THESIS

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dissertation entitled

CHARACTERIZATION AND MODELING OF A COMPACT ECR
PLASMA SOURCE DESIGNED FOR MATERIALS PROCESSING

presented by
Meng-Hua Tsai

has been accepted towards fulfillment
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Ph. D. degree in Electrical Eng

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CHARACTERIZATION AND MODELING OF A COMPACT ECR
PLASMA SOURCE DESIGNED FOR MATERIALS PROCESSING

By

MENG-HUA TSAI

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical and Computer Engineering

1999

ABSTRACT

CHARACTERIZATION AND MODELING OF A COMPACT ECR
PLASMA SOURCE DESIGNED FOR MATERIALS PROCESSING

By

MENG-HUA TSAI

A compact ECR ion/free—radical source designed for materials processing is experi-
mentally characterized and modeled under conditions typical for materials processing.
Specifically, argon, hydrogen and hydrogen-argon discharges are created and studied in a
compact microwave ECR (electron cyclotron resonance) plasma source operating in the

0.4-5 mTorr pressure range. The source has a discharge region of 3.6 cm in diameter and

an excitation volume of 50 cm3. The plasma species diffuse out of the source into a larger
materials processing region. The discharge is excited at a frequency of 2.45 GHz with
input power levels of 50-90 watts. Measurements of the ion density, electron temperature,
and absolute atomic hydrogen concentration are performed using Langmuir probes, OES
(optical emission spectroscopy) and actinometry. Measurements are performed in both the
source/excitation region and in the downstream processing region.

A global model for the discharges in this plasma source and a downstream charge par-
ticle diffusion model are also developed. The features of these plasma discharges deter-

mined by the combined measurements and models include a source region ion density in
the 10” cm'3 range, which corresponds to an ionization ratio of 0.001 to 0.01. For the
pure hydrogen discharges, the dominant ion species was Hf. Also, it was found for the

hydrogen discharge that the electron temperature was in the range 5-8 eV, the atomic

hydrogen molar fraction was measured as 25-34% with an increasing fraction at higher
pressures, and the surface recombination coefficient of atomic hydrogen was determined
to be approximately 0.005. Argon and argon-hydrogen discharge mixtures were also mod-
eled. The global model developed includes the effect of magnetic confinement/electron
heating enhancement by the ECR magnets in the source. Extensive comparisons are made

between the experimental measurements and the global model results.

Copyright by
Meng-Hua Tsai
1999

To my mother,
whose great personality has been a continuous

inspiration and encouragement to me.

COlli

expc

ACKNOWLEDGMENTS

I would like to thank Dr. Timothy Grotjohn, my academic advisor, for his
continuous support and encouragement during the study of this dissertation. His years of
experience in both modeling and experimenting in plasma sources has been a great help
and inspiration throughout the completion of this work. Special thanks to Dr. Jes
Asmussen for his support and many valuable suggestions during my years of study in
Michigan State University. I would also like to thank my committee members, Dr. Don
Reinhard and Dr. Norman Birdge, for their patience to proof-read my manuscript in such a
short time. Many thanks to my friends, Peng U. Mak, Bo K. Kim, Mark Perrin, and Robert
Pcionek for lots of useful discussions concerning the experiments and their wonderful
friendship. Finally, I would like to thank my family who always believe in my completion
of this dissertation. Thank you so much, my son, Jerry, who brought so much joys and

laughters to my life since you were born in summer 1998.

vi

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Table of Contents

List of Tables ......................................................... x

List of Figures ........................................................ xi
Chapter 1

Introduction .......................................................... 1

1.1 Motivation for High Density Plasma Sources .......................... l

1.2 Research Goals ................................................. 2

1.3 Dissertation Outline .............................................. 3
Chapter 2

High Density Plasma Sources in Material Processing and their Modeling ....... 5

2.1 Introduction .................................................... 5

2.2 Rf Powered High Density Plasma Sources ............................ 6

2.2.1 Planar Inductively Coupled Plasma Sources ..................... 6

2.2.2 Helical Resonator Plasma Sources ............................ 8

2.2.3 Helicon Plasma Sources ................................... 10

2.3 Microwave ECR Plasma Sources .................................. 12

2.3.1 Divergent-Field ECR Plasma Sources ......................... 12

2.3.2 Distributed ECR Plasma Sources (DECR) ..................... 16

2.3.3 Microwave Plasma Disk Reactor ............................ 18

2.4 Plasma Modeling and Simulation Methods ........................... 20
Chapter 3

The Experimental Diagnostic Techniques and Theories ..................... 25

3.1 Introduction ................................................... 25

vii

Chap!
Exper

4.?

3.2 Double Langmuir Probe Measurements ............................. 26

3.3 Single Langmuir Probe Measurements .............................. 31

3.4 Optical Emission Spectroscopy .................................... 35

3.4.1 Measurements for T6 ...................................... 35

3.4.2 Determination of Spectrometer Correction Factors .............. 39

3.4.3 Actinometry ............................................. 42
Chapter 4

Experimental Properties of MPDRTM 610 ................................. 46

4.1 Introduction of MPDR 610 ....................................... 46

4.2 System Configuration ........................................... 49

4.3 The System Parameter Spaces and Measured Quantities ................ 52

4.3.1 Plasma Source Region Measurements ......................... 53

4.3.2 Processing Chamber Region Measurements .................... 53

4.4 Calibration of the Source Pressure ................................. 54

4.5 Microwave Power Absorption in the Plasma ......................... 56

4.6 Ion Density ................................................... 58

4.7 Electron Energy ................................................ 65

4.8 Actinometry Results ............................................ 83
Chapter 5

Modeling of the MPDR 610 Plasma Source ............................... 84

5. 1 Introduction ................................................... 84

5.2 Discharge Global Models : A General Expression ..................... 85

5.3 Global Models Used in this Dissertation ............................. 90

5.3.1 Argon Discharge Global Models ............................. 91

5.3.2 Hydrogen Discharge Global Models .......................... 98

5.3.3 Argon-Hydrogen Mixture Discharge Global Models ............ 111

5.5.4 Downstream Diffusion Models ............................. 116

viii

Chapter 6

Modeling Results and Comparison ..................................... 117
6.1 Introduction .................................................. 117
6.2 Modeling Results in the Sources Region ............................ 117
6.2.1 Argon Plasma Model ..................................... 118
6.2.2 Hydrogen Plasma Model .................................. 120
6.2.3 Argon-Hydrogen Mixture Model ........................... 130
6.3 Modeling Results in the Downstream Region ........................ 133
6.4 Conclusions .................................................. 137
Chapter 7
Summary and Recommendations for Future Research ..................... 141
7.1 Summary of the Results ......................................... 141
7.1.1 .Ion Density. Measurements ................................ 141
7.1.2 Electron Energy in the Discharge ........................... 142
7.1.3 Optical Emission Spectroscopy (OES) ....................... 143
7.1.4 Modeling of the Compact ECR Plasma Source ....... . ........ 144
7.2 Recommendations for Future Research ............................. 145
Appendix A
Argon Plasma Global Model Program ..................................... 148
Appendix B
Hydrogen Discharge Global Model Program ............................... 151
Appendix C
Hydrogen/Argon Mixture Discharge Global Model Program ................... 157
Bibliography ........................................................ 16S

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Tab

Tab.

List of Tables

Chapter 3

Table 3.1 The cross section data, branching ratios, threshold energies of the observed
transitions [631-[65] and the spectrometer correction factors. .......... 37

Table 3.2 The measurement results for the ORIEL spectrometer ................. 41

Chapter 4

Table 4.1 Results of hydrogen flow rate test. The flow rate is set at 30 sccm. ...... 55

Table 4.2 Results of argon flow rate test. The flow rate is set at 8 seem. .......... 55

Table 4.3 Results of argon flow rate test. The flow rate is set at 20 sccm. ......... 56

Table 4.4 The emission ratios of H (486.1nm) to Ar (750.4nm) in H2 discharges from
actinometry (5% of argon gas added). ............................. 83

Chapter 5

Table 5.1 List of reactions used in the hydrogen discharge model. Reaction rates are
taken from Ref. [95] for reactions 14 and 7, 9-12. Reaction rates for reaction 5
and 8 are from Ref. [96] and Ref. [93]. Reaction 6 is developed as found in Ref.
[81]. ...................................................... 101

Table 5.2 Reactions that generates ions or neutrals in the Ar-H2 mixture discharge
models. Species considered: e, Ar, Ar+, H2, H, H3+ Hf, H+, ArH+. . . . . 110

Table 5.3 Lists of reactions not included in the discharge model due to small reaction
cross sections and therefore large mean free path compared to the dimension of
the source ................................................. 111

Ch;

List of Figures

Chapter 2
Figure 2.1 Schematic of a planar inductively coupled plasma source with an independent
rf biased substrate for material processing. ........................ 7
Figure 2.2 Schematic of a helical resonator plasma source ..................... 9
Figure 2.3 Schematic illustration of a helicon wave plasma source [20]. ......... 11
Figure 2.4 Divergent field ECR plasma source ............................. 13
Figure 2.5 The microwave circuit of an ECR plasma source. .................. 15
Figure 2.6 Distributed ECR plasma source (DECR). ........................ 17
Figure 2.7 An example of the MPDR system .............................. 19
Chapter 3
Figure 3.1 Double Langmuir Probe Circuit ................................ 27

Figure 3.2 A typical I-V characteristic of a double Langmuir probe. Plasma conditions
are Pin = 90W, 3 mTorr, and flow rate = 8 seem. The data is collected at 2 cm

downstream. ............................................... 29
Figure 3.3 Single Langmuir Probe Circuit Set Up. .......................... 32
Figure 3.4 The set up for CBS .......................................... 36
Figure 3.5 The emission spectrum of the tungsten lamp ...................... 40
Chapter 4
Figure 4.1 MPDR 610 cross section ..................................... 48

Figure 4.2 The vacuum system configuration .............................. 51

xi

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Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

Figure 4.17

Power absorption in the argon discharge. Input power = 91W, flow = 8 seem.
.......................................................... 57

The ion saturation currents in argon and hydrogen plasmas where Pin=90W,
z=2 cm. H2 flow = 30 sccm, Ar flow = 8 seem ..................... 59

Hydrogen plasma ion saturation current at 2 cm downstream with various gas
flow rates. Pin: 90W. Pressure = 1 mTorr. ....................... 60

The ion densities of argon plasma measured at 2 cm downstream. Input
powers are 90, and 72 W. Flow rate is 8 seem. .................... 62

Hydrogen ion densities measured at 2 cm downstream. The input powers are
90, 63, and 54 W. Flow rate = 30 sccm. The ion assumed for this density
determination is H24". ........................................ 63

Downstream ion densities of argon and hydrogen plasmas. The input power
is 90W, and the pressure is 1 mTorr. ............................ 64

Argon plasma electron temperatures at 2 cm downstream. Pin: 90W, and
flow rate = 8 seem. .......................................... 66

Hydrogen plasma electron temperature. Pin: 90 (*) and 63 W (0). Gas flow
rate = 30 sccm. ............................................. 67

The plasma potentials in Ar and H2 discharges measured at 2 cm downstream.
Pin: 90W. H2 flow rate: 30 sccm. Ar flow rate: 8 sccm. ........... 69

The electron temperatures in Ar/Hz mixture plasma taken at 2 cm
downstream. The input power is Pm=90W. The gas composition is varied.
.......................................................... 70

Argon plasma emission spectrum at 90W input power, 0.3 mTorr, and flow
rate of 3 seem. Monochromator slit width is 50 micron. ............. 71

The EEDF of Ar plasma. Pin = 90W. Pressure = l mTorr and z = 2, 3, 4, 6, 10
cm with <Ee>= 4.25, 3.50, 2.95, 2.37, 1.81 eV, respectively. ......... 73

Electron temperatures of Ar plasma at downstream positions 2 = 2—10 cm.
The extrapolating line to 2:0 cm indiates an estimation of the electron

temperature in the source being 5.9 eV . ......................... 74

The EEDF of H2 plasma. Pin: 90W. Pressure = l mTorr and z = 2, 3, 4, 6 cm
with <Ee>= 5.48, 4.46, 3.68, 2.39 eV, respectively. ................ 75

Electron temperatures of H2 plasma at downstream positions 2 = 2— 10 cm.

xii

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Figure 4.18

Figure 4.19

Figure 4.20

Figure 4.21

Figure 4.22

Chapter 5

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

. Figure 5.5

Chapter 6

Figure 6.1

The extrapolating line to 2:0 cm indiates an estimation of the electron
temperature in the source being 7.8 eV. .......................... 76

The EEDF of Ar plasma plotted on a logarithm scale. Pin=90 W, pressure = 2
mTorr, z = 2 cm, and flow rate = 8 seem. The electron temperature is 3.22 eV.
.......................................................... 77

The EEDF of hydrogen plasma plotted on a logarithm scale. Pin: 90 W,
pressure = 0.4 mTorr, and z = 3 cm. ............................ 78

Emission ratio of the observed emission line intensities to the calculated line
intensities (Iobs/Ical) for the hydrogen plasma. The operating pressure is 3
mTorr. The ratios are done for five different Te’s= 1.2, 2, 3, 5, 7.5eV. . .80

Emission ratio of the observed line intensities to calculated line intensities
(Iobs/Ical) for a hydrogen plasma(excluding xenon emissions). The operating
pressure is 3 mTorr. The ratios are done for five different Te’s = 2, 3, 5, 8, 12
eV ........................................................ 81

Plot of the fitted slopes of the emission ratio lines at five different Te’s(circles

: xenon emissions included, squares: xenon emissions excluded). Zero slope
indicates the electron temperature in the source. ................... 82

Illustration of power balance in the plasma volume (a), electron-neutral
collision process (b), and charged particle balance in the discharge volume
(0). ...................................................... 86

The universal plot of Te vs. ngdeff in an argon gas discharge. ......... 93

The collisional energy loss per electron-ion pair created vs. TC in the argon

plasma ..................................................... 97
The elastic scattering rate of electron collisions with hydrogen molecule.

......................................................... 107
The downstream particle diffusion simulation geometry. ........... 115

The electron temperatures in Ar plasmas. The dashed line is from the model
using constant f term over the pressure range (Model 1), and the solid line is
from Model 2 .............................................. 1 l9

xiii

fig

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

Figure 6.10

Figure 6.11

Figure 6.12

Figure 6.13

The electron densities in Ar plasma. The solid line with (*) is the model using
flOSS=0.40 and c=0. 17 and dash line with (x) is the model using constant floss
over the pressure. Both models use the absorbed power corresponding to the
experimental conditions. The density with a uniform absorbed power of 80 W
is also shown. ............................................. 121

Electron temperatures of hydrogen plasma. The solid line is from the model
prediction and the dashed line from the probe measurements taken at 2 cm
downstream. The discharge conditions are 90 W of input power and 30 sccm
of flow rate. .............................................. 122

The collisional energy loss per electron-ion pair created in the hydrogen
plasma. E14,] is the electron and atomic hydrogen collision energy loss, and
EL,2 is the electron and molecular hydrogen collision energy loss. . . . . 124

The power absorbed in the plasma volume by various collision processes
including dissociation, excitation, ionization, dissociative attachments,

and elastic collisions. The discharge conditions are 90 W of input power
and 30 sccm of flow rate. The model used floss = 0.38, c = 6.5, and 7: 0.005.
......................................................... 125

Determination of the atomic hydrogen surface recombination coefficient 7 in
the hydrogen plasma model from actinometry. The discharge conditions are
90 W of input power and 30 sccm of flow rate. The model used floss = 0.38,
c = 6.5. .................................................. 127

Neutral and ion densities in the H2 plasma with input power of 90 W. The
model used floss = 0.38, c = 6.5, and 7: 0.005. ................... 128

Atomic hydrogen dissociation percentage in the hydrogen plasma predicted
by the model using Pin = 90 W, floss = 0.38, c = 6.5, and 7: 0.005. . . . 129

Electron temperatures in the Ar—H2 mixture plasma. ............... 131

Neutral and ion densities in the Ar-HZ plasma at 0.6 mTorr with input power
of 90 W. ................................................. 132

Neutral and ion densities in the Ar-H2 mixture plasma at 4 mTorr with input
power of 90 W. ............................................ 134

Ar plasma electron density at downstream z = 2 cm. The dashed line is the
model 2 prediction and the solid line is model 1 prediction. The (*) points are
from experimental measurements. The input power is 90 W and flow rate is
8 seem .................................................... 135

Comparison of argon plasma diffusion model with experimental data taken in

xiv

the downstream region. Plasma conditions: Pin = 90 W, l mTorr, ans 8 seem.
......................................................... 136

Figure 6.14 Comparison of hydrogen discharge model with experimental data at z = 2 cm.
The input power is 90 W. Flow rate: 30 sccm. ................... 138

Figure 6.15 Comparison of hydrogen plasma diffusion model with experimental data
taken in the downstream region. Plasma conditions: Pin = 90 W, l mTorr, and
30 sccm. ................................................. 139

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CHAPTER 1

Introduction

1.1 Motivation for High Density Plasma Sources

In recent years, plasma-aided dry etching has been widely used for fine pattern
transfer in the fabrication of very large scale integrated (VLSI) circuits [1]. As integrated
circuit device dimensions continue to drop below a quarter micron, a low-pressure, high-
density plasma source (LPHD) is required for material processing to achieve more precise
dimension control, faster processing rates, and lower substrate temperatures during the
processing period.

Microwave electron cyclotron resonance (ECR) plasma sources are one possible
LPHD source type available to fulfill this increasing demand of semiconductor
manufacturing. They are able to provide a low pressure, low neutral gas temperature
plasma which has low ion energies that minimizes the possible damage (dopant
redistribution, thermal stress) to the substrate during the processing. In a low pressure (no
more than a few mTorr) discharge, the collision frequency is low, the chemical interactions
between the particles are minimized, and hence one of the primary interactions is between
the charged particles and the processing surfaces. With these features, microwave ECR

plasma sources may provide some useful applications in plasma-assisted etching,

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sputtering, implantation, and deposition in integrated circuit fabrication. Specifically.
studies have demonstrated their use in more precise etching line profile control [2H4].
and in higher etch rates for polymers [5]. Other examples include lower oxidation

temperatures in the conversion of Si to SiOz [6], low temperature homoepitaxial growth of

Si by plasma enhanced chemical vapor deposition [7], and ECR plasma etching for
selective and uniform high rate etching of polysilicon [8]. Additionally, as radiation
damage induced by ion bombardment is becoming an issue, the lower ion energies

produced in a microwave plasma may also offer some benefit in reduced damage.

1.2 Research Goals

Historically, plasma processes were often developed using a repeated trial/
incremental improvement approach. However, with increasing demands on the control and
repeatability of plasma processes, a different approach relying on careful measurements
and accurate models is becoming more necessary. Toward this end, this thesis work
carefully characterizes and models a compact ECR ion/free radical source used for a range
of material processing applications. The measurements and models developed in this
thesis will be useful in directing the future applications and design modifications of ECR
sources.

The first objective in this thesis is to understand via experimental characterization
the plasma generation in an ECR plasma source. This characterization includes the ion
density, electron density, electron temperature, neutral radical density, and electron energy

distribution function (EEDF). Both noble and molecular gas discharges will be studied

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including argon, hydrogen, and argon-hydrogen mixture discharges. The second objective
is to develop appropriate models which predict the discharge properties for ECR plasma
sources. The particular structure of the plasma reactor in this thesis is the MPDR 610
compact ECR ion/free radical source.

The research plan to achieve the objective of better understanding and controlling
of the plasma during material processing begins by dividing the study of the source into
sub-problems based on various plasma reactor regions including the plasma generation
region, the processing chamber region, and the immediate substrate region (i.e., the
substrate and its sheath). The first two regions will be studied in this work using a
combined approach of experimental diagnostics and modeling.

Another technique that will allow the objective to be reached without tackling the
most difficult problem immediately is to start with a single gas type discharge and work to
the more complex gas mixture discharges which are usually the case in plasma processing.

The discharges to be considered include argon, hydrogen, and argon-hydrogen mixtures.

1.3 Dissertation Outline

This dissertation starts with a brief introduction of the demanding trend for high
density-low pressure plasma sources in the semiconductor industry. Chapter 2 gives a brief
review of the some of the high density plasma sources (rf and microwave powered). The
applications for these plasma sources in material processing are also introduced. Chapter 3
and 4 discuss the experimental approach to the diagnosis of the discharge characteristics.

The experimental diagnostic techniques are described in Chapter 3. The techniques

employed in this study include single and double Langmuir probes and optical emission
spectroscopy (OES). Chapter 4 presents the experimental results utilizing the
characterization techniques described in Chapter 3. The plasma source prOperties
measured are electron temperature, neutral and ion density, and electron energy
distribution function. In addition to experimental diagnosis of the discharge properties, in
Chapter 5 and 6, a spatially-averaged global model is developed to characterize and
predict the behavior of the discharge species under low pressure conditions (no more than
a few mTorr). Chapter 5 describes the theory of the global model applied to a compact
ECR plasma source. Consideration of the magnetic confinement in the discharge chamber
and the species reactions in argon, hydrogen, and argon-hydrogen mixture are made in the
model. The model considers two basic equations including charged particle balance and
power balance. Chapter 6 presents the modeling results and their comparison with some of
the experimental data presented in Chapter 4. Finally, Chapter 7 concludes this work with

a summary of the important results and suggestions for future research.

2.1

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CHAPTER 2
High Density Plasma Sources in Material Pro-

cessing and their Modeling

2.1 Introduction

Plasma-aided manufacturing has proven to be an effective and efficient processing
method in a vast range of industrial applications including thin-film sputter deposition,
plasma polymerization, microcircuit fabrication, welding, tool hardening, arc melting,
synthesis of pure, ultrafine powders, plasma spraying, plasma sintering, and microwave
generation [9]. Of particular interest in this work is its applications in the deposition and
etching of thin films for the fabrication of semiconductor microelectronic devices. The
continuous demand for smaller device features requires a high density plasma source for a
fast processing rate that also operates at a low pressure to obtain small device dimensions
and prevent possible damage on the wafer. This chapter describes some of the high density
plasma sources which are commonly used as etching and deposition tools. The modeling

and simulation of these sources are later introduced.

2.2 Rf Powered High Density Plasma Sources

In this section, three high density plasma sources which utilize rf power coupling to
the plasma will be briefly introduced and some of their applications to material processing
will also be discussed. The three rf plasma sources presented include the planar
inductively coupled plasma source, helical resonator plasma source and the helicon
plasma source. The helicon sources require a dc magnetic field for efficient power
coupling, but the excitation of inductive coupled plasma sources and helical resonant

sources requires no magnetic field.

2.2.1 Planar Inductively Coupled Plasma Sources

A schematic of a typical planar inductive coupled plasma (ICP) source is shown in Fig.
2.1. It consists of a cylindrical discharge region with a diameter similar or larger than its
length. The reason for having such a geometry is the design requirement to have a large
area and uniform density plasma source for material surface processing in industry. The
induction planar coil is a flat spiral and usually placed on top of the vacuum chamber. The
coil is separated from the plasma source by a dielectric window in order to prevent
possible metallic contamination from the coil. This is especially important during
semiconductor microelectronic circuit processing. The dielectric window is typically
made of quartz or alumina. Multipole permanent magnets can be used around the process
chamber circumference to increase the radial plasma uniformity. This source type was
developed initially for rapid, high efficiency sample etching, because the ion densities are
about two orders of magnitude higher than in conventional rf parallel plate discharges

[22],[23]. ICP also provides independent control of discharge power and substrate bias

13.56 MHz Rf Power

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Figure 2.1 Schematic of a planar inductively coupled plasma source with an
independent rf biased substrate for material processing.

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voltage. Substrates are placed on a rf (13.56 MHz) driven electrode which is used to
control the ion bombardment energy.

Inductively coupled plasma sources are being applied to a wide range of applications.
An example of one such application is DLC thin film growth using a rf inductively
coupled plasma (RFICP) source operating with an argon and methane gas mixture
discharge as described by Pappas and Hopwood [21]. The typical processing pressure was

around 10 mTorr.

