modeling of wave heated discharges used in plasma
TRANSCRIPT
MODELING OF WAVE HEATED DISCHARGES USED INPLASMA PROCESSING REACTORS
BY
RONALD LEONEL KINDER OXOM
B.S. University of Illinois at Urbana-Champaign, 1997B.S. University of Illinois at Urbana-Champaign, 1997M.S. University of Illinois at Urbana-Champaign, 1998
THESIS
Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Nuclear Engineering
in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2001
Urbana, Illinois
ii
RED BORDER FORM
iii
MODELING OF WAVE HEATED DISCHARGES USED IN PLASMA PROCESSINGREACTORS
Ronald Leonel Kinder Oxom, Ph.D.Department of Nuclear Engineering
University of Illinois at Urbana-Champaign, 2001Mark J. Kushner, Adviser
Magnetically enhanced inductively coupled plasma (MEICP) and helicon plasma
sources typically have a higher plasma density for a given power deposition than
conventional inductively coupled plasma (ICP) sources. In industrial plasma sources
where magnetic fields typically span a large range of values and modes are likely not to
be pure, power deposition likely has contributions from both non-collisional heating and
electrostatic damping. Mechanisms for power deposition and electron energy transport in
MEICPs have been computationally investigated using 2-D and 3-D plasma equipment
models. In the 2-D model, 3-D components of the inductively coupled electric field are
produced from an m = 0 antenna and 2-D applied magnetic fields. These fields are then
used in Monte Carlo simulations to generate electron energy distributions (EEDs),
transport coefficients, and electron impact source functions. The electrostatic component
of the wave equation is resolved by estimating the charge density using a oscillatory
perturbed electron density.
Results for process relevant gas mixtures are examined, and the dependence on
magnetic field strength and field configuration is discussed. Standing wave patterns in the
electric fields result in power deposition within the volume if the plasma. As the static
magnetic field is increased, the electric field propagation follows magnetic flux lines, and
significant power can be deposited downstream. However, the ability to deposit power
iv
downstream is limited by the wavelength of the helicon wave, which depends on the
plasma density. If the plasma is significantly electronegative in the low power–high
magnetic field regime, power deposition will resemble ICP behavior.
Collisional damping may be the dominant heating mechanism at moderate
pressures (>2 mTorr). However, at lower pressures, where resonant electrons have
velocities near the wave phase velocity, Landau damping may be an important heating
mechanism. Landau damping may occur over a broad range of energies (10-100 eV), in
contrast to earlier predictions of narrow energetic beams.
The tails of the EEDs are enhanced in the downstream region, indicating some
amount of electron trapping. This results from noncollisional heating by the axial electric
field for electrons which have long mean-free-paths. Results indicate that the effect of
the electrostatic term in Maxwell’s equations is to structure the power deposition near the
coils. At low magnetic fields, the electrostatic term and the helicon term are strongly
coupled. However, the propagation of the helicon component is little affected at large
magnetic fields where the electrostatic term is damped.
Asymmetric antennas (m = +1,-1) produce 3-D components of the electric field
lacking any significant symmetries and so must be fully resolved in 3-D. To investigate
these processes, a 3-D plasma equipment model was improved to resolve 3-D
components of the electric field produced by m = +1,-1 antennas in solenoidal magnetic
fields. For magnetic fields of 10-600 G, rotation of the electric field was observed
downstream of the antenna where significant power deposition also occurs.
v
ACKNOWLEDGEMENTS
Uno propone y Dios dispone
Gracias le doy a mi Creador por los detalles mas dimutos de mi vida. Tratare de
continuar el camino que El me ha aluminado atravez de esta oportunidad, para mejorar
las vidas to todos.
I would like to thank my adviser/friend, Prof. Mark J. Kushner for his unbounded
patience and understanding. His ability to apply the appropriate pressure on a lump of
coal, has transformed that lump into something of value.
I would like to gratefully acknowledge the National Science Foundation,
Semiconductor Research Corporation, Defense Advanced Research Projects Agency
(DARPA)/Air Force Office of Scientific Research (AFOSR), Applied Materials, Inc.,
LAM Research, Inc., Novellus, Inc. for sponsoring this work.
I am grateful to my fellow, past and present, ODP members for helping see the
light. Rajesh Dorai, Arvind Sankaran, Pramod Subramonium, Kapil Rajaraman, Vivek
Vyas, Dr. Sang-Hoon Cho, Kelly Voyles, Dr. Junqing, Lu, Dr. Da Zhang, Dr. Xudong
Xu, Dr. Eric Keiter, Dr. Shahid Rauf, Dr. Robert Hoeskstra, Dr. Michael Grapperhaus,
and Dr. Helen Hwang. I am also grateful to the people in the Prof. Ruzic’s NRL group. I
would also like to thank Dean Paul Parker, Heidi Rockwood, Barb Niepert, and the rest
of the College of Engineering staff.
A mi querida family, que aunque la distancia nos separe, vivimos juntos dia tras
dia. Papa Guayo, tia Pati, David, Ana, Mami y Papi Chavez, la family Caceros (que es
vi
imposible nombrarlos todos aqui), Carlos Ivan del uno al tres, tia Carmela, Oscar, Aldo,
Vero (y mis adoradas sobrinas/o). Tia Anabela/primas/o, tia Waleska, tia Patricia Elvira,
el pueblo Majus que se encuentran en la capital, Coban, y California, los Palencia. A la
familia Morales por permitirme compartir en lo mejor de sus vidas. A mi familia
Venegas-Pizzaro. A Carlos y Magdy y los otros verdaderos amigos de mi madre que le
han dado muchas horas del alegria y compasion.
A mis amistades y queridos amigos que han hecho mi carrera academica un poco
mas larga pero definitivamente mas alegre; Marcelo, Luis, Iggy, Pancho, Dave and Ang,
Ketan, Jean, Larry, Miguel, Carlos, Javier, Manny, Isabel, Roxy, Jason, Shawn, Tuan, the
805 associates and most who went through it, Teresa, Opa, Consuelo, and Derick Garcia.
To Jenny and Ruby for trying to help her sister keep me in line.
A mi “negra linda”, Livia, por su amor, por su soporte y por soportarme. Por las
felicidades del presente y por los suenos del futuro.
A mi adorada madre, Elvira Oxom, sacando este exito juntos … simepre. Gracias
por la vida y por darme razon para vivir.
TABLE OF CONTENTS
CHAPTER
1 INTRODUCTION..……………………………………….……………….1.1 Introduction……………….………....…………………………………….1.2 References…………………….…………………………………………...
2 TWO DIMENSIONAL MODEL DESCRIPTION.…………...……..……2.1 Hybrid Plasma Equipment Model (HPEM)..………………….…………..2.2 The ElectroMagnetics Module (EMM)…………………………..………..2.3 The Electron Energy Transport Module (EETM)..…………………..……2.4 The Electron Monte Carlo Simulation (EMCS).………………………..…2.5 The Fluid-Chemical Kinetics Module (FKM)……..………………………2.6 References...………………………………………….……………………
3 WAVE PROPAGATION AND POWER DEPOSITION INMAGNETICALLY ENHANCED INDUCTIVELY COUPLEDPLASMA AND HELICON PLASMA SYSTEMS…….…………………
3.1 Introduction……………………………………………….……………….3.2 Propagation in a Solenoidal Geometry………………………..…………...3.3 Plasma Heating and Power Deposition…………….……………………...3.4 Simulations of a Trikon Helicon Plasma Source.……..…………………...3.5 Conclusions……………………………………………….……………….3.6 References…………………………………………………….…………...
4 NONCOLLISONAL HEATING AND ELECTRON ENERGYDISTRIBUTIONS…………………………………………………………
4.1 Introduction……………………………………………………….……….4.2 Nonlocal Heating by Axial Component of the Wave………...………..…..4.3 Effects of the Electrostatic Term on Propagation and Heating……..……..4.4 Conclusions…………………………………………………………….….4.5 References…………………………………………………………….…...
5 THREE DIMENSIONAL SIMUATIONS OF WAVE HEATEDDISCHARGES…………………………………………………………….
5.1 Introduction…………………………………………………………….….5.2 Propagation of an m = +1 Mode in a Solenoidal Geometry……..…...……5.3 Results for an Experimental Helicon Tool…………………………..…….5.4 Conclusions…………………………………………………………….….5.5 References…………………………………………………………….…...
6 CONCLUSIONS AND FUTURE WORK…………………………………6.1 Conclusions………………………………………………………………..
VITA……………………………………………………………………………
PAGE
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1
1 INTRODUCTION
1.1 Introduction
As the semiconductor industry transitions to larger wafer sizes (≥300 mm) plasma
sources capable of maintaining process uniformity over large areas will be required.
Magnetically enhanced inductively coupled plasma (MEICP) and helicon sources have
been proposed as possible alternatives to conventional inductively coupled plasma (ICP)
sources due to their high ionization efficiency and their ability to deposit power within
the volume of the plasma.1-8 The location of power deposition within the process reactor
couples strongly with etch and deposition rates and uniformity at the substrate surface.
Operation of MEICP and helicon sources at low magnetic fields (< 100 G) is not only
economically attractive, but may enable greater ion flux uniformity to the substrate than
would high magnetic field devices such as electron cyclotron resonance sources since
ions are only moderately magnetized. Further benefits include the absence of particulate
formation at low pressures9 and an enhanced etch rate from negative-ion formation in the
afterglow when these sources are pulsed.10,11
Chen and Boswell, and Cheetham and Rayner have identified several modes of
operation for MEICPs: electrostatic, inductive, helicon, and high pressure helicon
modes.12-16 Inductive fields are evanescent and decrease in amplitude with a classical
skin depth, whereas helicons are basically low-frequency whistler waves (also called R-
waves) where the frequency lies between the lower hybrid frequency and the electron
cyclotron frequency and lies well below the plasma frequency.17 The basic dispersion
relation for helicon waves is the same as that for low-frequency whistlers confined to a
cylinder with an axial static magnetic field. A typical chamber configuration is shown in
2
Fig. 1.1.18 (All figures are at the end of each chapter.) The main features of a helicon
reactor are the antenna and the solenoid, which generates a constant axial magnetic field.
The antenna can have one of many configurations, but all types fit around the cylinder
and usually have azimuthal and axial components. Electromagnetic waves propagate in
the presence of a magnetostatic field, creating a higher degree of complexity in the
electric field structure than obtained from an ICP reactor. The antenna design determines
the mode of operation.
Neglecting ion current and electron mass, Maxwell’s equations then lead to the
following vector Helmholtz equation for the wave magnetic field,
B B α=×∇ , (1.1)
and
0 B B 22 =+∇ α , (1.2)
where,
o
e
B
n
koµω
α = , (1.3)
where ne, ω, µo, k, and Bo, are the electron density, angular frequency, permeability,
frequency wavenumber, and static magnetic field, respectively. In the bounded system,
kα is not a free parameter but will have eigenvalues set by the permitted angles of
propagation corresponding to different radial modes. The solution of Eqs. (1.1) and (1.2)
in a cylinder of radius is given by,
3
( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]
( )TriAJ
TrJkTrJkiA
TrJkTrJkA
m
mm
mm
2B
B
B
z
11
11r
−=−−+=
−++=
+−
+−
αααα
θ , (1.4)
where the Jm are Bessel functions and
T2 = α2 + k2. (1.5)
The boundary conditions set the possible values of α to be approximately
α ≈ pmn / a, (1.6)
where pmn is the nth root of Jm corresponding to the nth radial mode and the mth
azimuthal mode and a is the cylinder radius. The m = 0 mode is the easiest to
analytically analyze, because it only has azimuthal and radial component of the electric
field for a given axial magnetic field. Most experiments are designed for the m = ±1
modes since they couple to the plasma more efficiently than the m = 0. The m = +1 mode
is right-hand (RH) circularly polarized, and the m =–1 mode is left-hand (LH) polarized,
when viewed along Bo. The transverse electric field patterns and propagation along the
axial direction are shown in Fig. 1.2.18 The antenna design in Chapters 3 and 4, in an
axial static magnetic field, generates an m = 0 mode. However, due to radial gradients in
the static magnetic field, these modes are not pure as will be shown in Chapter 3.
Simulations using a Nagoya type III antenna, which generates an m = +1 mode, are
discussed in Chapter 5.
4
Propagation of helicon waves has been detected using magnetic probes by several
authors. For example, Boswell19 measured the axial variations of the radial and axial
magnetic fields in a solenoidal geometry, shown in Fig. 1.3. The most obvious feature is
the appearance of amplitude modulations caused by standing waves due to reflections
from the endplates. However the ability to propagate away from the antenna is strongly
influenced by the strength and the configuration of the static magnetic fields. Arnush and
Peskoff20 conducted theoretical investigations for helicon waves incident into a region
where the bounding magnetic field lines form a parabola. The axial magnetic field as a
function of axial position for varying static magnetic field divergence is shown in Fig.
1.4. For a strongly divergent static magnetic field (b = 10), the axial magnetic field is
strongly damped, whereas for a less divergent static magnetic field (b = 40), the axial
magnetic field propagates further away from the antenna source.
It is often observed in helicon discharges that the density takes one or two sharp
jumps as either the rf power or the magnetic field is increased, as shown in Fig. 1.5.2
These “jumps” are attributed to transitions from capacitively coupled to ICP to helicon
mode. Degeling et al. 21 have suggested that the transition from an ICP to a helicon mode
occurs as a result of a positive feedback as the skin depth increases to the scale length of
the system. However, the “jumps” may in fact be a result of changes in the distribution
of the plasma in the vessel due to changes in the modal electromagnetic wave patterns
that in turn determine the location of the power deposition. This effect will be discussed
in greater detail in Chapter 3.
MEICP and helicon sources typically have a higher plasma density for a given
power deposition than conventional ICP sources.4,6,22,23 Chen has conducted extensive
5
experimental measurements on several types of helicon devices. An example of such a
device is shown in Fig. 1.6. Briefly stated, the plasma is contained by a 5 cm diameter,
165 cm long glass tube, surrounded by solenoidal coils that produce an axial magnetic
field. Chen and Sudit24 measured the axial distribution of ionized argon light intensity at
various magnetic fields, shown in Fig. 17. At Bo = 0 G, a faint discharge is localized to
the antenna region. Above 300 G, a helicon peak rises up on only one side of the antenna
(in the direction of propagation of the m = +1 mode). Eventually this peak grows to 20-
30 times the height of the ICP peak. The mechanisms through which more efficient
heating of electrons occurs in these systems are not well understood.
One common characteristic of many MEICP and helicon devices is their ability to
produce a maximum in the plasma density in the downstream region of processing
chambers (remote from the antenna), which implies that substantial power deposition also
occurs downstream.25,26 Computational investigations were conducted here to quantify
this heating and determine the conditions for which power can be deposited in the
downstream region of MEICP devices. For typical process conditions (10 mTorr, 1 kW
ICP) and magnetic fields above 40 G, radial and axial electric fields exhibit nodal
structure consistent with helicon behavior. As the magnetic fields are increased, axial
standing wave patterns occur with substantial power deposition downstream. The ability
to deposit power downstream with increasing magnetic field is ultimately limited by the
increasing wavelength. For example, if the plasma is significantly electronegative in the
low power-high magnetic field regime, power deposition resembles conventional ICP due
to the helicon wavelength exceeding the reactor.
6
Carter and Khachan27 experimentally investigated an m = 0 helicon source
sustained in argon. They showed that the electric fields in the downstream region are a
superposition of higher order radial modes and their reflections from the endplate. The
on-axis ion density, measured with Langmuir probes, generally increased with increasing
distance from the antenna. For a constant power deposition, the highest ion density was
obtained at low magnetic fields (~15 G) where the phase velocity of the electromagnetic
wave was commensurate with the thermal speed of electrons. They also found that the
ion density, peaking on axis at low magnetic fields, had off-axis peaks at high magnetic
fields. Borg and Kamenski28 showed that for low magnetic field plasma sources, the
response of the electron energy distribution (EED) near the antenna indicated a strong a
wave-electron interaction. They proposed that for helicon-wave-driven plasma sources,
the dominant collisionless wave-particle interaction mechanism is electron acceleration
by the parallel component of the electric field. The heating of electrons by wave
damping is dominant in the far-field.
Landau damping has been proposed as a mechanism through which more efficient
heating may occur.29 In this process, energetic primary electrons are produced through
trapping and acceleration by a helicon wave. The electrons produce ionizations, lowering
their energy and generating a low energy secondary. The wave reaccelerates electrons
after each ionization event. Gui and Scharer30 performed simulations of electron
trajectories in an m = +1 helicon plasma source sustained in argon. They found that
trapped electrons appeared as the magnetic field amplitude increased. The EED
displayed a bunching of particles with energies higher than the ionization potential of the
gas. Recent measurements of the phase of the optical emission from high-lying, short-
7
lived excited states of Ar+ in an m = +1 helicon source showed the measurements to be
well correlated with the phase velocity of the helicon wave.31 Simulations by Degeling
and Boswell32 demonstrated that in an argon plasma maxima in the ionization rate
traveled away from the antenna at the phase velocity of the wave. This implies that
resonant electrons are trapped in the wave reference frame. The ionization rate was
highest when the phase velocity of the wave was 2-3 × 108 cm-s-1. This value is
commensurate with the thermal speed of electrons that have energies just above the
ionization threshold. These observations support the proposal that electron trapping in
the axial electric field is the underlying mechanism that drives high density,
collisionlessly heated plasmas.
Additional support comes from measurements by Molvik33 et al. who, using an
electron energy analyzer, demonstrated that electrons having energies above the plasma
potential are correlated with the rf phase. Their results showed that transit-time heating
of electrons is sufficient to account for enhancements of the tail of the EED. They
observed energy deposition downstream of the antenna if the phase velocity of the axial
wave is in the range corresponding to electron energies of about 25 eV. Collisions have a
strong effect on reducing the wave-particle phase coherence, and so these effects should
be less pronounced at higher pressures.
In more recent work, Chen and Blackwell34 found that there may be too few
phased fast electrons to account for the majority of the ionization that occurs through
Landau damping. However, Chen and Blackwell could not rule out heating by nonlinear
processes under and near the antenna at low electron densities, as advocated by
others.35,36 Indeed, fast electrons oscillating in standing waves under the antenna or
8
injected into the plasma just past the antenna are subjected to beam-plasma instabilities
and could thermalize rapidly. More recently, Kwak37 suggested that much of the electron
heating comes from an electrostatic component of the helicon wave.
When a finite electron mass is taken into account in a cold plasma model, another
solution to the wave equation appears in bounded geometries at frequencies above the
lower hybrid. This is referred to as the electrostatic Trivelpiece-Gould (TG) and was
identified by Trivelpiece and Gould38 as the cavity eigenmode of a cold plasma, space
charge wave in a cylinder. Being nearly electrostatic and of short radial wavelength,
these waves are strongly absorbed as they propagate perpendicular to the externally
applied static magnetic field lines. The electrostatic TG wave can be resolved by
including a finite electron mass in Maxwell’s dispersion equation
Including the effect of finite electron mass, the wave magnetic field for helicon
waves follows the equation
( )0 B B B =+×∇−×∇×∇
+kk
i
c
αω
νω, (1.7)
where ωc and ν are the cyclotron frequency and the effective collision frequency. The
general solution is 21 B B B += , where B1 and B2 satisfy,
0 B B 111 =+∇ 22 β , (1.8)
and
0 B B 222 =+∇ 22 β , (1.9)
9
and β1 and β2 are the solutions of,
( )
0 2 =+−
+kk
i
c
αββω
νω, (1.10)
namely,
( )( )
+−
+=
2
1
22,1
411
2 c
c
k
ik
i
k
ωνωα
νωω
β m . (1.11)
The upper sign gives the helicon (H) branch β1 and the lower sign the Trivelpiece-
Gould (TG) branch β2. The nature of the normal modes can be seen by neglecting the
effective collision frequency. Propagating modes then require
skkk δ2 min ≡> , (1.12)
where δ = ω/ωc and ks = ωp/c, and where ωp and c are the plasma frequency and the
speed of light, respectively. For a uniform plasma of radius a, solutions to Eq. (1.7) can
be expressed in terms of Bessel functions Jm previously described, where T is given by,
Tj2 = βj
2 - k2, j = 1, 2, 3,… (1.13)
Real Tj requires k2 < βj2 and Eq. (1.12) then gives
10
skkk2
1
max 1
−≡≤
δδ
. (1.14)
At high magnetic fields, the H and TG branches are well separated, with β2 >> β1,
showing that the TG mode has a short radial wavelength. The transition to an ICP
discharge occurs in a complicated way. As Bo decreases δ increases, kmin and kmax
approach each other, so that the H and TG modes are strongly coupled. For increasingly
smaller values of Bo, the H mode is evanescent, with Tj2 < 0, the Bessel functions Jm are
replaced by Im functions. The TG mode is the only propagating wave, but the coupling to
the evanescent H branch must still be included to satisfy all the boundary conditions. As
Bo is further reduced toward zero, Eq. (1.11) can be written for large δ as,
δδβ
2
41
2
1
22
2
2,1
k
k
kik
s
s +
−= m . (1.15)
Ignoring the propagating part representing the remnants of the TG mode, from Eq. (1.15),
T' 2
1 T2
1
22
2
ik
kik
s
s ≡
−≈
δ (1.16)
Here the positive square root was taken, since there is no energy source for spatial
growth in the –r direction. The helicon solutions Jm(Tr) are then replaced by Im(T′r). For
11
z >> 1, the functions Im(z) vary as ez/(2πz)½. Thus, for 1/ks << a, the wave fields decay
exponentially as,
( )
−≈
2
1
22
2
m2
1exp T's
sk
krkrI
δ. (1.17)
The first term in the square root is the usual skin depth in an inductively coupled
plasma. The second term gives the increase in penetration because of the magnetization
of the electrons, preventing them from short-circuiting the transverse electric field.
