by:
ˆ  B
ˆ  a 2x
Eˆ   BS
where the hats (^) mean peak values.
The resistance of the loop neglecting the radial paths is given by:
Page - 118 -
Design, Fabrication and Modeling of mICP Sources
R  2
a
 dx
Appendix VIII
where  is the skin depth
The power loss due to the eddy current in each loop is given by:
dP 
ˆ 2 a x 2
1 Eˆ 2 2 B

 dx
2 R

Therefore the power loss due to the eddy currents in coil is given by:
Pcoil  2 Phalf coil
ˆ 2 a 
2 B
2

rout
4

0
ˆ 2 r 4 
2 B
out
x dx 
128 
2
The only thing that we need to estimate is the magnetic field generated by the
current in the coil.
B
The magnetic field intensity created at
I
a = ¾ rout
point A by an electrical dipole (Idl) is given
rout
by [15]:
A
z
r

a

x
y
ˆ  u j  Î dlsin  1  1  e jr
dH
z


4r 
j r 
Idl
Since the dimensions we are considering
are much smaller than the wavelength of the excitation (r <<1) the current can be
Page - 119 -
Design, Fabrication and Modeling of mICP Sources
Appendix VIII
considered constant along the coil and the above expression reduces to:
ˆ   u z Î dlsin 
dH
4  r2
Therefore the total magnetic field intensity in point A can be calculated by adding
the magnetic field intensity created by each electrical dipole:
 ˆ

ˆI sin     
I dlsin 
0.22 Iˆ
4
2
ˆ
|H|  

a d 
2
2
4r
a
0
0 4   2 a 1  sin   
And the magnetic field is:
ˆ
ˆ
ˆ | |H
ˆ |  0.22  I  0.29  I
|B
a
 rout
Thus the power loss due to the eddy currents in the coil is given by:
Pcoil
2
2 0.292 2 ˆI 2 rout


128 
Which corresponds to an equivalent resistance of:
R Eddy 
2
P
2 0.292 2 rout


1 ˆ2
64 
I
2
Page - 120 -