2.2.2 Helical Resonator Plasma Sources

A typical helical resonator is shown in Fig. 2.2. The source consists of a helix coil
connected to a rf power (3-30 MHz). The coil is wrapped around the quartz tube inside
which the plasma is generated. The entire source is covered by a grounded metallic
cylinder to shield the rf energy. An rf matching network is located externally to adjust for
different plasma load impedance matching. This structure becomes resonant when the
length of the helix approximately equals to an integral number of quarter wavelengths of
the if field [10]. The electromagnetic field within the helical resonator can sustain a
plasma with low match loss at low gas pressures. The quartz tube is connected with the
processing chamber near where the substrate is located. The substrate can also be biased
with rf power to independently control the ion energy bombarding the substrate.

J. M. Cook et al. [11] used the helical resonator structure operated at radio frequencies
for submicron polysilicon etching in C12 plasmas at low pressures (< 10'3 Torr). With 1%

of 02 added to the C12 plasma and 75 W of the input resonator power, a polysilicon etch

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rate of 16.2 nm/min was obtained with good material selectivity over SiOz/photoresist.

The etch rate can be increased considerably by scaling the size of the resonator to
accommodate larger diameter discharge tubes and by adding an axial magnetic field

confinement [12].

2.2.3 Helicon Plasma Sources

Helicon generation of plasmas in high efficiency was first employed by Boswell in
1970 [13]. The study of Chen in 1985 [14] and subsequently in 1991 [15] suggested that
electrons accelerated by the Landau/collisionless damping mechanism can be used to
produce primary electrons at the optimum energy for ionization. The typically driving rf
frequency for this high density plasma source is 1-50 MHz, with 13.56 MHz commonly

used for processing discharges. The magnetic fields vary from 20 to 200 Gauss for

processing discharges. Plasma densities range from 1011-1014 cm'3, with 10”-1012 cm'3

typical for processing. For a more detailed study of the helicon discharges, the reader
should refer to the research work conducted by Boswell et a1. [16][17], Chen et a1. [18],
and Shoji et al. [19].

Characteristics of the helicon wave plasma employing C12 gas for aluminum etching
was studied by Jiwari et a1. [20] using a m = 0 mode antenna powered with a 13.56 MHz rf
power. A schematic illustration of the experimental apparatus is shown in Fig. 2.3. The
magnetic field at the quartz tube was 50 G and at the reactor was varied to establish an
optimum condition. A wafer stage was installed in the reactor at a distance of 10 cm from

the edge of the quartz tube. The stage was biased negatively by an rf power supply of 100

10

 

 

Matching

 

Box

 

 

 

13.56MHz

Gas —>

 

 

 

 

 

 

 

 

 

 

 

 

Quartz
Tube

Source
Solenoid

 

 

Chamber

:5: Solenoid
\ Insulator

 

 

 

 

 

 

 

 

 

 

‘ RP

 

 

 

 

Roughing
Pump

 

TMP

 

 

 

l ,e.

 

Substrate

 

Water-cooling

Turbo-Molecular
Pump

 

Matching
Box 100kHz

 

 

Figure 2.3 Schematic illustration of a helicon wave plasma source [20].

I9
34..»

freq;

Where

 

tltctnv
imam
ECR j

l'DECi

kHz and cooled by water. The typical etching condition were 2 mTorr, 1200W Rf power,

and 200 V bias. The electron density, electron temperature, and ion saturation current in

the C12 plasma at 2 mTorr were 5x1010 cm'3, 5.5 eV, and 15 mA/cmz.

2.3 Microwave ECR Plasma Sources

The electron cyclotron resonance (ECR) condition occurs when the excitation wave

frequency (1) equals the electron cyclotron frequency a)“, in a static magnetic field.
(.0 = a). = — (2-1)

where B is the magnetic field intensity, e = 1.6x10'19 coul, and me = 9.1 1x10'31 kg is the
electron mass. For an microwave frequency of f = 2.45 GHz, the required magnetic field
intensity is approximately 875 Gauss. This section introduces three types of microwave
ECR plasma sources including the divergent field ECR source, the distributed ECR

(DECR) source and the microwave plasma disk reactor (MPDR) source.

2.3.1 Divergent-Field ECR Plasma Sources

In the 1970’s, Musil [25] and Suzuki [24] conducted the pioneering works for
semiconductor deposition and etching utilizing divergent field ECR plasma sources. Since
then, the research work regarding the source operation and applications on thin film
deposition and etching became widespread [26]—[29]. Here a brief description of this type

0f ECR plasma sources is presented.

12

 

 

 

 

 

 

 

 

 

 

 

 

Microwave —> Vacuum Divergent
input window magnetic
field lines
//// //// .
Coils '
\N Source
/ chamber

 

 

Plasma

\ Process
\ / chamber

 

 

 

 

 

 

 

 

 

 

 

 

Optional
coils

 

 

 

 

 

 

 

 

\ Substrate

\

To pump

Figure 2.4 Divergent field ECR plasma source

luau
mic:-
then
Inuit
mag

Wm:

A typical divergent field ECR processing source is shown in Fig. 2.4. The system
isoften divided into a source region, where most of the plasma is created, and a process
chamber where the substrate is located. A steady magnetic field is generated by one, or
more electromagnets surrounding the source chamber. The magnetic field is typically
1000 Gauss or more in the source chamber and tapers off to tens or hundreds of gauss in
the process chamber as the field lines diverge to larger radii in the process chamber.

Microwaves are introduced into the source chamber through a dielectric vacuum
window (usually quartz or alumina) by a coupling structure. The typical microwave circuit
is shown in Fig. 2.5 [30]. It consists a microwave source (2.45 GHz usually), a ferrite
microwave circulator to protect the source by shunting the reflected power to a dummy
load, a reflected power monitor, tuning stubs, and the plasma coupler. The reflected
microwave is sampled through a directional coupler and converted to a DC signal, and
then connected to a power meter. Three tuning stubs allow adjustment of the impedance to
match the plasma load. When the plasma is generated, electrons and ions diffuse along the
magnetic field lines that diverge as moving to the process chamber. Additional coils are
sometimes placed around the process chamber to create converging field lines which help
confine the charged particles in the process chamber where the substrate is located. This
arrangement also helps control the ions incident angle on the substrate which is important
for anisotropic etching.

One drawback of conventional divergent field ECR reactors is the use of
electromagnets with their associated costs of coils, power supplies, power, and cooling.
The development of ECR sources using permanent magnets to produce the required field

strength for ECR is aimed to solve this problem [31],[32]. Mantei et. a]. reported an ECR

 

 

Microwave
power

surply

 

 

Ferrite Directional
Circu ator coupler

 

Stub tuners
fl {‘1 Fl

——r—I

 

 

 

 

 

 

 

 

n

 

 

 

 

 

 

 

 

Reflected
Dummy POW?
load monitor

 

 

 

Coupler

 

/

Vacuum
window

 

Plasma

 

Figure 2.5 The microwave circuit of an ECR plasma source.

 

per

IDd

p0\

mic

C111

2.11

’9
be

IEIII
Fun
mdi:
CD61
muiz

fime:

permanent magnets located behind the microwave window to produce a divergent
magnetic field. They used a 15 x 15 x 9 cm, 18 kg block Nd-Fe-B magnet to generate a
magnetic field which falls to 875 G at a distance of 8 cm from the magnet face. Microwave
power is applied from the side into the space between the magnetic face and the
microwave window. Plasma uniformity was improved by surrounding the process

chamber with an additional permanent magnet bucket. Plasma densities greater than

2x10ll cm"3 in an argon discharge at 500 W and 1 mTorr were achieved.

2.3.2 Distributed ECR Plasma Sources (DECR)

In a common ECR microwave sustained discharge, the plasma density is limited by the
ratio of the surface area of the source to the volume of the reactor into which it diffuses.
Furthermore, this type of source has the drawback that conducting or semiconducting
materials can get deposited on the dielectric window which transmits the microwave
energy into the discharge. To solve these problems, a new reactor concept, based on a
multipolar magnetic field confinement structure, termed distributed ECR (DECR) [33]
emerged. This new design allows integration of the microwave field applicator to the
multipolar magnetic confinement structure such that the ECR conditions (approximately
875 G at a 2.45 GHz excitation frequency) are met. Such a scheme permits one to adjust
the plasma source dimension to the required plasma size.

Fig. 2.6 is a schematic representation of the microwave field applicator in a cylindrical
reactor. It consists of a linear conductor of cylindrical cross-section, called the antenna,
placed a few millimeters above the ground-plane constituted by the reactor wall.

Permanent magnets are arranged in a similar way as for classical multipolar magnetic field

Antenna
/

 

 

 

 

 

 

 

 

 

 

Multipolar
Plasma ‘/ bar
' magnets
A I‘ ii ‘i‘ A,
Chamber wall
Coolant
To
Pump
Antenna
ECR zone Magnetic Field Line Lobe

 

 

 

 

 

 

A,

Figure 2.6 Distributed ECR plasma source (DECR).

17

leng

mu}:

 

2.3.3
Si.

IMPD

confinement. These magnets provide the required 875 G (the ECR condition at a 2.45 GHz
excitation frequency) isomagnetic surface in the vicinity of the antenna along its entire
length. The magnetic field is closed by two adjacent magnet rows as in a conventional
multipolar field. A magnetic field configuration is thus created whereby a microwave field,
applied at 2.45 GHz through a coaxial feedthrough, results in ECR coupling along the full
antenna length. The gas is ionized by electrons in the lobes and the produced plasma then
diffuses to the center of the chamber.

The DECR plasma source can be easily scaled up by increasing the number of
antenna-magnet bar pairs. However the direct exposure of antennas to the discharge may
cause possible contamination from sputtering antenna material during reactive etching
processes. During the deposition processes, the antennas can also be deposited with non-
conducting material on their surfaces, causing abnormal operation. In addition, in the
presence of plasma, microwaves cannot propagate along the antennas without absorption,

and this may lead to a non-uniformity of the plasma along the axial direction.

2.3.3 Microwave Plasma Disk Reactor

Similar to the DECR system described previously, the microwave plasma disk reactor
(MPDR) uses a multipolar magnet configuration surrounding the discharge region. Rather
than using a discrete wave applicator for each magnetic cusp, the MPDR uses a resonant
cavity applicator allowing electromagnetic resonance in a single mode. Fig. 2.7 shows an
example of an MPDR utilized in the processing of material. The system has a cavity with
a sliding short for an adjustable cavity length and a tunable microwave coupling probe

which allows the impedance matching of the source with the plasma load and enables

18

Sliding Short

  
 

 

 

 

   

 

       

 

 

 

 

 

 

 

 

 

 

 

Plasma
Region
Microwave
/ Input Probe
Magnets 3:]
E
Gas Feed . ___{ [Z
t i
j
g —> To Turbo
A Pump
5 \
t t
z \

 

 

 

 

 

 

 

Rf Power

 

 

 

 

Figure 2.7 An example of the MPDR system

19

lfldg

til-r

 

and
discl

prod

magi
lm'cs
stud;

andl

efficient power coupling to the plasma. The cylindrical bell jar surrounded by multipolar
magnets and located in the lower end of the cavity is the plasma generation region. This
type of applicator enables specific and prechosen electromagnetic modes to be impressed
and efficiently coupled into the discharge chamber. The resonant cavity facilitates simple
discharge ignition, and by length and microwave coupling probe tuning of the cavity, it
produces rapid automatic impedance matching of the plasma machine as the process
variables are changed [34],[35]. The MPDR is capable of matching microwave power into
disk or cylindrical discharge volumes over pressure regimes of sub mTorr to over 300 Torr
[36]. The processing wafer stage is located in the downstream region away from the strong
magnetic field. Rf power maybe applied on the substrate to provide a substrate bias. The
investigated plasma source performance of the MPDR 610 generated plasma source is
studied in this thesis. A more detail description of this low pressure ECR plasma source

and its applications is given in Chapter 4 of this thesis.

2.4 Plasma Modeling and Simulation Methods

There are a variety of discharge models in a number of papers in the literature ranging
from zero-dimensional global models to two and three-dimensional models. In general, a
zero-dimensional global model assumes a uniform steady state plasma and solves the
plasma parameters by considering two basic equations: (1) charged particle conservation,
i.e., the number of ions created in the plasma by collisions is equal to the loss of ions by
diffusion to the wall, and (2) the power absorbed by the electrons must be equal to the

power loss from the electron gas by electron-neutral collisions, inelastic and elastic ion-

20

[NC

by .l

aid;

can t

 

 

 

axera
consc
hdam
the g!
mannt
162112111

(h
“Umcr

densn;

neutral collisions, and the loss of electron and ion kinetic energies as they recombine on
the wall. The global model assumes a Maxwellian electron energy distribution described

by a temperature of T6 and the electron collision rates are constants in the plasma volume.

In addition, at the low pressures of this study, the electron-ion recombination occurs only
at the walls and is absent in the plasma volume. A detail description of this global model
can be seen in Ref. [51] and the model is further described in Chapter 5 of this thesis.

M. Meyyappan and T. R. Govindan developed a model to predict the spatially-
averaged plasma characteristics of ECR reactors [37]. The model consists of global
conservation equations for species concentration, electron density and energy. A gas
balance is used to predict the neutral temperature self-consistently. The model provides
the global plasma characteristics as a function of system variables in a very efficient
manner and can be used in a sensitivity study to identify the contribution of various
reactions to the overall system behavior.

Other than the global model which solves a uniform zero-dimensional plasma, a
numerical simulation model considers at least one-dimensional variation of the particle
densities in the discharge. One method to deal with the properties of the processing
discharges is by means of fluid Simulation, in which each of the charged particle species
(electrons, and positive and negative ions) is treated as a separate fluid, characterized by its
temporally and spatially varying density, average velocity, and average energy. A
fundamental assumption in this model is that the particle distribution functions of the
various species are known. They are usually taken to be drifting Maxwellians.

Examples of applying the fluid model in plasma simulation are briefly described

below. A one-dimensional fluid model of the microwave ECR discharge, which includes

21

P0“ ’~
Short
Lymt
With 1
phflfli

chem]

 

 

dfinsflf
lnCICdx

1h? dc;

the inhomogeneity effects of the external magnetic field, was developed by N. S. Yoon et
al. [38]. For the efficiency of the simulation, the plasma body and the sheath regions were
separately modeled. For the argon discharge, various quantities such as the axial profiles
of plasma density, electron temperature, and microwave power deposition were obtained.
Another simulation is of low pressure inductive plasma sources using a time averaged
two-dimensional fluid model including an electromagnetic module with self-consistent
power deposition [39]. Comparison with the experiment and previous simulation results
showed that the fluid model is feasible in a certain range of gas pressure. D. P.
Lymberopoulos et a1. studied the two-dimensional fluid simulation of polysilicon etching
with chlorine in an inductively-coupled high density plasma source [40]. The complex
plasma chemical reactions (involving electrons, ions, and neutrals) as well as surface
chemistry were included in the simulation. Quantities such as power deposition, species
density and flux, and etch rate and uniformity were calculated. As power deposition was
increased, the electron density increased linearly, the plasma became less electronegative,
the degree of gas dissociation increased, and the plasma potential remained constant. The
radial uniformity of the Cl atom flux was better than that of the ion flux.

Another numerical simulation used is the particle-in-cell (PIC) model [41],[42]. The
Lorentz force on each particle determines its self-consistent motion, and the charges and
currents generated by the moving, charged particles determined the self-consistent fields
through Maxwell’s equations. Monte Carlo techniques are used to determine the short-
range collisional process (ionization, scattering, and so on). This technique yields time-
and space-resolved information on the charged particle velocity distribution functions in a

discharge from which fluxes and generation rates can be calculated.

22

dis

uni
dl N
sen

5C3

Una
elec
Spat
with

“in

Ineth
Nil]
circa
held.
ElE‘Ctr

CODDQ.

Surendra et a1. [41] used the PIC simulation to study the structure of rf glow
discharges in helium. The differences between discharges in which secondary electrons
play a key role in sustaining the discharge and those in which secondary electrons are
unimportant were examined in three cases which illustrate the importance of the
discharge-sustaining mechanisms. Electron energy distribution were found to be, in
general, non-Maxwellian. The simulation also indicated that the ion power deposition
scaled as the square of the applied voltage, while electron power deposition scaled
approximately linearly with applied voltage. Kuo et a1. [43] simulated the electron energy
distribution (EED) in an ECR microwave discharge via the Monte Carlo technique. The
time averaged, spatially dependent EED was computed self-consistently by integrating
electron trajectories subjected to the microwave field, the divergent magnetic field, the
space charge field, and the sheath field. The electron-electron collisions and collisions
with the neutral hydrogen atoms were considered. At low pressures (0.5 mTorr), the EED
was spatially independent due to the large electron mean free path.

Other simulations such as the hybrid model used particle ion and fluid electron
methods to simulate charged particle behavior in the high density ECR plasmas [44],[45].
N. H. Choi et al. used the one-dimensional hybrid model which included both the ECR
electron heating phenomena and the transport of ions along the divergent axial magnetic
field lines. Microwave power was considered as an energy flow attenuated by the thermal
electron fluid. Electron motions were coupled to the ions through ambipolarity. The results
showed a strong effect of the distributed ionization on the ion energy distribution.

The application of global models to the MPDR 610 plasma source with magnetic

confinement in the source region for three different gas discharges is considered in this

23

WEI

ofs

thesis. Previous modelings of this plasma source include a two-dimensional numerical
model using a Monte Carlo particle method coupled with a solution of the Maxwell
equations and a three-dimensional electromagnetic particle-in-cell (PIC) model [46]-[48].
The particles in the source move subject to the Lorentz force equation and to the
appropriate elastic and inelastic collision processes. The electric fields and time-varying
magnetic fields were solved using a finite-difference time-domain (FDTD) technique. The
electromagnetic fields and the plasma dynamics were solved in a self-consistent manner.
The static magnetic field produced by the permanent magnets, microwave electric fields at
2.45 GHz, microwave power absorption, and electron energy distribution in the source
were modeled for argon and helium discharges. The difficulty with these models is the use
of small times in the solutions. The time steps are small enough that many of the slower
chemical reactions can not be simulated in reasonable simulation times. An alternative is
to use global models which allow all the reactions to be included at the expense of not

having information on the detailed spatial variations in the plasma source.

24

 

3.1

3 HUI

€11ng

laser

CHAPTER 3
The Experimental Diagnostic Techniques and

Theories

3.1 Introduction

In order to understand and analyze the plasma properties during the actual processing,
a number of diagnostic techniques have been developed [49]. These techniques can be
categorized as intrusive such as Langmuir probe measurements and non-intrusive such as
optical emission spectroscopy(OES), actinometry, laser induced fluorescence and infrared
laser absorption.

In this chapter, we are going to discuss the plasma diagnostic techniques which were
used in this thesis. They include Langmuir probe measurements, DES, and actinometry.
Double Langmuir probe (DLP) were used to characterize positive ion concentrations and
to estimate the electron temperature. Single Langmuir probe (SLP) were used to obtain
electron temperatures and electron energy distribution function (EEDF). OES was
employed to measure the electron temperature in the source. Finally, actinometry is

utilized to obtain the density of neutral radical species.

25

 

pro

3551

0111]

the

[cm

pro]

3.2 Double Langmuir Probe Measurements

One of the most widely used plasma diagnostic techniques is the measurement of
positive ion densities using a double Langmuir probe (DLP). The DLP measurement also
provides the electron temperature of the high energy tail of the distribution under the
assumption that the electron energy distribution function (EEDF) is Maxwellian.

The set up of the DLP measurement circuit is shown in Figure 3.1. Two probes made
of tungsten are covered by a pyrex tube on the outside to isolate all but the probe tips from
the plasma source. Each probe tip has a diameter of 0.11 cm and a length of 0.64 cm. The
entire circuit is electrically floated. A dc voltage is applied between the two probes. The
output of the DC voltage supply and the current reading from the multimeter are sent to
the computer through a series connection of GPIB cables for further analysis. The electron
temperature and ion density are then determined from the I-V characteristics curve of the
probe.

Figure 3.2 shows a typical DLP I-V characteristics generated from the compact plasma
source, MPDR610. The applied voltage on the probe is from -50V to +50V. When the
applied voltage between the electrodes are zero, both the probes collect equal amount of
electrons and ions. Hence the current in the circuit is zero. When there is a potential
difference between the two probes, the more negative one will begin to repel electrons and
collect more ions. While the other probe (more positive one) will collect more electrons
than ions. The current flow will no longer be zero. As the differential potential starts to
increase, the current flow will also increase and eventually the more negative probe will

draw the ion saturation current, which is balanced by the net electron current drawn

26

 

Computer-
Controlled

Station

 

 

GPIB

 

 

ll

 

 

 

GPIB

 

 

 

 

Picoammeter
Keithley 485

 

 

 

 

 

 

DC Power Supply
HP 6634A
0-100V 0- l A

 

 

 

I DC Power Supply
HP E3612A

0—60 V 0—0.5A

 

 

 

 

 

Figure 3.1 Double Langmuir Probe Circuit

27

hCDL

11151.".

proht

 

In ter

Assu

the e

to the other probe. Notice that the net current never exceeds the ion saturation current, and
hence the electron current collected will be only from the high energy tail of the electron
distribution.

The theory of the DLP is explained as follows [51]. The ion and electron currents to

probes 1 and 2 are defined as I”, 16] and la , 162. The condition that the circuit float is

111 +112 '16] '182 = 0 (3.1)
In terms of net current in the loop I, Eqn. (3.1) can be rearranged as

Iel “Iil : Ii2'Ie2 = 1 (3-2)

Assuming a Maxwellian distribution of electron energy, and from Boltzmann’s relation,

the electron current can be expressed as

V T
= Allle '/ ”and (3.3)

el

vz/T,

Asze (3.4)

lez
where J l and .12 are electron random current densities to the probes, A1 and A2 are the
probe collecting areas, and V1 and V2 are the probe potentials with respect to the plasma
potential. Note that the electron temperature Te here is expressed in eV. Using V=Vl-V2

and substituting Eqns.(3.3), (3.4) into Eqn.(3.2), we have

1+1“ A, m-
= — ’ 3.5

 

If A1=A2, then In =Ii2 =Ii. Eqn. (3.5) can be simplified to

28

l(nl/\)

-(

Fl;
c0"dir.

 

0.2 - % .4

0.1 — 1i _

 

 

 

 

 

 

E o
:7
—0.1 - n
-0.2 “- -
‘0 3 1' l 1 1 1 1 1 1 1 1 L 1 7
-50 -40 -30 —20 -1 0 0 1 0 20 30 40 50
V (volt)

Figure 3.2 A typical I-V characteristic of a double Langmuir probe. Plasma
conditions are Pin = 90W, 3 mTorr, and flow rate = 8 seem. The data is collected at 2

cm downstream.

29

One

 

files

It here
1:11 2e

the d

it here
lhe 101'
Probe5

Deb,-

,t‘

 

l = [rev/T T1 or
6 '+1
V
I=I.t h .
.an (2T) (36)

One simpler way to obtain Te from the experimental I-V characteristic is by taking the

derivative of the above equation with respect to V, and we have

 

d1 ’1' V 2
(7V — 2Te(SCChTe:) (3.7)

The slope of the I-V plot at V=0 can then be related to T8 as

I.
:11 = —' (3.8)
dV V=0 2T

6

where Ii is the ion saturation current and is determined by the intersection point of the

tangential lines as shown in Fig. 3.2. Once the electron temperature is determined, from

the development of Chen [50], the ion density is related to the ion saturation current (It) by

kT 1/2
J (3.9) '

1. = 0.6n.eA {—5
l l p ml,

where “i is the ion density, e is the electron charge, Ap is the effective probe area, and mi is

the ion mass. The effective probe area represents the increased ion collecting area of the
probes due to the plasma sheath. It is approximated by the actual probe dimension plus the

Debye length, 21D, where

AD = 69 /T,/n, (m) (3.10)

30

31'

p1.

3.3

ECI
ener
eicn

under

vi.