Damping of the TG mode has also been proposed for power deposition in helicon
sources.39 In this regard, it has been suggested that helicon waves deposit power by
coupling to TG waves at the radial boundary. Strongly damped electrostatic waves can
reach the plasma core at low magnetic fields, while at high fields they deposit power at
the periphery of the plasma column. Power deposition in the volume of plasmas occurs
at high magnetic fields in special antiresonance regimes when the excitation of the
electrostatic wave is suppressed.40 There is still discussion as to the influence of these
mechanisms on electron heating. Borg and Boswell41 have suggested that for conditions
where the rf frequency is near the lower hybrid frequency, the TG mode does not lead to
a significant increase in antenna coupling in a helicon plasma. However, the TG mode
may enhance wave damping due to its high amplitude electric field in the presence of
high electron collision rates.
In industrial plasma sources where magnetic fields typically span a large range of
values and modes are likely not to be pure, power deposition likely has contributions
12
from both mechanisms. For example, Mouzouris and Scharer42 proposed that the
electrostatic TG mode may dominate electron heating at low magnetic fields where
power is deposited near the edge region. At higher magnetic fields (>80 G), the
propagating helicon mode then deposits power in the core of the plasma away from the
antenna. Collisional damping may be the dominant heating mechanism at moderate
pressures (>2 mTorr) and higher densities (≥2 × 1012 cm-3). However, at lower pressures
(<2 mTorr), Landau damping may be an important heating mechanism, provided that
resonant electrons have velocities near the wave phase velocity. Landau damping may
also occur over a broad range of energies (10 - 100 eV), in contrast to earlier predictions
of narrow energetic beams.43
To investigate the coupling of the electromagnetic radiation to the plasma in
MEICPs, algorithms were developed for wave propagation in the presence of static
magnetic fields using the 2-D Hybrid Plasma Equipment Model (HPEM) and the 3-D
HPEM.44-49 Simulations were conducted on a solenoidal geometry similar to that used by
Chen and a commercial helicon source, shown in Fig. 1.8. This source uses an m = 0
antenna comprising of two rings with opposing currents. This source has been
demonstrated to give uniformity over a large area, high ion flux, and high selectivity and
anisotropy when etching silicon, dielectrics, or metals. A full tensor conductivity was
added to the ElectroMagnetics Module (EMM), which enables one to calculate 3-D
components of the inductively coupled electric field based on 2-D applied magnetostatic
fields. Electromagnetic fields were obtained by solving the 3-D wave equation. These 3-
D fields were used in the Electron Monte Carlo Simulation (EMCS) of the HPEM to
obtain EEDs as a function of position.
13
This study was divided into severals parts. In the first part, plasma neutrality was
enforced in the solution of Maxwell’s equations and so the effects of the TG mode on
plasma heating were ignored. This separation of the two heating mechanism components
is valid for the m = 0 analysis performed here. Unlike higher order modes, such as the m
= ± 1, where a 3-D coil design can generate significant electrostatic fields, it is possible
to suppress the TG mode in an m = 0 design. The purpose of these investigations was to
determine the effect of helicon heating and the ability to deposit power in the downstream
region of helicon devices. An effective collision frequency for Landau damping was also
included and is most influential in the low electron density or high magnetic field
regimes. However, it was observed that this collision frequency only has a minimal
effect on power deposition efficiency. Results for an argon plasma excited by an m = 0
mode field at 13.65 MHz shows a resonant peak in the plasma density occurring at low
magnetic fields, which is attributed to off-resonant cyclotron heating. At higher magnetic
fields (>150 G), radial and axial electric fields exhibit downstream wave patterns
consistent with helicon behavior. The results agree with experiments in which the plasma
density increases as the magnetic field is increased, an effect attributed to the onset of a
propagating helicon wave or to a change in the helicon wave eigenmode.7 The transition
from inductive coupling to helicon mode appears to occur when the fraction of the power
deposited through radial and axial fields dominates. These results will be discussed in
Chapter 4.
The second part of this investigation was to determine the effects of helicon
waves on the EED and on the ability to deposit power downstream. We found that in the
absence of the TG mode, electric field propagation progressively follows magnetic field
14
lines and significant power can indeed be deposited downstream. The tails of the EEDs
are enhanced in the downstream region, indicating some amount of electron trapping.
This results from noncollisional heating by the axial electric field for electrons having
long mean-free-paths. These electrons typically reside in the tail of the EED while low
energy electrons are more collisional.
The third part of the study focused on resolving the TG mode by including the
divergence term in the solution of the wave equation. The electrostatic term was
approximated by a harmonically driven perturbation of the electron density. Results
indicate that the effect of the TG mode is to restructure the power deposition profile near
the coils. However, the propagation of the helicon component is little affected,
particularly at large magnetic fields where the TG mode is damped. For an m = 0 mode,
the TG mode does not significantly contribute as a noncollisional heating mechanism.
These results will be discussed in Chapter 4.
Finally, improvements for helicon propagation have been incorporated into the 3-
D Hybrid Plasma Equipment Model (HPEM-3D) to investigate propagation of
asymmetric modes of operation and antenna design.50-51 A tensor conductivity was used
to couple the components while solving the wave equation in the frequency domain using
an iterative, sparse matrix technique. For magnetic fields of 10-600 G, rotation of the
electric field was observed downstream of the antenna where significant power
deposition also occurs.
15
Figure 1.1 Typical helicon configuration. Plasma is confined in a quartz cylindersurrounded by magnetic coils that produce axial magnetic fields.18
Figure 1.2 The antenna design determines the type of mode propagating in the reactor.Transverse electric fields of helicon modes at different axial positions for a (a) m = 0, (b)m = +1.18
16
Figure 1.3 Axial variations of the radial and axial magnetic fields of a propagatinghelicon wave. Phase measurements show a standing wave created by reflection from theendplates.19
Figure 1.4 Axial magnetic field of a propagating helicon wave for increasingly divergentmagnetostatic field. As the static magnetic field becomes more divergent (lowernumber), the propagating wave becomes increasingly damped.20
17
Ele
ctro
n D
ensi
ty (
1011
cm
-3) 15
10
5
00 500 1000
Magnetic Field (G)
1500
a)
Ele
ctro
n D
ensi
ty (
1011
cm
-3) 3
2
1
00 400
Power (W)
800 1200
b)
Figure 1.5 Density jumps as (a) static magnetic field is increased and (b) input rf poweris increased.2,21
18
RightHelicalAntenna
to 27.12MHz
amplifier
4.7 cm ODquartz tube
End CoilsMagnetic Field Coils
Gas Feed
End Coils
Figure 1.6 Schematic of experimental apparatus used by Chen et al.21
19
Ar+
Lig
ht In
tens
ity
300 G200 G100 G
0 G
Ar+
Lig
ht In
tens
ity
0 20
Distance from center of antenna (cm)
40 60
0 - 900 G
Coilsa)
b)
Figure 1.7 The axial distribution of ionized argon light intensity at various magneticfield. The bottom four curves in (b) are the same as those in (a); the curves are 100 Gapart.21
20
Electromagnet
Process ChamberWafer
Permanent Magnets
Antenna
Figure 1.8 Schematic of Trikon M0RI helicon source. The quartz bell jar is surroundedby electromagnets which produce a solenoidal magnetic field. The system is powered bytwo ring coils surrounding the bell jar, operating at 13.56 MHz and are 180° out of phase.
21
1.2 References
1 A. J. Perry and R. W. Boswell, Appl. Phys. Lett. 55, 148 (1989).
2 A. J. Perry, D. Vender, R. W. Boswell, J. Vac. Sci. Technol. 9, 310 (1991).
3 N. Jiwari, H. Iwasawa, A. Narai, H. Sakaue, H. Shindo, T. Shoji and Y. Horike, JpnJ. Appl. Phys. 32, 3019 (1993).
4 G. Perry, D. Vender, R. W. Boswell, J. Vac. Sci. Technol. 9, 310 (1991).
5 N. Jiwari, H. Iwasawa, A. Narai, H. Sakaue, H. Shindo, T. Shoji and Y. Horike, JpnJ. Appl. Phys. 32, 3019 (1993).
6 J. E. Stevens, M. J. Sowa and J. L. Cecchi, J. Vac. Sci. Technol. A 13, 2476 (1995).
7 G. R. Tynan, A. D. Bailey III, G. A. Campbel, R. Charatan, A. de Chambrier, G.Gibson, D. J. Hemker, K. Jones, A. Kuthi, C. Lee, T. Shoji and M. Wilcoxson, J.Vac. Sci. Technol. A 15, 2885 (1997).
8 F. F. Chen, X. Jiang, and J. Evans, J. Vac. Sci. Technol. 18, 2108 (2000).
9 G. S. Selwyn and A. D. Bailey III, J. Vac. Sci. Technol. A 14, 649 (1996).
10 R. W. Boswell and R. K. Porteous, J. Appl. Phys. 62, 3123 (1989).
11 T. Mieno, T. Kamo, D. Hayashi, T. Shoji, and K. Kadota, Appl. Phys. Lett. 69, 617(1996).
12 R. W. Boswell and F. F. Chen, IEEE Trans. Plasma Sci. 25, 1229 (1997).
13 F. F. Chen and R. W. Boswell, IEEE Trans. Plasma Sci. 25, 1245 (1997).
14 F. F. Chen, Phys. Plasm. 5, 1239 (1998).
15 D. Cheetham and J. P. Rayner, J. Vac. Sci. Technol. A 16, 2777 (1998).
16 J. P. Rayner and A. D. Cheetham, Plasma Sour. Sci. Technol. 8, 79 (1999).
17 F. F. Chen, Plasma Physics and Controlled Fusion (Plenum Press, New York, 1984).
18 M. A. Liberman and A. J. Lichtenberg, Principles of Plasma Discharges andMaterials Processing, (John Wiley & Sons, Inc., New York, 1994).
19 R. W. Boswell, Plasma Phys. Control. Fusion 26, 1147 (1984).
22
20 D. Arnush and A. Peskoff, Institute of Plasma and Fusion Research, PPG 1538,UCLA, (1995).
21 A. Degeling, N. Mikhelson, R. W. Boswell and N. Sageghi, Phys. Plasma 5, 572(1998).
22 S. S. Kim, C. S. Chang, N. S. Yoon and Ki-Woong Whang, Phys. Plasma 6, 2926(1999).
23 K. N. Ostrikov, S. Xu and M. Y. Yu, J. Appl. Phys. 88, 2268 (2000).
24 I. D. Sudit and F. F. Chen, Plasma Sour. Sci. Technol. 4, 43 (1996).
25 D. G. Milak and F. F. Chen, Plasma Sour. Sci. Technol. 7, 61 (1998).
26 S. Yun, K. Taylor and G. R. Tynan, Phys. Plasma 7, 3448 (2000).
27 C. Carter and J Khachan, Plasma Sour. Sci. Technol. 8, 432 (1999).
28 G. Borg and I. Kamenski, Phys. Plasma 4, 529 (1997).
29 F. F. Chen, Plasma Phys. Control. Fusion 33, 339 (1991).
30 H. Gui and J. E. Scharer, 1999 IEEE International Conference on Plasma Science,141 (1999).
31 A. Degeling, J. E. Scharer, and R. W. Boswell 2000 IEEE International Conferenceon Plasma Science, 226 (2000).
32 A. Degeling and R. Boswell, Phys. Plasma 4, 2748 (1997).
33 A. Molvik, T. Rognlien, J. Byers, R. Cohen, A. Ellingboe, E. Hooper, H. McLean, B.Stallard, and P. Vitello, J. Vac. Sci. Technol. 14, 984 (1996).
34 F. F. Chen and D. D. Blackwell, Phys. Rev. Lett. 82, 2677 (1999).
35 A. Ellingboe, R. Boswell, J. Booth, and N. Sadeghi, Phys. Plasma 2, 1807 (1995).
36 A. Degeling, C. Jung, R. Boswell, and A. Ellingboe, Phys. Plasma 3, 2788 (1996).
37 J. G. Kwak, Phys. Plasma 4, 1463 (1997).
38 W. Trivelpiece and R. W. Gould, J. Appl. Phys. 30, 1784 (1959).
39 F. F. Chen and D. Arnush, Phys. Plasma 4, 3411 (1997).
23
40 K. P. Shamrai and V. B. Taranov, Plasma Sour. Sci. Technol. 5, 474 (1996).
41 G. G. Borg and R. W. Boswell, Phys. Plasma 5, 564 (1998).
42 Y. Mouzouris and J. Scharer, Phys. Plasma 5, 4253 (1998).
43 D. Arnush, Phys. Plasma 4, 2748 (1997).
44 M. J. Grapperhaus and M. J. Kushner, J. Appl. Phys. 81, 569 (1997).
45 S. Rauf and M. J. Kushner, J. Appl. Phys. 81, 5966 (1997).
46 M. J. Grapperhaus, Z. Krivokapic and M. J. Kushner, J. Appl. Phys. 83, 35 (1998).
47 J. Lu and M. J. Kushner, J. Appl. Phys. 89, 878 (2000).
48 R. Kinder and M. Kushner, J. Vac. Sci. Technol. A 19, 76 (2001).
49 D. Zhang and M. J. Kushner, J. Vac. Sci. Technol. A 19, 524 (2001).
50 M. J. Kushner, W. Z. Collison, M. J. Grapperhaus, J. P. Holland, and M. S. Barnes,J. Appl. Phys. 80, 1337 (1996).
51 M. J. Kushner, J. Appl. Phys. 82, 5312 (1997).
24
2 TWO DIMENSIONAL MODEL DESCRIPTION
2.1 Hybrid Plasma Equipment Model (HPEM)
The HPEM is a 2-D, plasma equipment model developed at the University of
Illinois.1-7 The HPEM can model complex reactor geometries and a wide variety of
operating conditions. The HPEM allows for a variety of plasma heating sources and gas
chemistries. The base 2-D HPEM consists of an electromagnetic module (EMM), an
electron energy transport module (EETM), an electron Monte Carlo simulation (EMCS),
and a fluid kinetics module (FKM). Electromagnetic fields and corresponding phases are
calculated in the EMM. Specifics on the EMM module will be discussed in Section 2.2
Electromagnetic fields calculated in the EMM are used in the EETM to generate electron
energy distribution functions as a function of position and phase. Methods of
determining the electron distribution function will be discussed in Section 2.3. The
electron distribution functions are used to generate sources for electron impact processes
and electron transport coefficients. Parameters determined in the EETM are transferred
to the FKM where momentum and continuity equations are solved for all heavy particles.
Transport properties and EEDs can also be obtained from the EMCS, described in
Section 2.4. A drift diffusion formulation is used for electrons to enable an implicit
solution of Poisson's equation for the electric potential. The FKM solves for species
densities and fluxes. Details of the FKM will be discussed in Section 2.5. The species
densities and electrostatic fields produced in the FKM are transferred to the EETM and
the EMM. These modules are iterated until a converged solution is obtained. A
flowchart of the HPEM is shown in Fig. 2.1. Note the HPEM has numerous other
modules that are described in greater detail elsewhere.1-7
25
2.2 The ElectroMagnetics Module (EMM)
The EMM portion of the plasma model was improved to resolve 3-D components
of the inductively coupled electric field based on 2-D applied magnetostatic fields and the
azimuthal antenna currents. The results discussed here are for a 2-D (r,z) azimuthally
symmetric geometry. The fluid equations for continuity, momentum, and energy
transport are therefore solved in 2-D. However, given azimuthal antenna currents and
(r,z) magnetostatic fields, all three components of the inductively coupled electric field
(r,θ,z) are generated, and we therefore solve for these three components. Local power
deposition is computed in 2-D from ( ) EJEJrP ⋅=⋅= σ, , where EJ and , ,σ are the 3-
D current density, tensor conductivity (see below), and electric field, respectively. This
2-D power deposition is then used in the electron energy equation to obtain the electron
temperature, source functions and transport coefficients. Previously, plasma neutrality
was enforced when solving the wave equation. However, in order to resolve the TG
mode, the electrostatic term in the wave equation must be taken into account. The
electromagnetic fields E are obtained by solving the following form of the wave
equation,
( )EJiEEE ⋅+=+
∇⋅∇−
⋅∇∇ σωεω
µµ
1
1 2 2 , (2.1)
where J , ω, ε, µ, and σ are the external antenna current density, angular
electromagnetic frequency, permittivity, permeability, and tensor conductivity,
26
respectively. The ion current in solution of Eq. (2.1) is ignored due to the low mobility of
ions. The conduction current is addressed by a warm plasma tensor described in Ref. 4.
The leading divergence term in Eq. (2.1) is included by using a perturbation form
of Poisson’s equation. For a quasi-neutral plasma, neglecting ion mobility over the rf
cycle, the divergence of the electric field is equal to the perturbation in the electron
density from neutrality, defined as,
εεεερ eee
jj
jnqnqqnqNqN
nq
E∆
=∆+++
===⋅∇ −+∑
(2.2)
where ρ, nj, N+, N-, ne, and ∆ne are the charge density, density of the jth charge species,
total positive charge density, total negative charge density, unperturbed electron density,
and perturbation to the electron density, respectively. Substituting Eq. 2.2 into Eq. 2.1
gives,
( ) ( )∑ ∆∇−⋅+−=
∇⋅∇−
jjj nqEJiEE
o
2 1
1
µεσωεω
µ2 . (2.3)
The electrostatic term appears as a source term in the wave equation. On the time
scale of the electromagnetic field, the total electron density ne(t) is the sum of the steady
state electron density ne and the perturbed electron density ∆neexp(iωt):
( ) ( )( ) ( )tinwitinntt
tneee
e ωωω expexp ∆=∆+∂∂
=∂
∂ (2.4)
27
The magnitude of the perturbed electron density is obtained by solving the
continuity equation for the electron density, with an appropriate damping term:
( ) ( )τ
ee
e nn
t
tn ∆−⋅∇−=
∂∂
v , (2.5)
τσ
ω ee
n
q
Eni
∆−
⋅⋅∇−=∆ , (2.6)
+
⋅⋅∇−
=∆ω
τ
σ
i
q
E
ne
1, (2.7)
where v and τ are the electron velocity and damping factor, respectively. The damping
factor takes into account the average time a perturbed electron returns to the steady state.
The propagation of the electrostatic wave perpendicular to the static magnetic field lines
it is limited by a factor proportional to the cyclotron frequency and the plasma frequency.
The current density has contributions from both the external antenna current and
the conduction current generated in the plasma due to the electromagnetic wave. The
conduction current is addressed by a cold plasma tensor:
++−++++−+−++
+=
2z
2rr
r22
rz
rrz2
r2
22
e
B BB BBB B
BB BB BB B
BB BBB BB
B
1qn
ααααααααα
αασ
θθ
θθθ
θθ
zz
z
z
(2.8)
and,
28
( )ωνα i q
m e
e += , (2.9)
where σ , q, ne, B , Br, Bθ, Bz, me, and νe are, respectively, the conductivity tensor,
electron charge, electron density, total magnetic field intensity, radial, azimuthal and
axial magnetic field components, electron mass, and effective electron momentum
transfer collision frequency. An analogous full tensor mobility is used for electron
transport in the EETM and FKM, as in the following sections. An analogous full tensor
mobility is used for electron transport in the electron energy equation option EETM and
in the FKM.
