Effect of low frequency voltage waveform
on plasma uniformity in a dual-frequency
capacitively coupled plasma
Cite as: J. Vac. Sci. Technol. B 40, 032202 (2022); https://doi.org/10.1116/6.0001732
Submitted: 30 December 2021 • Accepted: 15 March 2022 • Published Online: 04 April 2022
Shahid Rauf, Peng Tian, Jason Kenney, et al.
COLLECTIONS
Paper published as part of the special topic on Plasma Processing for Advanced Microelectronics
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J. Vac. Sci. Technol. B 40, 032202 (2022); https://doi.org/10.1116/6.0001732
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avs.scitation.org/journal/jvb
Effect of low frequency voltage waveform
on plasma uniformity in a dual-frequency
capacitively coupled plasma
Cite as: J. Vac. Sci. Technol. B 40, 032202 (2022); doi: 10.1116/6.0001732
Submitted: 30 December 2021 · Accepted: 15 March 2022 ·
Published Online: 4 April 2022
Shahid Rauf,a)
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Peng Tian, Jason Kenney, and Leonid Dorf
AFFILIATIONS
Applied Materials, Inc., 1140 E. Arques Ave., Sunnyvale, California 94085
Note: This paper is a part of the Special Topic Collection on Plasma Processing for Advanced Microelectronics.
a)
Author to whom correspondence should be addressed: shahid_rauf@amat.com
ABSTRACT
In a dual-frequency capacitively coupled plasma (CCP) with disparate frequencies, the low frequency (LF) voltage usually has a strong influence on the ion energy distribution function (IEDF) but contributes less to plasma generation. It is well-known that rectangular LF voltage
waveform with a small positive period yields a narrow, nearly monoenergetic IEDF. This paper focuses on the effect of the LF voltage waveform on plasma uniformity in a low-pressure dual-frequency (40 + 0.8 MHz) CCP. A two-dimensional particle-in-cell model is used for this
investigation, and the effect of LF voltage amplitude on plasma uniformity is investigated for sinusoidal and rectangular voltage waveforms.
When the LF voltage is low, the peak in plasma density is at the chamber center due to ample diffusion at the low pressure considered
(20 mTorr) and higher losses to the chamber walls. As the LF voltage is increased, the sheath gets thicker at the powered electrode and
charged species densities decrease for a constant 40 MHz voltage. The plasma profile, however, evolves differently for the two LF voltage
waveforms. With sinusoidal LF voltage, the plasma spreads out between the electrodes. On the other hand, with rectangular LF voltage
waveform, the plasma splits into two regions: a density peak at the chamber center and another peak near the electrode edge. This doublepeaked density profile with a rectangular wave can be attributed to the location and timing of plasma generation. 40 MHz produces plasma
most efficiently when the LF rectangular wave is positive and the sheath at the powered electrode is thin (frequency coupling). This plasma
is produced uniformly between the electrodes, but only for a short period. When the LF voltage becomes negative, the sheath expands at the
powered electrode and the plasma is produced near the electrode edge where the sheath is thinner and the electric field is stronger.
Published under an exclusive license by the AVS. https://doi.org/10.1116/6.0001732
I. INTRODUCTION
Low-pressure multifrequency capacitively coupled plasmas
(CCPs)1 are used for many plasma processing applications in the
semiconductor industry. A common methodology is to use two disparate frequencies where the higher frequency primarily contributes to plasma production while the low frequency (LF) is used to
control the ion energy distribution function (IEDF).2,3
Considerable attention has been paid recently to the use of nonsinusoidal LF waveforms to control the IEDF. In particular, it has
been demonstrated that rectangular voltage waveform with a small
positive period yields a narrow, nearly monoenergetic IEDF.4 Even
though the LF source might not significantly contribute to plasma
production directly, it has considerable influence on the plasma
J. Vac. Sci. Technol. B 40(3) May/Jun 2022; doi: 10.1116/6.0001732
Published under an exclusive license by the AVS
through the control of the sheaths in a CCP. The goal of this article
is to use a two-dimensional cylindrical geometry particle-in-cell
(PIC) plasma model to understand how the LF waveform influences the plasma spatial properties in an axisymmetric cylindrical
plasma reactor. PIC model has been found necessary for this study
as electron kinetic effects, which are important in all low-pressure
CCPs, become even more significant for rectangular voltage waveform with sharp changes in voltage.