 

appIOXI

Ofllhtr

 

which is also a function of plasma density and electron temperature. Thus several
iterations between Eqn’s. (3.9) and (3.10) were applied in the probe I-V characteristics
analyzing computer program. The termination of the iteration is set whenever the new

plasma density falls within 1% of the previous one.

3.3 Single Langmuir Probe Measurements

In an ECR discharge, electrons gain energy from the electromagnetic field through
ECR excitation or through ohmic collisional heating. The electrons then lose their
energies by collisions with ions, neutrals, and the wall. Since the electron impact
excitation is the major source of reactive species in low pressure plasmas, it is important to
understand the distribution of electron energy.

In 1930, Druyvesteyn [52] showed that the EEDF is proportional to the second
derivative of the I-V characteristic of a single Langmuir probe in the region where the

applied voltage is below the plasma potential:

(121

f(E)°< /Vp—V 2‘? (3.11)

dV

 

where Vp is the plasma potential, V is the voltage applied to the probe, Ie is the

electron current drawn by the probe, and E = Vp — V. The second derivative is

approximated with dZI/dV2 where I is the total probe current and it equals to the sum of I6

and ion currents drawn to the probe. This approximation is justified if the applied voltage

on the probe is less than Vp in the regions where the ion currents have small variations.

31

 

Multimeter
HP 3478A

 

 

 

 

‘1

ll

 

 

HP 65 1B
Function
Generator

 

(121/de
Output

 

Lock-in
“3f- Amplifier

 

 

 

E

 

 

 

DC Power Supply -

HP 6634A
0 - 100V

" ""_ i 100 ohm
_i i————A/\/\/\

PAR 128A

 

 

 

 

 

 

 

 

 

 

l
GPIB

 

Cable

 

 

 

 

 

Computer— Wall

Controled
Station

 

 

Figure 3.3 Single Langmuir Probe Circuit Set Up.

32

lilL‘

COIT.

(Elli.

 

The plasma potential adopted in this thesis is from the conventional definition in Ref.
[53],[54] which is taken to be the voltage where the maximum of the second derivative of

the Langmuir probe I-V characteristic occurs. The EEDF, f(E), is normalized according to

0'0

[f(EVlE = 1 (3.12)
0

The set up of the single Langmuir probe (SLP) measurement is comparably more
complicated than that of the DLP circuit. And the result shows how the electron energy is
actually distributed instead of the assumption of Maxwellian distribution used in the DLP
measurements. Figure 3.3 shows the set up circuit for SLP measurements. A single
Langmuir probe of a tungsten wire protruding from a pyrex tube is inserted in the plasma.
It has a cylindrical electron current collecting area of 1.1 mm in diameter and 6.3 mm in
length. A power supply with a GPIB interface is used to provide computer control of the
sweeping DC voltage applied to the probe. The output of the dc power supply is varied
from ~15 to 20 volts with a step voltage of 0.5 volts. A small sine wave with a frequency of
1 kHz and amplitude of no more than 0.5 volts is superimposed on the DC probe voltage
through a transformer. This will produce a probe current that has both DC and AC

components:

I=I{V+a(sin (or )} (3.13)

Expanding this function as a Taylor series, we have

— + ...:l + A10“.l + Alac2 + higher order harmonics (3.14)

33

where

3 3
AI . = [afl+g—fl+...:|sinu)t (3.15)
8dV3

AI _ gfdfi afifl
“”2- 4dV2 48dV4

—+———+ ...](—cos2(1)t) (3-16)

are the first and second harmonic terms. The second harmonic current is proportional to

the second derivative d21(V)/dV2 if the amplitude of the AC signal is small enough
compared to the DC value and the interference from the higher order differential terms is
minimal. Therefore those terms can be ignored without considerable error.

A Princeton Applied Research (PAR-128A) lock-in amplifier referenced with the
oscillator frequency differentially measures the voltage across a current-sampling resistor

with its input locked on twice the oscillator frequency. The filtered output of the lock-in

amplifier which is proportional to dZI/dV2 at the applied DC potential is sent to the
multimeter. Both the readings from the multimeter and DC power supply are recorded by
the computer through the GPIB bus for later analysis. Note that the entire probe circuit is
grounded at the processing chamber wall to provide a loop for electron current drawn from
the probe. Another computer program is used to modify the raw data taken from the probe
measurement and generate a plot of EEDF with normalization according to Eqn.(3.11).

The average of the electron temperature is then calculated from the EEDF using:

(a) = er(5)d5' (317)
0

34

 

d1:

3.4

 

3.4 Optical Emission Spectroscopy

Both the SLP and the DLP methods are intrusive techniques and hence they are
difficult to apply and interpret within the plasma source where the static magnetic fields
are present. Within the source region, optical emission spectroscopy (OES) may serve as a
more appropriate technique in plasma diagnosis. The set up of OES adopted in this thesis

is shown in Fig. 3.4.

3.4.1 Measurements for Te

The first OES technique applied in this thesis for measuring the electron temperature is
by using the emission spectrum originating from a range of excited state energy levels of
noble gases including argon, krypton, and xenon [55]. A small amount (less than 5%) of
these noble gases are added to the discharge using a second gas flow channel on the
plasma source system. The spectra of emission across the range from 750 nm to 850 nm
from the source region is then recorded with a ORIEL 1/4 m monochromator (Model
77100) which has self-scanning photodiode arrays for light detecting. The associated
threshold energies of the excited states observed range from 9.82 eV to 13.5 eV. A total of
19 emission lines were observed in this study. A list of the transitions with their associated

branching ratios, threshold energies, and electron excitation cross sections are

35

 

 

Plasma

 

 

 

 

 

 

 

 

 

ORIEL 77100
Spectrometer

 

 

 

 

 

 

 

Processing Chamber

 

 

 

 

 

Diode Array
Detector

 

 

 

 

 

Computerized
Emission Data
Recording

 

 

 

 

 

 

 

 

 

 

 

Figure 3.4 The set up for OES

36

summarized in Table 3.1.
To determine the electron temperature, the expected emission intensities are calculated
at each experimentally observed wavelength for various electron temperatures. The

assumption is made in this study that the population of the emitting states come mostly

Table 3.1 The cross section data, branching ratios, threshold energies of the observed

transitions [63]-[65] and the spectrometer correction factors.

 

 

 

Gas A j, k bj’k Em Emax Gmax Q
(nm) (eV) (x10‘19 cmz)

750.4 1.0 13.5 21.0 114 .6289

751.4 1.0 13.3 23.0 48 .6325

763.5 0.41 13.2 23.0 132 .6850

842.4 0.55 13.1 23.5 202 1.3751
811.5 1.0 13.1 22.0 195 1.0506

768.5 1.0 12.3 20.5 20 .7167
826.3 0.94 12.2 21.0 109 1.1734
785.5 0.55 12.1 20.5 51 .8282
806.0 0.38 12.1 19.5 76 1.0161
850.9 0.62 12.1 19.5 76 1.4596
758.7 1 0 11.7 20.0 88 .6621

760.1 0.57 l 1.5 20.0 160 .6693

819.0 0.43 11.5 20.0 160 1.1225
769.5 0.13 11.5 20.5 124 .7204

829.8 0.87 1 1.5 20.5 124 1.2003
810.4 0.25 11.4 20.0 215 1.0496
877.7 0.75 11.4 20.0 215 1.6249
811.3 1.0 11.4 20.0 108 1.0506

25555555555252.5533???

Xe 834.7 0.97 11.06 18.0 33 1.2829
Xe 828.0 1.0 9.94 27.0 83 1.1905
Xe 823.1 0.62 9.82 14.5 96 1.1552

 

37

from the ground state excitations instead of from metastable state excitations. This
assumption is made based on the electron energy being 4 - 8 eV for the pressure range in

this work. The energy range is high enough that extensive excitation occurs directly from

the ground state. The intensity of emission at wavelength hf, k accompanying the state

transition A k ——> A j is given by

1]., 1. = 4rtQ_l(AJ-,k)ngbj, 4:68. k(v)v3fe(v)dv (3.18)

where Q01.j k) is the spectrometer correction factor at 71. ng is the radical density at the

j. k ’
ground state, 0'8, k(v) is the effective cross section at electron speed v for electron impact
excitation from the ground state g to state k, and v0 is the minimum electron speed for
excitation. The determination of the Q’s values will be discussed in the following

subsection. The branching ratio is denoted as b“. for transition A k —> A j and it is defined

as

I.
b. = J—”‘ (3.19)
M 21,

where Ij,k is the relative intensity of emissions from state k to state j, and the summation in

the denominator represents the allowed transition from state k to all lower states. A

Maxwellian electron distribution fe is assumed in this technique. The electron temperature
is found by comparing the calculated emission intensities (Ical) at various assumed
electron temperatures to the observed emission intensities (Iobs)~ This comparison is done
as a function of the threshold energy (Em) of each of the excitations. First, dividing 10bS by

Ical and the log of this ratio is plotted versus Em. A least square line is then fitted to these

38

data [66]. Each fitted line corresponds to an assumed electron temperature. The slopes of

these lines in the assumed range of Te’s are then plotted versus Te. The electron

temperature in the source will then be found at zero slope.

3.4.2 Determination of Spectrometer Correction Factors

Although each spectrometer has its own scanning range, the output from the
photodetector may not be accurate without considering the spectrometer correction factor.
This difference between the measured output and the real spectrum of the incoming light
is usually due to the grating and photodetector sensitivity versus wavelength. It is
therefore necessary to obtain a correcting factor table for each different spectrometer
before taking the spectrum measurements.

The way to obtain the correction factor is stated as follows. A 45W quartz-halogen
tungsten lamp (Optronic Lab., Model 245C) with known calibration intensity over a range
of wavelengths was used and put in front of the Spectrometer that would be used for
measurements. During the calibration, the current flow through the lamp was fixed at 6.5
Amps in order to generate constant emissions from the lamp during the calibration
process. The outputs from the spectrometer were then recorded over the same wavelength

range of the known lamp spectrum (Ilamp)' Fig. 3.5 plots the spectrum of the tungsten

lamp. The summarized measurement results for the ORIEL spectrometer are shown in

Table 3.2. Once the measured spectrum (Imeasured) is known, the correction factor, Q can

be calculated from

Intensity (a.u.)

 

900

700—)

600

T

400

1003.

 

 

 

 

200

i 1 '
300 400 500 600 700 800
Wavelength (nm)

Figure 3.5 The emission spectrum of the tungsten lamp

4O

Table 3.2 The measurement results for the ORIEL spectrometer.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Wavfilingth Intensity Wazzlfiglgth Intensity
450 419 650 1498
460 448 660 1501
470 501 670 2042
480 525 680 2240
490 566 690 2010
500 595 700 1660
510 620 710 1505
520 710 720 1393
530 782 730 1329
540 900 740 1257
550 983 750 1195
560 1020 760 1136
570 1081 770 1108
580 1026 780 1075
590 1175 790 974
600 1233 800 906
610 1319 810 851
620 1403 820 623
630 1467 830 417
640 1492 840 388

 

 

 

 

 

 

41

The v

3.43

In
many
tonce
actint

Mudv

lllGl
\V

emlfisj

“id th.

IIamp< k)

l. = ——
Q( ) [measured(}')

(3.20)

The values of Q are seen to be varied for different wavelengths and shown in Table 3.1.

3.4.3 Actinometry

In a hydrogen discharge, hydrogen molecules dissociate into atomic hydrogen and also
many different kinds of charged species by inelastic collisions. In order to figure out the
concentrations of different species in this relatively complicated discharge, the
actinometry technique is applied to find out the relative atomic hydrogen density. In this

study, actinometry will be performed in H2 discharges for neutral density measurements.

(1) General Expression of Actinometry [56,57]
We begin with a general expression of actinometry as follows: the intensity of an

emission line may be expressed as

I ocv = N (vo ,) = N r vo’ f(E)EdE (3.21)
x ex x ex x E ex
ex
and the intensity of an actinometer as
I ocv = N (v0 ) = N r v0" f(E)EdE (3.22)
a ea a ea a E ea
ea

where o is the excitation cross section, N is atomic gas concentration, Ecx and Eea are

42

excitation threshold energies, f(E) is electron energy distribution function, and v is

excitation frequency. Dividing Eqn. (3.21) by Eqn. (3.22), we have

I =N r vo f(E)EdE
X X E ex

ex

 

(3.23)
I =N r v0 f(E)EdE
a a E ea
ea
AssumingO’ =0 andE 1=E ,then we have
ex ea ex ea
Ix Nx
]— : N— (3.24)
a a

The criteria of choosing an actinometer is (a) it is relatively inert to the desired species,
and (b) the energies of the involved excited states are similar. From Eqn. (3.24) we
conclude that the desired atomic species concentration is proportional to the ratio of

desired species emission intensity to the actinometer emission intensity.

(2) Modified Actinometry Theory in this Thesis

This modified actinometry technique is used to determine the absolute H-atom
concentration in the discharge following the work in [58],[59]. Argon is chosen as the
actinometry gas in this study since it satisfies the two criteria mentioned above. The
emission lines observed were 750.4 nm for argon (energy level = 13.4 eV) and 486.1 nm
for hydrogen (energy level=12.75 eV). As noted in section 3.4.1, the contribution of argon
emission at 750.4 nm due to excitation from metastable states is small. Hence only the

excitations from ground states are considered in this thesis.

43

The emission intensity from argon can be expressed as

—1 oo 2
1,, = Q (1A,)bArnArjvmoA,(v)vf,(v)4nv .112 = a(xA,)bA,nA,KA, (3.25)

where Q is the spectrometer correction factor at wavelength 21A,, 11 Ar is the argon neutral
density, b A, is the argon branching ratio for the line at AA, [55], and GA,(v) is the electron

excitation cross-section of argon ground state atoms by electrons of velocity v. A
Maxwellian assumption for the electron distribution is assumed and the integral is solved

as
KAr(Te) = I: 0Ar(v)vfe(v)41tv2dv (3.26)

Next consider the emission from H(486.l nm)

—1

where again a Maxwellian approximation to the electron distribution function is used. “H
is the atomic hydrogen density and nm is the molecular hydrogen density. bill} is the
branching ratio for the line at AHB [62]. The excitation rate constants K H15 and KHZ are
calculated using the same equation as in Eqn. (3.26) by replacing 6A, with O'HB and 0H2
respectively. Here the cross section data of CH5 and 5H2 are from [60][61]. Here we

consider both the direct electron excitation of atomic hydrogen and the dissociative
excitation of H2. The dissociative excitation becomes more important as the electron
temperature in the discharge goes higher. The electron temperatures in the source as
shown later in Chapter 6 range from 6 to 8 eV. It is found that for electron temperatures in

this range the effect of dissociative excitation is not negligible and therefore is included in

44

Eqn. (3.27). Dividing Eqn. (3.27) by Eqn. (3.25), we have

 

 

—l
l—— = _l K (3.28)
Ar Q (AAr)bA,— "Ar Ar
Rearranging this equation gives
—1
"H Km Q (Aypllnfi bHBKHznflz .
— = K b —1 T.— K n__ (3.29)
"Ar H9 1vB Q (km) Ar Ar Ar

which yields the absolute atomic hydrogen density based on actinometry results and on
the model of the associated electron excitation rates calculated from the electron

temperature in the discharge.

45

CHAPTER 4

Experimental Properties of MPDRTM 610

4.1 Introduction of MPDR 610

In this chapter the properties of the plasma discharges generated in the compact
microwave plasma disk reactor are experimentally investigated. The source studied is
designed for materials processing and synthesis applications, including use as an ion and
free radical source injector in MBE deposition processes [67]. Some materials processing
applications that have been investigated using this source include its use as an atomic
nitrogen source in III-V nitride growth [68],[69], as an atomic oxygen source in high-
temperature superconductor deposition [70],[71], and as an atomic hydrogen source for
III-V substrate cleaning [72]. In the past this source has been characterized [73]-[77] for a
number of discharge types including Ar, He, N2 and 02, but little work has been reported
on hydrogen discharge characteristics. This chapter experimentally characterizes the
Wavemat MPDRTM 610 [78] plasma source with argon, hydrogen, and argon-hydrogen
mixture discharges running at the pressure range of 0.4-5 mTorr.

A sketch of the MPDR 610 source is shown in Figure 4.1. The cylindrical source is
made of stainless steel with an outer diameter of 5.8 cm. The vacuum sea] at one end of

the source is made by a standard 4.5 inch Conflat flange with the entire length of the

46

source inserted into the processing chamber. At the other end of the source is a UHV metal
to quartz vacuum seal. The discharge region at this end is confined within a cylindrical
quartz tube which has a 3.6 cm diameter and 3.0 cm length. The quartz tube is surrounded
by three ring shaped axially magnetized permanent magnets which provide a static
magnetic field within the plasma discharge chamber for plasma confinement and for ECR
plasma heating. The magnets have an outer diameter of 4.95 cm, an inner diameter of 4.32
cm, and a height of 1.27 cm. The three magnets are aligned with the same poles facing
each other, as shown in Fig. 4.1. This magnet arrangement creates a multicusp magnetic
field in the axial direction and tends to push the 875 Gauss ECR zone away from the
quartz discharge wall. The quartz walls and the magnets are cooled by compressed air
flowing through the center conductor and blowing onto the quartz top. The center
conductor and the sliding short permit optimal microwave power coupling to the discharge
region. A small loop antenna is attached to the sliding short, and this loop excites the
electromagnetic modes in the microwave cavity. The region between the discharge and
sliding short is a waveguide section with the first half section a coaxial waveguide section
and the second half the evanescent circular waveguide region. The reason for the
evanescent electromagnetic wave is the diameter of the waveguide region (5.6 cm) is too
small to sustain any electromagnetic propagating modes for the 2.45 GHz excitation
frequency. The lowest frequency EM propagating mode - TE” requires a diameter of at
least 7.2 cm at the frequency of 2.45 GHz. For more detail description of the MPDR 610
source, the reader should refer to Ref. [79].

Previous experimental measurements and characterizations of the MPDR 610 source

done by A. K. Srivastava are summarized as the following:

47

.553 aeo Se man—2 3. 2&3

$0:me mo souauswccoo

 

 

_..__m 22 m7. 2_

 

 

 

 

 

 

 

:5 7:2 flmfl

 

882250
8:50

 

23.50
m team. :65 we
Enzm \ @3956

 

 

 

 

an sawed (f ,
N Al_ mesa \1 m

 

 

 

 

 

 

@2000 he.

 

 

 

L

 

 

 

 

 

 

 

 

 

 

MD

 

 

 

 

 

 

\ \

6:: 30 «58:2 @004

Enema—2 0:5 EU
“accustom

poem £0

 

 

»

WI 880550
o>m3832
\ 2a:

 

48

(1) Ion density versus pressure (0.1 - 0.8 mTorr) at different input microwave powers (66,
82, 123, 164Watt).
(2) Ion current density versus pressure (0.1 - 0.8 mTorr) at different input microwave
powers (66, 82, 123, 164Watt).
(3) Electron temperature versus pressure (0.1 - 0.8 mTorr) at different input microwave
powers (66, 82, 123, 164Watt).
(4) Ion current density versus radial position (r=0 - 2 cm) at different downstream
positions (2:1, 3 cm) and different input microwave powers (123, 164Watt).
(5) Electron energy distribution function at some specified plasma conditions.
(6) Ion energy distribution function at different input microwave powers (82, 123,
164Watt).
(7) Ion energy distribution function at different pressures (0.24, 0.43, and 0.72 mTorr).
(8) Plasma potential versus pressure (0.1 - 0.8 mTorr) at some specified plasma conditions.
(9) Ion energy distribution function at different downstream positions (2:5, 7, 9 cm).

It is noted that the pressures specified in this earlier work were the pump stack
pressures. The pressure in the chamber was measured later to be about twice that in the
pump stack. And, most of the measurements were taken in argon discharges. For nitrogen

discharges, only measurements (5), (6), (7), (8), (9) were taken.

4.2 System Configuration

Fig. 4.2 shows the vacuum system used in the experiments. The processing chamber

(1) with diameter of 19.5” and length of 10” is capped by two stainless steel plates (2)

49

with thickness of 2.5 cm on both its top and bottom ends. The vacuum seals on both ends
are made with two O-rings. The plates are equipped with electrical motorized control to
raise and lower the plates (2) . This provides an easy access to the processing chamber for
plasma diagnostics set up or processing substrate positioning. On the sides of the chamber
there are four 10” diameter flanges each of 90 degrees apart. The MPDR 610 source (3) is
inserted through one of the openings. Another opening at 90 degrees from the source is
connected to a diffusion pump (4). The passage is controlled by a high vacuum gate valve
(6) and then a throttling valve (7). The throttling valve is controlled by a MKS-651
pressure controller for a more precise control of chamber pressure. The diffusion pump is
backed by a mechanical pump (5) which is purged by dry nitrogen at the exhaust when
operating with a flammable gas such as hydrogen. Two thermocouple vacuum gauges (8)
measure the roughing and foreline pressures which range from several mTorr to a few
Torr. To measure the chamber pressure, a MKS—627 absolute pressure transducer (9) is

installed at the bottom of the chamber and connected to the MKS-651 pressure controller
which measures the pressure down to 1x10“5 Torr. In addition, an ionization vacuum
gauge (SensaVacTM 919) is located at the passage to the diffusion pump. This hot cathode

vacuum gauge is capable of pressure readings down to 1x10'10 Torr and is used to
calibrate the reading of the MKS-627.

High purity(99.999%) gas cylinders are connected to appropriate pressure regulators.
Then the gases are flowed through 1/4” stainless steel tubing to a MKS-1159 flow
controller (10) which is able to operate at a maximum flow of 100 sccm. The flow

controller is connected to a MKS-247C 4-channel readout for flow rate settings.

50

 

 

Rte—@— 10 x
X 10 x \\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\\ M"
q/

 

 

 

fl 0
- .

® : Vacuum valve switch

 

 

 

i\\

 

 

 

Figure 4.2 The vacuum system configuration

(1)Processing Chamber (2)Stain1ess Steel Plate (3)MPDR 610
(4)Diffusion Pump (5)Mechanica1 Pump (6)Gate Valve

(7)Throttling Valve (8)Thermocouple Vacuum Gauge (9)Absolute Pressure Transducer
(10)Flow Controller.

51

Microwave energy is supplied by a microwave power generator (Raytheon PGM-lO).
It outputs 2.45 GHz microwave power up to a maximum power of 100W. The microwave
circuit includes a three-port circulator and a dummy load to protect the power supply. Two
dual directional couplers each connected to attenuators and then two HP-432A power
meters measure the incident and reflected power. The power meter for incident power
reading was calibrated to have a correction factor of 0.909. That is, a 10 mWatt reading on
the meter have an actual power of 9.09 Watt. The correction factor of the power meter for
reflected power was also measured to be 0.902. The coaxial cable which transmits the
microwave power to the MPDR 610 source has been measured to have a loss of 0.89dB.

That is, 19% of the incident power is consumed into heat as it transmits down the cable.

4.3 The System Parameter Spaces and Measured Quantities

In order to have a better understanding and controllability in a complex plasma
process, the processing system variables are categorized as three types of variables: the
input variables, internal variables, and output variables. The input variables are defined to
be able to be independently controlled and/or set at the beginning of a processing
experiment, for example the operating power, pressure, gas flow rates, and substrate
temperature. The internal variables are the dependent variables which are related and
controlled by the different combinations of input variables. They can, for example, be
plasma electron temperature, ion density, and neutral density. The output variables are the
desired outputs from a process.

The following subsections describe the input variables and internal variables

52

COD

pen
diff

late

4.3

4.3

Pro

considered in this thesis. The diagnostic measurements of the internal variables were
performed in two regions: plasma source region and chamber (downstream) region with
different settings of input variables. The results of these measurements will be presented

later in this chapter.

4.3.1 Plasma Source Region Measurements
The input variables and observed internal variables defined in the measurements of the
plasma source region are :

(1) Independent input variables:
(a) Pressure, variable: P = 0.4 - 5.0 mTorr.