The procedure results in 2-D partial differential equations from Eq. (2.3)
discritized in the form shown in Equations (2.10), (2.11), and (2.12). In the initial
implementation of the HPEM, the EMM solved the matrix equation via successive-over-
relaxation (SOR). Unfortunately, this proved to be unreliable for problems that included
a static magnetic field. Without the magnetic field, the solution for the electric field
always resulted in a wave that was strongly attenuated and had a wavelength that was
much longer than the reactor dimension. The addition of the magnetic field allowed for
very short wavelengths in the electric field and possibly less absorption. The effect this
has on the matrix equation is to render it an ill-conditioned problem, which essentially
means that a small change in the input parameters will result in a relatively large change
in the answer. For a problem of this size, a direct solver is impractical, so iterative
methods were the only ones practically available for this problem:
29
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( ) ( )r
o
jiejiejizjijirjir
jiji
ji
jiji
jirjir
jiji
i
jiji
jirjir
jiji
ji
jiji
jirjir
jiji
ji
jiji
jirjir
Jir
nnqEEEiE
zrrrr
zrr
z
EE
zrrrr
zrr
z
EE
zrrrr
zrr
r
EE
zrrrr
zrr
r
EE
ωµε
σσσωεω
π
πµµ
π
πµµ
π
πµµ
π
πµµ
θ −=∆
∆−∆++++−
∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
+∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
−+
−
−
+
+
−
−
+
+
2
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
),1(),1(),(13),(12),(11),(
2
2),(
2),(
),(
)1,(),(
)1,(),(
2),(
2),(
)(
),()1,(
),()1,(
2),(
2),(
),(
),1(),(
),1(),(
2),(
2),(
),(
),(),1(
),(),1(
(2.10)
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( ) θθθ
θθ
θθ
θθ
θθ
ωσσσωεω
π
πµµ
π
πµµ
π
πµµ
π
πµµ
JiEEEiE
zrrrr
zrr
z
EE
zrrrr
zrr
z
EE
zrrrr
zrr
r
EE
zrrrr
zrr
r
EE
jizjijirji
jiji
ji
jiji
jiji
jiji
i
jiji
jiji
jiji
ji
jiji
jiji
jiji
ji
jiji
jiji
−=+++−
∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
+∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
−
−
+
+
−
−
+
+
),(23),(22),(21),(2
2),(
2),(
),(
)1,(),(
)1,(),(
2),(
2),(
)(
),()1,(
),()1,(
2),(
2),(
),(
),1(),(
),1(),(
2),(
2),(
),(
),(),1(
),(),1(
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
(2.11)
30
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( ) ( )z
o
jiejiejizjijirjiz
jiji
ji
jiji
jizjiz
jiji
i
jiji
jizjiz
jiji
ji
jiji
jizjiz
jiji
ji
jiji
jizjiz
Jiz
nnqEEEiE
zrrrr
zrr
z
EE
zrrrr
zrr
z
EE
zrrrr
zrr
r
EE
zrrrr
zrr
r
EE
ωµε
σσσωεω
π
πµµ
π
πµµ
π
πµµ
π
πµµ
θ −=∆
∆−∆++++−
∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
+∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
−+
−
−
+
+
−
−
+
+
2
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
)1,()1,(),(33),(32),(31),(
2
2),(
2),(
),(
)1,(),(
)1,(),(
2),(
2),(
)(
),()1,(
),()1,(
2),(
2),(
),(
),1(),(
),1(),(
2),(
2),(
),(
),(),1(
),(),1(
(2.12)
where Er(i,j), Eθ(i,j), and Ez(i,j) are the radial, azimuthal, and axial components of the electric
field, and Jr(i,j), Jθ(i,j), and Jz(i,j) are the radial, azimthal, and axial external currents at
positions i and j. r(i,j), ∆ne(i,j), ∆r, and ∆z, are the radial position and electron perturbation
at position i and j and the radial and axial distance of the computational cell, respectively.
The terms ε, µ, ω, and σ are the permitivitty, permeability, input angular frequency, and
the tensor conductivity as defined by Eq. (2.8). Equations (2.10)-(2.12) can be reduced to
( ) ( )0
2 ),1(),1(
),(13),(12
)1,()1,(),(),1(),1(
=∆
∆−∆+++
⋅+⋅+⋅−⋅+⋅
−+
−+−+
r
nnqEEi
EDFACECFACEGFACEBFACEAFAC
o
jiejiejizji
jirjirjirjirjir
µεσσω θ
(2.13)
( ) HFACEEi
EDFACECFACEGFACEBFACEAFAC
jizjir
jijijijiji
−=++
⋅+⋅+⋅−⋅+⋅ −+−+
),(23),(21
)1,()1,(),(),1(),1(
σσωθθθθθ
(2.14)
31
( ) ( )0
2 )1,()1,(
),(32),(31
)1,()1,(),(),1(),1(
=∆
∆−∆+++
⋅+⋅+⋅−⋅+⋅
−+
−+−+
z
nnqEEi
EDFACECFACEGFACEBFACEAFAC
o
jiejiejijir
jizjizjizjizjiz
µεσσω θ
(2.15)
Equations (2.13), (2.14), and (2.15) are solved for the electromagnetic fields using a
sparse matrix conjugate gradient method.8 The previous equations are written in matrix
form in Eq. (2.16):
-G’A’
B’C’
D’
=
AFAC-GFACθ
BFACCFAC
DFAC
-G’A’
B’C’
D’
-G’A’
B’C’
D’
AFAC-GFACR
BFACCFAC
DFAC
AFAC-GFACZ
BFACCFAC
DFAC
iωσ21 iωσ23
iωσ12 iωσ13
iωσ32iωσ31
-1
q/2∆r
-q/2∆r
-q/2∆z
q/2∆z
Er(i,j)
Er(i-1,j)
Er(i,j-1)
Er(i+1,j)
Er(i,j+1)
Eθ(i,j)
Eθ(i-1,j)
Eθ(i,j-1)
Eθ(i+1,j)
Eθ(i,j+1)
Ez(i,j)
Ez(i-1,j)
Ez(i,j-1)
Ez(i+1,j)
Ez(i,j+1)
ne(i,j)
ne(i-1,j)
ne(i,j-1)
ne(i+1,j)
ne(i,j+1)
0
iωJθ
0
0
(2.16)
32
where
(2.31) '
(2.30) '
(2.29) '
(2.28) ))()(('
(2.27) '
(2.26) '
(2.25) '
(2.24) '
(2.23) ))()(('
(2.22) '
(2.21) '
(2.20) '
(2.19) '
(2.18) ))()(('
(2.17) '
133
133
131
13331
131
132
132
112
13212
112
113
113
111
11311
111
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
⋅⋅⋅−=
⋅⋅⋅=
⋅⋅⋅−=
⋅⋅−+−=
⋅⋅⋅=
⋅⋅⋅−=
⋅⋅⋅=
⋅⋅⋅−=
⋅⋅−+−=
⋅⋅⋅=
⋅⋅⋅−=
⋅⋅⋅=
⋅⋅⋅−=
⋅⋅−+−=
⋅⋅⋅=
MnDFACO
MnCFACN
MnBFACM
MnDFACCFACBFACAFACL
MnAFACK
MnDFACJ
MnCFACI
MnBFACH
MnDFACCFACBFACAFACG
MnAFACF
MnDFACE
MnCFACD
MnBFACC
MnDFACCFACBFACAFACB
MnAFACA
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
σ
σ
σ
σσ
σ
σ
σ
σ
σσ
σ
σ
σ
σ
σσ
σ
and
M = α2 + B2, (2.32)
where B is the total magnetic field intensity.
2.3 The Electron-Energy Transport Module (EETM)
The EETM solves for electron impact sources and electron transport properties by
using electric and magnetic fields computed in the EMM and FKM. There are two
methods for determining these parameters. The first method determines the electron
temperature by solving the electron energy equation. The second method uses a Monte
33
Carlo simulation for electron transport to gather statistics used to generate the EED as a
function of position.
The electron energy equation method first solves the zero-order Boltzman
equation for a range of predetermined Townsend values to create a table that provides an
EED for each Townsend value. Once the EED is obtained, an average temperature
(defined as 3
2 <ε>, where <ε> is the average energy) is computed from the EED. Electron
mobility, thermal conductivity, energy loss rates due to collisions, and electron impact
rate coefficients are also determined from the EED’s.
For a plasma with weak interparticle collisions, the Boltzmann equation describes
its kinetics. The Boltzmann equation is expressed as
collision
ee
ee
e
t
ff
m
ef
t
f
=∇⋅
×+−∇⋅+
∂∂
δδ
vr)BvE(
v (2.33)
where fe = fe(t, r, v) is the electron distribution function, r∇ is the spatial gradient, v∇ is
the velocity gradient, me is the electron mass, and collision
e
t
f
δδ
represents the effect of
collisions. Results of the zero-dimensional Boltzmann equation are then used to provide
transport coefficients as a function of average electron energy to solve the electron
energy equation:
lossheatingeee PPTT −=Γ⋅∇+∇∇ )(κ , (2.34)
34
where κ is the thermal conductivity, Te is the electron temperature, Γe is the electron flux,
Pheating is the electron heating due to deposition, and Ploss is the power loss due to inelastic
collisions. Pheating is computed from the time averaged value of J ⋅ E , where J is the
electron current obtained from the FKM, and E is the electric field due to both
inductively and capacitively coupled effects. Equation (2.34) is discretized and solved by
successive-over-relaxation (SOR), with transport coefficients updated based on the local
electron temperature.6
2.4 The Electron Monte Carlo Simulation (EMCS)
Electron transport properties and EEDs are obtained from the EMCS. The EMCS
integrates electron trajectories from electric and magnetic fields obtained using the EMM
and FKM, and employs Monte Carlo techniques for collisions. The electrons are initially
given a Maxwellian EED and placed in the reactor using a distribution weighted by the
local electron density obtained from the FKM. Pseudoparticle trajectories are advanced
using the Lorentz equation,
( )B v + Ev
×e
e
m
q
dt
d = , (2.35)
and v =dt
rd, (2.36)
where v, E, and B are the electron velocity, local electric field, and magnetic field
respectively. Equations. (2.35) and (2.36) are updated using an implicit integration
technique that enables a single timestep to span a large fraction of the cyclotron period.3
35
The range of electron energies of interest is divided into discrete energy bins. Energy
bins have constant widths over a specified energy range to simplify gathering statistical
data while resolving structure in electron impact cross sections. Typically 300-500 total
bins are used with energy ranges (100 bins/range depending on the chemistry) of 0-5, 5-
15, 15-50, and 50-200 eV. Within an energy bin, the total collision frequency νi is
computed by summing all the possible collisions:
νε
σii
eijk j
j,k
2
m N=
∑
1
2
, (2.37)
where εi is the average energy within the bin, σijk is the cross section at energy i for
species j and collision process k, and Nj is the number density of species j. Null collision
cross sections are employed over the larger energy ranges to provide a constant collision
frequency. The integration time-step for an electron in a given energy range is then the
minimum of the randomly chosen free flight time, τ = iν
1 − ln(r), the time required to cross
a specified fraction of the local computational cell, a specified fraction of the rf period,
and a specified fraction of the local cyclotron period. Here, r, is a random number
distributed on (0, 1). Psuedoparticles are allowed to diverge in time until they reach a
specified future time. When a psuedoparticle reaches that time, it is no longer advanced
until all other particles catch up. After the free-flight time, the type of collision is
determined by choosing a random number. Should the selected collision be null, the
pseudoparticle proceeds unhindered. For a real collision, additional random numbers are
chosen to determine the type of collision that occurs (and hence the electron energy loss)
36
and the scattering angles. The final velocity is then determined by applying the scattering
matrix
( )( )( )θαφθα
ϕθβθαβφθαβϕθβθαβφθαβ
coscoscossinsin
sinsincoscossinsincossincossin
sinsinsincossincoscossincoscos
⋅+⋅⋅−⋅=
⋅⋅+⋅⋅+⋅⋅⋅⋅=⋅⋅−⋅⋅+⋅⋅⋅⋅=
VV
VV
VV
z
y
x
(2.38)
where α and β are the polar and azimuthal Eularian angles prior to the collision; θ and φ
are the polar and azimuthal scattering angles, and V is the electron speed after the
collision. Assuming azimuthal symmetry for the collision, φ is randomly chosen from the
interval (0, 2π). Unless experimental data is available, θ is chosen by specifying a
scattering parameter γ where the polar scattering probability is given by cosγ(θ/2), where
γ = 0 provides for isotropic scattering and γ >> 1 provides for forward scattering. The
randomly selected scattering angle is then
( )[ ]
+− −= γθ 2
11 1cos2 r (2.39)
where r is a random number distributed (0, 1).
Statistics are collected for every particle on every time step. The particles are
binned by energy and location with a weighting proportional to the product of the number
of electrons each psuedoparticle represents and the last time step. Particle trajectories are
integrated for ≈ 100 rf cycles for each call of the EETM. Statistics are typically gathered
37
for only the latter two-thirds of those cycles to allow transients which occur at the
beginning of each iteration to damp out.
At the end of a given iteration, the EED at each spatial location is obtained by
normalizing the statistics such that
( ) ( )∑∑ =∆=i
iiii
i fF 1 r r 2
1εε , (2.40)
where Fi ( )r is the sum of the psuedoparticles’ weightings at r for energy bin i having
energy εi, fi ( )r (eV-3/2) is the EED at r , and ∆εi is the bin width. Electron impact rate
coefficients for process j at location r are determined by convolving the EED with the
process cross section
( ) ( ) ( ) jjje
j
jjj m
fk εεσε
ε ∆
= ∑
2
1
2
1
j
2 r r , (2.41)
where σj is the energy dependent cross section for process j. Source functions for
electron impact processes (or more properly collision frequency per atom or molecule)
are then generated for the current iteration l of the HPEM by
( ) ( ) ( )rr r 1e
−= lj
lj nkS , (2.42)
38
where nel-1 ( )r is the electron density obtained from the FKM in the previous iteration.
The source functions which are actually transferred to the FKM, ljoS , may be back
averaged over previous iterations
( ) 1 1−+−= lj
lj
ljo SSS αα , (2.43)
followed by ljo
lj SS = ,where α is a back averaging coefficient. Typically α ≈ 0.3-0.5.
2.5 The Fluid-Chemical Kinetics Model (FKM)
The fluid continuity, momentum and energy equations are time integrated in the
FKM to provide species densities, fluxes and temperatures, and Poisson's equation is
solved for the electrostatic potential. Electron transport coefficients and electron impact
sources are obtained from the EETM. The species densities are derived from the
continuity equation,
( ) HFACEEi
EDFACECFACEGFACEBFACEAFAC
jizjir
jijijijiji
−=++
⋅+⋅+⋅−⋅+⋅ −+−+
),(23),(21
)1,()1,(),(),1(),1(
σσωθθθθθ
, (2.44)
where Ni, Γi, and Si are the species density, flux, and source for species i. The flux for
electrons is obtained using a drift-diffusion formulation to enable a semi-implicit solution
of Poisson's equation, described below. The electron flux is given by:
39
∇−⋅=Γ e
e
eseee N
q
kTENq rr
µe , (2.45)
where eµ is the electron tensor mobility having a form analogous to Eq. (2.8), Te is the
electron temperature, and Es is the electrostatic field. Fluxes for heavy particles (neutrals
and ions) are individually obtained from their momentum equations
( ) ( ) ( )
( ) ijjijij ji
j
isii
iiiiii
i
v - vNNm + m
m -
Bv ENm
q + vv N - kTN
m
1- =
t
νν
∂∂
∑⋅∇
−×+⋅∇∇Γ
i
i
, (2.46)
where Ti is the temperature, qi is the charge, v i is velocity, iν is the viscosity tensor (used
only for neutral species), and νij is the collision frequency between species i and species j.
The heavy particle temperature is determined by solving the energy equation,
( ) 222
2
)()( E
m
qNvNvP-T
t
TcN
ii
iiiiiiiii
iii
ων
νεκ
∂∂
++⋅∇−⋅∇∇⋅∇=
rr
∑±∑ −+
++j
jBijjij
ijBijjiji
ijs
ii
ii TkRNNTTkNNmm
mE
m
qN3)(32
2ν
ν, (2.47)
where ci is the heat capacity, κi is the thermal conductivity, Pi is the partial pressure, and
Rij is rate coefficient for formation of the species by collisions between heavy particles.
40
There are heating contributions for charged particles from both the electrostatic and
electromagnetic fields.
The electrostatic field is obtained from a semi-implicit solution of Poisson's
equation. The potential for use at time t + ∆t, Φ, is obtained from an estimate of the
charge density at that time which consists of the charge density ρo at time t, incremented
by the integral of the divergence of fluxes and sources over the next time interval.
−Γ⋅∇∑∆−=∆+=Φ∇⋅∇− ii
ioo qtdt
dt ρ
ρρε
ii
iee
eeee SqtNq
kTNq tq ∑∆+
∇−Φ∇−⋅⋅∇∆ µ (2.48)
The first sum is over the divergence of ion fluxes (as obtained from Eq. 2.45).
The following term accounts for the electron flux and contains the potential, thereby
providing the implicitness. The last term accounts for independent sources of charge
which result from processes such as collision, photoionization, secondary electron
emission or electron beam injection. Equation 2.48 is solved by the SOR technique.
The second method for determining the electric potential uses an ambipolar
approximation. Using this assumption, the electron density is computed assuming that the
plasma is quasi-neutral at all points. The flux conservation equation can be written, after
substituting the drift diffusion formulation,
( )∑ ∑=∇φ∇µ⋅∇i i
iiiiiiii Sq n D- nqq , (2.49)
41
where Si is the electron source function. Equation (2.4.6) can be rewritten to give a
Poisson-like equation for the electrostatic potential:
( ) ∑∑∑ +∇=
φ∇µ⋅∇
ii1
iii1
iii
21 Sq nDq nq , (2.50)
where the summation is now taken over all the charged species, including electrons. This
Poisson-like equation is discretized and solved using a SOR method. By solving for the
electrostatic potential using the ambipolar approximation the time step is only limited by
the Courant limit.
42
MATCH BOX-COIL CIRCUIT MODEL
ELECTRO- MAGNETICS
FREQUENCY DOMAIN
ELECTRO-MAGNETICS
FDTD
MAGNETO- STATICS MODULE
ELECTRONMONTE CARLO
SIMULATION
ELECTRONBEAM MODULE
ELECTRON ENERGY
EQUATION
BOLTZMANN MODULE
NON-COLLISIONAL
HEATING
ON-THE-FLY FREQUENCY
DOMAIN EXTERNALCIRCUITMODULE
PLASMACHEMISTRY
MONTE CARLOSIMULATION
MESO-SCALEMODULE
SURFACECHEMISTRY
MODULE
CONTINUITY
MOMENTUM
ENERGY
SHEATH MODULE
LONG MEANFREE PATH
(MONTE CARLO)
SIMPLE CIRCUIT MODULE
POISSON ELECTRO- STATICS
AMBIPOLAR ELECTRO- STATICS
SPUTTER MODULE
E(r,θ,z,φ)
B(r,θ,z,φ)
B(r,z)
S(r,z,φ)
Te(r,z,φ)
µ(r,z,φ)
Es(r,z,φ) N(r,z)
σ(r,z)
V(rf),V(dc)
Φ(r,z,φ)
Φ(r,z,φ)
s(r,z)
Es(r,z,φ)
S(r,z,φ)
J(r,z,φ)
ID(coils)
MONTE CARLO FEATUREPROFILE MODEL
IAD(r,z) IED(r,z)
VPEM: SENSORS, CONTROLLERS, ACTUATORS
Figure 2.1 Flowchart of Hybrid Plasma Equipment Model (HPEM).
43
2.6 References
1. M. J. Grapperhaus and M. J. Kushner, J. Appl. Phys. 81, 569 (1997).
2. S. Rauf and M. J. Kushner, J. Appl. Phys. 81, 5966 (1997).
3. M. J. Grapperhaus, Z. Krivokapic and M. J. Kushner, J. Appl. Phys. 83, 35 (1998).
4. R. Kinder and M. Kushner, J. Vac. Sci. Technol. A 17, 2421 (1999).
5. J. Lu and M. J. Kushner, J. Appl. Phys. 89, 878 (2000).
6. R. Kinder and M. Kushner, J. Vac. Sci. Technol. A 19, 76 (2001).
7. D. Zhang and M. J. Kushner, J. Vac. Sci. Technol. A 19, 524 (2001).
8. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, NumericalRecipes: The art of Scientific Computing, (Cambridge University Press, Cambridge,1986).
44
3 WAVE PROPAGATION AND POWER DEPOSITION IN MAGNETICALLY ENHANCED INDUCTIVELY COUPLED AND HELICON PLASMA SYSTEMS
3.1 Introduction
To investigate the coupling of the electromagnetic radiation to the plasma in
MEICPs, the 2-D HPEM was used to analyze the wave propagation and power deposition
in the presence of static magnetic fields. By neglecting the divergence term in the
solution of Maxwell’s equations, the effects of the electrostatic term on plasma heating
are ignored. Plasma properties were determined by solving Boltzmann’s equation
coupled to the electron energy equation. The purpose of these investigations was to
determine the effect of helicon heating and the ability to deposit power in the downstream
region of helicon devices. An effective collision frequency for Landau damping was also
included and is most influential in the low electron density or high magnetic field
regimes. However, it was observed that it has only a minimal effect on power deposition
efficiency. Results for an argon plasma excited by an m = 0 mode field at 13.65 MHz
show a resonant peak in the plasma density occurring at low magnetic fields which is
attributed to off-resonant cyclotron heating. At higher magnetic fields (>150 G), radial
and axial electric fields exhibit downstream wave patterns consistent with helicon
behavior. The results agree with experiments in which the plasma density increases as
the magnetic field is increased, an effect attributed to the onset of a propagating helicon
wave or to a change in the helicon wave eigenmode. The transition from inductive
coupling to helicon mode appears to occur when the fraction of the power deposited
through radial and axial fields dominates.