There is considerable literature on the design and modeling of
multifrequency CCPs as well as their PIC modeling. We limit our
literature review in this article to two pertinent topics: the use of
nonsinusoidal voltage waveforms to control the IEDF and 2D PIC
modeling of CCPs. The early work on the use of nonsinusoidal
voltage waveforms to control the IEDF was done in inductively
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coupled plasmas.4,5 Wang and Wendt4 experimentally demonstrated
that a rectangular voltage waveform can be used to get a narrow
IEDF. They also showed how these IEDFs can be used to control
selectivity during plasma etching,6 a feature that is particularly
helpful for atomic layer etching.7 Nonsinusoidal voltage waveforms
can be obtained by combining multiple harmonics. Czarnetzki et al.8
described this approach in the context of the electrical asymmetry
effect in CCPs and showed that multiple harmonics could be used to
obtain a narrow IEDF. Schüngel et al.9 elaborated on the method to
generate single peak ion flux energy distribution using waveform tailoring. Delattre et al.10 presented detailed diagnostics and modeling
results to examine the effect of waveform tailoring on Ar plasma
characteristics at low pressures. Lafleur’s review on tailored waveform
excitation of CCPs (Ref. 11) excellently summarizes the use of waveforms generated using a combination of harmonics to control
plasma properties, in particular the DC self-bias. Franek et al.12
described a new radio frequency (RF) power supply and impedance
matching method suitable for customized voltage waveforms.
Schmidt et al.13 and Wang et al.14 further developed multifrequency
matchboxes based on series and parallel LC circuits that are suitable
for waveform tailoring. Brandt et al.15,16 experimentally and numerically investigated electron heating in CF4 and Ar/CF4 CCP discharges excited using multiple consecutive harmonics of 13.56 MHz.
Derszi et al.17 investigated power coupling in O2 CCP discharge generated using a combination of harmonics. Hartmann et al.18 have
recently modeled CCPs with high voltage tailored voltages and
shown that tailored waveforms can also be used to control the electron velocity distribution, especially at high energies. These energetic
electrons generated during electric field reversal in the sheath can be
useful for high aspect ratio dielectric etching, as discussed by Krüger
et al.19,20
Many previous studies have addressed 2D PIC modeling of
CCPs. Kim et al.21 modeled high-pressure CCPs in Cartesian
geometry highlighting the plasma nonuniformities at the electrode
edges. 2D simulations are more challenging in cylindrical geometry
due to small differential volume and consequent large statistical
error near the axis. The early 2D cylindrical-geometry model of
dual-frequency CCP by Wakayama and Nanbu22 is noteworthy in
this regard. Their published results illustrate the effect of interelectrode spacing on plasma uniformity, especially near the axis. Wang
et al.23–25 published a series of articles about a 2D implicit PIC
model of CCPs. They examined issues related to plasma uniformity
and compared PIC and fluid plasma modeling results. Rauf26
described a 2D PIC model for an axisymmetric cylindrical
chamber, validated modeling results using experiments and examined kinetic effects influencing plasma transport. Wang et al.27–29
used a 2D PIC model to examine the electric asymmetry effect in a
geometrically asymmetric CCP reactor and investigated the effect
of plasma reactor geometrical asymmetry.
This article is organized in the following manner. The computational model is described in Sec. II. Modeling results are discussed in Sec. III, and Sec. IV includes a summary.
II. COMPUTATIONAL MODEL
The investigation in this article has been conducted using a
modified version of the plasma model described in Ref. 26. In this
J. Vac. Sci. Technol. B 40(3) May/Jun 2022; doi: 10.1116/6.0001732
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model, charged species are simulated using PIC while neutral
species are treated as a fluid. To briefly recap the modeling methodology, we start with assumed initial densities, connect appropriate RF sources to the electrodes, and simulate the evolution of
charged and neutral species until all major quantities converge to
steady-state values. The PIC model (of charged species) and fluid
model (for neutral species) are coupled at each time step. The PIC
model is electrostatic and electromagnetic standing wave effect30
has been neglected as the chamber dimensions are significantly
smaller than the vacuum wavelength at the relevant frequencies
(0.8 and 40 MHz). Our PIC model for charged species generally
follows the methodology outlined in Refs. 31–34. This PIC model
uses the Monte Carlo method to account for collisions. Current at
the powered electrode is used in a circuit model to compute voltage
drop across the external blocking capacitor. Continuity equation is
solved for neutral species, which includes diffusion and production/destruction. This methodology makes it practical to simulate
plasmas of molecular gases using this plasma modeling code where
neutral radicals can be an important constituent of the gas mixture.
For the main gas considered in this article (Ar), one benefit of considering the neutral continuity equation is to take account of the
two-step ionization process (Ar → Ar* → Ar+), with Ar* density
being self-consistently computed. This two-step process is responsible for 1.7%–3% of electron production in the simulations reported
in this article. Solution of neutral continuity equations includes
numerical acceleration to speed up evolution to steady-state conditions. The Poisson equation is solved using iterative sparse matrix
techniques, utilizing the PETSC library.35 The code has been parallelized using MPI.36
All simulations reported in this article have been done for Ar.