(b) Input microwave power, variable: Pm = 60 - 90 Watt.
(c) Gas feed, fixed: H2.
((1) Gas flow rate, fixed: f = 50 sccm.

(e) Microwave cavity tuning (Sliding short), tuned for minimum Pref.

(f) Microwave cavity tuning (Probe position), tuned for minimum Pref.

(2) Dependent internal variables:

(a) Reflected power, Pref.
(b) Electron temperature, Te.

(c) Neutral density: [H].

4.3.2 Processing Chamber Region Measurements
The input variables and observed internal variables defined in the measurements of the

processing chamber region are :

53

(1) Independent input variables:
(a) Pressure, variable: P = 0.4 - 5.0 mTorr.

(b) Input microwave power, variable: Pin = 60 - 100 Watt.
(c) Gas feed, variable: Ar, H2, Ar/Hz .
((1) Gas flow rate, variable: f = 5 - 50 sccm.

(e) Microwave cavity tuning (Sliding short), tuned for minimum Pref.

(f) Microwave cavity tuning (Probe position), tuned for minimum Pref.

(2) Dependent internal variables:

(a) Electron temperature, T6 and EEDF.
(b) Electron density, [Ne].

(c) Ion densities: [ArI], [H+], [Hf], [Hf].

4.4 Calibration of the Source Pressure

A pressure gradient was found to exist between the plasma source and the processing
chamber region where the pressure transducer (9) is installed for chamber pressures
sensing. The measured pressures in the chamber is usually less than the actual pressure in
the MPDR 610 source. In order to find out the difference between them, an experiment
was conducted. The discharge source region was connected directly to the pressure
transducer by a plastic tube. The throttling valve position was changed from 100 to 1 %
open and at each valve position, the actual pressure reading in the source is recorded.
Next, the tube connecting the source region to the pressure transducer was removed. This
is the setting under which the actual diagnostic measurements were taken. The test was
then repeated. The chamber pressure was recorded. Table 4.1 lists the results of the

hydrogen flow rate test. The H2 gas flow was set at 30 seem, a typical value used in plasma

54

parameters diagnostics in hydrogen discharges. Table 4.2 and 4.3 list the results of argon

flow rate test with its flow rate set at 8 and 20 sccm, respectively.

Table 4.1 Results of hydrogen flow rate test. The flow rate is set at 30 sccm.

 

 

 

Valve position Pressure (mTorr) Difference

% In source In chamber

100 0.63 0.41 0.22
80 0.63 0.41 0.22
60 0.65 0.43 0.22
50 0.67 0.45 0.22
40 0.72 0.50 0.22
30 0.82 0.60 0.22
20 1.10 0.86 0.24
15 1.42 1.22 0.20
10 2.18 1.92 0.26

 

Table 4.2 Results of argon flow rate test. The flow rate is set at 8 seem.

 

 

 

Valve position Pressure (mTorr) Difference
% In source In chamber
100 0.73 0.41 0.32
70 0.73 0.42 0.31
50 0.80 0.50 0.30
30 1.04 0.73 0.31
20 1.45 1.17 0.28
15 1.96 1.70 0.26
10 3.02 2.82 0.20
5 4.95 4.81 0.15
1 5.81 5.65 0.17

 

55

the s
and

deer:
this t

SPfici.

Table 4.3 Results of argon flow rate test. The flow rate is set at 20 sccm.

 

 

 

Valve position Pressure (mTorr) Difference

% In source In chamber

100 1.70 0.81 0.89
70 1.75 0.86 0.89
50 1.90 1.02 0.88
30 2.42 1.56 0.86
20 3.40 2.59 0.81
15 4.62 3.88 0.74
10 7.24 6.56 0.68

 

From the above tables, it is seen that for argon flow at 8 sccm, the actual pressure in
the source is about 0.17 - 0.32 mTorr higher than the measured pressure in the chamber
and about 0.22 mTorr higher for hydrogen flow at 30 sccm. The pressure difference
decrease Slightly as the operating pressure increases. The experimental data presented in
this thesis therefore uses the corrected pressures in the plasma source unless otherwise

specified.

4.5 Microwave Power Absorption in the Plasma

Although the microwave reflected power is tuned to be minimum each time the
experiment on the MPDR 610 is running, the reflected powers in the discharges were

found tobe nonuniform in the argon discharge. Fig. 4.3 plots the absorbed power

56

100

90

80

70

60

Absorbed Power (Watt)

50

40

Figure 4.3 Power absorption in the argon discharge. Input power = 91W, flow = 8

 

l

l

 

1 1 1 1 l

 

30
0

0.5

1.5 2 2.5 3 3.5

Pressure (mTorr)

SCCIII.

57

 

(incidentpower - reflected power) in the argon plasma at a flow rate of 8 seem. The input
power is 91 W. During the experiment, the sliding short position and the center conductor
position were adjusted for minimum reflected power. Also note that the absorbed power
plotted in Fig. 4.3 did not account for the 19% power loss in the coaxial transmission line
as described in Section 4.2. Additionally, power absorbed by microwave cavity heating
(direct wall heating), which is estimated to be 13% of the incident power for the operation
conditions used in this work [79] needs to also be substracted to get the actual power
absorbed. This concludes that the actual absorbed power is 32% less the value shown in
Fig. 4.3. Unlike in the argon discharge, the typical reflected power observed in the
hydrogen discharge was usually zero at lower pressures and less than 5 Watt at higher

pressures (>3 mTorr).

4.6 Ion Density

The ion saturation current collected using a double Langmuir probe at pressures 0.5 -
4.0 mTorr in argon and hydrogen plasmas is shown in Fig. 4.4. The probe position was at
2 cm downstream. Fig. 4.5 shows the ion saturation current of the hydrogen discharge
measured at 2 cm downstream with different flow rates (10-55 sccm). The input power is
90 W and the chamber pressure is fixed at l mTorr. The measurements were taken twice
and the averaged value is plotted here. The saturation current appeared to be a weak
function of the gas flow rate. The maximum saturation current occurred at 30 sccm. From
Eqn. (3.9) in Chapter 3, we can see that the ion saturation current is directly proportional

to the density of the plasma. Hence it can be used as an estimation of the ion densities.

58

 

9 1°
.1 U" N on

Saturation Current Density (mA/cmz)

.0
0'1

 

Ar

1 1 1 l 1 1 1

 

0.5 1 1.5 2 2.5 3 3.5

Pressure (mTorr)

Figure 4.4 The ion saturation currents in argon and hydrogen plasmas where

 

Pin=90W, z=2 cm. H2 flow = 30 sccm, Ar flow = 8 sccm.

59

Saturation Current Density (mA/cmz)

 

213*

l

2.8

2.6 —

2.5 r

2.4 r

l

2.3

 

 

 

2 l 1 1 l 1
0 1 0 20 30 4O 50

Flow Rate (sccm)

Figure 4.5 Hydrogen plasma ion saturation current at 2 cm downstream with
various gas flow rates. Pin: 90W. Pressure = l mTorr.

60

60

The ior
stated in 5‘
input mi't‘“
versus pr 6‘:
taken at 3 C
this definil
measuremer

Fig, 4.7
(2:2ch wit
Note that in I
in Chapter I
increases as
90 watt input
powers. This
to the dissoci
of the ion dc
detail analysi
following Clltt

Fig. 4.8 sh
wnuld indicatt
predicted iron,

Species decreas

the wall. Note 1

The ion densities reported in this thesis are from the results of DLP measurements as
stated in Section 3.2. The operating pressure ranges from 0.5 mTorr to 4 mTorr and the
input microwave powers are 90W and 72W. The ion densities of the argon plasma plotted
versus pressures is shown in Fig. 4.6. The gas flow rate is 8 sccm. The measurements were
taken at 2 cm downstream from the end of the source to the center of the probe. Note that
this definition of downstream distance will be adopted for the Langmuir probe
measurements presented through this thesis.

Fig. 4.7 shows the hydrogen ion density versus pressure at the same probe position

(z=2cm) with flow rate of 30 sccm. Three input power levels (90, 72, and 54W) were used.
Note that in the calculation of ion density, we chose the H; ion mass which as shown later

in Chapter 6 is the dominant ion species in the hydrogen discharge. The ion density
increases as the pressure increases up to about 1.5 mTorr and then it starts to decrease at
90 watt input power. Similar density curves also occur at 72 and 54 watt microwave input
powers. This may be due to at higher pressures more of the electron collision energy goes
to the dissociation of molecular hydrogen instead of ionization. Therefore slight decreases
of the ion densities were observed as the pressure increased above 1.5 mTorr. A more
detail analysis of the power dissipation of the absorbed power will become clear in the
following chapters where the modeling of the discharge source is presented.

Fig. 4.8 shows the downstream ion density variation. A linear line on this semilog plot
would indicate an exponential decay of the densities along the downstream region. As is
predicted from the ambipolar diffusion of charged species, the densities of the charged
species decrease in the power of exponential as they diffuse away from the source and to

the wall. Note here that to account for the possible increase of sheath thickness due to the

61

loH
f

ATE—UV ICECDQ Enchant—mu

109 L

“unit

 

10 7 1 l l l 1

 

 

 

I I
90w
"3" * r
E, l
3? r r ... H I I
ii ..
Q 10‘0— * x +
S ... x \ 72w
*3
.33
LL]
.
109 1 l i l I I 1 I
0.5 1 1.5 2 2.5 3 3.5 4 4.5

Pressure (mTorr)

5.5

Figure 4.6 The ion densities of argon plasma measured at 2 cm downstream. Input
powers are 90, and 72 W. Flow rate is 8 seem.

62

CO~
-~Jv kite-0AA
Ar- . . .

0.5

Hunt71

are 90,(

 

 

 

 

10 I r I I I I T
are 316 3“ as
x are 3"
are
A "‘ 90w .
m
. x
x
S ...
V
>» x
3:: X o
8 3" o 72w
a x
t: o 54W
0
III!
109 J 1 1 1 1 l l
0.5 1 1.5 2 2.5 3 3.5 4 4.5

Pressure (mTorr)

Figure 4.7 Hydrogen ion densities measured at 2 cm downstream. The input powers
are 90, 63, and 54 W. Flow rate = 30 sccm. The ion assumed for this density

determination is Hf.

63

 

A

r

-23: 57:0: .__.,.-_.u.,:2

n,

 

Electron Density (cm'3)

 

 

 

   

 

1O 1 1 1
0 1 2 3 4 5 6 7

Downstream Position (cm)

Figure 4.8 Downstream ion densities of argon and hydrogen plasmas. The input
power is 90W, and the pressure is 1 mTorr.

increase of electron Debye length, we used a Child law sheath [51] to calculate sheath
thickness. The equation for the sheath thickness is given as
2V
5 = “/51 {—3) (4.1)

306T

e

where ADC is the electron Debye length, and V0 is the potential difference between the
plasma and probe surface. The effective probe collecting area Ap is then corrected using a
cylindrical surface with radius R = r + s, where r is the probe radius. The values of Ap

become much larger as the plasma densities continue to decrease further downstream in

the chamber region.

4.7 Electron Energy

The electron energy distribution function (EEDF) and average electron energy in the
downstream region were obtained from single Langmuir probe (SLP) measurements. For
downstream spatial variation of the electron temperatures, an in vacuum stepper motor
was used to remote control the position of the probe. This arrangement has the advantage
of not having to open the chamber to re-position the probe and pump down the processing
chamber to vacuum again. Since the plasma stayed on during the measurements, it
prevented possible variations of the operation condition if the plasma was regenerated
every time the probe position was changed. The electron temperature in the source was
also measured using OES with small amounts of noble gases (Ar, Kr, Xe) added in the
hydrogen discharge.

Figs. 4.9 and 4.10 show the electron temperature dependence on pressure in argon and

65

 

4_5_,. . .

Electron Temperature (eV)

25.. ........ .......... ........ ._

 

 

 

Pressure (mTorr)

Figure 4.9 Argon plasma electron temperatures at 2 cm downstream. Pin: 90W, and
flow rate = 8 sccm.

66

 

9

3

D , ,

g 11%. _ ‘ ' i

o. .* . . a ; 2

E 4* : 90W ' j

[2 5_5_......O. , .. . ......... / ........... . ........... ....... ._
g o * _. é ; 2

f: , O . : . :

g ' ;

E 5 ._ .............................. .0, ..................... .9. ..................... .1

 

 

.rs
_..
e

..

_

_

_.

_

 

Pressure (mTorr)

Figure 4.10 Hydrogen plasma electron temperature. Pin: 90 (*) and 63 W (0). Gas
flow rate = 30 sccm.

67

hydrogen discharges. Two input powers 90 and 63W were used in Fig. 4.10. The single
Langmuir probe used for this measurement was positioned at 2 cm downstream from the
end of the source for both argon and hydrogen plasmas. The data shows an electron
temperature increase as the pressure goes down. This is the expected behavior since in
order to sustain the discharge as the pressure drOps, the electron temperature must increase
enough to maintain sufficient ionization. Fig. 4.9 shows the electron temperatures in an Ar
plasma drops from 5.0 eV to 3.2 eV as the chamber pressures changed from 0.5 to 4.7
mTorr. In Fig. 4.10 the hydrogen electron temperatures had about 1.0 eV change in the
pressure range of 0.5 -3 mTorr and 90 W input power. The plasma potentials determined
from the maximum of the second derivative of SLP I-V characteristic are shown in Fig.
4.11.

Fig. 4.12 plots the electron temperature in the Ar-Hz mixture discharge with various

gas compositions. At 0.6 mTorr, the electron temperature increases from 5.4 eV to 6.2 eV
as the hydrogen partial pressure ratio increases from 0.5 to 5. As we compare the results
with argon and hydrogen discharges at the same pressure, it is found that at a low ratio

(Pm/Pm), the electron temperature in the mixture discharge is close to that in the argon

discharge and at a high ratio, it is close to that in the hydrogen discharge within reasonable
experimental error.

The emission spectrum of an argon plasma at a low pressure of 0.3 mTorr with an
input power of 90 W and a flow rate of 3 seem was also observed and it is shown in Fig.
4.13. Two argon doubly ionized emission lines were found. They are lines at 328.59 nm
and 331.13 nm. This indicated the existence of high energy electrons (>25 eV) at a low

pressure, i.e. 0.3 mTorr. Or, that is to say, that the high energy tail of the EEDF can not be

68

Plasma Potential (V)

 

12 I 7 r I I I I I l

 

 

 

11 . a
10— D
9- :1
Ar
8' U D [1
7c
0
6..
O
51.
0 H2
4. o o o
O
3 I l J l I l 1 1 1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Pressure (mTorr)

Figure 4.11 The plasma potentials in Ar and H2 discharges measured at 2 cm
downstream. Pin: 90W. H2 flow rate: 30 sccm. Ar flow rate: 8 seem.

69

 

6.5

I
i

.U‘
01
I

P
01
I

 

I

3.5

 

 

 

Figure 4.12 The electron temperatures in Ar/I-Iz mixture plasma taken at 2 cm
downstream. The input power is Pin=90W. The gas composition is varied.

7O

 

10

5x I T 7 T T F F
Ar+3293.6

4.5— ._ .. q

4— a

3.5— ‘ ........}...Ar+33_07.2 ...... ........... _

3 .............................. _

Emission Intensity (a.u.)

 

 

 

 

 

 

 

 

l l l l l l 1

..1q

 

 

 

 

 

0
3280

3285 3290 3295 3300 3305 3310 3315

Wavelength (Angstrom)

3320

Figure 4.13 Argon plasma emission spectrum at 90W input power, 0.3 mTorr, and

flow rate of 35ccm. Monochromator slit width is 50 micron.

71

ignored.

The downstream EEDF of argon and hydrogen discharges are plotted in Figs. 4.14 and
4.16. It can be seen that at the same operating conditions, the electron temperature
decreased from 4.25 to 2.37 eV in argon plasma and from 5.48 to 2.39 eV in the hydrogen
plasma as the probe position was moved from 2 cm to 6 cm downstream from the plasma
source. The electrons gain energy in the source region through ECR heating and lose
energy by collisions with particles (neutrals, and ions) in the discharge. As the electrons
diffuse out of the source region they continue to lose their energies when collisions happen
and they gain no more energies in the downstream region. Hence the electron temperature
drops as electrons diffuse through the downstream region and toward the chamber wall.

Figs. 4.15 and 4.17 plot the electron temperature vs. downstream position in the argon
and hydrogen discharge respectively. An extrapolated dashed line is drawn to the 2 =0 cm
position. This gives an estimated electron temperature of the discharge in the source
region where the Langmuir probe technique is not applicable due to the static magnetic
fields in this region. The estimated electron temperature in the argon discharge region is
about 5.9 eV and in hydrogen discharge, it is 7.8 eV. The EEDF of argon and hydrogen
plasmas plotted on a logarithm scale is shown in Figs. 4.18 and 4.19. The data was taken at
an input power of 90W, pressure of 2 mTorr, and a downstream position of 2:2 cm for
argon discharge.

Within the plasma source the electron temperature of the hydrogen plasma is measured
using the DES technique as described earlier in Section 3.4.1. The ratio of observed
emission intensity to calculated intensity versus transition threshold energy is plotted in

Fig. 4.20 for a pressure of 3 mTorr. Various electron energies ranging from 1.2 eV to 7.5

72

 

(145 r 1 r 1

 

 

 

 

 

/ z=10 cm
0.4 » «
0.35 ~ z=6 cm -
(13- -
\ z=4 cm
u_ 0.25 - \ ~
8 Z: 3 cm
L“ \
(12~ ‘| ‘
‘\ 2:2 cm
(115 a
0.1 I ..
(105 _
0 L -_ -5“ 1
0 5 10 15 20 25
E (eV)

Figure 4.14 The EEDF of Ar plasma. Pin: 90W. Pressure = 1 mTorr and z = 2, 3, 4, 6,
10 cm with <Ee>= 4.25, 3.50, 2.95, 2.37, 1.81 eV respectively.

73

 

Te (eV)

 

 

 

1 l l l J l l l l l

 

0 l 2 3 4 5 6 7 8 9 10

Downstream Position (cm)

Figure 4.15 Electron temperatures of Ar plasma at downstream positions 2 = 2-10
cm. The extrapolating line to 2:0 cm indiates an estimation of the electron
temperature in the source being 5.9 eV .

74

 

I
l

0.35
/ z=6 cm

 

0.3 - a
2:4 cm
0.25 ~ -
z=3 cm
E 0.2 ~ -
LU z=2 cm
L“ \
0.15 \ -

 

0.1 ,I \ s

0.05 ' -

 

 

E (eV)

Figure 4.16 The EEDF of H2 plasma. Pin: 90W. Pressure = 1 mTorr and z = 2, 3, 4, 6
cm with <Ee>= 5.48, 4.46, 3.68, 2.39 eV respectively.

75

 

Te (eV)

 

 

 

 

Downstream Position (cm)

Figure 4.17 Electron temperatures of H2 plasma at downstream positions z = 2-10

cm. The extrapolating line to 2:0 cm indiates an estimation of the electron
temperature in the source being 7.8 eV.

76

 

 

 

 

 

E (eV)

Figure 4.18 The EEDF of Ar plasma plotted on a logarithm scale. Pin=90 W,

pressure = 2 mTorr, z = 2 cm, and flow rate = 8 sccm. The electron temperature is
3.22 eV.

77

 

10 .. r I T I I
b

Log f(E)

 

 

 

 

E (eV)

Figure 4.19 The EEDF of hydrogen plasma plotted on a logarithm scale. Pin: 90 W,
pressure = 0.4 mTorr, and z = 3 cm.

78

eV were used for the calculated emission value, Ical‘ A least squares line fit of each [obs/Ical

data set versus threshold energy is used to obtain the slope for the different assumed
electron energies. The electron temperature was estimated to be between 1.2 and 2 eV
where the slope of fitted line would be zero. This value was low compared with the results
from Langmuir probe measurements(~4.5 eV). The possible reasons for the difference
maybe due to the following. The emission lines observed here have threshold energies
ranging from 9.8 to 13.5 eV. The excitation and ionization energies of molecular hydrogen
are known to be mostly in the range of 11.2 - 15 eV. Since only a small amount of noble
gases are added in the discharge, most of the electronic collisions will happen with
hydrogen neutrals. This could make the electron distribution depleted in the energy range
above about 1 1 eV from Maxwellian distribution. For example, refer to the earlier electron
temperature measurement shown in Fig. 4.19. Here the calculated emission intensities

(Ical) would be estimated to be larger for a Maxwellian distribution and therefore their
ratios Iobs/Ical would be smaller. If we look at the emission lines used again, most of them

have energies of 1 1.2 - 13.5 eV except for the xenon emission lines.

An alternative way is ignoring the xenon emission data and replotting Fig. 4.20 in Fig.
4.21. The electron temperatures used here were from 2 to 12 eV and the threshold energies
from 11.4 to 13.5 eV. The slopes of the fitted lines for each electron temperature value in
Fig. 4.21 are plotted in Fig. 4.22. For comparison, the fitted line slopes from Fig. 4.20
were also plotted at the same time. The electron temperature where the slope is zero

indicates lobs=IC31(Te). It is therefore this electron temperature of 6.6 eV from the

interpolation of Fig. 4.22 that is responsible for the observed source region emission.

79

21

 

 

 

 

 

 

10 : r 1 1 l I I I I :
I 1
L - Te=l.2eV ‘
102°? , I 1
O . . O . 1
o : . ° 1
10‘9; -
’7 F
3, .
a: t
V X 2 CV
E 18 x x
U 10 ’- X x X i ‘
:8 1. A § xfi
o ‘ X § x x x
u—q
b O O O 4
1017:“ O o o 3 6V 3
E O 3
1
10‘s: 3
i
1
1015 1 1 1 1 1 1 1 1
9.5 10 10.5 11 11.5 12 12.5 13 13.5 14

Figure 4.20 Emission ratio of the observed emission line intensities to the calculated
line intensities (labs/lea.) for the hydrogen plasma. The operating pressure is 3 mTorr.

The ratios are done for five different Te’s= 1.2, 2, 3, 5, 7.5eV.

80

 

 

 

 

 

 

 

 

 

 

 

 

 

10 I 1 1 1 I
E
l
' Te=2eV
1018: . ' 1
l '0///’/—*/
l- . C
L o '
l ; . . -
r
:1 17 x
5:; 1° F x x x 38V ‘2
_ - x x #_— .
8 A ; X x
:0 § x x x
..D .
H0 J
o O 5 v
10115_ O O C _
L 6* 8 O I
L O O O O O
’ i' *. *
t a“ 1‘ 8eV ’
+ * 1* ¥ + r ‘l’ .*
1.; .L.
i + + + 12eV
1015:. + + + _.
l 1 1 1 1 a
11 11.5 12 12.5 13 13.5 14

Eth (CV)

Figure 4.21 Emission ratio of the observed line intensities to calculated line
intensities (labs/lea.) for a hydrogen plasma(excluding xenon emissions). The

operating pressure is 3 mTorr. The ratios are done for five different Te’s = 2, 3, 5, 8,
12 eV.

81

 

   

 

 

 

 

 

 

0.15 1 1
0.1 r— _ ............... 5
0.05 - L —
3 0 Without. Xenon -
CU I
s ‘ .
§ -0.05 - ~ a
I E
o :
& _0.1 _ ......... _
.2 :
m 3
_0.151_ ......................................... _
-002 h .. . \ iWith Xéhéh. -
.025 Y. 1 1 Y 1 1
0 2 4 6 8 10 12

Figure 4.22 Plot of the fitted slopes of the emission ratio lines at five different
Te’s(circles : xenon emissions included, squares: xenon emissions excluded). Zero

slope indicates the electron temperature in the source.

82

4.8 Actinometry Results

The actinometry was performed on hydrogen discharges with a small amount of argon
(less than 5%) added as the actinometer. The results are shown in Table 4.4 [80]. The
relative atomic hydrogen densities are proportional to the intensity ratios of H(486.1 nm)
to Ar(656.3 nm) multiplied by argon densities which increased as the pressures increased
from 1.1 to 5.3 mTorr. The absolute values of the atomic hydrogen densities can be found
using Eqn. (3.30) once the electron temperatures in the source are known. This will be

done in Chapter 6.