45
3.2 Propagation in a Solenoidal Geometry
Since helicon sources can have complex geometries, a solenoidal reactor was first
used as a demonstration platform and to provide validation. This geometry is
schematically shown in Fig. 3.1. The reactor is powered by a set of ring coils that are
driven at 13.56 MHz with currents 180° out of phase. Process gas is injected at the top of
a quartz tubular reactor through a shower head nozzle and flows out a pump port located
at the bottom. The reactor sits inside a solenoidal magnetic field having dominantly an
axial component with larger radial gradients near the ends of the solenoid. The base case
has operating conditions of Ar gas at 10 mTorr, 50 sccm, and a power deposition of 1
kW. The collisional processes included in the model are ionization, excitation, and
momentum transfer between electrons and neutral particles, Coulomb collisions between
electrons and ions, charge exchange collisions between ions and neutral particles, and
momentum transfer collisions among neutral particles.
The spatially dependent plasma properties are a sensitive function of magnetic
field strength and configuration. For example, azimuthal electric field amplitudes and
phases are shown in Fig. 3.1 for magnetic field intensities of 10-150 G. At 10 G the
azimuthal electric field peak near the coils at 15 V/cm and remains in an inductively
coupled mode where the amplitude decreases evanescently and is limited by the
conventional plasma skin depth. The phase distributions show a radially inward traveling
wave. (Wave propagation is perpendicular to the phase fronts.) As the magnetic field is
increased, further penetration of the azimuthal electric field into the plasma occurs. Once
the fields encounter a boundary or a counter propagating wave, a standing wave begins to
form. At 40 G, a node appears in the azimuthal field while propagation begins to occur
46
in the axial direction with an increasing axial wavelength. At 150 G there is a standing
wave pattern in the radial direction with a peak midway between the coil and the axis of
symmetry. As the static magnetic fields further increase, axial propagation of the
electromagnetic fields dominates and the wavelength increases.
The radial electric fields over the same range of magnetic fields are shown in Fig.
3.2. At 10 G, the radial electric field has weak penetration into the plasma, with a local
maximum close to the coils and a node on the axis of symmetry. As the magnetic field is
increased, further penetration occurs, and a second local maximum in the radial electric
field develops, indicating the onset of a standing wave in the radial direction. As the
magnetic field is increased further, the amplitude of the second peak increases, while that
of the first peak decreases. By 80 G, the first peak dissipates. At 10 G, propagation is
dominantly in the radial direction and is highly damped. As the static magnetic field is
increased, propagation changes from dominantly radial to dominantly axial, while the
radial electric field wavelength increases proportionally. The axial electric field, shown
in Fig. 3.3, behaves similarly. Note that the axial electric field intensity is two orders of
magnitude smaller than the radial electric field due to the smaller magnetic field gradients
in the radial direction compared to the axial direction.
The wavelength of the electromagnetic wave can be estimated from the phase
diagrams of Figs. 3.1-3.3. The axial wavelength as a function of static magnetic field
divided by the average electron density, β = B/ne, for several tube radii are shown in Fig.
3.4(a). For a magnetic field of 40 G, electron density of 1011 cm-3 and tube radius of 6
cm, the axial wavelength is approximately 10 cm. As the magnetic field increases or the
electron density decreases the axial wavelength increases. Similarly the axial wavelength
47
increases with a decrease in tube radius. These results can be numerically fitted for the
axial wavelength λz as,
0.63
3
6
)cm(
(Gauss)
)cm(
10 6.7 (cm)
×= −
ez n
B
Rλ (3.1)
where R is the radius of the tube. Using the dispersion relation for a helicon wave, an
estimate of the dependence of wavelength on plasma parameters can be obtained.1 The
axial wave number kz is proportional to the total wave number k through the dispersion
relation:
B
n q o
ezkk µω= . (3.2)
For an m = 0 mode, the radial wave number k⊥ is fixed by the tube radius,
Rk
3.83 =⊥ (3.3)
while the total wave number is defined by
222 kkk z =+⊥ . (3.4)
48
The theoretical axial wavelength for an m = 0 mode obtained by substituting Eqs. (3.3)
and (3.4) into Eq. (3.2), is shown in Fig. 3.4(b) as a function β for several tube radii. The
computed axial wavelength is roughly two thirds of the theoretical, most likely because
of the mixed mode nature of the computed wave and the finite axial extent of the plasma.
3.3 Plasma Heating and Power Deposition
Electron temperature and electron source rate are shown in Fig. 3.5 for different
solenoidal magnetic fields. At low magnetic fields (10 G) the electron temperature peaks
near the coils at 3.6 eV. As the static magnetic fields are increased, the electric field
propagates further into the volume of the plasma. The peak in the electron temperature
shifts from near the surface to the volume of the plasma. The lower peak electron
temperature at 150 G of 2.9 eV is due to a decrease of diffusional losses at higher
magnetic fields. Electron source rate is shown on the right side of Fig. 3.5. Typical
values are between 1016 and 1018 cm-3 s-1. Power deposition and electron density are
shown in Fig. 3.6 for different solenoidal magnetic fields. At 10 G, the electric fields are
still predominantly inductively coupled, with power deposition occurring near the coils
with a classical skin depth limited by the plasma conductivity. As the magnetic field is
increased, the power deposition penetrates further into the volume of the plasma, in
accordance with the electric fields shown in Figs. 3.1–3.3. At 40 G, the power deposition
displays nodal behavior reflecting the shorter wavelengths of the azimuthal and radial
electric fields. In all cases power deposition is off axis. The electron density is
maximum on axis in the low magnetic field regime. As the magnetic field is increased,
the electron density increases, reflecting a decrease in radial diffusion losses. At fields
49
larger than 150 G, the electron density is maximum off axis at the location of maximum
power deposition.
Measurements by Chen and Decker showed a peak in the plasma density in the
low magnetic field regime (20 – 60 G), showed in Fig. 3.7.2 This peak was attributed to
an electron cyclotron resonance (ECR), where the incident electromagnetic frequency is
of the order of the electron cyclotron frequency. Simulations of this low magnetic field
regime also produced a resonant peak in the plasma density in the downstream region.
This local maximum can be resolved in Fig. 3.8, below 100 G. The local maximum shifts
towards higher magnetic fields as the radius of the tube is decreased. The maxima are
similarly attributed to “off-resonant” electron cyclotron heating. The shift of the peak to
higher magnetic field results from a shift in the efficiency of depositing power to a higher
magnetic field as the effective electron collision frequency is increased. Normalized
power deposition, P = Re[J⋅⋅E*], as a function of magnetic field for several electron
collision frequencies is shown in Fig. 3.9 for a plasma density of 1012 cm-3 and ω/2π =
13.56 MHz. At νT = 107 s-1, the normalized power deposition has a resonant peak
occurring at around B = 15 G. As the effective collision frequency increases, the width of
the power deposition increases and the peak value decreases, while shifting the maximum
towards higher magnetic fields. At νT = 108 s-1, the normalized power deposition
maximum has shifted to ~ 30 G.
In a typical microwave ECR, operating at 2.45 GHz, the electron momentum
transfer collision frequency is significantly smaller than the incident electromagnetic
frequency and this off-resonant shift is unnoticeable. However, in an rf system the
collision frequency is of the order of the incident electromagnetic frequency, thereby
50
affecting the magnetic field at resonance and hence the location of the ECR zone. The
collision frequency increases due to an increase in electron temperature as the tube radius
decreases, as shown in Fig. 3.10.
There is considerable evidence that collisional absorption is too weak to account
for the high efficiency of energy deposition at low pressures.3-5 To this point the model is
strictly collisional. Noncollisional effects will be considered in Chapter 4. However, it is
instructive to consider an effective collisional damping which accounts for noncollisional
effects. The effective electron collision frequency includes a term for Landau damping,
νLD, as described by Chen,3
( )23LD exp4 ζζππν −= f (3.5)
where,
ee2
e
Tn2q
m
0µ
ωζ
B= (3.6)
where Te is the electron temperature. For ζ >> 1, νLD increases with increasing electron
density at a constant magnetic field. However, in typical helicon sources where ζ may be
less than unity, νLD can decrease with increasing electron density. The effective electron
collision frequency is then the sum of the conventional momentum transfer collision
frequency, νm, and the effective frequency for Landau damping, νe = νm + νLD.
The consequences of including collisional Landau damping on the electron
density, are shown in Fig. 3.11. The Landau damping term accounts for at most 15% of
51
the total collision frequency. The Landau damping term has a peak near β = 1010 cm-3 G-1
and has an approximate FWHM of β = 2 × 1010 cm-3 G-1. In the low electron density or
high magnetic field regimes, the increase in the effective collision frequency due to
Landau damping can shift the position of the peak power deposition (ECR resonance) to
higher magnetic fields. For example, the peak in the electron density at low magnetic
fields is removed when Landau damping is taken into account, as shown in Fig. 3.11.
The reactor averaged electron density as a function of magnetic field is shown in
Fig. 3.12. In the low magnetic field regime (<20 G), the electromagnetic fields are
inductively coupled to the plasma. There is no significant change in the electron density
as the magnetic field is increased. At 20–60 G, ECR heating begins to contribute and the
electron density increases. As the magnetic field is further increased, the power
deposited through the radial electric field begins to be comparable to the power deposited
through the azimuthal electric field. Power deposition through the radial electric fields
increases due to the higher order standing wave pattern. At higher magnetic fields, it is
expected that standing wave patterns with even higher order nodal values may appear as
eigenvalue solutions for the radial electric field. Perry and Boswell showed that the
electron density increases in “jumps” as the magnetic field and power is increased.6,7
Each density step may be attributed to the onset of a higher order standing wave
structure. Our simulations to date have not been able to resolve "jumps" in plasma
density as a function of power when considering reactor averaged densities. For
example, the electron density as function of incident power, shown in Fig. 3.13, is linear
with power, in agreement with experiments by Chen et al.2
52
3.4 Simulations of a Trikon Helicon Plasma Source
Simulations were also conducted in a geometry based on the Trikon
Technologies, Inc., Pinnacle 8000 helicon plasma source, shown in Fig. 3.14. The
simulation geometry is schematically shown in Fig. 3.15. Processing gas is injected
through a nozzle located below the electromagnets and is exhausted through a pump port
located around the outside diameter of the substrate. The quartz bell jar is surrounded by
electromagnets which produce a solenoidal magnetic field inside the bell jar with flaring
field lines in the downstream chamber region, as shown on the right side of Fig. 3.15.
The system is powered by two ring coils surrounding the bell jar. Each coil operates at
13.56 MHz and they are 180° out of phase. Base case results have operating conditions
of Ar gas at 10 mTorr, 50 sccm, and a power deposition of 1 kW. Experimental
measurements of the radial, azimuthal, and axial magnetic field profiles at z = 20 cm are
consistent with axis-symmetric helicon behavior, shown in Fig. 3.16(a).8 The calculated
radial and axial magnetic fields are shown in Fig. 3.16(b).
Parametric studies were conducted while varying the magnetic field intensity.
The cited magnetic fields are for an on axis location midway between the antennas.
Azimuthal electric field amplitudes and phases are shown in Fig. 3.17 for magnetic fields
of 20-300 G. For B = 20 G, the azimuthal electric field is inductively coupled with the
amplitude decreasing evanescently and is limited by the conventional plasma skin depth,
as shown in Fig. 3.17(a). Similar to the results for the solenoidal reactor, the phase
distributions show that a radial traveling wave dominates the propagation of the
electromagnetic fields through the bell jar region and away from the coil. As the
magnetic field is increased, further penetration of the azimuthal electric field into the
53
plasma occurs. However, as radial penetration increases, axial conductivity increases
thereby allowing the azimuthal electric field to propagate downstream, as shown in for B
= 100 G. When the propagating wave encounters a boundary condition, a standing wave
pattern of the azimuthal field in the axial direction begins to form. At 300 G, Fig.
3.17(d), there is an off axis downstream peak in the electric field. The propagation of the
electric field in the axial direction begins to dominate concurrent with an increase in the
wavelength. The distribution and propagation of the radial and axial electric fields,
shown in Fig. 3.18 and Fig. 3.19, follow a similar pattern. Initially, inductive coupling
dominates in the low magnetic field regime. Once the magnetic field is high enough
(>100 G), downstream standing wave patterns appear, and electromagnetic propagation
in the axial direction dominates. Note that the magnitude of the radial and axial fields are
of the same order, due to the diverging magnetic field lines which produce similar
gradients in the radial and axial directions.
Experimental results for the axial ion saturation in the Trikon tool are shown in
Fig. 3.20(a).8 At low static magnetic fields, the ion saturation current peaks in the bell jar
region of the reactor. As the magnetic fields are increased, the peak in the ion saturation
current and plasma density shifts downstream. The calculated ion densities follow
similar trends, shown in Fig. 3.20(b). At constant power, the computed downstream peak
in ion density decreases with increasing magnetic fields. This is due to the flaring
magnetic flux lines that cause the plasma to peak at a larger radius and produce a larger
plasma volume.
The electron density and power deposition are shown in Fig. 3.21. Power
deposition follows the electric field profiles in the reactor. At 20 G, the power deposition
54
peaks near the coils. As the magnetic fields are increased, there are nodal peaks in the
electric field in the downstream region of the reactor, thereby depositing most of the
power in the downstream region. This shift in the power deposition produces a shift in
the peak plasma density to the downstream chamber. An off axis peak in the plasma
density can be maintained at such low pressures because of the high magnetic field lines
which inhibit radial diffusion. The shift in power deposition increases the ion flux to the
substrate while, for this example, decreasing uniformity.
Simulations were also conducted using a Ar/Cl2 = 80/20 gas mixture at 10 mTorr.
The reaction chemistry for this case produces a large Cl- negative ion density that
significantly affects the power deposition and plasma distribution. The power deposition
and corresponding electron density for varying magnetic fields are shown in Fig. 3.22.
As with pure Ar, at low magnetic fields coupling is conventional ICP, and most of the
power deposition and the peak plasma density occurs in the bell jar. As the magnetic
field increases, propagation of the electromagnetic fields tends to follow magnetic fields
lines. The wavelength of the helicon-like wave increases along the direction of
propagation and a nodal structure in the electromagnetic fields develops. At 150 G most
of the power is deposited in the downstream region of the reactor. As the static magnetic
field is further increased, the power deposition shifts back to the bell jar region as does
the peak in plasma density. At 300 G the peak in the power deposition and ion density
again occurs in the bell jar region. This shift upstream occurs because of the increase in
the wavelength of the helicon-like wave as the static magnetic field is increased. At a
high enough magnetic field, the wavelength is too large to sustain a standing wave
pattern inside the chamber.
55
The wavelength of the propagating wave is proportional to the magnitude of the
static magnetic field and inversely proportional to the electron density. Noting that λz ~
β, computed β as a function of magnetic field and power are shown in Fig. 3.23 (0 < B <
300 G and 500 < P < 1500 W). The electron density for the calculation of β was at the
reference location of the magnetic field. At low power, the electron density is low due to
both a low ionization rate and a high rate of attachment to Cl2. As the power increases,
the Cl2 is more heavily dissociated resulting in a lowering of the rate of attachment and
increase in plasma density. In the low magnetic field–high power deposition regime, β is
small, and the axial wavelength is shorter than the chamber dimensions, allowing
standing waves to occur. However, as the deposition power is decreased, β and the
wavelength increase. Likewise, as the magnetic field is increased, there is an increase in
β and the wavelength. In the high magnetic field–low power regime, the wavelength is
larger than the dimensions of the chamber. Unable to sustain a standing wave pattern,
power deposition occurs near the coils in what resembles an inductively coupled mode.
The change in power deposition downstream, which occurs as a result of
changing the mode of operation, clearly affects system performance such as uniformity
and magnitude of reactive fluxes. Although the goal of this study was not to optimize
system performance such as the uniformity of ion flux, it is nevertheless instructive to
briefly examine such dependence. For example, the consequences of power deposition
and magnetic field on the uniformity of the ion flux to the substrate for the conditions of
Fig. 3.23 were quantified, using as a metric U = 100⋅(max – min)/(max + min), where
max/min are the maximum/minimum values of ion flux to the substrate. The results are
shown in Fig. 3.24. For a region of operation having wavelength of 10–75 cm, U is
56
largest, meaning less uniform. These conditions correspond to where significant electric
field propagation occurs downstream and where the power deposition occurs directly
above the substrate. Longer and shorter wavelengths have power deposition more remote
from the substrate, thereby allowing diffusion to help homogenize the fluxes.
3.5 Conclusions
Power deposition in MEICP sources was investigated using a 2-D plasma
equipment model. The purpose of these investigations was to determine the
consequences of helicon heating and the ability to deposit power in the downstream
region of MEICP devices. 3-D electromagnetic fields were obtained from 2-D
magnetostatic fields by solving the wave equation with a tensor conductivity using a
sparse matrix conjugate gradient method where plasma neutrality was enforced. An
effective collision frequency for Landau damping was included. Results for an argon
plasma in a solenoidal geometry showed the propagation of an electromagnetic wave that
exhibits properties generally described by the dispersion relation of an m = 0 mode. A
resonant peak in the downstream plasma density occurring at low magnetic fields is
attributed to off-resonant cyclotron heating. As the tube radius is decreased, this low
field maximum in plasma density shifts towards higher magnetic fields. This result can
be attributed to an increase in electron temperature and collision frequency as the tube
radius is decreased, which tends to shift the power deposition efficiency to higher
magnetic fields.
Results for a helicon plasma source sustained in Ar with an m = 0 azimuthal
antenna showed that as the magnetic field is increased to ≈100 G, electromagnetic field
57
propagation in the axial direction dominates. Once the propagating wave encounters a
boundary or intersects a counter propagating wave (as in the radial direction), standing
wave patterns may occur in the direction of propagation. Coincident in the axial
propagation, power deposition and the peak in the plasma density shifts downstream. A
downstream shift in the peak plasma density also occurs in an Ar/Cl2 gas mixture.
However, unlike the pure Ar case, the plasma density shifts back upstream at 300 G.
This shift upstream occurs due to the continual increase in the wavelength of the helicon-
like wave as the static magnetic field is increased. At a high enough magnetic field, the
wavelength is too large to sustain a standing wave pattern inside the chamber. If the
plasma is significantly electronegative in the low power-high magnetic field regime,
power deposition will resemble conventional ICP.
By setting 0=⋅∇ Er
, we ignore the consequences of the electrostatic TG mode on
plasma heating. The focus of this study is to investigate the propagation and coupling
mechanisms of the helicon component of the wave. In the next cahpter, we have
included the electrostatic component in the wave equation by approximating
oE ερ∆=⋅∇r
, where ∆ρ is a harmonically driven perturbation. As a preview, the
results of that study indicate that the effect of the electrostatic term is to restructure the
power deposition profile near the coils. However, the propagation of the helicon
component is little affected, particularly at large magnetic fields where the electrostatic is
damped.
58
Eθ Phase
Coils
Glass
Substrate
Solenoid
Radius (cm)010 10
a) b) c)
d) e)
Eθ Phase Eθ Phase
Phase(V / cm)2π
0
15
0.15
1.5 π
10 G 20 G 40 G
80 G 150 G
0
Hei
gth
(cm
)
40
20
0
Hei
gth
(cm
)
40
20
Radius (cm)010 10
Figure 3.1 Azimuthal electric field amplitude and corresponding phase in solenoidalgeometry for (a) 10 G, (b) 20 G, (c) 40 G, (d) 80 G, and (e) 150 G. The processconditions are Ar, 10 mTorr, 1 kW power deposition. Increasing magnetic field producespropagation in the axial direction.
59
ER Phase
Radius (cm)010 10
a) b) c)
d) e)
Phase(V / cm)5
0.05
0.5
2π
0
π
10 G 20 G 40 G
80 G 150 G
0
Hei
gth
(cm
)
40
20
0
Hei
gth
(cm
)
40
20
Radius (cm)010 10
ER Phase ER Phase
Figure 3.2 Radial electric field amplitude and corresponding phase in solenoidalgeometry for (a) 10 G, (b) 20 G, (c) 40 G, (d) 80 G, and (e) 150 G. The processconditions are Ar, 10 mTorr, 1 kW power deposition. Increasing magnetic field producesradial components of the electric field and propagation in the axial direction.
60
EZ Phase EZ PhaseEZ Phase
Radius (cm)010 10
a) b) c)
d) e)
Phase(V / cm)0.03
0.0003
0.003
2π
0
π
10 G 20 G 40 G
80 G 150 G
0
Hei
gth
(cm
)
40
20
0
Hei
gth
(cm
)
40
20
Radius (cm)010 10
Figure 3.3 Axial electric field amplitude and corresponding phase in solenoidal geometryfor (a) 10 G, (b) 20 G, (c) 40 G, (d) 80 G, and (e) 150 G. The process conditions are Ar,10 mTorr, 1 kW power deposition.