Our Ar plasma chemistry mechanism is described in Ref. 26. It
includes elastic, metastable excitation, and ionization collisions for
Ar; ionization and deexcitation reactions for Ar* metastable; and
momentum transfer collisions for Ar+ ions. Both back-scattering
and isotropic collisions are considered for Ar+ ions using the cross
sections proposed by Phelps.37 Surface processes have been shown
to be important for overall plasma dynamics, but their impact is
less significant at low pressure.38,39 To focus on the dynamics of
bulk plasma electrons and due to the low gas pressure, we have not
considered secondary electron emission in the current set of simulations. Electrons are assumed to get fully absorbed at the surfaces
with a reflection probability of 0.0.
One challenge with 2D PIC simulations in cylindrical geometry
is the small differential volume near the axis, which requires a large
number of computational particles to adequately resolve the plasma
characteristics near r = 0. To keep the total number of particles manageable, we have used nonuniform weight for particles, where weight
is defined as the actual number of electrons or ions per PIC macroparticle. Initial particle weight (wi) is proportional to W,
h
i2
r
W(r) ¼ (1 α) þ α ,
R
(1)
where R is the maximum radius in the geometry, α is a constant,
and r is the radial location of the particle. α = 0.05 for the 2D simulations reported in this article. As particles move to smaller radii
during the simulation and wi/W(r) exceeds 2, the particles are split
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into two particles with ½ the weight but otherwise identical particle
characteristics.
The smaller weight particles can move to larger radii as well,
and particle merging is important to avoid unrestricted growth in
the number of particles. We describe our particle merging algorithm in the Appendix. Some kinetic information is always lost
when particles are merged. To avoid energy distribution distortion
induced by particle merging, particles are merged sparingly: once
per 50 000 time steps in the simulations reported.
A slightly nonuniform mesh is used for the simulations
reported in this article with dx and dz of approximately 1 mm. In
the absence of energetic secondary electrons, we have used
dt = 50 ps, which satisfies the Courant condition for the electrons in
the plasma. The time-averaged modeling results presented in this
article have been averaged over the last 800 kHz cycle (=50 cycles
of 40 MHz) of the simulation.
III. MODELING RESULTS
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for sinusoidal and rectangular LF voltages in Fig 2(b) for the cases
VLF = 900 V; V40 = 250 V for the sinusoidal LF voltage simulation
and V40 = 400 V for the rectangular LF wave. These voltages have
been plotted for the final 1.25 μs of the simulation when plasma
and electrical properties have reached periodic steady-state. As the
chamber is electrically asymmetric, a DC bias also builds up at the
capacitor, which appears at the electrode. In addition, as the mean
voltage of the rectangular wave in Fig. 2(a) is not zero, the DC
component of the rectangular wave is also blocked by the capacitor.
All simulations reported in this article have been done for
250 μs (10 000 cycles of 40 MHz, 200 cycles of the LF waveform)
for VLF in the range of 500–900 V. V40 = 250 V for sinusoidal LF
bias and V40 = 400 V for rectangular wave LF bias for reasons
explained later in this article. The plasma and electrical characteristics are observed to reach periodic steady-state in 250 μs for all conditions. To illustrate convergence of the simulations, we have
plotted spatially averaged electron density (ne), Ar* density (nAr*),
and voltage across the blocking capacitor (VC) in Fig. 3. The results
The two-dimensional (2D) simulations in this article have
been done for the plasma reactor shown in Fig. 1. A combination
of fHF = 40 MHz and low frequency (fLF = 800 kHz) bias voltage
Vs(t) = V40 sin(2πfHFt) + VLFGLF(t) is applied to the bottom electrode. The top electrode and the chamber sidewalls are grounded.
A blocking capacitor C = 2 nF is also connected to the bottom electrode. The two electrodes have a radius of 8 cm and are separated
by 2.5 cm. All simulations have been done for Ar plasma at
20 mTorr.
The simulations have been done using the sinusoidal and rectangular LF voltage waveforms GLF(t) shown in Fig. 2(a). The duty
cycle of the rectangular wave (fraction of time at the most negative
voltage) is 80%, and the voltage rise and fall times are 50 ns. The
voltage magnitude is scaled to the level specified for individual simulations. The combined voltage at the powered electrode is shown
FIG. 1. Schematic of the capacitively coupled plasma reactor used for
simulations in this article.
J. Vac. Sci. Technol. B 40(3) May/Jun 2022; doi: 10.1116/6.0001732
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FIG. 2. (a) Low frequency 800 kHz voltage waveforms used in the simulations.
These waveforms are scaled to the voltages specified in individual simulations.
(b) Voltage at the powered electrode during the final 800 kHz time period of the
simulation. These waveforms correspond to 400 V 40 MHz + 900 V rectangular
wave and 250 V 40 MHz + 900 V sinusoidal wave simulations. Simulations have
been done for Ar plasma at 20 mTorr.