Table 4.4 The emission ratios of H (486.1nm) to Ar (750.4nm) in H2 discharges from

actinometry (5% of argon gas added).

 

 

 

 

 

 

 

 

 

 

 

 

 

1:33;: 63w 90w
1.14 .2134 .2401
1.4 .2068 .2081
1.6 .1844 .1954
2.0 .1759 .2201
2.4 .2921 .2073
2.8 .2938 .1908
3.5 .2939 .2896
5.3 .2893 .2988

 

83

 

CHAPTER 5

Modeling of the MPDR 610 Plasma Source

5.1 Introduction

The modeling of the plasma discharge in the source region is done using a global [81]
or zero-dimensional model in this thesis. This model considers charged particle balance
and energy balance in the sourceregion. This chapter begins with a description of the
global model and its general expression for most types of gas discharges. Next the models
for the plasma sources of different types of gases are developed followed by the
determination of some selected parameters in the model from the experimental data. After
the appropriate particle and energy balance equations are determined for each type of gas
discharge, and arranged in a linear system of equations, the desired discharge properties/
parameters are computed from the set of equations using a numerical method. In addition,
a model for the charged particle densities in the chamber region where the substrate is
usually located during the processing in a microwave plasma system is considered later in
this chapter. The model uses a two-dimensional steady state finite difference method to
solve the ambipolar diffusion equation. A comparison of the experimental data and model

results will be given and discussed in the next chapter.

84

5.2 Discharge Global Models : A General Expression

This model is based on the development of Lieberman et al. in 1995 [81]. The two
main sets of equations considered in the global models are the power balance and particle

balance for all species of interest. These equations are summarized as the following:

(1) Power Balance Equations

The general form of the total power balance equation is
Pabs = Pev + Piw + Pew (5.2)
where Pabs is the power absorbed by the system, Pev is the electron energy loss due to all
electron-neutral collision processes in the discharge volume, Piw is the ion energy loss to

the walls, and Pew is the electron energy loss to the walls. This power balance between the

system absorbed power which comes from the external power source (dc, rf, or
microwave) and power consumed in the plasma volume (electron-neutral collisions and
charged species energy loss to the wall) is illustrated in Fig. 5. la.

Ion energy is lost to the wall due to ions flowing to the walls at a characteristic velocity
which is the Bohm velocity uB=(eTe/M)“2 at the plasma sheath edge. With the ion
velocity known, then

Piw = eniSuBAel-w (5.3)

where “is is the ion sheath edge density. A is the surface area of the chamber wall, and 81w

is the ion kinetic energy loss per ion lost to the wall. Similarly, the electron energy loss to

the walls is

85

 

Pabs

> ' 0 P6:" . (a)

 

 

0\/0
O”b*.\

1011123111011

 

elastic coll.
'\/’O>\O
. 0" \6-2

W

(b)

 

/—f\\
#69 63

charge
, , exchange
recomblnatlon

96-) 69/0
EB

 

 

\/ O>\
V

630$.

electron
positive ion
neutral

negative ion

(C)

Figure 5.1 Illustration of power balance in the plasma volume (a), electron-neutral
collision process (b), and charged particle balance in the discharge volume (c).

86

Pew: eneSuBAeew (5.4)

where sew is the electron kinetic energy loss per electron lost to the walls, and ties is the
electron sheath edge density.

For an atomic gas, the energy loss EL per electron-ion pair created due to all electron-
neutral collision processes can be expressed as

N'“ 3 m T

e
VizEL - vizgiz + 2 vexc,k€exc,k + velas M (55)
k = 1

where v = (6v)nn is the appropriate collision frequency including ionization, excitation,

and elastic collision, (0v) is the rate coefficient, nn is the neutral density, and Nfixc is the

number of excitation energy loss channels. The first term on the right-hand side of Eqn.

(5.4) is the energy loss due to the ionization of neutral atoms with an ionization threshold

energy of 8,: , the second term represents the total energy loss due to excitation of neutral

atoms to various excited states with threshold energies gem, and the last term is the energy

loss due to electron-neutral elastic scattering. These three types of electron energy loss
mechanism due to electron-neutral collisions are illustrated in Fig. 5. lb.
For molecular gases, several additional considerations need be made including:
(a) The generation of multiple positive ions.
(b) Fragmentation of the neutral molecule can provide multiple neutral
sources for the generation of ions.
(0) Generation of negative ions.

(d) Additional energy loss channels such as dissociation, and particle loss

87

channels such as positive-negative ions recombination, need to be included.

These require the modification of Eqns. (5.1) and (5.4). Eqn. (5.1) is rewritten as

Pabs =Pev + 2 Pimj + I)ew (5-6)
I: 1

where r is the number of positive ion species generated in the system. For H2, r = 3 for the

generation of H+, Hf, Hf. Eqn. (5.4) is rewritten as

N

".1

N .
”"1 3mT
_ e
Viz, 18L, i - 2 [Via ijeiz, ij + 2 Vexc, kjeexc, kj + velas,j M . (57)
j: 1 k = 1 ’

where Nn.i is the number of neutral species that generate the ith ion. For Ar+, Nn.i = 1 (Ar),

and for H+, NnJ = 2 (H and H2). v is the ionization frequency for production of the ith

1'2, 1'}

ion from neutral species j, V1 is the total ionization frequency for production of the ith

2,1

ion, £-

12 i}. is the threshold ionization frequency for production of the ith ion from neutral

species j, and 8 L ,- is the total collisional energy loss per electron-ion pair created for the

ith ion. The sum over k includes all inelastic electron-neutral collisional processes that do
not produce positive ions; e.g., rotational, vibrational and electronic excitation,
dissociation, attachment and detachment. The total power loss in the volume therefore is

r
Pe = eneVE Viz, 18L,i (5.8)

V

i=1

(2) Particle Balance Equation

Using the continuity equation for the ith positive ion which includes ion diffusion loss

88

to the walls, volume loss due to positive-negative ion recombination, and asymmetric

charge exchange (for the case of mixtures), we have

NJ
Vn v.- . = Aefflinisugfi Vk .n.n_ + V Z kmjninj (5.9)

e 1...! I‘C’COIII,I l

j = 1
where V is the discharge reactor volume, kremmj is the recombination rate coefficient,

kexjj is the charge-exchange rate coefficient for asymmetric collisions between the ith ion

and jth neutral, and n' is the negative-ion density. The balance between the creation of the
ion and the loss of it due to the above loss channel in Eqn. (5.8) is illustrated in Fig. 5.1c.

Under the quasineutral condition, the densities at the sheath edge are related by

"es = 2 "is (5.10)

Substituting Eqn. (5.9) into Eqn. (5.3) and Eqn. (5.8) into Eqn. (5.7), the total power

balance of Eqn. (5 .5) becomes

r
s = z eni[Aeff8T,iuB.i+ krecom,in— [V + z, kex. ij ”jEL, i V] (5'11)
1:1 j— ._ I

where

2”
2nR +—5

i radial

n.
A = n—" 21tRL (5.12)

l .
axial

 

 

is the effective surface area for ion loss and

a“ = EL,i+siw+€ew (5.13)

89

The ratio of sheath edge density nis to the bulk average density ni is

L —1/2
:=CL86(3'+E§X:) (5.14)

i 1

 

axial

at the axial sheath edge (2:0 and z=L) and

—l/2
z 0.8(4 + {1) (5.15)

I

is

th—

 

’ radial
at the radial sheath edge (r = R). Here R and L are the radius and length of the discharge,
and 21,, = (ngo'i)l is the ion mean free path. Eqns.(5.l3) and (5.14) are valid for
electropositive discharge in low to intermediate pressure regime, where ZMIL 2 Ti/Te and
MR 2 Ti/I‘e respectively. For Ti = 0.5 eV, T6 = 5 eV, and discharge dimension of R = 3.6
cm and L = 3 cm, 1., should be no less than 0.15 cm for Eqns. (5.13) and (5.14) to be valid.

For the detailed derivation of Eqns.(5.l3) and (5.14), the reader should refer to Ref.

[100],[101].

5.3 Global Models Used in this Dissertation

Models for discharges at lower pressure [84],[85] in the range of 1 mTorr have been
created using either full electron energy distribution function (EEDF) solutions or
Maxwellian distribution function based solutions. A condition generally considered as
necessary for using the Maxwellian distribution is that elastic (electron-electron)

collisions dominate over inelastic collisions. The dominance of elastic collisions requires

ionization ratios of greater than 10‘4[85]. In the discharge considered in this study, the

90

ionization ratios are typically 10'2 to 103. This thesis therefore uses the assumption of a

Maxwellian distribution.

5.3.1 Argon Discharge Global Models

For argon plasmas, the ionization reaction considered in the model is

e+Ar—)Ar + +26 (5.16)

The electron ionization rate constant K12 in this reaction can be approximated to an

Arrhenius form over a limited range of Te (<100 eV) [82]

—Eiz/Te

Ki: z [(019 (5.17)

where K o 2 6x 10‘'4 m3/s is the pre-exponential factor for argon, and Eiz=15.76 eV is the

ionization energy of argon.

The discharge model is developed first by considering the particle balance in the argon
plasma. Three models of charged particle confinements are considered including: (1) no
magnetic confinement, (2) magnetic confinement at the side wall, and (3) both magnetic
confinement at the side wall and additional ionization created by primary electrons

trapped in the magnetic mirror. These three cases are discussed separately as follows:

(1) Particle Diffusion Without Magnetic Confinement
The argon discharge model for the plasma created with MPDR 610 followed from the
derivation in the previous section is explained as follows. The particle balance equation

considers equating the total species surface loss and the total volume generation of species

91

via ionization in the source, giving
"i“BAeff = VKl-Z(Te)neng (5.18)

where Aeff = 27tR(Rh,+ lhR) is the effective area accounting for possible different

plasma densities at axial and radial sheath edges. V is the total volume of the plasma

source. The plasma density in the source is denoted as ni, and ni = 11C in the argon

 

 

discharge. ng is the neutral density. Eqn.(5. 17) can be rewritten as
K . T
'Z( '3) = 1 (5.19)
“3(Te) n 861’ eff
where
d 1 RI (5.20)

eff ‘ 21%, + th
Using Eqn.(5. 16) for KiZ(Te), Fig. 5.2 plots Te vs. ngdcff for electron temperatures of l to

20 eV. The model without magnetic confinement is denoted as floss = l in Fig. 5.2. Once ng

and deff are given, the electron temperature can be determined from it.

(2) Particle Diffusion With Magnetic Confinement

When a static magnetic field exists around the wall of the quartz discharge region,
which is the case in the MPDR 610 plasma source, the charged particle diffusion to the
wall is reduced due to magnetic confinement. This reduced charge particle flux to the

walls is treated by multiplying a fractional loss term, floss, on the LHS of Eqn. (5.18),

giving

92

Te (eV)

 

 

 

 

 

Figure 5.2 The universal plot of Te vs. ngdefl' in an argon gas discharge.

93

flossniuBAeff = VKizning (5-21)

The floss term is introduced here to account for the effective fractional loss of electron-

ion pairs loss to the wall due to permanent magnets applied around the MPDR 610 source

for plasma confinement [105],[106]. The determination of floss value is based on the

arrangement of permanent magnets around the quartz discharge region of MPDR 610. The

charged particle loss fraction to the wall floss with magnetic field confinement is
determined by first considering a cylindrical discharge volume of radius R = 1.8 cm and
length L = 3.0 cm. It has a total surface diffusion area of At = 2n (R2 + RL) = 54.28 cmz.
The static magnetic field at the MPDR 610 discharge region provides side wall

confinement of the charged particles with an area equals to 33.93 cmz. For complete
confinement on the side wall, the fractional loss will be (54.28-33.93) / 54.28 = 0.37. If
there is a small leak of diffusion loss to the side wall, we will expect an adjustment of the

floss value. In the argon plasma model the value of floss is chosen to be 045 Similarly, Eqn.

(5.20) can be rearranged as

K AT.) _ f loss (5.22)

“3(Te) — ngdeff

 

where deff is as defined in Eqn. (5.19). Again using Eqn.(5.l6) for Kiz(Te), Fig. 5.2 plots
Te vs. ngdeff with floss = 0.4. The electron temperature can be determined from Fig. 5.2 if

ng and dcff are given.

(3) Particle Diffusion With Magnetic Confinement and Additional Ionization from

'D'apped Electrons

94

In addition to the reduced diffusion of charged particle to the wall, the existance of a
static magnetic field in the discharge region can provide much better trapping of ionization
electrons (the electrons at the high energy tail of its distribution) because of their much
smaller mass and high energies compared with those of ions. Therefore these electrons can
provide an additional ionization in the discharge. Here another term is added in the RHS

of Eqn. (5.20) to account for this ionization, giving
flossniuBAeff = VKt-Zneng-l- Vfl-zKl-zfeneng (5.23)

where fiz is the additional fractional rate of ionization caused by those electrons. fe is the

fraction of electrons trapped in the magnetic field with respect to the total number of

electrons created in the plasma. fiz can be approximated as proportional to the lifetime

ratio of trapped electrons to untrapped electrons, which is

1:tra ed Ae

untrappeed

where R is the radius of the discharge volume, which is a constant. Xe is the mean free path

of electrons which is inversely proposal to the discharge pressure. Eqn.(5.22) can then be

rearranged as

col

where 00 is an arbitrary constant. Combine fe, and CO/R to a constant c, Eqn. (5.24) can be

written as

VKiZning(l +2) = flossniuBAeff (5.26)

95

where p (in mTorr) is the gas pressure. The determination of c is based on the Langmuir
probe experimental results. Using the downstream electron temperature data as shown in
Fig. 4.15, the electron temperature in the source can be determined by an extrapolation of

the Te curve to z = 0 cm. For argon, the electron temperature in the source is about 5.9 eV

at l mTorr. The choosing of the c value is to match this temperature at the same pressure in
the model. In this argon discharge model, the value of c is determined to be 0.17.
For the power balance equation, since there is no negative-positive ion recombination

or charge exchange between ions and neutrals (the reaction rate constants Krecom = Kex =

0), Eqn. (5.10) is simplified to
Pabs = CniAefl' ET 1.113 (5.27)

for the case of without magnetic field confinement for charged particle. For a discharge

with a static magnetic field

Pabs : eni flossAeff ET uB (5°28)
and again ET = EL + Eiw + Eew. For Maxwellian electrons, the mean kinetic energy loss to

the walls per electron loss is Eew = 2T6. The mean kinetic energy loss per ion loss is the

sum of the ion energy entering the sheath and the energy that the ion gains as it travels

through the sheath. The ion velocity entering the sheath is uB, the Bohm velocity. It
corresponds to an energy of Te/2. The potential drop within the sheath between a plasma

and a floating wall can be expressed as [51]

1/2
M ) (5.29)

Vs = Te 111(7
For argon, VS = 4.7Te. Accounting for the ion energy when entering the sheath, we have

96

 

 

 

 

 

 

10 10 ‘0
Te (eV)

Figure 5.3 The collisional energy loss per electron-ion pair created vs. Te in the
argon plasma.

97

In the case of argon, Eiw = 5.2Te. Using the Te determined earlier from Fig. 5.2, the

collisional energy loss EL is obtained from Eqn. (3.5.8) in Ref. [83]

KiZEL = Kiinz+K E +K 3’"

exc exc el M

T. (5.31)

where Kexc(Te) is obtained by using the similar approximation for Kiz(Te) in Eqn. (5.16),

_ —E . T
KexcszlO 14e m/ ' (m3/sec) (5.32)

where Eexcz 11.55 eV is the excitation threshold energy for argon. And for elastic

scattering, Kel is approximated by

K3, z 1110"3 (m3/sec) (5.33)

for T6 > 1 eV. Fig. 5.3 plotted the collisional energy loss, EL vs. Tc for an argon discharge.

After Er is calculated, the plasma density can be found from Eqn. (5.27).

5.3.2 Hydrogen Discharge Global Models

Several hydrogen discharge kinetic and chemical reaction models were developed in
the past by various researchers [84-88]. They cover a wide pressure range from sub mTorr
- 100 Torr. In order to select the appropriate kinetics, species, and reactions for the low
pressure range and compact plasma source studied here, we will first consider typical
expected collision rates in the compact plasma source. Then these collision rates are

compared to the residence time of the species in the plasma which for a flow rate of 30

98

sccm, plasma discharge volume of 50 cm3, and a pressure of l mTorr is on the order of
10'4 seconds. First, an estimate of the neutral-neutral collision rate based on a gas
temperature of 400 K, a pressure of l mTorr and a collision cross section of 10"9 m2 [89]
gives a collision frequency on the order of 104 Hz. This corresponds to a mean time

between collisions also on the order of 10‘4 seconds. Next an estimate of the electron-
neutral collision frequency for producing excitations or ionization by electrons of energy
in the range of 10-30 eV is considered. In this energy range the inelastic collision cross

0'20 m2 [94] and the collision frequency is on the order of 105

section is in the range of 1
Hz. This is about 10 times larger than the neutral-neutral collision frequency. Hence the
first assumption made is that the primary reaction kinetics are electron-neutral collisions
because of the higher electron-neutral collision frequency and the short residence time of
neutrals which limits the possibility of neutral-neutral chemical reactions.

The next selection to be made in the model is the species to include. This selection is

made based on the species that occur due to direct electron excitation or ionization. The

species included are H2, H, H“, Hf, and Hf. The species specifically not included in the

model is H". In the case of H’, the dominate mechanism of H' creation for the conditions in

the compact source studied here is dissociative attachment [88]

e+H2(v) —)H(ls) +H' (v24)
The cross section for this process depends strongly on the vibrational state of the
hydrogen. For example, for the v=0 vibrational state the cross section is 2.8x10‘25 m'2 but

for the v>4 states the cross section is greater than 10'20 m2 [88]. The formation of H' ions

typically requires that the hydrogen molecules first be vibrational excited, then

99

dissociative attachment can occur. A check of the vibrational excitation cross sections [94]

0-23 2

gives values of a few 10'21 m2 for v=0 to v=l excitation and 1 m range for excitation

to the v>4 range for the 1-10 eV electron energy range. Therefore, the short residence time

of the hydrogen gas and the lower cross section for v>4 vibrational excitation prevents the

formation of any significant amounts of H' ions in the plasma source

Summarizing, the neutral and charged species included in this model are electron,
molecular hydrogen, atomic hydrogen, H+, H2+ and Hf. Table 5.1 lists the reactions used

in the hydrogen discharge model in the low pressure range (0.5-5 mTorr) where most of

the ionizations come from electron-neutral collisional ionizations except for Reaction (6),
the surface recombination of atomic hydrogen, and Reaction (8), the generation of H3+
which comes from the collision of H; with H2. Reactions (2), (3), and (9)-(12) account
for the electron impact excitations of H and H2. Also listed are the required threshold
energies of electrons, Em, for applicable reactions. K is the reaction rate constant. The

calculation of rate constant in Reaction (5) used a linear fit of Fig. 5 in Ref. [96]. For 3 eV

gTe<7eV,

log(Kh2_h) = 2.002(logTe - 0.477) - 14.657 (m3/sec) (5.34)

For7 eVgTe< 14 eV,

log(Kh2_h) = 0.808(1ogTe - 0.845) - 13.921 (m3/sec) (5.35)

For 14 eV 5 Te 5 100 eV, Kh2_h ~ 2.2x10‘14 m3/sec. For rate constants in Reactions (1)-

(4), (7), and (9)-( 12), a nine—term polynomial fit is used for K’s with the following formula

[95]

Table 5.1 List of reactions used in the hydrogen discharge model. Reaction rates are
taken from Ref. [95] for reactions 1-4 and 7, 9-12. Reaction rates for reaction 5 and 8

are from Ref. [96] and Ref. [93]. Reaction 6 is developed as found in Ref. [81].

 

 

 

Reactions Em (eV) K(m3/sec)

(1) 8+H—)€+H + +8 13.6 K12,“
(2) e+H(ls)—-)e+H*(2s) 10.2 Km.“
(3) e+H(ls)—>e+H*(2p) 10.2 chc,12
(4) e+H2—>H(ls)+H * +2e 18.0 Ki,12
(5) e+H2—>e+H(ls)+H 10.0 2.2—221110“15
(6) H+H—>H2 (wall recombination) ----- 'yDeff/A2(s")
(7) e+H2—>e+H2+ +6 15.4 Ki,2
(8) H; +H,—>H3" +H ————— 211111045
(9) e+H2(XlZg+ )—>e+H,(B‘>:u" ) 11.37 Km”
(10)e+H2(X]Zg+ )-—>e+H2(leIu) 11.7 chm

l + l + l +
(ll)e+H2(X 2g )—->e+H2(E 2g ,F 2g ) 12.2 Km.”

1 + 3 + 3 + 3
(12) e+H2(X 28 )—)e+H2(a 28 ,b 2“ ,6 fl“) 10.0 Kexc,24

 

101

8
an = Z bn(lnTe)n (cm3/sec) (5.36)
n = 0
The coefficients bn and the minimum temperature Tmin (in eV) that was fit for each
reactions are listed as follows

Reaction 1 1 ):

bO=-3.27140x10‘ b1=l.35366x10' b2=-5.73933 b3=1.56315 b4=-2.87706xlo'l

05:3.482561110'2 b6=-2.63 198x103 b7=1 .1 19541110“4 b8=-2.O3915X10-6
Tmin=2.00

Reaction ( 2):

b0=-2.81495x10'b1=l.00983xlO' b2=-4.77196 b3=1.46781 b4=-2.97980x10"

b5=3.86163x10’2 b6=-3.05169x10’3 67:1.33547x104 b8=-2.47609X10'6
Tmin=1.26

Reaction ( 3):

b0=-2.83326x10 b1=9.58736 b2=-4.83358 b3=1.41586 b4=-2.53789x10‘1

b5=2.8007x10‘2 b6=-l.87l4lx10‘3 b7=6.98667e-5 b8=-1.12376x10'6
Tmin=0' 10

Reaction (4):

b0=-3.83460x10 b,=1.42632x10 b2=-5.82647 b3=1.72794 b4=-3.59812x10'1

b5=4.82220x10’2 b6=-3.9094x10'3 b7=1.73878x10‘4 b8=-3.25284x10'6
Tmin=3.98

Reaction (7):

b0=—3.56864x10 bl=l.733469x10 b2=-7.76747 b3=2.21158 b4=-4.16984x10‘l

05:5.08829x10'2 b6=-3.83274x10'3 b7=1.61286x10'4 b3=-2.89339x10‘6
Tmin=2.00

Reaction (9 1:

102

bO=-3.0819x10 b,=1.03887x10 132-4-425977 b3=1.18123 b4=-2.27751x10'l
b5=2.90058x10‘2 b6=-2.28759x 10'3 b7=1.00435x104 b8=-1.86993x10‘6
T =2.00

min
Reaction (10):

bO=-3.3482x10 b1=l.37l7x10 b2=-5.92261 b3=1.70972 b4=-3.50523x10"

b5=4.83438x10'2 b6=-4.l3l4lx10'3 b7=l.94839x10‘4 b8=-3.85428x10‘6
T =2.00

min
Reaction (1 l):

bO=-3.64659x10 bl=1.43036x10 b2=-6.O7443 b3=1.67731 b4=-3.12871x10’1

b5=3.80542x10'2 b6=-2.86001x10’3 b7=l . 1996411104 b8=-2.14223x10'6
Tmin=3° 16

Reaction ( 12):

b0=-2.85801x10 b1=l.03854x10 b2=-5.38383 b3=l.95064 b4=-5.39367x10"

b5=1.00692x 10‘l b6=- 1 . 16076x10‘2 b7=7.41 162x10’4 b8=-2.00137x10'5
T =1.26

min

For reaction (6), which describes wall recombination of atomic hydrogen, the

hydrogen surface recombination frequency at the wall is computed by using [81]
D?
V” = y—A—gi (5") (5.37)

where y is the recombination coefficient, and

k7,,
De” = Mv

nn

(mz/s) (5.38)

 

is the effective diffusion coefficient with Th being the temperature of atomic hydrogen, M

being the hydrogen atom mass, and Vnn being the neutral-neutral collision frequency given

103

by

v = nHzofz (5.39)

"II
where 0’ is the collision cross section of H with H2. From Fig. 8 of Ref. [89], G=2x10719

m2. And

8kTh 1/2
p = [_] (5.40)
nmh

is the averaged atomic hydrogen speed assuming a Maxwellian distribution. Lastly, A the
effective diffusion length is given by

111—2 = (%§)2+(B2 (5.41)

The discharge region is a cylinder of 3.6 cm in diameter and 3.0 cm in length. The
discharge is assumed uniform with magnetic confinement on the side walls and
unrestricted diffusion to both ends of the cylinder. Two balance equations are considered
including the power balance equation and the particle balance equation. For power
balancing, the absorbed power in the plasma must equal to the sum of the power loss in the
discharge volume and the power loss by charged species diffusion to the wall. This can be

expressed as

Pabs = enlflossAeffJuBJETJ 1' eanlossAeff,2uB.2ET.2 1' en3flossAeff,3uB.3ET,3
+ VnHanKex’ 23EL, 2 (5.42)

where Aeff’l, Aefm, and Aefm are the effective surface areas at the sheath edge for H“,

Hf, and Hf. Here we also introduce floss, the fraction of diffusing electron-ion pairs loss

to the wall, which accounts for trapping of electrons and ions in the source by the
magnetic field lines produced by the three ring-shaped permanent magnets placed around

the discharge region. For the case of without magnetic field confinement, floss = 1. In a
discharge with magnetic field confinement, the value of floss for MPDR 610 is 0.38 as is

explained previously in Section 5.3.1. Eiw is the ion energy loss to the wall. From Eqn.