61
Axi
al W
avel
engt
h (c
m)
0
5
10
15
20
25
30Radius = 4 cm
6 cm
8.5 cm
1 2 3 4 5 6 7Magnetic Field / Electron Density
(G/cm-3 x 10-11)b)
a)
1 2 3 4 5 6 7Magnetic Field / Electron Density
(G/cm-3 x 10-11)
Axi
al W
avel
engt
h (c
m)
0
5
10
15
20
25
30
Radius = 4 cm
6 cm
8.5 cm
Figure 3.4 Axial wavelength of electromagnetic field as a function of static magneticfield divided by electron density. (a) Theoretical value from dispersion relation of an m =0 mode. (b) Computed value from simulations. The computed axial wavelengthresembles that of an m = 0 mode.
62
Te Se
Hei
gth
(cm
)
0
40Te Se Te Se
a) b) c)
Radius (cm)010 10
(1018 cm-3 s-1)(eV)3.6
0.36 0.01
d) e)
1.
Hei
gth
(cm
)
0
40
Figure 3.5 Electron temperature and electron source rate in solenoidal geometry for (a)10 G, (b) 20 G, (c) 40 G, (d) 80 G, and (e) 150 G. The process conditions are Ar, 10mTorr, 1 kW power deposition.
63
Power ElectronDensity
Radius (cm)010 10
a) b) c)
d) e)
[e](1012 cm-3)
Power(W/cm3)
6
0.06
0.6
7
0.07
0.7
10 G 20 G 40 G
80 G 150 G
0
Hei
gth
(cm
)
40
20
0
Hei
gth
(cm
)
40
20
Radius (cm)010 10
Power ElectronDensity Power Electron
Density
Figure 3.6 Power deposition and electron density in solenoidal geometry for (a) 10 G, (b)20 G, (c) 40 G, (d) 80 G, and (e) 150 G. The process conditions are Ar, 10 mTorr, 1 kWpower deposition.
64
0 120MAGNETIC FIELD (G)
10080604020
ELE
CT
RO
N D
EN
SIT
Y (
101
2 c
m-3
)
0
3
1
2
EXPERIMENT (Chen et al.)
MODEL
Figure 3.7 Experimental and calculated electron density as a function of static magneticfield in solenoidal geometry. Peaks at low magnetic field are attributed to off resonancecyclotron heating.
65
100 100010
Magnetic Field (G)
0
2
1
2.54.06.08.5
Radius (cm)
Ele
ctro
n D
ensi
ty (
1012
cm
-3)
Figure 3.8 On axis value of electron density at z = 8 cm for several tube radii as afunction of static magnetic field in solenoidal geometry. Peaks at low magnetic field areattributed to off resonance cyclotron heating.
66
1.0
0.5
0.00 20 40 60
Magnetic Field (G)
Nor
mal
ized
Pow
er D
epos
ition
νe = 107 s-1
108
5 x 107
Figure 3.9 Power deposition in the solenoidal geometry. Normalized power deposition(P = J⋅E*) as a function of magnetic field for several values of the effective electroncollision frequency.
67
3.50
3.25
3.00
Tube Radius (cm)
Ele
ctro
n T
empe
ratu
re (
eV)
B = 20 G
2 4 6 8 10
30 G
40 G50 G
Figure 3.10 Average electron temperature as a function of tube radius for varyingmagnetic field. The increase in the electron temperature with decreasing magnetic fieldproduces a shift in the resonant magnetic field.
68
Magnetic Field (G)20 30 40 5010
1.0
0.5
0.0
Without Landau Damping
With Landau Damping
Ele
ctro
n D
ensi
ty (
1012
cm
-3)
Figure 3.11 On axis value of electron density at z = 10 cm as a function of staticmagnetic field in solenoidal geometry with and without the collisional Landau dampingterm.
69
Magnetic Field (G)
Ele
ctro
n D
ensi
ty (
10
12 c
m-3
)
100 1000100
1
2
3
4
5
2.54.06.08.5
Radius (cm)
Figure 3.12 Average electron density as a function of static magnetic field for severaltube radii in the solenoidal geometry.
70
Power (W)
Ele
ctro
n D
ensi
ty (
1012
cm
-3)
2.5
5.0
7.5
1.0
0500 15000 1000
Experiment (Chen et al., Ref. 29)
Model
Figure 3.13 Experimental and computed average electron density in the solenoidalgeometry.
71
Figure 3.14 Commercial Trikon Technologies, Inc., Pinnacle 8000 helicon sourceplasma system used to validate simulation models.
72
Nozzle
Wafer
Bell Jar
Radius (cm)
Pump Port
Bref
20 200
Electro- magnets
Coil
Coil
Hei
ght (
cm)
50
0
25
F1 F2 F3
1
2
3
Figure 3.15 Schematic of the Trikon helicon plasma source. Streamlines representmagnetic flux lines in the reactor. Location of the reference magnetic field point is in thebell jar.
73
Figure 3.16 Experimental and calculated magnetic field profiles in Trikon heliconsource.
Bz
Br
Bθ
Experimental
0
1
MA
GN
ET
IC F
IELD
(a.
u.)
RADIUS (cm)
0
1M
AG
NE
TIC
FIE
LD (
a.u.
)
Bz
BrAr, 10 mTorr, 1kW, 20 sccm
RADIUS (cm)0
0
15
15
74
20 G
Eθ Phase
60 G
0
50
25
Hei
ght (
cm)
a)
0
50
25
Hei
ght (
cm)
20 200Radius (cm)
c)
100 G 300 G
20 200Radius (cm)
V / cmPhase
150.15 1.5
2π0 π
b)
d)
Eθ Phase
Figure 3.17 Azimuthal electric field amplitude and corresponding phase in the Trikontool for (a) 20 G, (b) 60 G, (c) 100 G, and (d) 300 G for an Ar plasma. Increasingmagnetic field produces axial propagating waves.
75
20 G
ER Phase
60 G
0
50
25
Hei
ght (
cm)
a)
0
50
25
Hei
ght (
cm)
20 200Radius (cm)
c)
100 G 300 G
20 200Radius (cm)
V / cmPhase
50.05 0.5
2π0 π
b)
d)
ER Phase
Figure 3.18 Radial electric field amplitude and corresponding phase in the Trikon toolfor (a) 20 G, (b) 60 G, (c) 100 G, and (d) 300 G for an Ar. plasma.
76
20 G
EZ Phase
60 G
0
50
25
Hei
ght (
cm)
a)
0
50
25
Hei
ght (
cm)
20 200Radius (cm)
c)
100 G 300 G
20 200Radius (cm)
V / cmPhase
10.01 0.1
2π0 π
b)
d)
EZ Phase
Figure 3.19 Axial electric field amplitude and corresponding phase in the Trikon tool for(a) 20 G, (b) 60 G, (c) 100 G, and (d) 300 G for an Ar plasma.
77
a)
b)
Ion
Sat
urat
ion
Cur
rent
(m
A/c
m2 )
0 G
25 G
Axial Position (cm)-5 0 5 10 15 20-10
50
0
10
20
30
40
200 G
DownstreamBell Jar
Nor
mal
ized
Ion
Den
sity 25 G 200 G
0 G
DownstreamBell Jar
-5 0 5 10 15 20-10Axial Position (cm)
1.0
0
0.5
Figure 3.20 Plasma properties as a function of axial location. (a) Experimental ionsaturation current for several values of the static magnetic field. (b) Computed values ofthe normalized ion density. When increasing the magnetic field, the peak in the plasmadensity shifts from the bell jar to downstream.
78
20 G
Power [e]
60 G
0
50
25
Hei
ght (
cm)
a)
0
50
25
Hei
ght (
cm)
20 200Radius (cm)
c)
100 G 300 G
20 200Radius (cm)
Power (W/cm3)
[e] (cm-3)
1.00.01 0.1
4 x 1010
b)
d)
Power [e]
4 x 1011 4 x 1012
Figure 3.21 Power deposition and electron density in the Trikon tool for (a) 20 G, (b) 60G, (c) 100 G, and (d) 300 G for an Ar plasma. Increasing magnetic field produces adownstream plasma source.
79
0 G 40 G
Hei
ght (
cm)
0
50
25
a)
20 200Radius (cm)
c)
150 G 250 G
b)
d)
Power [e] Power [e]
Hei
ght (
cm)
0
50
25
20 200Radius (cm)300 G
Hei
ght (
cm)
0
50
25
e)
Power(W/cm3)
[e](cm-3)
2
0.02
0.2
1012
1011
1010
Figure 3.22 Power deposition and electron density in the Trikon tool using an Ar/Cl2 =80/20 mixture for (a) 0 G, (b) 40 G, (c) 150 G, (d) 250 G, and (e) 300 G. At highmagnetic field, the plasma density again peaks in the bell jar due to the axial wavelengthexceeding the reactor dimensions.
80
Figure 3.23 Wavelength scaling. Static magnetic field divided by average electrondensity as a function of magnetic field and power using an Ar/Cl2 = 80/20 mixture.Wavelength scales with this quantity. The wavelength is large at high magnetic field andlow power.
500
1500
Pow
er (
W)
0
1000
100 200
Magnetic Field (G)
300
10
30
60
90
120
81
Figure 3.24 Uniformity parameter U for the ion flux to the substrate for the sameconditions. The uniformity is worse for high values of U where the power deposition isdirectly above the substrate.
500
1500
Pow
er (
W)
0
1000
100 200
Magnetic Field (G)
300
16
30
24
18 10
82
3.6 References
1. M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges andMaterials Processing (John Wiley and Sons, Inc., New York, 1994).
2. F. F. Chen, J. Vac. Sci. Technol. A 10, 1389 (1992).
3. F. F. Chen, Plasma Phys. Control. Fusion 33, 339 (1991).
4. H. Gui and J. E. Scharer, in IEEE International Conference on Plasma Science, 1999p. 141.
5. Degeling, J. E. Scharer, and R. W. Boswell, in IEEE International Conference onPlasma Science, 2000 p. 226.
6. J. Perry and R. W. Boswell, Appl. Phys. Lett. 55, 148 (1989).
7. G. Perry, D. Vender, R. W. Boswell, J. Vac. Sci. Technol. 9, 310 (1991).
8. PMT Corp., unpublished.
83
4 NONCOLLISIONAL HEATING AND ELECTRON ENERGY DISTRIBUTIONS
4.1 Introduction
To investigate the coupling of the electromagnetic waves to the plasma in MEICPs and
their effects on EEDs, the 2-D HPEM was used to solve for electromagnetic fields in the
presence of static magnetic fields. These 3-D fields were used in the Electron Monte Carlo
Simulation (EMCS) of the HPEM to obtain EEDs as a function of position.
This study was divided into two parts. In the first part, plasma neutrality was enforced in
the solution of Maxwell’s equations, ignoring the effects of the electrostatic term is on plasma
heating. This separation of the two heating mechanism components is valid for the m = 0
analysis performed here. Unlike higher-order modes, such as the m = ± 1, where a 3-d coil
design can generate significant electrostatic fields, it is possible to suppress the TG mode in an m
= 0 design. The purpose of this part of the investigation was to determine the effects of helicon
waves on the EED and on the ability to deposit power downstream of the antenna. We found
that in the absence of the electrostatic term, electric field propagation progressively follows
magnetic field lines, and significant power can indeed be deposited downstream. The tails of the
EEDs are enhanced in the downstream region, indicating some amount of electron trapping and
collisionless acceleration by the axial electric field for electrons having long mean-free-paths.
These electrons typically reside in the tail of the EED, while low energy electrons are more
collisional.
The second part of the study focused on resolving the TG mode by including the
divergence term in the solution of the wave equation. The electrostatic term was approximated
by a harmonically driven perturbation of the electron density. Results indicate that the effect of
84
the where the electrostatic term is damped is to restructure the power deposition profile near the
coils. However, the propagation of the helicon component is little affected, particularly at large
magnetic fields where the where the electrostatic term is damped is damped. For an m = 0 mode,
the TG mode does not significantly contribute as a noncollisional heating mechanism.
4.2 Nonlocal Heating by Axial Component of the Wave
The geometry used in this investigation is based on the Trikon Technologies, Inc.,
Pinnacle 8000 helicon plasma source, previously shown in Fig. 3.15 and described in Chapter 3.4
The cited magnetic field values are for an on-axis location midway between the antennas noted
by Bref in Fig. 3.15. Throughout the chapter, EEDs will be shown for three axial locations along
the flux lines labeled, F1, F2, and F3 as shown on the right side of Fig. 3.15. The axial positions
are labeled 1 (z = 27 cm), 2 (z = 18 cm) and 3 (z = 9 cm) on the left side of Fig. 3.15. Assuming
that electrons are magnetized in fields greater than 10s G, this technique samples approximately
the same population of electrons.
Azimuthal, radial and axial electric field amplitudes, and power deposition for the base
case are shown in Fig. 4.1 for magnetic fields of 20 and 300 G. For these results, transport
coefficients were obtained using the EMCS while neglecting the electrostatic term in Maxwell’s
equation. At B = 20 G, the azimuthal electric field is predominately inductively coupled with an
enhanced plasma skin depth, as shown in the left half of Fig. 4.1(a). Radial and axial electric
fields are generated by the tensor conductivity and indicate the onset of a radially traveling wave,
shown on the left side of Fig. 4.1(b) and 4.1(c). The skin depth has increased beyond the axis of
symmetry, and a standing wave pattern begins to develop. As the magnetic field is increased to
300 G, further penetration of the electromagnetic fields into the plasma occurs, as shown on the
85
right side of Fig. 4.1. However, as radial penetration increases, axial conductivity increases
thereby allowing the electromagnetic fields to propagate downstream. When the propagating
wave encounters a boundary, such as the substrate, a standing wave pattern forms in the axial
direction. The profiles for power deposition reflect those of the electromagnetic fields. At 300
G, significant power can be deposited downstream, as shown in Fig. 4.1(d). At 20 G, power
deposition peak at 2.1 W/cm3. At 300 G the same total power is deposited over a larger volume,
resulting in a lower peak power deposition of 1.1 W/cm3.
The electron temperature (defined by ε3
2), as derived from the EMCS, and the electron
density are shown in Fig. 4.2. The electron temperature for 20 G peaks in the bell jar at ≈ 4 eV
where most of the power deposition occurs. However, at 300 G the electron temperature peaks
downstream, as shown in the right side of Fig. 4.2(a) at about 3.6 eV. As will be discussed
below, the axial component of the electric field is responsible for heating the electrons
downstream. As electrons are accelerated in the axially propagating wave, they can absorb
substantial power from the field in spite of the plasma being somewhat collisional. At 10 mTorr,
the electron density peaks near the location of peak power deposition, shown in Fig. 4.2(b). At
20 G, the peak plasma density occurs in the bell jar at ≈ 4 × 1012 cm-3. At 300 G, the downstream
power deposition allows for a large plasma density, peaking at about 3.2 × 1012 cm-3.
The EED along the magnetic flux lines at the reference positions are shown in Fig. 4.3 for
a magnetic field of 20 G. The EEDs along the magnetic field line F1 shown in Fig. 4.3(a) are
closely spaced indicating that the electron temperature remains fairly constant. The EEDs along
F2 show a small increase in the high energy tail of the distribution at positions progressively
further away from the coil. The increase is not substantial enough to conclude that there is
significant axial heating downstream. The EEDs along F3 are similar for positions 1 (z = 27 cm)
86
and 2 (z = 18 cm). However, the EED at position 3 (z = 9 cm) has a lower high energy tail. The
field line F3 is significantly divergent, so electrons have physically traversed more gas to reach a
location that is fairly close to the chamber wall for position 3. Electrons reaching this point,
having traversed a larger column density of gas into a region of lower electric field, have a colder
EED.
The EEDs for a magnetic field of 300 G are shown in Fig. 4.4. Along F1, the EEDs at
the three axial locations overlap with a small increase in the high energy tail for position 3 (z = 9
cm), as shown in Fig. 4.4(a). Electrons on F1 are close to the line of symmetry where electric
fields are small (or zero) and so, on the average, are cooler. At the higher magnetic field,
electrons on F1 have sufficiently low transverse mobility that they do not sample the larger
electric fields at larger radii. However, along F2, there is a substantial increase in the high
energy tail of the EED at positions progressively further from the coil, as shown in Fig. 4.4(b).
This raising of the tail of the EED indicates that there is some amount of both continuous
acceleration and nonlocal heating of electrons downstream. Along the F3 field line, there is a
small amount of heating from position 1 (z = 27 cm) to position 2 (z = 18 cm), as shown in Fig.
4.4(c). Once again electrons at position 3 (z = 9 cm) have a colder high energy tail due to the
larger column density of gas they have traversed and smaller electric fields in the periphery of
the reactor. Comparing EEDs, on the three flux lines at a given axial position, the electron
population is typically hotter closer to the coil. For example, the high energy tail of the EEDs at
position 1 (z = 27 cm) increases from F1 to F3, transitioning from being near the center line of
symmetry where the electric field is low to near the coil where the electric field is higher. An
exception is position 3 (z = 9 cm) along F3, where the EED has a lower tail due to the longer
path traversed and lower electric fields in this region.
87
The consequences of noncollisional heating by the propagating axial wave should
become more pronounced with decreasing pressure due to the longer mean free paths of
electrons. For example, the average mean free path of electrons along F2 with energy between 5
and 15 eV and electrons above 15 eV are shown in Fig. 4.5. The mean free path λ for electrons
in these energy ranges is defined as,
( ) ( )( ) ( )
( )
1
E
E2
1
E
E2
1
gH
L
H
L
r,
r,rN r
−
=
∫
∫
εεε
εεεεσλ
df
dfT (4.1)
where Ng is the background gas density, σT is the total collision cross section, and EL and EH
bound the energy range of interest. Electrons with energy between 5-15 eV have mean free
paths ranging from 0.5 cm at 10 mTorr to 2 cm at 2 mTorr, as shown in Fig. 4.5(a). While
electrons with energy higher than 15 eV have mean free paths ranging from 2 cm at 10 mTorr to
12 cm at 2 mTorr, as shown in Fig. 4.5(b). Electrons below 5 eV have mean free paths of the
order of 0.01-0.1 cm in the 2–10 mTorr range.
The large mean free path of high energy electrons at low pressure allows for substantial
noncollisional heating to occur in the propagating electric field. For example, EEDs for 5 mTorr
and 2 mTorr at 300 G are shown in Figs. 4.6 and 4.7, respectively (the EEDs for 10 mTorr are in
Fig. 4.4). At 5 mTorr, the high energy tail of the EEDs increases with increasing distance from
the coil in the axial direction and decreases with increasing distance from the coil in the radial
direction. Both trends are consistent with continual acceleration and non-collisional heating by
axial electric fields. At 2 mTorr, these trends become even more pronounced. Along F1,
88
positions 1 and 2 produce a similar EED, while the EED at position 3 has a significantly higher
energy tail, shown in Fig. 4.7(a). Along F2, a progressive increase in the high energy tail is seen
with increasing axial distance from the coil, shown in Fig. 4.7(b). As seen previously, the high
energy tail of the EEDs increases on flux lines closer to the coil. Note that the mean free path at
2 mTorr increases with distance downstream. This trend results from a lifting of the tail of the
EED, shifting a larger fraction of electrons to higher energies, where the total scattering cross
section is smaller. This allows more efficient acceleration and is akin to a runaway electron
effect.
Radially averaged EEDs at 10 mTorr for several magnetic field strengths as a function of
axial position are shown in Fig. 4.8. The EED was averaged over a radius of 10 cm. The arrows
indicate the axial position of the rf coils. At 20 G, the EED is most extended (indicating a hot
tail) near the coils were most of the power deposition occurs as in conventional ICP systems. At
this magnetic field, the axial electric field is not significant, thereby limiting the amount of
noncollisional power deposition downstream. Hot electrons thermalize as they diffuse into the
downstream region, thereby decreasing the electron EED at 20 eV from 10-4 eV-3/2 near the coils
to 2 ×10-6 eV-3/2 near the substrate. As the magnetic field is increased to 150 G , the magnitude
of the axial electric field increases, radial diffusion is decreased and the relative conduction in
the axial direction is increased. An increase in axial mobility does not necessarily translate to a
monotonically higher energy tail downstream. At this pressure, there appears to be a local
equilibrium between acceleration by the axial electric field and the power absorbed by collisional
processes. This results in electrons accelerated by the axial electric field depositing their power
locally. As a result, hot electrons produced near the coils maintain their energy into the
downstream region. For example, at 300 G, the EED of electrons at 20 eV is nearly the same
89
near the coils and the substrate, shown in Fig. 4.8(c). At 10 mTorr, non-collisional heating does
not appear to be dominant.
As the pressure is reduced, the increase in mean free path and decrease in collisional
processes enables continuous acceleration by the axial electric field far into the downstream
region. For example, the radially averaged EED at 300 G for several pressures as a function of
axial position is shown in Fig. 4.9. As the pressure is decreased from 10 mTorr (Fig. 4.8(c)) to 5
mTorr and finally to 2 mTorr, the tail of the EED downstream is raised through noncollisional
heating. The population of electrons at 30 eV near the substrate increases from 5 × 10-6 eV-3/2 at
10 mTorr to 4 × 10-3 eV-3/2 at 2 mTorr.