(5.28) and (5.29), Eiw = 3.3Te for H+. For H; and 113+, Eiw = 3.7Tc and 3.9Te

respectively. ECW = 2Te is the electron energy loss to the wall for Maxwellian electrons. n1,
n2, and n3 are the densities of H+, H2+ and H3)“. “3.1, “8,21 and UB3 are the Bohm
velocities of W", Hf, and H3+ respectively. ET = EL+ Eiw+ Eew, where BL is the collision

energy loss per electron-ion pair created. For H+,

[ 2 3mT

e
viz. llEiz. II + Z vexc, leexc, lj + Velas,l Ml + Viz. 12Eiz, 12] (543)
1': 1

where vinlanKiZJ] is the ionization frequency corresponding to reaction (1) in Table
5.1, and E12.“ is the ionization threshold energy of the reaction. Similarly, vent“:
nHKinl and Vexc,12= nHKiz,12 are the excitation frequencies corresponding to reactions (2)
and (3) in Table 5.1, and Eeml and Eexc,12 correspond to the excitation threshold energies
of reactions (2) and (3), respectively. Velas.1= nHKelaSJ is the elastic collision frequency

between electron and atomic hydrogen. Here hard sphere collisions are assumed, giving

2
Kelas,l = “(0H) vavg (5.44)

105

with

 

8kT 1/2
=[ e] (5.45)

vavg 1m
(3

where aH = 0.53x10'l0 (m) is the radius of hydrogen atom. M] in Eqn. (5.42) is the mass of

H+. Viz.12 = “H2Kiz,12 is the ionization frequency corresponding to reaction (4) in Table

5.1, and 512,12 is the ionization threshold energy of that reaction. Viz,l = Viz,“ + Viz.12 is the

total ionization frequency for creation of H+. Similarly, for H2+

4
1 3mTe
EL. 2 = _ 2 Viz, 2Eiz. 2 + 2 vexc, ZjEexc, 2j + velar, 2 M2 (5-46)
lz .
7 J ___ l

where viz; = nHZKiz,2 is the ionization frequency corresponding to reaction (7) in Table
5.1, and 512.2 is the ionization threshold energy of the reaction. Similarly, Vexc,21 =
nHZKexc,21. Vexc,22 : “HZKCXC22’ Vexc,23 = nH2Kexc,23’ and Vexc,24 : “11ch111224 are the

excitation frequencies corresponding to Reactions (9), (10), (l 1), and (12) in Table 5.1,

and Eexc.21’ Eexc,22 , Eexc,23 , and Emmy, correspond to the excitation threshold energies of
Reactions (9), (10), (11), and (12), respectively. Velas,2 = nHzKela52 is the elastic collision
frequency between electron and hydrogen molecules. Here Kelas,2 = Kelas,2(Te) is found by
integrating the elastic scattering cross section of electrons and molecular hydrogen over

the electron energies of l to 103 eV assuming a Maxwellian distribution of electron.

 

m 3/2 1000 2 2
Kelas,2(Te) = (G(v)v)v = (21tkeT) JO 6(v)vexp[-2nl:—;)4nv dv
e e

106

Rate constant (m3/sec)

10

 

 

...........................................................................................

..................................................................................................

.........................................................................................

 

 

_
—
p.-
p—
)—

 

Te (eV)

Figure 5.4 The elastic scattering rate of electron collisions with hydrogen molecule.

107

 

me 3/2 81t 1000 E
_ (21:14,) [PJJO G(E)Eexp(—fi)dE (5.47)

e
e

The computed result of K612111206) from Eqn. (5.46) is plotted in Fig. 5.4. For Hf, E143 =
Ecw = 0 since there is no direct electron impact ionization with H3 neutrals which do not
exist. Hence B13 = Eew’3.

For particle balance, the generation of species in the discharge volume equals the sum

of the loss of species within the discharge and the loss of species by recombination on the

wall. For H, it is expressed as

KdissanZne = “H VH + Kiz,lnenH (5-48)
for H+,
VnanKiz,1 + vnenHZKiz.12 = flossAeffllnluBJ (5-49)
for H;
vnenHZKiz.2 = flossAeff,2n2uB,2 + vnH2n2Kex.25 (5-50)
and for H3,+
VnznnzKex25 = flossAeff.5n5uB.5 (551)

where KdiSSC is the dissociation rate constant of Hz to H. Kin and K122 are the ionization

rate constant of H and H2, and Kiz,l2 is the ionization rate constant of Hz to H+, and Kex,25
is the rate constant of Reaction (8). V“ is the surface recombination rate of hydrogen
atoms, V is the plasma source volume, and n6 is the electron density given by ne = n] + n2

+ n3. The densities of molecular hydrogen and atomic hydrogen are denoted as nm and

108

11”.
Considering another ionization channel from the collisions of trapped primary
electrons in the static magnetic field with the neutrals, the charged particle balance

equations can be rewritten as

KdiSHHzne (1 '1' C / p) = 11H VH + Kil,lnenH (1 '1' C / p) (5.52)

for H. And for H+,
vnenHKin (1 + C / P) + VnenHzKiz.12 (1 + C / P) = flossAeffJnluBJ (5-53)

for 112+
vnenHZKiz.2 (1 + C / P) = flossAeff.2n2uB.2 + vnH2n2K6x25 (5-54)

and lastly for H3+

VnznnzKex25 = flossAeff.5n5uB.5 (5-55)

The determination of the constant c is based on the Langmuir probe measurements and
CBS results for electron temperatures in the hydrogen discharge at some particular

discharge conditions. The electron temperature (Te) at 90 Watt input power and 3 mTorr
from CBS is about 6.5 eV (see Fig. 4.22), and from the downstream Te measurements of

single Langmuir probe, the electron temperature in the source at a pressure of 1 mTorr is

around 7.8 eV by extrapolating the Te curve in Fig. (4.17) to z = 0 cm. Fitting these values
of T6 in the hydrogen discharge model, we get c = 6.5.

For the energy balance and charged particle balance expressions used in the global
model, the value of the surface recombination coefficient of atomic hydrogen is difficult to

a prior determine. For the 7 quantity, atomic hydrogen recombination on surfaces depends

109

on many factors including material type, surface temperature, and surface cleanliness
[87,97]. Hence, the ability to just look up the 7 value from previous studies does not exist,
rather the literature just allows the prediction of a broad range of possible values. In this
study we will use measured atomic hydrogen densities together with the model to extract
the 7 value in the compact plasma source studied. The specific value of 7 used in the

hydrogen model is discussed in the next chapter.

Table 5.2 Reactions that generates ions or neutrals in the Ar-Hz mixture discharge

models. Species considered: e, Ar, Ar+, H2, H, H3+ Hf, H", ArH+.

 

 

 

Reaction Rate Constant (m3/s) Reference
(1) Ar+e—>Ar + +e+e 6x10446455”re 81
(2)1le+ +Ar—9ArH * +H 1.76111015 90
(3)e+H2—>H(ls)+H * +2e SeeTable5.l 95
(4)e+H2—-)e+H(ls)+H SeeTable5.l 96
(5)11,+ 44512—411,+ +H 211111045 93
(6)e+H—->e+H ” +e SeeTable5.l 96
(7)Ar+ +H2——)ArH+ +H 8.90x10'16 90
(8)/1m+ +e—)Ar+H(n= 2) 51110'5 91
(9)ArH" +H2—>H3+ +Ar 4.5111016 92

 

llO

Table 5.3 Lists of reactions not included in the discharge model due to small reaction

cross sections and therefore large mean free path compared to the dimension of the

 

 

 

source.
Reaction Cross Section T(eV) Reference
x10”20 (m2)
H+Ar—)Ar " +H‘ ~0 <20 90
H+H2—>H+H+H ~0 <20 89
H2+H2—>H+H+H2 ~O <17 89
H * +112 —>H2* +H <07 <50 89
H2+ +Ar—->Ar " +112 8.6-3.3 01-1 90
H + +Ar—>Ar * +H ~0 <40 90
Ar + +112 —>H2* +Ar 14.8-7.75 0.1-1 90

 

5.3.3 Argon-Hydrogen Mixture Discharge Global Models

In addition to the reactions considered in argon and hydrogen discharges, a new ionic

species, ArH+ is taken into account. Also the reactions of charge exchange between
positive ions and neutrals need to be considered. A summary of the reactions that generate
ions or neutral in the argon-hydrogen discharge model are listed in Table 5.2. These

include the direct electron impact ionization and dissociative attachment. The electron-

lll

neutral collisions that result in the excitation of hydrogen are the same as were discussed
in the hydrogen global model listed in Table 5.1. The rate constants in Reactions (2) and

(7) were calculated from the cross section data in Ref. [90] assuming ions of averaged

temperature of T1 = 0.5 eV. And therefore Kfix = <O’V> = C(2eTi/Mi)]/2.

Some neutral-neutral and ion-neutral collisions that were not included in this model
are summarized in Table 5.3. The reasons to exclude them in the model are these reactions
all have small enough collision cross sections and hence large enough mean free paths
(>3.0 cm) compared to the dimensions of the source that they do not occur in large

numbers in the discharge.
The particle balance equations for H, H+, Hf, Hf, Ar+, ArH+ were obtained by

equating the generation of species in the discharge to the sum of species loss due to charge
transfer/exchange to another species and species loss on wall recombination. For H, it is

written as

VKiz.12nH2ne + VKex.34nH2n3 + VKex.24nArn2 + Vch,25nH2n2 + VKrecomn4ne

= vnHVH + VKinnenH (5.56)
for H+ :
vnenHKin + VnenH2K12,12 = flossAeffJnluBJ (5-57)
for H; :
vnenHZKiz.2 = VnH2( nzKex,24 + nzKex,25) + flossAeff,2n2uB,2 (5-58)
for Hf:
Vn2UH2KeX.25 + Vnuz n4 KCX,45 = flossAeff,5n5uB,5 (5-59)

112

for Ar+:
vnenArKizAr = Vn3nH2ch34 + flossAeff,3n3uB,3 (5-60)
for ArH+z
Vn3nH2Kex,34 + VnZnArch24 : vnen4Krecom + VI111213415345 + flossAeff,4n4uB,4 (5°61)
and
nC=n1+n2+n3+n4+n5. (5.62)
where nl denotes the density of H+, n2 the density of Hf, n3 the density of Ar+, n4 the

density of ArH+, and n5 the density of Hf. n6 is the electron density. “H21 11“, and nAr

denote the neutral densities of molecular hydrogen, atomic hydrogen, and argon
respectively. Considering another ionization channel from the collision of trapped primary
electrons in the static magnetic field with the neutrals, the charged particle balance

equations can be rewritten as, for H

VKiz.12nH2ne (1 + C1 / P) + VKex.34 n112113 + VKex,24 “A102 + VKex,25 n112112 +VKrecomn4ne

= VnHvH + VKiZ’lnenHU + C] /p) (5.63)
for H’r :
VneHHKin (1 + Ci /P) + vnenHZKiz,12(1 + C1 /P) = flossAeffJnluBJ (5-64)
for Hf :
vnenHZKiz,2 (1 + Ci / P) = V0112 “2(KCX.24 + K6125) + flossAeff,2n2uB2 (5-65)
for H3+z

113

Vnznnchx25 + Vnuz n4 Kex.45 = flossAeff.5n5uB,5 (5-66)

for Ar+z
vnenArKiz,Ar (l + C2 / p) = VIl3nH2Kex.34 + flossAeff,3n3uB.3 (5'67)

for ArH+z

Vn3nH2Kex,34 + annArch24 = Vnen4Krecom + V“H2n4Kex,45 + floss’b‘eff,4n4uB,4 (5°68)
where c, = 6.5 and c2 = 0.17 as were determined in the hydrogen and argon discharge

models.
From Eqn.(5. 10), the power balance equation for the mixture model can be written as
S

Pabs 2' 2 eni(Aeff, iET, 1“B,i)

i=1

+ Vn2(nAer‘ + "H2K9_,-,25)EL, 2 + Vn3nH2K EL, 3 (5.69)

24 ex.34

Here EL] and E142 are the collisional energy loss per electron-ion pair created for electron
collisions with H and H2 and are as defined in Eqns. (5.42) and (5.45). E13 is the

collisional energy loss for electron collision with Ar and is determined from Fig. 5.3 once

the electron temperature in the discharge is known. Since there is no direct electron impact
ionization 0f AFH+ and 113+. EL.4 = ELS = 0. Hence the total energy loss ETA = Eiw‘4 =

5.19Te, and E15 = Eiw.5 = 3.89Te from Eqns. (5.28) and (5.29).

114

 

 

 

 

Discharge A
region \

 

3.0 cm 10 cm

 

 

 

 

1.7 cm z=30 cm

 

 

 

 

 

Chamber wall

 

 

 

Figure 5.5 The downstream particle diffusion simulation geometry.

115

5.3.4 Downstream Diffusion Models

A downstream model is constructed as shown in Fig. 5.5 to predict the plasma species
density in the region where the substrates being processed are located. A two-dimension
cylindrical coordinate system using the r and z coordinates is chosen for the downstream
model of charged particle densities. The charge species density is assumed to be
symmetric in the (I) direction. The total area simulated is with r = 15 cm and z = 30 cm.
The walls of the downstream chamber and the walls of the compact plasma source are
given the boundary conditions that the ion density is zero. The input ion flux to the region
is at the exit of the plasma source. The plasma density calculated with the plasma source
model is used as the input density value to the downstream model. In the downstream
region the ambipolar diffusion equation is solved [98],[99]. The ambipolar diffusion

equation can be expressed as

<

N.

Vzn = " (5.70)

 

b

D

where n is the positive ion density, viz is the ionization frequency, and D3 is the ambipolar

diffusion coefficient. This equation does not consider volume recombination processes

because of the low pressures.

116

CHAPTER 6

Modeling Results and Comparison

6.1 Introduction

The results applying the global model to the MPDR 610 plasma source with magnetic
confinement in the source region for three different gas discharges are discussed here.

Specifically, the modeling results of the Ar, H2, and Ar-Hz mixture plasmas using the

models developed in Chapter 5 will be presented. The modeled plasma properties are the
electron and ion densities, the neutral densities, and the electron temperatures. The power
absorption in the discharge by elastic and inelastic collisions (e. g. dissociation, excitation,
and ionization etc.) are also discussed. Lastly, the modeling results of particle diffusion in
the downstream region using ambipolar diffusion model are presented and compared with

the experimental measurements.

6.2 Modeling Results in the Sources Region

The results of discharge properties using the models discussed in the previous chapter
with magnetic confinement of charged particle and modified charged particle confinement

with additional ionization/dissociation processes due to trapped electrons in the static

ll7

magnetic field lines are discussed in the following three subsections each of which
corresponds to a different type of gas discharges.

Recall that from Section 5.2 in Chapter 5, the fractional loss term floss = 0.37 for

complete confinement on the side wall. If there is a small leak resulting to the diffusion

loss to the side wall, we will expect an adjustment of the floss value. In the argon plasma

model the value of f is chosen to be 0.4 to account for the possible particle loss to the wall.

For hydrogen models, the f value is chosen to be 0.38. The reason for a smaller floss in the

hydrogen model than that in the argon model is because argon ions have much larger mass
than those of hydrogen, about 20 times, the gyro-radius of the argon ions along the
magnetic field lines is larger than that of the hydrogen ions. Therefore the diffusion leak to
the wall in the argon plasma is expected to be larger than in the hydrogen plasma

[102],[103].

6.2.1 Argon Plasma Model

Fig. 6.1 plots the modeling results of electron temperatures in argon discharges. Two
models were used including Model 1: particle diffusion with magnetic field confinement
at the wall, and Model 2: particle diffusion with magnetic field confinement at the wall and
an additional ionization due to electrons trapped in the magnetic field lines. For the model

with a fixed floss value of 0.4, it predicts a much higher electron temperature at low

pressures (especially less than 1 mTorr) compared with the probe measurements done at
the downstream position of 2 cm from the source. In the second model, the constant c =

0.17 is used. This value is chosen for a good fit with the electron temperature of around 5.8

118

 

 

 

 

 

10 1 17 1 r 1 1 I r 7 1
\ , . 3
9_ .( : ' ‘ : I, J
1 : Model: floss=0.4 andc=0' .
\/ ;
\ Z I
8— \ t - p I -
\ : Model: floss = 0.4 and c :0. l7
1 :
\ .
A \i
>
i”, 6- —
F0
5- O ‘
4- 2
3_ Experimental 0 . 3 o o . _
Dataatz=2cm / -
2 l l l l 1 l 1 I l

1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Pressure (mTorr)

Figure 6.1 The electron temperatures in Ar plasmas. The dashed line is from the
model using constant f term over the pressure range (Model 1), and the solid line is
from Model 2.

119

eV at a discharge pressure of l mTorr from extrapolating the downstream probe

measurements of Te as shown earlier in Fig. 4.15. At higher pressures, the predicted

electron temperatures from the two models are very close since the effects of electron-
trapping in the magnetic field lines, that is, the (1 + c / p) term accounting for additional
ionization due to trapped electrons in the magnetic field lines in Eqns. (5.48)—(5.50), will
lessen with higher collision frequencies between the electrons and the neutrals.

The modeled electron densities in the argon plasma sources are plotted in Fig. 6.2. The
absorbed powers from the experimental conditions (see Fig. 4.3), which were not uniform

over the investigated pressure range, were used in computing the electron densities, ne , in

both of the models for a fixed input power of 90 W. A constant absorbed power of 80 W is

also plotted for comparison. In the model of fixed floss, the electron densities drop down

more quickly as the pressures go lower. While in the model of considering additional
ionizations from primary electrons, the electron densities are more uniform over the
investigated pressure range. This is because at lower pressures the confinement of primary
electrons are much better due to less collisions, and as a result, lower the averaged electron
temperature as is shown in Fig. 6.1. The ion energy loss to the wall is therefore less with
lower electron temperatures. Hence, from Eqn. (5.27) in Chapter 5, the ion densities will

become larger compared to the fixed floss value model (Model 1) for the same absorbed

power.

6.2.2 Hydrogen Plasma Model

Fig. 6.3 plots the electron temperatures of hydrogen plasmas over the pressures from

120

13

 

   
   

 

 

 

10 I I I I I I I I I I
L ......................................................................................
g ........ ConstantPabs=80W.....
21‘ i i ‘ i 3 : : t
'5 . - . ~ ~ _ : : __,_.;
5 ........ lilo-58:504.and_c..;o'l7...,. 1. . -a—r‘”"’flfi
Q 3 ; 3 . _ ’ ' E : 3'
c .
e .
8 .
2 ,E’
LL] ......................................... 1"? ..........................................
- j j ; : Experimental
. l’ g . , . g 3 Absorbed Power
2‘ j rm = 0.4 and c :10 ;
1012 1 1 1 1 1 1 1 1 1 1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Pressure (mTorr)

Figure 6.2 The electron densities in Ar plasma. The solid line with (*) is the model
using floss: 0.40 and c: 0.17 and dash line with (x) is the model using constant floss

over the pressure. Both models use the absorbed power corresponding to the
experimental conditions. The density with a uniform absorbed power 80 W is also

shown.

12]

 

  
    

8- _
. Model 1
7- . . ;4 V _. / 1,0,,=o.38,c=6.s
i ' ' i i A 1 ' ' '
3, 6_ , 1 1;, \ q
{—2) 3 *‘~ q, \ ; : . OES measurement
‘ g‘ 1.1 \ ; : , in the plasma source :
*———*-" ' ' .' I ‘

5......r . . ‘2‘-
: ' i 3 ' ”’1" : i : f

Langmuir probe measurement at z = 2 cm

 

 

3 J J l I l I l l l l

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

 

Pressure (mTorr)

Figure 6.3 Electron temperatures of hydrogen plasma. The solid line is from the
model prediction and the dashed line from the probe measurements taken at 2 cm
downstream. The discharge conditions are 90 W of input power and 30 sccm of flow
rate.

122

0.5 to 5 mTorr using the model with floss = 0.38 and c = 6.5. The determination of the loss
fraction floss is described in the beginning of Section 6.2. The constant value c is chosen so

that the electron temperatures of the model at pressures 1 mTorr and 3 mTorr and 90 W
input powers will match the experimental results both from the probe measurements and
optical emission spectroscopy (0138). As it is shown in Fig. 4.17, the source electron

temperature at 1 mTorr can be estimated by extrapolating the downstream Te vs. position

curve to z = 0 cm and from the DES of the hydrogen discharge at 3 mTorr as shown in Fig.

4.22. The model with the chosen c value gives T6 = 7.8 eV at 1 mTorr and T8 = 6.9 eV at 3

mTorr both agree in small error with the results from the experiments. For a constant value

of the loss fraction floss = 0.38 and c = 0, there is no reasonable Te (less than 20 eV) that

can be obtained in the model within the lower pressure range. The electron temperatures at
2 cm downstream shown in Fig. 4.11 is also replotted here in Fig. 6.3. The two curves
(model prediction and experimental results) show similar trend versus pressure variation
with an average difference of about 2 eV higher in the source than in the downstream
region. This reduction in electron temperature at the downstream position occurs because
(1) the electrons lose energy when they collide inelastically with other species and (2) the
higher energy electrons can reach the source walls through the sheath potential and thus
are lost via wall recombination.

The collisional energy loss per electron-ion pair created for electron collisions with

neutrals H and H2 is shown in Fig. 6.4. It is shown that the value of EL,2 ranged from 130

down to 58 eV at electron temperatures of 5 to 10 eV. These values for hydrogen are about
two to four times larger than those for argon discharge shown earlier in Fig. 5.3. This is

because additional energy loss channels are included in a molecular gas discharge, such as

123

 

‘40 l. I I l 1 1 1 1

EL (CV)

 

 

 

 

 

,0 111;..;;
5

10 1 5 20 25 30 35 40 45 50
Te (eV)

Figure 6.4 The collisional energy loss per electron-ion pair created in the hydrogen
plasma. ELJ is the electron and atomic hydrogen collision energy loss, and EL; is the

electron and molecular hydrogen collision energy loss.

124

 

18 T r l l l l l l T l

16

 

.5
h
I

..L
N
T

 

 

_L
O
l

Power (Watt)
00

 

 

 

 

0 __it=dl-_d x I: t x x
O 0.5 1 1.5 2 2.5 3 3.5

Pressure (mTorr)

Aer
P.
m
01
01
on

Figure 6.5 The power absorbed in the plasma volume by various collision processes
including dissociation, excitation, ionization, dissociative attachments, and elastic
collisions. The discharge conditions are 90 W of input power and 30 sccm of flow

‘ rate. The model used floss = 0.38, c = 6.5, and y = 0.005.

125

dissociation, and excitation (vibrational, rotational, and electronic).

Fig. 6.5 shows the power absorbed in the plasma by different collision processes
including dissociation, excitation, ionization, dissociative attachment (Reactions (8) in
Table 5.1), and elastic collisions. A constant absorbed power of 60 W in the plasma over
the pressure range is used in the hydrogen model. Note that the power absorbed by
dissociation first slowly goes up, peaks at around 2.7 mTorr, and then slowly goes down,
while the power absorbed by the dissociative attachment process goes up quickly within
the investigated pressure range. The power absorbed by elastic collisions is found to be
very small compared to all the other processes.

Fig. 6.6 shows the prediction of atomic hydrogen densities in the hydrogen plasma
from the model using two different atomic hydrogen surface recombination coefficients:
0.005 and 0.05. The measured atomic hydrogen density from actinometry is also plotted.
The atomic hydrogen densities were determined using the data shown earlier in Table 4.4
and Eqn. (3.29). The use of Eqn. (3.29) requires the calculation of the reaction rates, K’s,
which are each a function of the electron temperature. The electron temperatures utilized
in the calculation at various pressures are those from the hydrogen discharge model
plotted in Fig. 6.3. The predicted densities showed a good match with those of the
experiment at y = 0.005. Therefore this 7 value is adopted in the model.

Fig. 6.7 shows the neutral and ionic particle densities in the hydrogen plasma. The

dominant neutral species found in the hydrogen plasma source is H2 and the dominant

ionic species is Hf. Also noted in Fig. 6.7, the H3+ density increases quickly as the
pressure increases. From the prediction of the model, at pressures of 5 mTorr and above,

the H3+ density will become larger than those of H+ and continue to increase as the

126

14

 

 

 

 

 

10 f' I l l r I l l l T ‘
10‘3, -,
.5“ 3
E 1
3 .
3‘ .
'5
C‘.
Q.)
Q
1012 _ 1
1011 l l l l l l l l l
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Pressure (mTorr)

Figure 6.6 Determination of the atomic hydrogen surface recombination coefficient 7
in the hydrogen plasma model from actinometry. The discharge conditions are 90 W
of input power and 30 sccm of flow rate. The model used floss = 0.38, c = 6.5.

127

14

10

 

 

 

 

 

 

 

 

 

;
r
F
)-
13
10 ~ ‘2
12
«1‘ 10 :
18 i
O t
v
>~.
'27”. L A H+
A
8 11 E“ H A ’9‘ ‘9' A“ A A ‘A 2
L '1
1
1o
10 r 1
P
109 I l l l l l l l l l

 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Pressure (mTorr)

Figure 6.7 Neutral and ion densities in the H2 plasma with input power of 90 W. The
model used floss = 0.38, c = 6.5, and y = 0.005.

128

Dissociation Ratio %

 

40 I I I I T T r I I I

38- d

I
l

30

I
O
1

28

I
l

24

22- -

 

 

p—

l l l

2.5 3 3.5

 

20 l l l
0 0.5 1 1.5

2F

4.5 5 5.5

N

Pressure (mTorr)

Figure 6.8 Atomic hydrogen dissociation percentage in the hydrogen plasma
predicted by the model using Pin = 90 W, floss = 0.38, c = 6.5, and y = 0.005.

129

pressure increases.
The dissociation percentage in the hydrogen plasmas is calculated using the atomic
hydrogen density divided by the total gas density at various pressures and is shown in Fig.

6.8. The dissociation rate shows an overall increase with the pressure.

6.2.3 Argon-Hydrogen Mixture Model

The electron temperatures of the argon-hydrogen mixture discharges at different gas
combinations are shown in Fig. 6.9. The discharge at two pressures were modeled: 0.6 and
4 mTorr. Both had input powers of 90 W. The electron temperature increases as it goes
from a low hydrogen partial pressure ratio, P(H2)/P(Ar) to a high partial pressure ratio. For
example, at 4 mTorr, the electron temperature is 4.0 eV at hydrogen partial pressure ratio
of 0.2 and increases to 5.1 eV at pressure ratio of 5.0. This is expected since at a low

P(H2)/P(Ar), the mixture contains mostly the argon gas and little hydrogen gas, the
discharge hence will behave more close to an argon discharge with T6 = 3.7 eV at 4 mTorr

as shown earlier in Fig. 6.1. As the hydrogen partial pressure ratio increases to 5.0, the Ar-

H2 mixture discharge contains mostly the hydrogen gas, therefore the discharge has a T8
more close to that of the hydrogen discharge (Te = 6.5 eV) at the same pressure of 4 mTorr

which is shown in Fig. 6.3. The experimental data at the same plasma conditions are also
plotted for comparison. The measurements were taken with the probe positioned at 2 cm
downstream from the end of the MPDR 610 body.

Fig. 6.10 shows the modeled neutral and charged particle densities in the mixture

discharge at 0.6 mTorr. The dominant ion species is Ar+ at a hydrogen partial pressure

130

Te (eV)

 

 

 

 

   
 

T I I I I l I I l l

l EModel: 0.6 mTorr §

: z i= 2 cm , 0.6 mTorr E

' : : : __ __ _. —-x

,x ————————— f i

X’ ; 3 Model: 4 mTorr 3

f _ , Experiment , Z
f,.‘.3.-‘—'-—"+¥"z’=“25c‘m,4zmTorr‘ " " '5 ‘

*’ __ .— -' "’ . . . l

i i a" i i i L i i i

 

 

2.5 3 3.5 4

l’(I‘Izm’U’J)

Figure 6.9 Electron temperatures in the Ar-Hz mixture plasma.

131

5.5

 

 

 

 

 

 

 

 

1O » I I I I I I I I T I
E 9
: H i
n C} {J
10‘2~ —
1
p 10“— W \\1 +3
. Ar :
g 2
; + ”*T 4 4* H-I- .
.5; 1
g 101% ArH+ _
E A A 3
V +..
H3
109 ~ .
1
108 l l l l l l 41 J l l
o 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
P(H2)/P(Ar)

Figure 6.10 Neutral and ion densities in the Ar-H2 plasma at 0.6 mTorr with input
power of 90 W.

132

ratio smaller than 2.0. While at ratios greater than 2.0, the dominant ion species becomes
Hf. The densities of H3+ are relatively low compared with that of H+. This agrees with
the hydrogen model prediction at the same pressure of 0.6 mTorr. At a higher pressure of 4

mTorr, the density of Ar+ are dominant most of the time in the partial pressure ratio range

of 0.2 to 4.0. This is shown in Fig. 6.11.

6.3 Modeling Results in the Downstream Region

The downstream charged particle density prediction using the diffusion model was
described earlier in Section 5.3.4. The model used the plasma density in the source found
from the plasma source modeling as the input boundary condition and then solved the
ambipolar diffusion model to calculate the charged particle density in the downstream
region (2: 0 - 30 cm, r: 0 - 15 cm). In the source region, two models were used in the
calculation of electron densities in the argon discharges. The results are shown earlier in
Fig. 6.2. Using these values as the boundary values for the diffusion model, the electron
densities of the argon plasmas in the downstream region are computed for various
pressures.

Fig. 6.12 shows the simulated electron density in the argon plasma with a 90 W input
power at 2 cm downstream. The density measurements from double Langmuir probe as

shown in Fig. 4.6 are also plotted for comparison. The model using floss = 0.4 and c = 0

predicts a density curve that is more close to the experimental variation over pressures,

while the model using floss = 0.4 and c = 0.17 predicts a more uniform density over the

investigated pressure range.

I33

14

 

 

 

 

 

 

 

 

 

1o : 1 1 1 1 1
t
n ——s— 3 {I H
10”-
"PE 1012? AFH+
3 +
21‘ 3 /Ar
g 11. W
\ H2+
a
1o‘°-
109 l l J l l
0 1 2 3 4 5
P(H2)/P(Ar)

Figure 6.11 Neutral and ion densities in the Ar-Hz mixture plasma at 4 mTorr with

input power of 90 W.

134

 

11

10 I I I T I
b

 

l l .I L

 

Model 2: f=0.4, c=0, z: 2 cm

Experimental Data, 2 = 2 cm

Electron Densrty (cm 3)
ES
*

 

 

 

10 l l l
0 1 2 3 4 5 6

Pressure (mTorr)

Figure 6.12 Ar plasma electron density at downstream z = 2 cm. The dashed line is
the model 2 prediction and the solid line is model 1 prediction. The (*) points are
from experimental measurements. The input power is 90 W and flow rate is 8 sccm.

I35

 

 

 

 

 

10 L I I I I 1 r I

A 10
".‘E 10 r j
o r .
v b '1
>5 L .1
H
“a; 1 1
1:
a)
Q
C
0
La
H
o
.33 9
1.1.1 10 r j

1'

8

10 l l L L 1 l l
O 1 2 3 4 5 6 7 8

Downstream Position (cm)

Figure 6.13 Comparison of argon plasma diffusion model with experimental data
taken in the downstream region. Plasma conditions: Pin = 90 W, 1 mTorr, ans 8 sccm.

l36

Fig. 6.13 compares the experimental results of electron densities at 2 cm downstream from
Langmuir probe measurements with the results from the hydrogen discharge model using

floss = 0.38 and c = 6.5. The input power is also 90 W, the same as in Fig. 6.12. It is found

that at pressures higher than 2 mTorr, the model has a better fit with the experimental data
in magnitude, while at pressures less than 2 mTorr, the difference between the measured
and modeled results becomes larger.

The electron densities in the argon and hydrogen plasmas at various downstream
positions are plotted in Fig. 6.14 and Fig. 6.15, respectively. The input conditions for both
figures are 90 W input power and l mTorr pressure. It is shown that the model prediction
of the electron density in both of the discharges has a faster decreasing rate along the
downstream distance than those from the experimental measurements. This may be
explained as follows. In the downstream diffusion model, it assumes no ionization process
in the downstream region, hence the ionization frequency equals to zero. This may not be
true. The ionization processes are expected to continue in the downstream region until at

some distance the electron energy is too low to cause any ionization.

6.4 Conclusions

Global plasma source models have been developed for argon, hydrogen, and argon-
hydrogen discharges, The models incorporate the effects of both static magnetic field
confinement of charged particles and additional ionization produced by trapped electrons
in the magnetic fields. Comparison of the experimental measurements for the argon,

hydrogen, and argon-hydrogen discharges to the models indicates the importance of

l37

10

 

 

 

10 1 r r r r I I 1 r 1
1
i'
1 11 * *
411 *1“
* '1
1* u ——x
[ 4
MA M
I ¥
E p *
U
v
>\
.2
CD *-
C
Q)
Q
C
O
1..
H
0
2
I-I-I 1
109 1 1 1 1 1 1 1 1 1 1

 

 

O 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
Pressure (mTorr)

Figure 6.14 Comparison of hydrogen discharge model with experimental data at z =
2 cm. The input power is 90 W. Flow rate: 30 sccm.

I38

11

 

 

 

 

 

10 I I I I I I I
1' .
J
1
a? 10
E 10 _ ‘j
3
3‘
'3
C:
Q)
Q
C
8
‘5
2 9
LL] 10 '- -"
108 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8

Downstream Position (cm)

Figure 6.15 Comparison of hydrogen plasma diffusion model with experimental
data taken in the downstream region. Plasma conditions: Pin = 90 W, 1 mTorr, and 30

sccm.

139

considering both of these magnetic field influences in the plasma source for predicting the
electron temperature. Overall the electron temperature in the discharges can be reasonably
predicted by comparing with the experimental results The determination of the charged
species densities showed general agreement at downstream z = 2 cm with a factor of 2 or

less difference between the measurements and models.

l40

CHAPTER 7
Summary and Recommendations for Future

Research

7.1 Summary of the Results

The study of the neutral and charged species properties in the electron cyclotron
resonance microwave plasma disk reactor (MPDR) presented in this work provides a more
complete understanding of the low pressure, high density plasma behavior in both atomic
and molecular gas discharges. Knowledge of the macrosc0pic plasma properties such as
ion densities, electron energy distribution function, and neutral density allows better
control and integration of plasma processing, as well as, improvement in discharge
operation (uniformity, deposition/etch rate, substrate damage, ...). A spatially-averaged
discharge model is developed based on the chosen experimental results for better
understanding and predicting of the discharge characteristics in atomic, molecular, and

mixture gas discharges.

7.1.1 Ion Density Measurements

The ion density in the MPDR 610 discharge is measured with a double Langmuir

l4l

probe in the downstream region at 2 - 10 cm from the source. The ion density at z = 2 cm
in the argon discharges generally increases with the pressure (0.5-5 mTorr) except at
pressures of around 1.2 mTorr where the microwave reflected power jumps from 10 W to

24 W. With a decreased absorbed power versus pressure, the ion density has little change

3

in this range of pressures. The peak ion density of 3.08x1010 cm' occurs at pressures

around 3 mTorr with a microwave input power of 90 W and a flow of 8 sccm (see Fig.4.6).

The ion density in the hydrogen discharges at z = 2 cm has a peak of 7.5x109 cm’3 at a
pressure around 1.5 mTorr, 90 W input power, and flow of 30 sccm (Fig. 4.7). The
microwave reflected power, unlike the case in argon discharges, is stable and very small
(less than 5 W) in the hydrogen plasma. The ion density along the downstream direction is
also measured for both argon and hydrogen discharges (Fig. 4.8). It is found to decrease
less quickly versus downstream distance that predicted by a pure ambipolar diffusion

model.

7.1.2 Electron Energy in the Discharge

The electron energy and its distribution function in the MPDR created discharges are
investigated via single Langmuir probe. The probe is usually positioned at 2 cm
downstream away from the discharge generated region. The measured electron energy in
the argon discharge at an input power of 90 W varies from 5.0 to 3.2 eV for the pressure
range of 0.5 - 4.7 mTorr. The electron energy in hydrogen plasma at the same input power
varies from 5.8 to 4.9 eV for the pressure range of 0.5 - 3 mTorr. The electron energy

distribution functions for both argon and hydrogen discharges show a Maxwellian like

142

distribution at energies less than 10 eV and have some deviations from the Maxwellian at
the tail of the distributions. The downstream electron energies are also recorded at
positions 2 = 2 to 10 cm for both of the discharges and are used to estimate the electron
temperature in the source by extrapolating the temperature data taken at various
downstream positions. For the argon plasma at 1 mTorr, 90 W input power, and 8 sccm
flow rate, the predicted electron temperature in the source is about 5.8 eV and is 7.8 eV for
the hydrogen discharge at the same input conditions at flow rate of 30 sccm. The electron
temperature in the argon-hydrogen mixture discharge is measured at three pressures
including 0.6, 1.0, and 4 mTorr. The mixture gas composition is varied at each pressure.
The electron temperature at l mTorr and 90 W input power varies from 4.8 to 5.7 eV for

hydrogen partial pressure ratio of l to 4.

7 .1.3 Optical Emission Spectroscopy (OES)

Optical emission from three noble gases including Ar, Kr, and Xe added in the
hydrogen discharge are observed and recorded to obtain the electron temperature in the
discharge source region. The observed emission lines have threshold energies from 9.82 to
13.5 eV. The results are discussed for the case of with and without xenon emission
included. The electron temperature in the hydrogen discharge determined from 0138 at 3
mTorr and 90 W input power is 6.5 eV. Argon emission from its doubly ionized atoms are
observed at 328.59 nm and 331.13 nm (Fig. 4.12) which suggests the existence of an non
ignorable number of high energy electrons in the argon discharge. Actinometry is
performed to obtain the information on the atomic hydrogen concentration in the

hydrogen discharge. The actinometry data is further analyzed in Chapter 6 using the

[43

electron temperature in the source so that the density of atomic hydrogen is obtained.

7.1.4 Modeling of the Compact ECR Plasma Source

A spatially-averaged (global) model is developed for low pressure discharge
conditions. The model uses charged particle conservation by equating the generation of
the ions in the plasma volume to the diffusion loss on the wall, and it uses power balance
of electrons which absorb energy from the input microwave power and lose energy due to

collisions with the neutral (e.g. Ar, H2, and H). The collision processes can be elastic or

inelastic, such as ionization, excitation, dissociation. Radial diffusion of the charged
species in the plasma is restricted due to the arrangement of permanent magnets around
the discharge chamber in MPDR 610. Electron temperature, neutral and charged species
densities are solved in argon, hydrogen, and argon-hydrogen mixture discharges.
Determination of atomic hydrogen surface recombination coefficient is done by
comparing the modeled hydrogen atom concentrations of different recombination
coefficients with the experimental results of actinometry (see Fig.6.6). The model provides
a good prediction of the electron temperature in all three discharges when compared with
the experimental results. For the argon discharge, the model also have a good agreement of
the ion densities with the experiments within reasonable error. However, the density
profile from the model of the hydrogen discharge does not predict a peak at a certain range

of pressure as the experiment indicates.

144

7.2 Recommendations for Future Research

In addition to the plasma properties (electron temperature, neutral and ion densities)
investigated in this thesis, there are still some properties left experimentally unknown,
such as the neutral temperature and ion energy in the discharge, which can play a critical
role in surface processing. Positive ion bombardment on the substrate is important in
achieving anisotropic etching with high aspect ratio necessary for microelectronic
manufacturing in deep submicron features. Electrostatic energy [108] or optical Doppler
shift measurement using optical emission spectroscopy (OES) [109] or Laser-induced
fluorescence (LIF) [110],[111] technique is suggested for the study of ion energies or gas
temperature in the discharge. In a molecular gas or gas mixture discharge, it is also useful
to use a mass spectrometer to measure the relative number of various ions with different
charge or mass. While it is not possible to distinguish the individual density of each
charged species when utilizing a Langmuir probe to measure the charged particle density
in the plasma.

In the modeling of the discharge with magnetic confinement applied on the discharge
chamber, the model developed in this work provides a good overall agreement with the
experiment within the pressure range investigated. The ion density profile in the hydrogen
model, however, does not show a similarity at a certain pressure range as the experiment
result. A more complex model of the magnetic confinement of the charged particle should
be studied for improvement.

The developed model has successfully predict the electron temperature, neutral and
ion densities in the discharge (atomic, molecular, and mixture gas discharges). Future

extended work should go to the modeling of other molecular or mixture discharges (C12,

145

Ar/CH4, 112/€114, ...) which is frequently used in the deposition and etching processes.

146

APPENDICES

147

Appendix A

Argon Plasma Global Model Program

%************************************************************* %

% Argon plasma modeling with consideration of plasma confinement, f_loss %

% and additional ionization from trapped electrons. %
% This program calculates the electron denity at various absorbed powers %
% and pressures by first generating Te vs ng*d plots with and without a %
% f_loss term and using it to determine Te. %

%*31‘*********************************************************** %

KizO=6e-14; %m3/s
Eiz= 15.76; %eV
e= l .6e-19;
mi=40*1.67e-27; %kg

i=0;

for Te=1 :0. l :20;
i=i+l;
Kiz=KizO*exp(-Eiz/T e);
ub=sqrt(e*Te/mi);
left=Kiz/ub;
x(i)=0.4/left;
y(i)=1/1eft;

end

Te=1 :0. 1:20;

semilogx(x,Te,y,Te,’--’),grid

xlabel(’n_gd_eff (m-2)’)

ylabel(’Te (eV)’)

title(’Ar plasma with f_l=l, 0.4 /model/argon/te-nd.eps’)
axis([1e17 le21 1 20])

148

R=l.8; L=3; %cm, cylindrical discharge dimension
k=1.38e-23;

Pin=. 1; %mW, input power
Pr=[1 1 2.4 2.6 3.8 4 4 4 4 4]; %0.1W, reflected power
p=[0.5 0.75 l 1.5 2 2.5 3 3.5 4 5]; %mTorr .

f_loss=.4; %fraction of diffusion loss to the wall

clear T

clear nd

for j=l:10

Tar=420; %Kelvin, argon gas temperature

ng=p(i)*le-3*133.32/(k*Tar) %m-3, neutral gas density at a specified pressure

1ampda=1/(ng* le-6)/1e-14 %cm, ion collision cross section=le-14 cm2
if lampda>L
hl=.4;
hl=.4;
else
hl=.86/sqrt(3+L/2/lampda);
hr=.8/sqrt(4+R/lampda);
end

Aeff=f_loss*2*pi*R*(R*hl+L*hr)*1e-4; %m2
deff=R*L/(f_1oss/(1+.17/p(j))*2*(R*hl+L*hr))* 1e-2; %m
nd0)=ng*deff %m-2

 

% -------------- Determine Te in the plasma %
for i=1 : 191
if (nd0)<y<i)> & (nd0>>y<i+1»
T0): 1+0. 1 *i
break
end
end

ub=sqrt(e*T(j)/mi); %m/s
%Determine electron collisional energy loss, Ec

if T(j)>=l & T(j)<=2
Ec(i)=800-(800-88)*(T(i)- l)

elseif T(j)<=3
Ec(j)=88-(88-50)*(T(j)-2)

elseif T(j)<=4
Ec(j)=50-(50-38)*(T(j)-3)

149

elseif T(j)<=5
Ec(j)=38-(38-31)*(T(j)-4)
elseif T(j)<=6
Ec(j)=3l-(3 l-28)*(T(j)-5)
elseif T(j)<=7
Ec(j)=28-(28-25)*(T(j)-6)
elseif T(j)<=8
Ec(j)=25-(25-23)*(T(j)-7)
elseif T(j)<=9
Ec(i)=23—(23-22)*(T(j)-8)
elseif T(j)<=lO
Ec(j)=22-(22-21)*(T(j)-9)
elseif T(j)<=20
Ec(j)=21-(21-20)*(T(j)-10)
elseif T(j)>20
Ec0)=20
end

% Total energy loss per electron-ion pair created, eV
Et=Ec(j)+7.2*T(j);

% The absorbed power by plasma, Watt
Pabs=Pin*909—Pr(j)*9.015;

% Calculated electron density, m-3
ne(j)=Pabs/(e*ub*Aeff*Et);

end
ne

150

Appendix B

Hydrogen Discharge Global Model Program

 

 

 

% %
% This program calculates the neutral, electron, and ion densities of %
% hydrogen discharges at low pressures (0.5-4 mTorr). The neutral and %
% ionic species considered include H2, H, H+, H2+, and H3+. The reactions %
% included are mostly electron-neutral collisions. %
% %

 

%Constant values

k=1.38e-23;

e=1.6e-19;

a_h=.53e-10; %meter
m_h=1.67e—27; %kg

 

 

% Plasma source condition %
R=l .8; L=3; %cm, cylindrical discharge dimension
Th=420; Th2=420; %Kelvin; temperature of neutrals

p=5; %mTorr, chamber pressure

Th=Th2;

Ng=p* le-3* l33.32/(k*Th2);

N_h2=Ng %mA-3

Ne=3.0e17; %m"-3

gamma=0.005; %hydrogen surface recombination coefficient
fl_h=0.38; %fraction of diffusion loss to the wall

lampda_1=1/(N_h2)/1e-l9; %m, H+ collision cross section=le-19 m2
lampda_2=l/(N_h2)/ 1e-18; %m, H2+ collision cross section=le-18 m2

if lampda_l >L* le-2

151

hl=0.4; hr=0.4;
else
hl=.86/sqrt(3+L* le-2/2/lampda_1);
h1=.8/sqrt(4+R* le-2/1ampda_ l );
end

Al=fl__h*2*pi*R*(R*hl+L*hr)*le-4; %m2

if lampda_2 >L* le-2
hl=0.4; hr=0.4;
else
hl=.86/sqrt(3+L* le-2/2/lampda__2);
h1=.8/sqrt(4+R* 1e-2/1ampda_2);
end ‘