The ability to deposit power nonlocally through Landau damping depends on matching
the parallel phase velocity of the propagating wave with the thermal velocity of a large fraction
of the electron population. The more closely matched these velocities are for a larger fraction of
the electron populaton, the more likely electrons are to be resonantly accelerated. The parallel
phase velocity of the electric fields is proportional to the rf frequency. For example, radially
averaged EEDs at 300 G and 2 mTorr for several rf frequencies as a function of axial position are
shown in Fig. 4.10. As the rf frequency is decreased from 27.1 MHz to 1.36 MHz,
noncollisional heating throughout the reactor becomes more prevalent due to better phase
matching with thermal electrons. The population of electrons at 30 eV near the substrate
increases from 10-6 eV-3/2 at 27.1 MHz to 3 ×10-4 eV-3/2 at 1.36 MHz. The proposal is that this
increased heating is due to better phase matching.
To determine if the increase in the high energy tail was indeed a result of better phase
matching of thermal electrons to the wave phase velocity or simply a skin depth effect, the
population of electrons that are capable of being continuously heated through Landau damping
90
was estimated. The electrons which are likely to be continuously accelerated are those having a
thermal speed within a factor of Lz
2 λ
of the phase velocity, where λz is the axial wavelength of
the electric field and L is the length of the reactor. The criterion limits electrons to at worst a π/2
phase slip relative to the wave. This fraction, β, is obtained from the EED at each location in the
reactor from,
( ) ( )∫
−
= φ
φλβ
v
2L 1v z
v r,v r df (4.2)
where vφ is the phase velocity of the axial wave. The axial wavelength is obtained through a
numerical fit of the propagating phase fronts, as described in Chapter 4.1. The parallel phase
velocity, vφ, is defined as the product of the frequency and the axial wavelength.
The fraction of electrons in phase with the electromagnetic field for rf frequencies of
1.36, 13.56 and 27.1 MHz is shown in Fig. 4.11. At 27.1 MHz, the population of the electrons in
phase with the field peaks at about 15% near the substrate. In general, electrons are too cool
(slow) to phase match with the electric field. At 1.36 MHz, the fraction of the electrons capable
of phase matching can reach up to 35% near the coils, so electrons are most efficiently heated in
the bell jar region. This heating then extends significantly downstream. Due to the large mean
free path of electrons above 15 eV, noncollisional power absorbed in the bell jar can be
deposited downstream. Electron temperatures for these rf frequencies are shown in Fig. 4.12.
For frequencies varying from 1.36–27.1 MHz, the mean free path for electrons above 15 eV is
between 10 and 20 cm. At 27.1 MHz, electrons are not well matched with the propagating wave,
and noncollisional absorption is minimal. Electron temperatures peak near the coils where
91
power deposition is limited by the skin depth, as shown in Fig. 4.12(a). There is a small increase
in the electron temperature near the bottom of the chamber. In this region electrons begin to be
better phase matched to the propagating wave, as shown in Fig. 4.11(a). At 1.36 MHz, electrons
are well matched in the bell jar, and substantial noncollisional acceleration occurs. This
acceleration enables phase matching downstream. The energy gained from the propagating wave
is deposited further downstream due to the long mean free paths.
Several investigators have measured abrupt increases in plasma density as the static
magnetic field increases.1-3 For example, Boswell and Chen1 made measurements of ion density
in a 10 cm diameter glass tube using a double Langmuir probe oriented along the magnetic field.
The observed increase in the ion density with magnetic field was attributed to a narrowing of the
plasma column and to an increase in the ionization efficiency by an improved coupling of the
electrons to the helicon wave. The magnitudes of the jumps in plasma density often measured in
these experiments are factors of 2 to 10 while the total power deposition is constant. If the
change in density is attributed solely to an increase in efficiency, then the change in power
dissipation mechanisms from nonionizing to ionizing must have similarly increased by factors of
2 to 10. In rare gases such as Ar, it is difficult to find such mechanisms. As has been observed
in many inductively coupled systems operating in an H-mode, the total electron population
(density integrated over the volume) largely depends only on input power and so should remain
somewhat constant with increasing magnetic field. Small increases in the total plasma density
with increasing magnetic fields can be attributed to a decrease in perpendicular diffusional
losses, but not necessarily with a change in ionization mechanism.
It is important to note that most experimental measurements in which plasma density
jumps are observed were taken at a single point in the reactor as the magnetic field was varied.
92
The “jumps” may in fact be a result of changes in the distribution of the plasma in the vessel due
to changes in the modal electromagnetic wave patterns that in turn determine the location of the
power deposition. For example, it was previously shown that as the magnetic field is increased
axial propagation of the electromagnetic wave dominates and significant power can be deposited
downstream.3 The peak in the electron density shifts from being near the coils to being located
in the downstream chamber. If monitored at a single point, this shift in the plasma density gives
the appearance of a large increase in the ionization while the total inventory of electrons does not
significantly change.
To demonstrate this effect the electron densities on F2 at position 1, 2, and 3 are shown
in Fig. 4.13 for Ar at 10 mTorr as a function of magnetic field while keeping the total power
deposition constant. The average electron density (total electron inventory divided by the volume
of the chamber) is also shown. At 50 G, the electron density at position 1 peaks at 1.45 × 1012
cm-3, while at position 3 the electron density is only 4.5 × 1011 cm-3. As the magnetic field is
increased, the peak plasma density shifts downstream. At 300 G, the electron density at position
1 is 3.0 × 1011 cm-3, while at position 3 it is 6.5 × 1011 cm-3. While these shifts are occurring, the
average density is nearly constant at 3.0 × 1011 cm-3 from 25 G to 300 G. A small increase in
average density from 0-25 G is likely due to confinement.
4.3 Effects of the Electrostatic Term on Propagation and Heating
In order to resolve the TG mode, the divergence term (electrostatic term) in Eq. (2.1) was
incorporated into the solution for the electromagnetic fields as described in Section II. Base case
conditions are the same as described in Section III. The azimuthal, radial, axial electric field,
and power deposition are shown in Fig. 4.14 for 20 G. At this low value of the static magnetic
93
field, the electrostatic term significantly changes the propagation and structure of the
electromagnetic fields. The azimuthal electric field peaks near the coils and penetrates further
into the chamber than in the absence of the electrostatic term. The radial and axial components
also show an increased skin depth, however standing wave structures are not as evident. Both Eθ
and Er have maxima near the center of the chamber. The structure of the radial and axial electric
fields outside of the bell jar region also changes. This is due to new sources of the electric field
which are generated in the bell jar and subsequently propagate away from the sources.
Electromagnetic propagation in enhanced in the axial direction, but falls off rapidly at about z ≈
12 cm. At this location propagating wave encounters an ECR zone where substantial power
absorption occurs. At 13.56 MHz, ECR occurs at 4.8 G. For this magnetic field configuration,
the 5 G surface lies between z = 12 cm on axis and z = 20 cm near the chamber wall. Although
power deposition peaks near the location of peak electric fields upstream, significant power
deposition also occurs in the ECR zone. This ECR heating was not observed at 20 G in the
absence of the TG mode because the electromagnetic field was largely absorbed upstream, and
there was less propagation of the electric field downstream to encounter the ECR surface.
The azimuthal, radial, axial electric field, and power deposition with the electrostatic
term are shown in Fig. 4.15 for 300 G. The results are quite similar to those shown in Fig. 4.1 in
the absence of the electrostatic term. These results indicate that propagation of the electric fields
is not significantly altered by the electrostatic term at higher static magnetic fields. Note that for
these magnetic fields, an ECR zone does not occur in the chamber and so does not affect the
propagation of the fields.
The manifestation of the TG mode can be seen through the harmonically driven
perturbation in the electron density shown in the left side of Fig. 4.16. At 20 G, the perturbed
94
electron density has a small maxima near the bell jar-plasma interface. There is significant
propagation into the chamber with a peak near the axis. Axial propagation occurs up to the ECR
zone where strong absorption occurs. The electron density at 20 G, shown on the right side of
Fig. 4.16(a) is similar to the pure helicon case shown in Fig. 4.2(b). The electrostatic term case
has a lower peak plasma density because power is deposited in a large volume, thereby
distributing plasma over a somewhat larger volume.
At 300 G, the electrostatic term is strongly absorbed near the bell jar-plasma interface, as
shown on the left side of Fig. 4.16(b). There is minimal penetration into the chamber volume at
these higher magnetic fields. These trends agree with Shamrai and Taranov4 who suggested that
strongly damped electrostatic waves can reach the plasma core at low magnetic fields, while at
high fields they deposit power at the periphery of the plasma column. Since the electromagnetic
fields are little affected by the electrostatic term at 300 G, the electron density profiles are similar
to the pure helicon case. There is a local maximum in the electron density near the coils and near
the substrate where significant power deposition. The peak electron density in the electrostatic
term case is smaller than for the pure helicon case, but the reactor averaged electron density is
higher for the TG case. For the pure helicon case at 300 G and 2 mTorr, the average electron
density is 2.35 × 1011 cm-3, while for the electrostatic term case it is 7.74 × 1011 cm-3.
The EEDs for 20 G at 10 mTorr, and 300 G at 2 mTorr with the electrostatic term along
F2 are shown in Fig. 4.17. These EEDs should be compared to Figs. 4.3 and 4.7 for results
without the TG mode. At 20 G, there is a significant raising of the tail of the EED from position
1 (z = 24 cm) to position 2 (z = 16 cm) indicating some additional downstream heating, perhaps
due to the nearby ECR zone, which was not observed in the pure helicon case. Past the ECR
zone, at position 3 (z = 8 cm), the electron population begins to cool, and there is a decrease in
95
the tail of the EED. At 300 G, the tails of the EEDs progressively increase further away from the
coil. The TG mode produces only a small increase in the tail of the EED in comparison with the
pure helicon case shown in Fig. 4.7. Radially averaged EEDs for 20 G at 10 mTorr and 300 G at
2 mTorr as a function of axial position are shown in Fig. 4.18. These radially averaged EEDs
should be compared to Figs. 4.8(a) and 4.10(c) for results without the electrostatic term.
Comparing with the pure helicon case at 20 G, Fig. 4.8(a) there is an increase in the EED
population for the TG mode near the coils at 20 eV from 7 × 10-5 eV-3/2 to 5 ×10-4 eV-3/2, shown
in Fig. 4.18(a). In contrast, at 300 G, the radially averaged EED with the electrostatic term,
shown in Fig. 4.18(b), is nearly the same as the pure helicon case, shown in Fig. 4.10(c). These
results again indicate that the electrostatic term plays a more dominant role in the power
deposition at lower magnetic fields. The effect of the electrostatic term is to restructure the
power deposition profile, especially near the coils. However, the propagation of the helicon
component is little affected, particularly at large magnetic fields where the electrostatic term is
damped.
4.4 Conclusions
Using a tensor conductivity in the solution of Maxwell’s equations, 3-D components of
the inductively coupled electric field were computed from an m = 0 antenna and 2-D applied
magnetic fields. These fields were used in a Monte Carlo Simulation to generate EEDs,
transport coefficients and electron impact source functions. The tail of the EEDs were found to
increase in magnitude in the downstream chamber for B >150 G and P <5 mTorr, indicating
some amount of electron trapping and wave heating. This heating results from noncollisional
acceleration by the axial electric field for electrons in the tail end of the EED which have long
96
mean-free-paths, while low energy electrons are still somewhat collisional. The mean free path
for electrons above 15 eV is between 10 and 20 cm at 2 mTorr. The increase in the high energy
tail of the EED was a result of better phase matching of thermal electrons to the phase velocity of
the wave. Electrons whose speeds are close to the phase velocity are efficiently heated and
subsequently deposit their power downstream through collisions with the background gas. These
results indicate that the helicon component of the wave produces significant nonlocal heating.
The TG mode was resolved by including the divergence electrostatic term in the solution
of the wave equation. The electrostatic component in the wave equation was approximated by a
harmonically driven perturbation of the electron density. Results indicate that strongly damped
electrostatic waves can reach the plasma core at low magnetic fields (<20 G), while at high
magnetic fields (>150 G) the electrostatic waves deposit power primarily at the periphery of the
plasma column. These results indicate that the TG mode plays a more important role in the
power deposition mechanisms at lower magnetic fields by restructuring the power deposition
profile. However at larger magnetic fields, where the electrostatic term is damped, the
propagation of the helicon component and nonlocal heating is little affected.
97
20 G
Eθ
a)
300 G
(V / cm)
(W / cm3)
200.02 1.1
20.02 0.2
b)
Er
Hei
ght (
cm)
0
50
25
20 G 300 G
20 200Radius (cm)
c)
20 200Radius (cm)
d)
Hei
ght (
cm)
0
50
25
20 G 300 G20 G 300 G
Ez Power
.
Figure 4.1 Electric field amplitudes and power deposition for a static magnetic field of 20 G and300 G (a) Azimuthal electric field, (b) radial electric field, (c) axial electric field, (d) powerdeposition. The process conditions are Ar, 10 mTorr and 1 kW. Increasing magnetic fieldproduces radial components of the electric field and propagation in the axial direction.
98
20 G
[e]
b)
300 G
Hei
ght (
cm)
0
50
25
a)
Te
20 G 300 G
(eV)
(cm-3)
40.4 2.2
4 x 1011 4 x 10122.2 x 1012
Hei
ght (
cm)
0
50
25
Figure 4.2 Plasma properties for a static magnetic field of 20 G and 300 G. (a) Electrontemperature and (b) electron density. The process conditions are Ar, 10 mTorr and 1 kW powerdeposition. Increasing magnetic field allows for peak temperatures and electron densitiesdownstream.
99
2
1
3
F1100
10-2
10-4
10-6
EE
D (
eV-3
/2)
21
3
F2
100
10-2
10-4
10-6
EE
D (
eV-3
/2)
2
1
3
0 10 20 30 40Energy (eV)
F3
50
100
10-2
10-4
10-6
EE
D (
eV-3
/2)
20 G, 10 mTorr
c)
a)
b)
Figure 4.3 EEDs along the magnetic flux lines (a) F1, (b) F2 and (c) F3. EEDs are shown forreference axial positions 1 (z = 27 cm), 2 (z = 18 cm) and 3 (z = 9 cm) for 20 G and 10 mTorr.
100
2
1
3
F1100
10-2
10-4
10-6
2
1
3
F2
100
10-2
10-4
10-6
21
3
0 10 20 30 40Energy (eV)
F3
50
100
10-2
10-4
10-6
300 G, 10 mTorr
EE
D (
eV-3
/2)
EE
D (
eV-3
/2)
EE
D (
eV-3
/2)
c)
a)
b)
Figure 4.4 EEDs along the magnetic flux lines (a) F1, (b) F2 and (c) F3. EEDs are shown forreference axial positions 1 (z = 27 cm), 2 (z = 18 cm), and 3 (z = 9 cm) for 300 G and 10 mTorr.The tail of the EEDs raised downstream.
101
25 20 15 10 0Axial Position (cm)
0
5
10
15
λ (
cm)
10 mTorr
2 mTorr5 mTorr
5b)
0
1
2 λ
(cm
)
10 mTorr
2 mTorr
5 mTorr
a)
> 15 eV
5-15 eV
Figure 4.5 Mean free path as a function of axial position for 2, 5, and 10 mTorr. (a) Forelectrons with energy between 5 and 15 eV and (b) for electrons with energy above 15 eV. Lowenergy electrons have a shorter mean free path and are more collisional, while higher energyelectrons have longer mean free paths and are more non-collisional.
102
213
213
2
1
3
F1
100
10-2
10-4
10-6
0 10 20 30 40Energy (eV)
F2
F3
50
100
10-2
10-4
10-6
100
10-2
10-4
10-6
300 G, 5 mTorr
EE
D (
eV-3
/2)
EE
D (
eV-3
/2)
EE
D (
eV-3
/2)
c)
a)
b)
Figure 4.6 EEDs along the magnetic flux lines (a) F1, (b) F2 and (c) F3. EEDs are shown at thereference axial positions 1 (z = 27 cm), 2 (z = 18 cm), and 3 (z = 9 cm) for 300 G and 5 mTorr.There is continual lifting of the tail further downstream along F2.
103
21
3
213
1
32
F1100
10-2
10-4
10-6
0 10 20 30 40Energy (eV)
F2
F3
50
100
10-2
10-4
10-6
100
10-2
10-4
10-6
300 G, 2 mTorr
EE
D (
eV-3
/2)
EE
D (
eV-3
/2)
EE
D (
eV-3
/2)
c)
a)
b)
Figure 4.7 EEDs along the magnetic flux lines (a) F1, (b) F2 and (c) F3. EEDs are shown at thereference axial positions 1 (z = 27 cm), 2 (z = 18 cm), and 3 (z = 9 cm) for 300 G and 2 mTorr.The longer mean free paths are lower pressure produces more non-collisional heating.
104
Energy (eV)0 10 20 30 40
10-210-4
10-6
Hei
ght (
cm)
c)
10-2
10-4
10-6
b)
Hei
ght (
cm)
10-2
10-410-6
a)
Hei
ght (
cm)
0
50
25
0
50
25
0
50
25
150 G
20 G
300 G
10 mTorr
Figure 4.8 Radially averaged EED as a function of axial position for 10 mTorr. (a) 20 G, (b)150 G and (c) 300 G. A near equilibrium may occur between acceleration and collisionaldamping.
105
Energy (eV)
0 10 20 30 40
10-2
10-4
b)
10-2 10-410-6
Hei
ght (
cm)
0
25
50a)
Hei
ght (
cm)
0
25
50
5 mTorr
2 mTorr
300 G
Figure 4.9 Radially averaged EED as a function of axial position for 300 G. (a) 5 mTorr and (b)2 mTorr. Significant heating occurs downstream.
106
Energy (eV)0 10 20 30 40
10-2
10-4
Hei
ght (
cm)
0
50
25
b)
10-31.36 MHz
10-2
10-4
10-6
a)
Hei
ght (
cm)
0
50
25 27.1 MHz
300 G, 2 mTorr
Figure 4.10 Radially averaged EED as a function of axial position for 300 G, 2 mTorr. (a) 27.1MHz and (b) 1.36 MHz.
107
200Radius (cm)
0
50
c)
0.4
0.2
0.05
0.01
a)0.15
0.05
0.01
b)
0.25
0.15
0.20.01
0.1
0.1
0.25
0
50
0
50
27.1 MHz
13.56 MHz
1.36 MHz
100
Hei
ght (
cm)
25
Hei
ght (
cm)
25
Hei
ght (
cm)
25
Fraction Phase Matched
Figure 4.11 Fraction of electrons that are in phase with the rf electric field for 300 G and 2mTorr. (a) 27.1 MHz, (b) 13.56 MHz, and (c) 1.36 MHz.
108
1.5
2.5
4.5
7.5
2.11.0
4.5
0.7
2.2
2.0
1.7
1.5
200Radius (cm)
0
50
0
50
0
50
27.1 MHz
13.56 MHz
1.36 MHz
c)
a)
b)
Hei
ght (
cm)
25
Hei
ght (
cm)
25
Hei
ght (
cm)
25
ElectronTemperature (eV)
10
Figure 4.12 Electron temperature for 300 G, 2 mTorr. (a) 27.1 MHz, (b) 13.56 MHz, and (c)1.36 MHz.
109
0 100 200 300
Ele
ctro
n D
ensi
ty (
10
12 c
m-3
)
0.5
1.0
1.5
0
Magnetic Field (G)
1
2
3
Average
Figure 4.13 Electron density as a function of static magnetic field at the reference axial positions1 (z = 27 cm), 2 (z = 18 cm), and 3 (z = 9 cm) along F2 for Ar at 10 mTorr for constant powerdeposition.
110
(V / cm)
(W / cm3)
200.02 1.1
20.02 0.2
Eθ
a)
Er
Hei
ght (
cm)
0
50
25
20 200Radius (cm)
b)
Ez Power
Hei
ght (
cm)
0
50
25
Figure 4.14 Plasma properties obtained when including the electrostatic term in Maxwell’sequations for Ar, 20 G, 2 mTorr, 1 kW power deposition. (a) Azimuthal and radial electric fieldamplitude and (b) axial electric field amplitude and power deposition.
111
(V / cm)
(W / cm3)
200.02 1.1
20.02 0.2
Eθ
a)
Er
Hei
ght (
cm)
0
50
25
20 200Radius (cm)
b)
Ez Power
Hei
ght (
cm)
0
50
25
Figure 4.15 Plasma properties while including the elctrostatic term in Maxwell’s equation forAr, 300 G, 2 mTorr and 1 kW power deposition. (a) Azimuthal and radial electric field amplitudeand (b) axial electric field amplitude and power deposition.