A2=fl_h*2*pi*R*(R*hl+L*hr)*le-4; %m2
1ampda_5=1/(N_h2*1e-6)/1e-15; %cm, H3+ collision cross section=le—15 cm2

if lampda_S >L
hl=0.4; hr=0.4;
else
hl=.86/sqrt(3+L* le-2/2/lampda_5);
hr=.8/sqrt(4+R* le-2/1ampda_5);
end

A5=fl_h*2*pi*R*(R*hl+L*hr)* le-4; %m2

Te=input(’lnput the electron temperature (eV), Te =’);

V=pi*R"2*L* le-6; %m"3
v_avg=sqrt(8*k*Th/pi/m_h); %m/s
z=(pi/L)"2+(2.405/R)"2; %cmA-2

% Calculate ionization rate constants Kiz_l l, Kiz_2, Kiz_12;

b1=[-3.27140e1 1.35366el -5.73933 1.56315 -2.87706e-1 3.48256e-2 -2.63198e-
3 1.11954e-4 -2.03915e-6];

b2=[-3.56864e1 1.733469el -7.76747 2.21158 -4.l6984e-1 5.08829e-2 -
3.83274e-3 1.612866-4 -2.89339e-6];

b3=[-3.8346061 1.42632e1 -5.82647 1.72794 -3.59812e-1 4.82220e—2 —3.9094e-3
1.73878e-4 -3.25284e-6];

152

ln__KizZ=0;

1n_Kizl 1 =0;

1n_Ki212=0;

for i=1 :9
1n_Kizl l=ln_Kizl 1+b 1 (i)*log(Te)"(i— l );
ln_Kiz2=1n_Ki22+b2(i)*log(Te)"(i-1);
1n_Kizl2=1n_Kiz12+b3(i)*log(Te)"(i-1 );

end

Kiz_l 1=exp(ln_Kizl 1)*1e-6; %m"3/s
Kiz_2=exp(1n_Ki22)* 1e-6; %m"3/s
Kiz_12=exp(1n_Ki212)* le-6; %mA3/s

% Determine hydrogen dissociation rate constant
if Te<7 & Te>=3

m=log 10(12/2.2)/log10(7/3);
lg_Kh=m*(log10(Te)—log10(3))+log10(2.2e- 15);
Kh2_h=10"(lg_Kh); %m"3/s

elseif Te<=14
m=log10(2.l/l.2)/10g10(14/7);
lg_Kh=m*(log10(Te)-log10(7))+log10( 1.2e- 14);
Kh2_h= 10"(1g_Kh);

else
Kh2_h=2.2e- l4; %mA3/s

end

% Rate constant for creation of H3+
Kex_25=2.l le- 15; %mA3/s

% Ionization threshold energies, eV
Eiz_11=13.6; Eiz_2=15.4; Eiz_12=18;

ubl=sqrt( 1 .6e-19*Te/m_h); %m/s
ub2=sqrt( 1 .6e- l9*Te/2/m_h); %m/s
ub5=sqrt( l .6e-19*Te/3/m_h); %m/s

% ------------ Solving for densities of H+,H2+,H3+ %

 

153

errl=10;
c=6.5;

while errl > 5

A(1,:)=[V*gamma*k*Th*z*le4/m_h/(2e-19*v_avg)/N_h2+V*Kiz__l1*Ne*(l+c/
p) O -V*Kex_25*N_h2 0];

A(2,:)=[-V*Ne*Kiz_l1*(l+c/p) Al*ub1 0 0];

A(3,:)=[0 0 V*N_h2*Kex_25+A2*ub2 0];

A(4,:)=[0 0 -V*N_h2*Kex_25 A5*ub5];

b=[V*Kh2_h*N_h2*Ne; V*Ne*N_h2*Kiz_12; V*N_h2*Ne*Kiz_2; 0]*(1+c/p);
X=A\b;

N_h=X( 1);

N_1=X(2); N_2=X(3); N_3=X(4);
new=Ng-N_h;
errl=abs(new-N_h2)/N_h2* 100;
N_h2=new;

end
%Checking for reasonable Te assumption

c=sum(X)-X( 1);
err=abs(Ne-c)/Ne* 100

 

% Computing the power absorbed by plasma %

 

b1=[-2.81495el 1.00983e1 -4.77196 1.46781 -2.97980e-1 3.86163e-2 -3.05169e-
3 1.33547e-4 -2.47609e-6];

b2=[-2.83326e1 9.58736 -4.83358 1.41586 -2.53789e-1 2.8007le-2 -1.87l4le-3
6.98667e-5 -1. 12376e-6];

b3=[-3.0819el 1.03887el -4.25977 1.18123 -2.27751e-l 2.90058e-2 -2.28759e-3
1.00435e—4 -l.86993e-6];

b4=[-3.3482e1 1.371761 -5.92261 1.70972 -3.50523e-1 4.834386-2 -4.13l41e-3
1.94839e-4 -3.854286-6];

b5=[-3.64659el 1.43036el -6.07443 1.67731 -3.1287le-l 3.80542e-2 -2.86001e-
3 1.19964e-4 -2. 14223e-6];

b6=[-2.85801e1 1.03854el -5.38383 1.95064 -5.39367e-1 1.00692e-1 -1.16076e-

154

 

2 7.41162e-4 -2.00137e-5];

1n_Kexc 1 1 =0; 1n_Kexc 12:0;
ln_Kexc21=0; ln_Kexc22=0; ln_Kexc23=0; ln_Kexc24=0; ln_Kexc25=0;

for i=1 :9

1n_Kexcl l=ln_Kexc1 1+bl(i)*log(Te)"(i-1);
1n_Kexc12=ln_l(exc12+b2(i)*log(Te)"(i-l);
1n_Kexc2 l=ln_Kexc21+b3(i)*log(Te)"(i-l );
1n_Kexc22=ln_Kexc22+b4(i)*log(Te)"(i-1);
ln_Kexc23=1n_Kexc23+b5(i)*log(Te)"(i- l );
ln_Kexc24=ln_Kexc24+b6(i)*log(Te)"(i- l );

end

Kexc_] l=exp(ln_Kexc 1 1)* 1e-6; %m"3/s
Kexc_12=exp(ln_Kexc12)* 1e—6; %mA3/s
Kexc_21=exp(ln_l(exc21)* le-6; %mA3/s
Kexc_22=exp(ln_Kexc22)* le-6; %m"3/s
Kexc_23=exp(ln_Kexc23)* le-6; %m"3/s
Kexc_24=exp(ln_Kexc24)* le-6; %mA3/s
Kexc_25=Kh2_h; %m"3/s

Kelas_l=pi*(a_h"2)*sqrt(8* 1.6e-19*Te/pi/9.1 le-31); %mA3/s

if Te<l3

Kelas_2=] .5e-13; %mA3/s
else

Kelas_2=1 .2e- 1 3;
end

%Excitation threshold energy, eV

Eexc_l 1:102; Eexc_l2=10.2;

Eexc_2 1:1 1.37; Eexc_22=l 1.7; Eexc_23=12.2; Eexc_24=10;

Eexc_25= 10;

Eexc 1=Kexc_l 1 *Eexc_l l +Kexc_l 2*Eexc_ 12;

Eexc2= Kexc_2 1 * Eexc_21+Kexc_22*Eexc_22+Kexc_23 * Eexc_2 3+Kexc_24* Ee
xc_24+ Kexc_25 *Eexc_25;

viz_l=N_h*Kiz_l l+N_h2*Kiz_12;
viz_2=N_h2*Kiz_2;

Eexc=[N_h*Eexc1/viz_1 N_h2*(Eexc2-Kexc_25*Eexc_25)/viz_2];

155

Eelas=[N_h*Kelas_l *3*9.1 le-31*Te/m_h/viz_l N_h2*Kelas_2*3*9.l1e—31*Te/
2/m_h/viz_2];

Eion=[(N_h*Kiz_l 1*Eiz_1 l+N_h2*Kiz_l2*Eiz_12)/viz_l N_h2*Kiz_2*Eiz_2/
viz_2];

Edissc=N_h2*Kexc_25*Eexc_25/viz_2;

Pexc=e*[Eexc(1)*N_1*A1*ubl Eexc(2)*N_2*A2*ub2];
Pelas=e*[Eelas(l)*N_1*Al*ub1 Eelas(2)*N_2*A2*ub2];
Pion=e*[Eion(1)*N_l*Al*ubl Eion(2)*N_2*A2*ub2];
Pdissc=e*Edissc*N_2*A2*ub2;

El_l=(N_h*(Kiz_l 1*Eiz_1 1+Eexc 1+Kelas_1 *3*9.l le-31*Te/
m_h)+N_h2*Kiz_l 2*Eiz_12)/(N_h*Kiz_l l+N_h2*Kiz_12);

El_2=N_h2*(Kiz_2*Eiz_2+Eexc2+Kelas_2*3*9.l le-31*Te/2/m__h)/viz_2; %eV
e_iz=N_h2*Kiz_2*Eiz_2/viz_2;

e_exc=N_h2*Eexc2/viz_2;

e_e1as=N_h2*Kelas_2*3*9.l 1e-31*Te/2/m_h/viz_2;

Eew=2*Te;

Eiw l =3.3*Te;Eiw2=3.7*Te;Eiw3=3.9*Te;

Et_ 1=El_1+Eew+Eiw 1 ; Et_2=El_2+Eew+Eiw2;

Et_5=Eiw3;

E1=[El_1 El_2];

Pex=e*V*N_2*(N__h2*Kex_25)*El_2;

% Compute absorbed power by plasma, Watt
Pabs=e*(N__l *Al *Et_l *ub1+N_2*A2*Et_2*ub2+N_3*A5*Et_5*ub5)+Pex

%(Reset Ne value for actual absorbed power, Pabs)
N=[N_h2 N_h Ne N_l N_2 N_3];
Pw=[Pelas(l)+Pelas(2) Pdissc Pexc( 1)+Pexc(2) Pion(l)+Pion(2) Pex Pabs];

N_h,N (4:6)

156

Appendix C

Hydrogen/Argon Mixture Discharge Global

Model Program

 

%

 

% This program calculates the neutral, electron, and ion densities of
% H2-Ar discharges at low pressures (0.5-4 mTorr). The neutral and
% ionic species considered include H2, H, Ar, Ar+, H+, H2+, and H3+.

% The reactions included are electron-neutral and ion—neutral collisions.

 

%

 

%Constant values
k=1.38e-23;

e: 1 .6e- 19;

a_h=.53e-10; %meter
m_h=1.67e-27; %kg

 

% Plasma source condition %
R=l.8; L=3; %cm, cylindrical discharge dimension
Th=420; Th2=420; %Kelvin; temperature of neutrals
Tar=420;

 

p=4; %mTorr, total chamber pressure

ng=p*1e-3* 133.32/k/Th;

r__flow=5; %H2/Ar flow(pressure) ratio
N_a1=1/(l+r_flow)*ng %m"-3

N_h2=r_flow/( l+r_flow)*n g %m"-3

p2=N_h2;

Ne=4.0el7; %m"-3

gamma=0.005; %hydrogen surface recombination coefficient
fl___ar=0.4; %fraction of diffusion loss to the wall
fl_h=.38;

157

%
%
%
%
%
%

1ampda_l=l/(N_h2)/le-l9; %m, H-I- collision cross section=1e-19 m2
lampda_2=1/(N_h2)/1e- l 8; %m, H2+ collision cross section=1e-18 m2
lampda_3=1/(N_ar)/ 1e-18; %m, Ar+ collision cross section=le-14 cm2
lampda_4=l/(N_ar)/1e-l9; %m, ArH+ collision cross section=le-15 cm2
lampda_5=1/(N_h2)/ le-l9; %m, H3+ collision cross section=le-15 cm2

lampda=[lampda_l lampda_2 1ampda_3 lampda_4 lampda_S];
for i=1 :5

if lampda(i) >L
hl=0.4; hl=0.4;
else
h1=.86/sqrt(3+L*le-2/2/lampda(i));
hr=.8/sqrt(4+R* 1e-2/lampda(i));
end
if i<=2
Aeff(i)=fl_h*2*pi*R*(R*hl+L*hr)*le-4; %m2
elseifi<=4
Aeff(i)=fl_ar*2*pi*R*(R*hl+L*hr)*1e-4; %m2
else
Aeff(i)=fl_h*2*pi*R*(R*hl+L*hr)*1e-4; %m2
end

end

A1=Aeff(l); A2=Aeff(2); A3=Aeff(3); A4=Aeff(4); A5=Aeff(5);
Te=input(’Input the electron temperature (eV), Te =’);
V=pi*R"2*L*1e-6; %m"3

v_avg=sqrt(8*k*Th/pi/m_h); %m/s
z=(pi/L)"2+(2.405/R)"2; %cmA-Z

 

% -------------- Calculate Kiz_l 1, Kiz_2, Kiz_12 %
bl=[-3.27l40el 1.35366el -5.73933 1.56315 -2.87706e-1 3.48256e-2 -2.63198e-
3 1.11954e-4 -2.03915e-6];

b2=[-3.56864e1 1.733469e1 -7.76747 2.21158 -4.l6984e-1 5.08829e-2 -
3.83274e-3 1.61286e-4 -2.89339e-6];

b3=[-3.83460e1 1.42632e1 -5.82647 1.72794 -3.59812e-1 4.82220e-2 -3.9094e-3
1.73878e-4 -3.25284e-6];

158

 

ln_Kiz2=0;
1n_Kizl 1:0;
1n_Kiz12=0;

for i=1 :9
1n_Kizl l=ln_Kizl l+b1(i)*log(Te)"(i-1);
ln_Kiz2=ln_KizZ+b2(i)*log(Te)"(i- 1 );
1n_Ki212=ln_Kiz 12+b3(i)*log(Te)"(i- l );
end

Kiz_l l=exp(ln_Kizl 1)*1e-6; %m"3/s
Kiz_2=exp(ln_Kiz2)* 1e-6; %mA3/s
Kiz_12=exp(ln_Ki212)*le-6; %mA3/s
Kiz_ar=6e-l4*exp(-15.76/'I'e); %m3/s

% Determine hydrogen dissociation rate constant

if Te<7 & Te>=3
m=log10(12/2.2)/log10(7/3);
lg_Kh=m*(log10(Te)-loglO(3))+loglO(2.2e-15);
Kh2_h=10"(lg_l(h); %m"3/s

elseif Te<=14
m=log10(2. l/l .2)/log10(14/7);
lg_Kh=m*(log10(Te)-log10(7))+log10(l.2e-14);
Kh2_h= 10"(1g_Kh);

else
Kh2_h=2.2e- 14;

end

Kex_24=1.76e-15; %m3/s, Rate constant for creation of ArH+
Kex_34=8.9e-16; %m3/s, Rate constant for creation of ArH+
Kex_23=0; %m3/s

Kex_32=0; %m3/s

Kex_25=2.l le-15; %m3/s, Rate constant for creation of H3+
Kex_45=4.5e-16; %m3/s, Rate constant for creation of H3+
Krecom=5e-15; %m3/s, Rate constant for e-Arl-I+ recombination

% Ionization threshold energies, eV
Eiz_11=13.6; Eiz_2=15.4; Eiz_12=18;
Eiz_ar=15.76;

ubl=sqrt(1.6e-19*Te/m_h); %m/s
ub2=sqrt( 1 .6e-19*Te/2/m_h); %m/s
ub3=sqrt( l.6e-19*Te/40/m_h); %m/s
ub4=sqrt(1.6e-19*Te/41/m_h); %m/s
ub5=sqrt( 1.6e- 19*Te/3/m_h); %m/s

159

%Solving for densities of H+,H2+,Ar+,ArH+,H3+

errl=10;
cl=6.5; c2=.17;

while errl >2

A( 1,:)=[V*gamma*k*Th*z* 1e4/m_h/(2e-l9*v_avg)/

N_h2+V*Kiz_l 1*Ne*(1+cl/p) 0 -V*(Kex_24*N_ar+Kex_25*N_h2) -
V*Kex_34*N_h2 -V*Krecom*Ne 0];

A(2,:)=[-V*Ne*Kiz_1 1*(1+cl/p) Al*ubl 0 0 0 0];

A(3,:)=[0 0 V*N_ar*(Kex_24+Kex_23)+A2*ub2 -V*N_h2*Kex_32 0 0];
A(4,:)=[O 0 -V*N_ar*Kex_23 V*N_h2*(Kex_32+Kex_34)+A3*ub3 O 0];
A(5,:)=[0 0 -V*N_ar*Kex_24 -V*N_h2*Kex_34 A4*ub4+V*Ne*Krecom 0];
A(6,:)=[0 0 -V*N_h2*Kex_25 0 0 A5*ub5];

b=[V*Kh2_h*N_h2*Ne; V*Ne*N_h2*Kiz_12; V*N_h2*Ne*Kiz_2;
V*Ne*N_ar*Kiz_ar*(l+c2/p)/( 1+cl/p); 0; 0]*( l+c l/p)];

X=A\b;
N_h=X( 1);
N_l=X(2);N_2=X(3);N_3=X(4);N_4=X(5); N_5=X(6);

new=p2-N_h;
errl=abs(new-N_h2)/N_h2* 100;
N_h2=new;

end
%Checking for reasonable Te assumption

ni=sum(X)-X( l);
err=abs(Ne-ni)/Ne* 100

% ------------- Computing the power absorbed by plasma ------------ %

bl=[-2.81495el 1.00983e1 -4.77196 1.46781 -2.979806-1 3.86163e-2 -3.05169e-
3 1.33547e-4 -2.47609e-6];

b2=[-2.83326e1 9.58736 -4.83358 1.41586 -2.53789e-1 2.8007le-2 -l.87141e-3
6.98667e-5 -1.12376e-6];

b3=[-3.0819e1 1.03887el -4.25977 1.18123 -2.27751e—1 2.90058e-2 -2.28759e-3
1.00435e-4 - 1.86993e-6];

b4=[-3.3482e1 1.3717e1 -5.92261 1.70972 -3.50523e-1 4.83438e-2 -4.l3141e-3
1.94839e-4 -3.85428e-6];

b5=[-3.64659el 1.43036e1 -6.07443 1.67731 -3.1287le-1 3.80542e-2 -2.86001e-

160

3 1.19964e-4 -2.14223e-6];
b6=[-2.85801e1 1.03854e1 -5.38383 1.95064 -5.39367e-1 1.00692e-l -l.l6076e-
2 7.41162e-4 -2.00137e-5];

1n_Kexc1 1:0; 1n_Kexc12=0;
ln_Kexc21=0; ln_Kexc22=0; ln_Kexc23=0; 1n_Kexc24=0; ln_Kexc25=0;

for i=1 :9
1n_Kexcl l=ln_Kexcl 1 +b1(i)*log(Te)"(i- l );
1n_Kexc12=ln_Kexc12+b2(i)*log(Te)"(i-l);
1n_Kexc2l=ln_Kexc21+b3(i)*log(Te)"(i-l);
ln_Kexc22=ln_Kexc22+b4(i)*log(Te)"(i-1);
1n_Kexc23=1n_Kexc23+b5(i)*log(Te)"(i-1 );
1n_Kexc24=ln_Kexc24+b6(i)*log(Te)"(i-1);

end

Kexc_] l=exp(ln_Kexcl l)* 1e-6; %m"3/s
Kexc_l2=exp(ln_Kexc12)* 1e-6; %m"3/s
Kexc_21=exp(ln_Kexc2 l)* 1e-6; %m"3/s
Kexc_22=exp(1n_Kexc22)* le-6; %m"3/s
Kexc_23=exp(1n_Kexc23)* 1e-6; %mA3/s
Kexc_24=exp(ln_Kexc24)* 1e-6; %m"3/s
Kexc_25=Kh2_h; %mA3/s

Kelas_l=pi*(a_h"2)*sqrt(8* 1 .6e- 19*Te/pi/9.1 1e-31); %mA3/s
if Te<l3
Kelas_2= 1 .Se- 1 3; %mA3/s
else
Kelas_2= l 26- 13;
end

%Excitation threshold energy, eV

Eexc_ll=10.2; Eexc_12=10.2;
Eexc_21=11.37; Eexc_22=1 1.7; Eexc_23=12.2; Eexc_24=10;
Eexc_25=10;

Eexc 1=Kexc_l 1 *Eexc_l l+Kexc_12*Eexc_12;
Eexc2=Kexc_2 1 *Eexc_2 l +Kexc_22*Eexc_22+Kexc_23 *Eexc_23+Kexc_24*Ee
xc_24+Kexc_25*Eexc_25;

viz_l=N_h*Kiz_l l+N_h2*Kiz_12;

viz_2=N_h2*Kiz_2;

Eexc=[N_h*Eexc l/viz_l N_h2*(Eexc2-Kexc_25*Eexc_25)/viz_2];
Eelas=[N_h*Kelas_l *3*9.1 le-31*Te/m_h/viz_l N_h2*Kelas_2*3*9.1 1e-31*Te/
2/m_h/viz_2];

161

 

Eion=[(N_h*Kiz_l 1*Eiz_l 1+N_h2*Kiz_l2*Eiz_12)/viz_1 N_h2*Kiz_2*Eiz_2/
viz_2];
Edissc=N_h2*Kexc_25*Eexc_25/viz_2;

Pexc=e*[Eexc( l)*N_1*A1*ubl Eexc(2)*N_2*A2*ub2];
Pelas=e*[Eelas(1)*N_1*Al*ubl Eelas(2)*N_2*A2*ub2];

Pion=e*[Eion( l)*N_1*A1 *ubl Eion(2)*N_2*A2*ub2 Eiz_ar*N_3*A3*ub3];
Pdissc=e*Edissc*N_2*A2*ub2;

El_l=(N_h*(Kiz_1 1*Eiz_1 1+Eexc1+Kelas_1*3*9.1 1e-31*Te/
m_h)+N_h2*Kiz_12*Eiz_l2)/(N_h*Kiz_l 1+N_h2*Kiz_12);
El_2=N_h2*(Kjz_2*Eiz_2+Eexc2+Kelas_2*3*9. l 1e-31 *Te/2/m_h)/
(N_h2*Kiz_2); %eV

%Find the electron collisional energy loss of creating Ar+
if Te>=l & Te<=2
El_3=800-(800-88)*(Te- 1)
elseif Te<=3
El_3=88-(88-50)*(Te-2)
elseif Te<=4
El_3=50-(50-38)*(Te-3)
elseif Te<=5
El_3=38-(38-31)*(Te—4)
elseif Te<=6
El_3=31-(3l-28)*(Te-5)
elseif Te<=7
El_3=28-(28-25)*(Te-6)
elseif Te<=8
El_3=25-(25-23)*(Te-7)
elseif Te<=9
El_3=23-(23-22)*(Te-8)
elseif Te<=10
El_3=22-(22-21)*(Te-9)
elseif Te<=20
El_3=21-(21—20)*(Te- 10)
elseif Te>20

El_3=20
end

Eew=2*Te; %eV

Eiw1=3.3*Te; Eiw2=3.7*Te; Eiw3=4.7*Te; Eiw4=5.2*Te; Eiw5=3.9*Te;
Et_ 1 =El_l +Eew+Eiw 1 ; Et_2=El_2+Eew+Eiw2; Et_3=El_3+Eew+Eiw3;
Et_4=Eew+Eiw4; Et_5=Eew+Eiw5;

Pex=e*V* [N_2*(N_ar* Kex_23+N_ar*Kex_24+N_h2*Kex_25)*El_2

162

N_3*N_h2*(Kex_32+Kex_34)*El_3]

% Compute absorbed power by plasma, Watt
Pabs=e*(N_l *A l *Et_1 *ub 1 +N_2*A2*Et_2*ub2+N_3*A3*Et_3*ub3+N_4*A4*
Et_4*ub4+N_5 *A5*Et_5 *ub5)+Pex( 1)+Pex(2)

%(Reset Ne value for actual absorbed power Pabs)

N=[N_ar N_h2 N_h Ne N_l N_2 N_3 N_4 N_S];

Pw=[Pelas(l)+Pelas(2) Pdissc Pexc(1)+Pexc(2) Pion(1)+Pion(2) Pion(3) Pex
Pabs];

N(5:9)

163

BIBLIOGRAPHY

164

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