112
Hei
ght (
cm)
0
50
25
b)20 200
Radius (cm)
a)
∆ne [e]
[e] (cm-3) 4 x 1011 4 x 10122.2 x 1012
∆ne (cm-3) 102 1075 x 105
20 G
300 G
Hei
ght (
cm)
0
50
25
Figure 4.16 The manifestation of the TG mode can be seen through the harmonically drivenperturbation in the electron density shown on the left side for (a) 20 G and (b) 300 G. Theelectron density is shown in the right side for (a) 20 G and (b) 300 G.
113
100
10-2
10-4
10-6
EE
D (
eV-3
/2)
1 2
3
20 G
0 10 20 30 40Energy (eV)
50
213
300 G
100
10-2
10-4
10-6
EE
D (
eV-3
/2)
b)
a)
Figure 4.17 EEDs along the magnetic flux line F2 at the reference axial positions 1 (z = 27 cm),2 (z = 18 cm), and 3 (z = 9 cm) with electrostatic term. (a) 20 G (10 mTorr) and (b) 300 G (2mTorr).
114
Hei
ght (
cm)
Energy (eV)0 10 20 30 40
10-2
10-4
0
50
25
10-2 10-4
10-3
a)
b)
0
50
25
Hei
ght (
cm)
Figure 4.18 Radially averaged EED with the electrostatic term as a function of axial position for2 mTorr and (a) 20 G and (b) 300 G.
115
4.5 References
1. R. W. Boswell and F. F. Chen, IEEE Trans. Plasma Sci. 25, 1229 (1997).
2. F. Chen, X. Jiang, and J. Evans, J. Vac. Sci. Technol. 18, 2108 (2000).
3. S. Yun, K. Taylor and G. R. Tynan, Phys. Plasma 7, 3448 (2000).
4. K. P. Shamrai and V. B. Taranov, Plasm. Sour. Sci. Technol. 5, 474 (1996).
116
5 THREE DIMENSIONAL SIMULATIONS OF WAVE HEATED DISCHARGES
5.1 Introduction
In this chapter, we investigate the effects of antennas with azimuthal asymmetries. Most
experiments are designed for the m = ±1 modes since they couple to the plasma more efficiently
than the m = 0 mode, where m is the azimuthal mode number as described in Chapter 1. The m =
+1 mode is right-hand circularly polarized, and the m = -1 mode is left-hand circularly polarized,
when viewed along the axial magnetic field. Chen and colleagues1-3 have shown that helical
antennas exciting a right-hand circularly polarized wave gives the highest electron densities. The
rise in the density downstream from the antenna has been explained by a pressure balance, along
the magnetic field lines, as the electron temperature decays from the antenna. The downstream
peak was also observed in the nonuniform magnetic field of a processing reactor.4
The model used in this investigation is a 3-D extension of the HPEM called the HPEM-
3D.5,6 The formulism is the same as discussed in Chapter 2 with the addition of the third
dimension to account for azimuthal dependencies. The HPEM-3D was improved to resolve 3-D
components of the electric field produced by m = +1 antennas in solenoidal magnetic fields. The
EMM of the HPEM-3D solves for the (r,θ,z) components of the complex inductively coupled
electric field as given by Eq. (2.1). A tensor conductivity was used to couple the components
while solving the wave equation in the frequency domain. For these simulations plasma
neutrality was enforced, thereby neglecting effects from the electrostatic fields. We acknowledge
that for an m = +1 mode, the 3-d coil design can generate significant electrostatic fields. These
fields may significantly change the structure of the electromagnetic field near the antenna
location. However, from the previous results of Chapters 3 and 4, we can assume that the
117
electrostatic term has little effect on either the propagation far from the antenna or nonlocal
downstream heating. Furthermore, this analysis was conducted as a proof of principle of
simulation of magnetized plasmas in three dimensions. With the addition of the azimuthal
dependencies, the resulting matrix for solving Maxwell’s equations is significantly sparse and
the problem becomes ill-conditioned.
Recall that in the initial implementation of the HPEM, the EMM solved the matrix
equation via successive-over-relaxation (SOR). Unfortunately, this approach proved to be
unreliable for problems that included static magnetic field with azimuthal dependencies.
Without the magnetic field, the solution for the electric field always resulted in a wave that was
strongly attenuated and had a wavelength that was much longer than the reactor dimension. The
addition of the magnetic field allowed for short wavelengths in the electric field producing a far
field solution, and possibly less absorption. These effects become more pronounced in the ECR
region. In the SOR method, the conduction current created in a cell may grow at a faster rate
than is allowed to diffuse out of the computational cell. The conjugate gradient method used to
solve the electromagnetic fields in the 2D-HPEM proved to be unstable in the HPEM-3D. The
largest varying field in the reactor reached a convergence of 1 × 10-3, before becoming unstable
and diverging.
Fortunately, an iterative sparse matrix solver with a quasi-minimal residual method
provided the satisfactory convergence of 1 × 10-6. The procedure results in 3-D partial
differential equations for the electric field, which is solved in the frequency domain similar to
that described in Chapter 2. The discritized equations are shown in Eq. (5.1), (5.2), and (5.3).
The routine implemented the coupled two term recurrence variants of the look-ahead Lanczos
algorithms and the solution of linear systems with a quasi-minimal residual method.7
118
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( ) rjizjijirjir
jiji
ji
jiji
jirjir
jiji
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jiji
ji
jiji
jirjir
jiji
ji
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jirjir
jiji
ji
jiji
jirjir
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ji
jiji
jirjir
JiEEEiE
zrrrr
zrr
z
EE
zrrrr
zrr
z
EE
zrrrr
zrrEE
zrrrr
zrrEE
zrrrr
zrr
r
EE
zrrrr
zrr
r
EE
ωσσσωεω
π
πµµ
π
πµµ
π
πµµθ
π
πµµθ
π
πµµ
π
πµµ
θθθθθ
θθ
θ
θθ
θθ
θθ
θ
θθ
θθ
θθ
θ
θθ
θθ
θθ
θ
θθ
θθ
θθ
θ
θθ
θθ
θθ
θ
θθ
θθ
−=+++−
∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
+∆∆−−∆+
∆∆−⋅
+⋅
∆−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
+∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆−
−
−
+
+
−
+
+
−
−
+
+
),,(13),,(12),,(11),,(2
2),,(
2),,(
),,(
),1,(),,(
),1,(),,(
2),,(
2),,(
),,(
),,(),1,(
),,(),1,(
2),,(
2),,(
),,(
),,(),,(
)1,,(),,(
2),,(
2),,(
),,(
),,()1,,(
),,()1,,(
2),,(
2),,(
),,(
),,1(),,(
),,1(),,(
2),,(
2),,(
),,(
),,(),,1(
),,(),,1(
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
(5.0.1)
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( ) θθθθθθθ
θθ
θ
θθ
θθθθ
θθ
θ
θθ
θθθθ
θθ
θ
θθ
θθθθ
θθ
θ
θθ
θθθθ
θθ
θ
θθ
θθθθ
θθ
θ
θθ
θθθθ
ωσσσωεω
π
πµµ
π
πµµ
π
πµµθ
π
πµµθ
π
πµµ
π
πµµ
JiEEEiE
zrrrr
zrr
z
EE
zrrrr
zrr
z
EE
zrrrr
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zrrrr
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zrrrr
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r
EE
zrrrr
zrr
r
EE
jizjijirji
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ji
jiji
jiji
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ji
jiji
jiji
jiji
ji
jiji
jiji
jiji
ji
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jiji
−=+++−
∆∆−−∆+
∆∆−⋅
+⋅
∆−
−∆∆−−∆+
∆∆+⋅
+⋅
∆−
+∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
+∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
−
−
+
+
−
+
+
−
−
+
+
),,(23),,(22),,(21),,(2
2),,(
2),,(
),,(
),1,(),,(
),1,(),,(
2),,(
2),,(
),,(
),,(),1,(
),,(),1,(
2),,(
2),,(
),,(
),,(),,(
)1,,(),,(
2),,(
2),,(
),,(
),,()1,,(
),,()1,,(
2),,(
2),,(
),,(
),,1(),,(
),,1(),,(
2),,(
2),,(
),,(
),,(),,1(
),,(),,1(
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
(5.0.2)
119
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( ) ( )( )
( ) zjizjijirjiz
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ji
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jizjiz
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zrrrr
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zrrrr
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r
EE
zrrrr
zrr
r
EE
ωσσσωεω
π
πµµ
π
πµµ
π
πµµθ
π
πµµθ
π
πµµ
π
πµµ
θθθθθ
θ
θ
θθ
θθ
θθ
θ
θθ
θθ
θθ
θ
θθ
θθ
θθ
θ
θθ
θθ
θθ
θ
θθ
θθ
θθ
θ
θθ
θθ
−=+++−
∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆−
+∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
+∆∆−−∆+
∆∆−⋅
+⋅
∆
−
−∆∆−−∆+
∆∆+⋅
+⋅
∆
−
−
−
+
+
−
+
+
−
−
+
+
),,(33),,(32),,(31),,(2
2),(
2),,(
),,(
),1,(),,(
),1,(),,(
2),,(
2),,(
),,(
),,(),1,(
),,(),1,(
2),,(
2),,(
),,(
),,(),,(
)1,,(),,(
2),,(
2),,(
),,(
),,()1,,(
),,()1,,(
2),,(
2),,(
),,(
),,1(),,(
),,1(),,(
2),,(
2),,(
),,(
),,(),,1(
),,(),,1(
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
5.05.0
5.02
2
1
(5.0.3)
where Er(i,j), Eθ(i,j), and Ez(i,j) are the radial, azimuthal, and axial components of the electric field,
respectively, and Jr(i,j), Jθ(i,j), and Jz(i,j) are the radial, azimthal, and axial external currents,
respectively, at positions i and j. The terms r(i,j), ∆ne(i,j), ∆r, and ∆z, are the radial position and
electron perturbation at position i and j and the radial and axial distance of the computational
cell, respectively. While ε, µ, ω, and σ are the permitivitty, permeability, input angular
frequency, and the tensor conductivity as defined by Eq. (2.8). Furthermore, Eqs. (5.1), (5.2), and
(5.3) can be reduced to the following:
( ) rjizjijirjir
jirjirjirjirjir
JiEEiEFFACEEFAC
EDFACECFACEGFACREBFACEAFAC
ωσσω θθθθθ
θθθθθ
−=+++⋅+⋅
⋅+⋅+⋅−⋅+⋅
−+
−+−+
),,(13),,(12)1,,()1,,(
),1,(),1,(),,(),,1(),,1(
(5.4)
120
( ) θθθθθθθ
θθθθθθθθθθ
ωσσω
θ
JiEEiEFFACEEFAC
EDFACECFACEGFACEBFACEAFAC
jizjirjiji
jijijijiji
−=+++⋅+⋅
⋅+⋅+⋅−⋅+⋅
−+
−+−+
),,(23),,(21)1,,()1,,(
),1,(),1,(),,(),,1(),,1(
(5.5)
( ) zjijirjizjiz
jizjizjizjizjiz
JiEEiEFFACEEFAC
EDFACECFACEGFACZEBFACEAFAC
ωσσω θθθθθ
θθθθθ
−=+++⋅+⋅
⋅+⋅+⋅−⋅+⋅
−+
−+−+
),,(32),,(31)1,,()1,,(
),1,(),1,(),,(),,1(),,1(
(5.6)
Equations (5.4), (5.5), and (5.6). are written in matrix form in Eq. (5.7).
=
Er(i,j)
Er(i-1,j)
Er(i,j-1)
Er(i+1,j)
Er(i,j+1)
Eθ(i,j)
Eθ(i-1,j)
Eθ(i,j-1)
Eθ(i+1,j)
Eθ(i,j+1)
Ez(i,j)
Ez(i-1,j)
Ez(i,j-1)
Ez(i+1,j)
Ez(i,j+1)
iωJθAFAC
-GFACθBFAC
CFAC
DFAC
AFAC-GFACR
BFACCFAC
DFAC
AFAC-GFACZ
BFACCFAC
DFAC
iωσ21 iωσ23
iωσ12 iωσ13
iωσ32iωσ31
Ez(i,j,θ+1)
Ez(i,j,θ−1)
Er(i,j,θ−1)
Er(i,j,θ+1)
Eθ(i,j,θ−1)
Eθ(i,j,θ+1)
iωJr
iωJz
FFAC
EFAC
FFAC
EFAC
FFAC
EFAC
(5.7)
121
Electron impact sources and electron transport properties are solved in the EETM and
EMCS as described in Sections 2.3 and 2.4 with the added azimuthal dimensionality. There are
two methods for determining these parameters. The first method determines the electron
temperature by solving the electron energy equation. The second method uses a Monte Carlo
simulation for electron transport to gather statistics used to generate the electron energy
distribution (EED) as a function of position. These values are transferred to the FKM in which
densities for all charged and neutral species are obtained, and Poisson’s equation is solved for
electrostatic fields as described in Section 2.5 with added dimensionality. Momentum equations
were solved for ions and neutral species, thereby accounting for inertial effects and gas flow.
We have not solved the species’ energy equations, and so we assume isothermal conditions
throughout the reactor. The densities, conductivities, and electrostatic fields obtained from the
FKM are then transferred to the EMM and the EETM/EMCS. This iterative cycle is repeated
until a converged solution is obtained.
5.2 Propagation of an m = +1 Mode in a Solenoidal Geometry
A solenoidal reactor was used as a demonstration platform and to provide a means for
validating the model. Two types of coil configurations were used, schematically shown in Fig.
5.1. The first reactor, shown in Fig. 5.1(a), is powered by a set of ring coils that are driven at
13.56 MHz with currents 180° out of phase. This two turn antenna produces an m = 0 mode of
propagation. The second reactor, shown in Fig. 5.1(b), is powered by a Nagoya Type III coil
which has both azimuthal and axial antenna components that produce an m = +1 mode. Process
gas is injected at the top of a quartz tubular reactor through a shower head nozzle and flows out a
pump port located at the bottom. The reactor sits inside a solenoidal magnetic field having
122
dominantly an axial component with larger radial gradients near the ends of the solenoid. The
base case has operating conditions of Ar gas at 2 mTorr, 150 sccm, 13.56 MHz, and a power
deposition of 1 kW. Plasma properties will be shown at different axial positions in the reactor
labeled in Fig. 5.1 as the “upstream slice”, “antenna slice”, and “downstream slice”.
The resulting electric field pattern predicted by Chen et al.3 for an m = +1 mode using a
solenoidal geometry in a uniform plasma is shown in Fig. 5.2(a). The dark solid lines represent
the electric field lines, while the dotted lines represent the magnetic field lines. The total electric
field pattern obtained from the HPEM-3D for the geometry shown in Fig. 5.1(b), is shown in Fig.
5.2(b). The field pattern is at a position 20 cm upstream from the antenna at a magnetic field
strength 100 G. At this position the field profile is determined by the azimuthal segments of the
antenna with two peaks off axis. Results from Chen have the two off axis peaks closer to the
axis of symmetry due to their assumption of there being a uniform plasma. The total electric
field amplitudes at the three positions in the cylindrical tube indicated in Fig. 5.1(b) are shown in
Fig. 5.3 for varying static magnetic fields strengths. Near the coils the total electric field peaks
near the axial segments of the antenna at a value of about 12 V/cm. Far from the coils, the
electric field profile is determined by the azimuthal segments of the antenna. The magnitudes of
the up- or downstream electric field are strongly dependent on the strength of the static magnetic
field. A standing wave with two nodes near the axis begins to form at the antenna position as the
static magnetic field is increased.
The axial electric field amplitudes at three positions in the cylindrical tube are shown in
Fig. 5.4 for varying static magnetic field strength. Peak values near the coils can reach 2 V/cm.
At low static magnetic fields there is a strong coupling of the axial electric field to the azimuthal
electric field. Between 50 and 150 G, the axial electric field amplitude follows the azimuthal
123
electric field profile and is determined by the azimuthal segments of the antenna. At higher
magnetic fields the axial electric fields follows the axial antenna profile. The electron density at
the three positions is shown in Fig. 5.5 for varying static magnetic fields. For low magnetic
fields, the electron density peaks on axis near the antenna at about 3.4 × 1011 cm-3. As the static
magnetic field increases, the electron mobility decreases and the electron density profiles reflect
the asymmetry of the electric field intensities. At 300 G, the electron density peaks near the
antenna at about 2.2 × 1012 cm-3.
From the basic theory of helicon waves, it is expected that the electric field for an m = 1
mode rotates in space without changing shape. A stationary observer would see the m =1 pattern
rotating counterclockwise in time when viewed from the direction of the static magnetic field.
The phase of the axial electric field for 50 G at the “antenna” slice is shown in Fig. 5.6(a). While
three consecutive “upstream” slices are shown in Fig. 5.6(b), (c), and (d) . The phase fronts
show the direction of propagation. At the antenna height, Fig. 5.6(a), the axial electric field
propagates radially inward and there is no significant azimuthal variation. While at the upstream
slices, the propagation is not only axial, along the magnetic field lines, but also exhibits an
azimuthal rotation. The dark circle in Fig. 5.6(b), (c) and (d) represents a position of constant
phase. At positions progressively further from the antenna, Fig. 5.6(c) and (d), rotation of the
axial phase occurs in a counterclockwise direction, as viewed along the magnetic field line. At
higher static magnetic fields rotation of the axial field occurs further from the antenna. The
phase of the axial electric field for 300 G at the “antenna” slice is shown in Fig. 5.7(a). Again
near the antenna the propagation of the wave is radially inward. At three consecutive “upstream”
slices, shown in Fig. 5.7(b), (c), and (d), significant azimuthal rotation occurs. Stronger
magnetic fields produce a similar angular rotational speed, as indicated by the final position of
124
the dark circle in Fig. 5.7(c). However the structure in the azimuthal variation is more complex,
than the 50 G case.
The reactor averaged electron densities as a function of magnetic field for an m = 0 and
and m = 1 modes are shown in Fig. 5.8(a). The average electron density for an m = 1 mode at
300 G is approximately 1.5 × 1012 cm-3. The average electron density for an m = 0 mode is
typically 1.5 times larger than the m =1 mode. Typically the bulk plasma is heated through the
azimuthal electric field component. For a constant total power deposition and a solenoidal
magnetic field, most of the power is transferred to the plasma by the azimuthal and radial electric
fields for an m = 0 mode. The amplitudes are typically higher than those produced by an m = +1
mode. However, the m = +1 mode produces a stronger axial electric field than an m = 0 mode at
the cost of reducing the radial and azimuthal electric field amplitudes. The larger azimuthal and
radial fields in the m = 0 mode allow for a larger electron density.
The electron density 20 cm downstream of the antenna as a function of magnetic field for
a m = 0 and m = 1 mode is shown in Fig. 5.8(b). Here the electron density varies between 1 and
2 × 1012 cm-3 over a 600 G range for both modes. The m = 1 mode tends to produce a slightly
higher electron density than the m = 0 mode. The axial electric field determines the amount of
power deposited downstream. For the m = +1 mode significant propagation occurs downstream
with complex azimuthal rotation with increasing magnetic field as shown in Figs. 5.6 and 5.7.
5.3 Results for an Experimental Helicon Tool
Simulations were also conducted in a geometry based on the PlasmaQuest helicon tool
which uses a Trikon Technologies, Inc., M0RI helicon plasma source, shown in Fig. 5.9. The
simulation geometry is schematically shown in Fig. 5.10. Processing gas is injected through a
125
nozzle located below the electromagnets and is exhausted through a pump port located around
the outside diameter of a grounded substrate. The quartz bell jar is surrounded by
electromagnets that produce a solenoidal magnetic field inside the bell. A lower coil is placed
downstream to provide an axial magnetic field throughout the chamber. Magnetic flux lines are
shown on the right side of Fig. 5.10. The cited magnetic fields are for an on axis location
midway between the antennas. The system is powered by two ring coils surrounding the bell jar.
Each coil operates at 13.56 MHz and are 180° out of phase. Base case results have operating
conditions of Ar gas at 3 mTorr, 50 sccm, and a power deposition of 1 kW.
The grid for the simulation was 52 × 30 × 88 in (r,θ,z) covering a range of 20 cm × 90 cm
in (r,z). For the sake of computational time and memory allocation, the resolution used was very
coarse in the axial direction (1.02 cm/grid cell). When simulating wave effects sufficient grid
points are needed to resolve wave propagation, typically 40-50 points per wavelength. For a
chamber this size, this is not practical due to memory allocation. The amount of memory used to
store the non-zero elements in the ‘A’ matrix (shown in Eq. (5.7)) is 52 × 30 × 88 grid points × 3
electric field components × 9 non-zero elements in the ‘A’ matrix/per variable (shown in Eq.
(5.7)) × 16 bytes/variable = 60 MB. The results shown are converged to within 1% and are used
to simply illustrate the difference in electric fields produced by the different modes of operation.
The total electric field amplitude and power deposition is shown in Fig. 5.11(a) for
magnetic fields of 10 G. The total electric field is inductively coupled with the amplitude
decreasing evanescently and being limited by the conventional plasma skin depth, as shown in
the left side of Fig. 5.11(a). Power deposition follows electric field propagation, as shown in the
right side of Fig. 5.11(a). The axial electric field and the corresponding phase is shown in Fig.
5.11(b). The axial field is at least an order of magnitude smaller than the azimuthal electric field,
126
with a peak near the coils of 0.3 V/cm. Similar to the results for the solenoidal reactor, the phase
distributions show that a radial traveling wave dominates the propagation of the electromagnetic
fields through the bell jar region and away from the coil. As the magnetic field is increased to
300 G, further penetration of the total electric field into the plasma occurs, as shown in the left
side of Fig. 5.12(a). Power deposition, shown in the right side of Fig. 5.12(a), penetrates further
into the bell jar region. The axial electric field, shown in the left side of Fig. 5.12(b), increases
within the volume of the plasma. Downstream, the axial and azimuthal electric fields are
comparable, but not large enough to deposit sufficient power. The phase shows significant
propagation far from the antenna region, as shown in the right side of Fig. 5.12(b).
The electron temperatures and electron densities are shown for 10 G and 300 G in Fig.
5.13. At 10 G, the electron temperature peaks at 4 eV near the antenna, at the position of peak
power deposition. The electron density, shown in the right side of Fig. 5.13(a), peaks at 1.1 ×
1012 cm-3 in the bell jar region and diffuses into the downstream chamber region. As the
magnetic field is increased to 300 G, the electron temperature peaks in the volume of the bell jar
at a value of 3.2 eV. As the magnetic field increases the electron density increases reflecting a
decrease in radial diffusion losses. Furthermore, due to increased axial conductivity, the off axis
peak is maintained 15 cm away from the antenna location into the downstream chamber, as
shown in the right side of Fig. 5.13(b).
Ruzic and Norman8 have measured radial electron densities and electron temperatures
from the PlasmaQuest tool, shown in Fig. 5.13. Data was taken using a PMT FastProbe, which
is located 16.2 cm below the top of the main chamber, shown in Fig. 5.9. Experiments were
conducted at 3 mTorr, 0.8-1.0 kW, and 50 sccm. Coil currents were changed to create a
divergent magnetic field and a solenoidal magnetic field. The divergent magnetic field was
127
created having 40 A through the inner top magnetic coil and no current in the other coils. The
solenoidal magnetic field, similar to that shown in Fig. 5.10, was created by having 40 A through
both upper magnetic coils and 80 A through the lower magnetic coil. Experimental and
simulation results are also shown in Fig. 5.14 for a N2 plasma. For a divergent field, electron
densities and temperatures tend to be more radially uniform. As the magnetic field is increased
and the magnetic flux lines become more axial, electron densities and temperatures tend to
maintain radial non-uniformities in the downstream region.
Simulations were again conducted in the PlasmaQuest geometry previously described,
however the helicon source was replaced with a Nagoya Type III antenna of the kind shown in
Fig. 5.1(b). . Such an antenna has both azimuthal and axial antenna components which produce
an m = +1 mode. The total electric field and axial electric field, at 300 G, are shown in Fig. 5.15
at several axial positions in the reactor. The “antenna” slice is cross section of (r,θ) located
axially at the antenna position. The slices shown are cross sections progressively downstream 15
cm apart, were the bottom slice is 60 cm from the antenna. The total electric field at the
“antenna” slice peaks at the axial segments of the antenna, with a smaller peak at the azimuthal
segments of the antenna. The total electric field profile in the progressively downstream slices
follows the azimuthal antenna profile and can propagate downstream having a value of 0.07
V/cm at an axial position 60 cm away from the antenna. The axial electric field profile follows
the total electric field profile and can propagate into the downstream region.
The power deposition and corresponding electron temperature are shown in Fig. 5.16 at
several axial positions in the reactor. Power deposition follows electric field profiles and peaks
near the axial segment of the antenna at about 4.8 W/cm-3. However, downstream, the power
deposition profile is determined by the azimuthal segment of the antenna. Due to the increased
128
axial conductivity, significant power can be deposited downstream. The electron temperature
peaks near the azimuthal segments of the antenna at approximately 5 eV and maintains this
asymmetry into the downstream region. The excited Ar* species and the electron density are
shown in Fig. 5.17. Both reach peak values of 2 × 1012 cm-3 at the antenna location. However
the Ar* density profile has four azimuthal peaks, having contributions to both axial and
azimuthal segments of the antenna. However the electron density reflects the electron
temperature profile and has two asymmetric peaks where the electron temperature is lowest. At
300 G, all species tend to maintain their asymmetric profiles far into the downstream.
5.4 Conclusions
Electromagnetic wave propagation in magnetically enhanced inductively coupled
plasmas (MEICPs) enables power deposition to occur remotely from the coils and at locations
beyond the classical skin depth. Three dimensional, azimuthally symmetric components of the
electric field can be produced by an azimuthally symmetric (m = 0) antenna solenoidal static
magnetic fields. Asymmetric antennas (m = +1) produce 3-D components of the electric field
lacking any significant symmetries, and so must be fully resolved in three dimensions.
To investigate these processes, a 3-D plasma equipment model was improved to resolve
3-D components of the electric field produced by m = +1 antennas in solenoidal magnetic fields.
A tensor conductivity was used to couple the components while solving the wave equation in the
frequency domain using an iterative, sparse matrix technique. For magnetic fields of 10-600 G,
rotation of the electric field was observed downstream of the antenna where significant power
deposition also occurs. Feedback from the plasma that produces local extrema in conductivity
129
(e.g., ionization rates and electron temperatures peak where fields are largest) result in the
electric field patterns not having pure modal content.
Simulations in process chambers show significant propagation of the electric fields far from
the antenna for both symmetric and asymmetric antennas at high magnetic field (150 G). The
asymmetric antenna (m = +1) produces larger downstream plasma densities than the azimuthally
symmetric (m = 0) antenna. This was attributed to the enhanced rotation of the axial fields
downstream. The higher densities can at the cost of increased non-uniformity, since the m = +1
maintained significant asymmetries in the species’ profiles downstream.
130
RADIUS (cm)* (Not to Scale)
0 66
HE
IGH
T (c
m)
0
130
Coils
Glass
Substrate
Solenoid
Nozzle
RADIUS (cm)* (Not to Scale)
0 66
HE
IGH
T (c
m)
0
130
Coils
Glass
Substrate
Solenoid
Nozzle
m = 0 m = 1
Antenna Slice
Upstream Slice
Downstream Slice
17 cm
17 cm
a) b)
Figure 5.1 The computational geometry for 3-d studies consists of a cylindrical quartz tubeimmersed in a solenoidal magnetic field. (a) The two turn antenna with opposing currentsproduces an m = 0 helicon mode. (b) The Nagoya Type III antenna produces an m = 1 heliconmode. The arrows show the direction of current in the antenna.
131
a)
AZIMUTHALANTENNA
AXIALANTENNA
AZIMUTHALANTENNA
0.5
0.3
0.1
b)
Figure 5.2 Patterns of the electric field lines for the m = 1 mode in a cylindrical tube (a) fromChen in a uniform plasma [1], (b) from HPEM-3D simulations with electric field intensities inV/cm.
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100 G50 G0 G
a) b) c)
300 G 600 G
d) e)
(V/cm )
20
0.02
Figure 5.3 The total electric field amplitude at three positions in the cylindrical tube shown inFig. 5.1(b) for an m = 1 mode at a static magnetic field of (a) 0 G, (b) 50 G, (c) 100 G, (d) 300 G,and (e) 600 G.
133
100 G50 G0 G
a) b) c)
300 G 600 G
d) e)
(V/cm )
2
0.002
Figure 5.4 The axial electric field amplitude at three positions in the cylindrical tube shown inFig. 5.1(b) for an m = 1 mode at a static magnetic field of (a) 0 G, (b) 50 G, (c) 100 G, (d) 300 G,and (e) 600 G.
134
100 G50 G0 G
a) b) c)
300 G 600 G
d) e)
(cm-3)
10 12
1010
Figure 5.5 The electron density at three positions in the cylindrical tube shown in Fig. 5.1(b) foran m = 1 mode at a static magnetic field of (a) 0 G, (b) 50 G, (c) 100 G, (d) 300 G, and (e) 600G.
135
a) b)
c) d)
6.280
Radians
Figure 5.6 Phase of axial electric field in solenoidal geometry for an m = 1 mode at 50 G at (a)antenna slice and at three consecutive upstream slices (b) 17 cm, (c) 17.6 cm and (d) 18.2 cm.The dark circle represents a position of constant phase, which rotates counterclockwise as theslices progress further from the antenna.
136
a) b)
c)d)
6.280
Radians
Figure 5.7 Phase of axial electric field in solenoidal geometry for an m = 1 mode at 300 G at (a)antenna slice and at three consecutive upstream slices (b) 17 cm, (c) 17.6 cm and (d) 18.2 cm.The dark circle represents a position of constant phase, which rotates counterclockwise as theslices progress further from the antenna
137
1012
(cm
-3)
Magnetic Field (G)0 200 400 600
0
1
2
3
m = 1
m = 0
a)
Magnetic Field (G)
1012
(cm
-3)
0 200 400 6000
1
2
3
m = 1
m = 0
b)
Figure 5.8 (a) The average electron density in the cylindrical tube for an m=0 and m=1 modes.(b) Electron density at 20 cm downstream from the antenna for an m=0 and m=1 modes.
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MAGNETICCOILS
RF COILS
QUARTZ
LOWERMAGNET
SUBSTRATE
PUMPPORT
Figure 5.9 Schematic of experimental helicon tool. The PlasmaQuest helicon tool uses a TrikonM0RI helicon source operating at 13.56 MHz.
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RF COILS
MAGNETICCOILS
LOWERCOIL
INLET
PUMPPORT
SUBSTRATE
GLASS
Figure 5.10 Schematic of simulation geometry based on the PlasmaQuest diagram shown in Fig.5.10. Streamlines represents magnetic flux lines in the reactor. Contour lines (red) representmagnetic field strength.
140
90
020 0 20
Radius (cm)
Hei
ght (
cm)
V/cm30
5 0.05W/cm3
0.03
ELECTRIC FIELD POWER
a)
90
020 0 20
Radius (cm)
Hei
ght (
cm)
V/cm0.3
2π 0Radians
0.003
AXIAL FIELD PHASE
b)
Figure 5.11 a) Total electric field amplitude and power deposition. b) Total axial electric fieldamplitude and axial electric field phase in the PlasmaQuest reactor for 10 G, Ar at 3 mTorr
141
90
020 0 20
Radius (cm)
Hei
ght (
cm)
V/cm20
5 0.05W/cm3
0.02
ELECTRIC FIELD POWER
a)
90
020 0 20
Radius (cm)
Hei
ght (
cm)
V/cm0.3
2π 0Radians
0.003
AXIAL FIELD PHASE
b)
Figure 5.12 a) Total electric field amplitude and power deposition. b) Total axial electric fieldamplitude and axial electric field phase in the PlasmaQuest reactor for 300 G, Ar at 3 mTorr
142
90
020 0 20
Radius (cm)
Hei
ght (
cm)
eV4 0.4
ELECTRON TEMP.
ELECTRON DENSITY
1.3 x 1012 1.3 x 1010cm-3
a)
90
020 0 20
Radius (cm)
Hei
ght (
cm)
eV4 0.4
ELECTRON TEMP.
ELECTRON DENSITY
1.3 x 1012 1.3 x 1010cm-3
b)
Figure 5.13 The electron temperature and electron density in the PlasmaQuest reactor for a) 10G and b) 300 G, Ar at 3 mTorr.
143
8
7
6
5
4
3
2
1
00 5 10 15
0
1
2
3
4
5
6
Electron T
emperature (eV
) E
lect
ron
Den
sity
(cm
-3 x
101
0)
Radius (cm)
Experimental TeExperimental ne
Simulation TeSimulation ne
a)
10
8
6
4
00 5 10 15
0
1
2
3
4
5
6
Electron T
emperature (eV
) E
lect
ron
Den
sity
(cm
-3 x
101
0)
Radius (cm)
2
Experimental TeExperimental ne
Simulation TeSimulation ne
b)
Figure 5.14 Experimental measurements (Ruzic and Norman)8 and simulation results of theelectron temperature and electron density in the PlasmaQuest reactor for a) divergent magneticflux and b) solenoidal magnetic flux. The error on the measured electron densities is 12%, whilefor the electron temperature it is 6%.
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V/cm30 0.03
Antenna
Slice 1
Slice 2
Slice 3
Slice 4
Total Electric Field Axial Electric Field
Figure 5.15 The total electric field and the axial electric field in the PlasmaQuest reactor using aNagoya Type III (m = +1) antenna at 300 G, Ar at 3 mTorr. Slices are 15 cm apart axially.
145
W/cm-35 0.05
Antenna
Slice 1
Slice 2
Slice 3
Slice 4
6 0.6eV
Power Deposition Electron Temperature
Figure 5.16 The power deposition and the electron temperature in the PlasmaQuest reactor usinga Nagoya Type III (m = +1) antenna at 300 G, Ar at 3 mTorr. Slices are 15 cm apart axially.
146
cm-32 x 1012
Antenna
Slice 1
Slice 2
Slice 3
Slice 4
2 x 1010
Ar* Density [e] Density
Figure 5.17 The excited Ar* density and the electron density in the PlasmaQuest reactor using aNagoya Type III (m = +1) antenna at 300 G, Ar at 3 mTorr. Slices are 15 cm apart axially.
147
5.5 References
1. F. F. Chen, J. Vac. Sci. Technol. A 10, 1389 (1992).
2. F. F. Chen and R. W. Boswell, IEEE Trans. Plasma Sci. 25, 1245 (1997).
3. F. F. Chen, X. Jiang and J. D. Evans, J. Vac. Sci. Technol. A 18, 2108 (2000).
4. D. G. Miljak and F. F. Chen, Plasma Sources Sci. Technol. 7, 61 (1998).
5. M. J. Kushner, W. Z. Collison, M. J. Grapperhaus, J. P. Holland and M. S. Barnes, J. Appl.Phys. 80, 1337 (1996).
6. M. J. Kushner, J. Appl. Phys. 82, 5312 (1997).
7. R. W. Freynd and N. M. Nachtigal, Siam J. Scientific Computing 15, 313 (1994).
8. D. Ruzic and J. Norman, Nuclear Radiation Laboratory, Department of Nuclear Engineering,University of Illinois-Urbana, (private communications), 2001.
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6 CONCLUSIONS AND FUTURE WORK
6.1 Conclusions
Magnetically Enhanced Inductively Coupled Plasma (MEICP) and helicon
sources for materials processing are of interest because of their ability to deposit power
within the volume of the plasma beyond the classical skin depth. The location and
manner of power deposition can vary substantially depending on the mode of operation
and reactor conditions. The coupling of electromagnetic fields to the plasma typically
occurs through two channels; a weakly damped helicon-like wave that penetrates into the
bulk plasma and an electrostatic wave. The electrostatic wave can often be suppressed
resulting in the helicon component being responsible for the majority of the power
deposition.
A computational investigation was conducted to quantify this heating and
determine the conditions for which power can be deposited in the downstream region of
MEICP devices. To investigate the coupling of the electromagnetic radiation to the
plasma in MEICPs, algorithms were developed for wave propagation in the presence of
static magnetic fields using the two dimensional Hybrid Plasma Equipment Model
(HPEM). A full tensor conductivity was added to the Electromagnetics Module which
enables one to calculate three dimensional components of the inductively coupled electric
field based on two dimensional applied magnetostatic fields. Electromagnetic fields were
obtained by solving the wave equation where plasma neutrality was enforced. The
purpose of these investigations was to determine the effect of helicon heating and the
ability to deposit power in the downstream region of helicon devices. An effective
149
collision frequency for Landau damping was included.
For typical process conditions (10 mTorr, 1 kW ICP), in the absence of the
electrostatic term, electric field propagation progressively follows magnetic flux lines and
significant power can be deposited downstream. When the propagating wave encounters
a boundary, such as the substrate, a standing wave pattern in the direction of propagation
forms. Power deposition profiles reflect electromagnetic profiles in the reactor. The
ability to deposit power downstream with increasing B-field is ultimately limited by the
increasing wavelength. For example, if the plasma is significantly electronegative in the
low power-high magnetic field regime, power deposition resembles conventional ICP due
to the helicon wavelength exceeding the reactor.
MEICPs typically have a higher plasma density for a given power deposition than
ICP sources. Landau damping has been proposed as one mechanism through which more
efficient heating may occur. In this process, energetic primary electrons are produced
through trapping and acceleration by a helicon wave. The electrons produce ionization,
lowering their energy and generating a low energy secondary. The wave reaccelerates
electrons after each ionization event. To investigate the coupling of the electromagnetic
waves to the plasma in MEICPs and their effects on electron energy distributions
algorithms were developed. Three dimensional electromagnetic fields generated in the
EMM were used in the Electron Monte Carlo Simulation (EMCS) of the HPEM. These
fields were used in a EMCS to generate EEDs, transport coefficients and electron impact
source functions
The tail of the EEDs were found to increase in magnitude in the downstream
chamber for B > 150 G and P < 5 mTorr indicating some amount of electron trapping and
150
wave heating. This heating results from non-collisional acceleration by the axial electric
field for electrons in the tail end of the EED which have long mean-free-paths while low
energy electrons are still somewhat collisional. The mean free path for electrons above
15 eV is between 10 –20 cm at 2 mTorr. The increase in the high energy tail of the EED
was a result of better phase matching of thermal electrons to the phase velocity of the
wave. Electrons whose speeds are close to the phase velocity are efficiently heated and
subsequently deposit their power downstream through collisions with the background
gas. These results indicate that the helicon component of the wave produces significant
nonlocal heating.
The TG mode was resolved by including the divergence electrostatic term in the
solution of the wave equation. The electrostatic component in the wave equation was
approximated by a harmonically driven perturbation of the electron density. Results
indicate that strongly damped electrostatic waves can reach the plasma core at low
magnetic fields (< 20 G), while at high magnetic fields (>150 G) the electrostatic waves
deposit power primarily at the periphery of the plasma column. These results indicate
that the electrostatic component plays a more important role in the power deposition
mechanisms at lower magnetic fields by restructuring the power deposition profile. The
propagation of the helicon component is little affected, particularly at large magnetic
fields where the electrostatic component is damped.
Asymmetric antennas (m = +1) produce 3-D components of the electric field
lacking any significant symmetries, and so must be fully resolved in three dimensions.
To investigate these processes, a 3-D plasma equipment model was improved to resolve
3-D components of the electric field produced by m = +1 antennas in solenoidal magnetic
151
fields. A tensor conductivity was used to couple the components while solving the wave
equation in the frequency domain using an iterative, sparse matrix technique. For
magnetic fields of 10-600 G, rotation of the electric field was observed downstream of
the antenna where significant power deposition also occurs. Feedback from the plasma
that produces local extrema in conductivity result in the electric field patterns not having
pure modal content.
Simulations in process chambers show significant propagation of the electric fields
far from the antenna for both symmetric and asymmetric antennas at high magnetic field
(150 G). The asymmetric antenna (m = +1) produces larger downstream plasma densities
than the azimuthally symmetric (m = 0) antenna. This was attributed to the enhanced
rotation of the axial fields downstream. The higher densities can at the cost of increased
non-uniformity, since the m = +1 maintained significant asymmetries in the species’
profiles downstream.
To fully understand wave coupling to plasma properties investigations of asymmetric
antenna designs and continual collaboration with experimental facility must continue.
Validation of the model can be readily obtained from the experimental tool located in the
Nuclear Engineering Department at the University of Illinois at Urbana-Champaign.
Furthermore, improvement of HPEM-3D to include electrostatic term in solution of wave
equation must be completed to obtain a self-consistent solution of the wave propagation
of an m = +1 mode.
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VITA
Ronald Leonel Kinder Oxom was born in Chicago, Illinois in 1973. He attended
the University of Illinois at Urbana-Champaign, where he received two B. S. degrees, in
Engineering Physics and Nuclear Engineering, in 1997. He also received his M. S. at the
University of Illinois at Urbana-Champaign in 1998. In 1997, he received the
Semiconductor Research Corporation Master’s Scholarship. Under the direction of Prof.
Mark J. Kushner he develop theoretical 2- and 3-dimensional simulations of low
temperature plasmas used in etching and deposition for the fabrication of
microelectronics. Projects involved simulations of downstream processing systems
sustained by inductively coupled and microwave discharges. His work has resulted in
four reviewed papers and eleven conference presentations. He has also worked as a
teaching assistant in several engineering courses and as an industrial consultant were he
holds one patent. He is a member of IEEE, American Vacuum Society, Society of
Hispanic Professional Engineers and Phi Beta Kappa honor society. During his time at
the University of Illinois at Urbana-Champaign he worked at the College of Engineering
as Assistant to the Dean. He also worked in the Minority Engineering Program as an
academic adviser, at the Latino Cultural Center as graduate student counselor and headed
several tutoring projects at the Office of Minority Student Affairs. In the summer of
2001, he will begin work at Novellus Systems, Inc., in San Jose, California.