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FIG. 3. Spatially averaged electron density (ne), density of Ar* (nAr*), and
voltage on the capacitor (VC) for (a) 400 V 40 MHz + rectangular 700 V LF bias
and (b) 250 V 40 MHz + 700 V sinusoidal LF voltage. These simulations have
been done for Ar plasma at 20 mTorr.
in Fig. 3(a) are for 400 V 40 MHz + rectangular 700 V voltage, and
Fig. 3(b) results are for 250 V 40 MHz + 700 V sinusoidal LF
voltage. VC is negative for the sinusoidal LF bias due to the smaller
size of the powered electrode compared to the grounded surfaces.
VC is negative but smaller for the rectangular LF bias as the DC
component of the bias waveform drops across the blocking capacitor as well.
One of the main reasons for modifying the LF voltage waveform is to control the IEDF. We have shown the IEDF at both the
powered and grounded electrodes for rectangular wave LF bias in
Fig. 4 and for sinusoidal wave LF voltage in Fig. 5. Results are
shown for VLF = 500, 700, and 900 V. The IEDFs computed by our
PIC model have been previously validated against experimental
measurements,40 albeit at lower voltages. The IEDF at the powered
electrode has a sharp peak with rectangular wave LF bias
[Fig. 4(a)], and the location of this peak shifts to higher energy
with increasing bias voltage VLF. There is also significant population of low energy ions, which are produced during the positive
voltage phase of the LF waveform and through collisions in the
thick sheath during the negative LF voltage phase. For similar LF
voltages, the IEDF is wider with sinusoidal bias voltage [Fig. 5(a)]
and the low energy population is also higher than the rectangular
wave. These changes can be attributed to the constantly evolving
nature of the powered electrode sheath voltage, which results in a
bimodal IEDF at low frequencies. With a thick sheath, ion-neutral
collisions are, however, making the low energy peak broader. The
J. Vac. Sci. Technol. B 40(3) May/Jun 2022; doi: 10.1116/6.0001732
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FIG. 4. IEDF at the (a) powered electrode and (b) grounded electrode. These
results are for Ar plasma at 20 mTorr and 400 V 40 MHz. The 800 kHz LF
voltage is rectangular with an amplitude of 500, 700, or 900 V.
IEDF at the opposite grounded electrode is wider with rectangular
LF bias [Fig. 4(b)] compared to sinusoidal bias [Fig. 5(b)]. The
high energy ions for rectangular wave bias bombard the grounded
electrode during the positive phase of the rectangular wave while
the low energy ions leave the plasma when the powered electrode
LF voltage is negative.
We next discuss the 2D simulation results. Results are first
presented for both the rectangular and sinusoidal wave LF bias,
which is followed by a detailed physics discussion. The timeaveraged potential Φave is shown in Fig. 6 for a range of voltages
for the rectangular wave LF bias. These results are for V40 = 400 V
and 20 mTorr in Ar. The DC voltage at the powered electrode is
mentioned in the figure, and it increases in magnitude with
increasing VLF. The potential in the plasma bulk is reasonably
uniform. The radial gradient in the potential, especially for the
highest VLF simulations, results in a radial electric field. This radial
field is a consequence of the reactor geometry and plasma uniformity, but, in turn, influences plasma uniformity through ambipolar
diffusion. The sheath at the powered electrode gets thicker with
increasing VLF. Corresponding time-averaged electron density ne is
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FIG. 5. IEDF at the (a) powered electrode and (b) grounded electrode. These
results are for Ar plasma at 20 mTorr and 250 V 40 MHz. The 800 kHz LF
voltage is sinusoidal with an amplitude of 500, 700, or 900 V.
shown in Fig. 7. The electron (and Ar+) density decreases with
increasing VLF, which is consistent with previous data in the literature.18 Density reduction is due to plasma production in a smaller
region at larger VLF and enhanced plasma loss at the powered electrode. Spatially, the plasma density peaks at the chamber center
when VLF = 500 V. As VLF is increased, the plasma spreads out in
the radial direction, but the charged species densities become
uniform only for a narrow range of VLF. With increasing VLF, the
plasma splits into two regions, and ne peaks at both the chamber
center and near the electrode edge. The source for the production
of electrons is shown in Fig. 8. The spatial distribution of the
source is generally like the electron density profile, except that we
can observe plasma production in the time-averaged sheath region
as well. Production in this region mainly occurs when the LF
voltage is positive and the sheath collapses in front of the powered
electrode. During this time, plasma is efficiently produced through
stochastic heating by the 40 MHz source adjacent to the powered
electrode.
The time-averaged potential for sinusoidal LF waveform is
shown in Fig. 9 for a range of bias voltages. These results are for
J. Vac. Sci. Technol. B 40(3) May/Jun 2022; doi: 10.1116/6.0001732
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FIG. 6. Time-averaged plasma potential for 800 kHz rectangular wave voltage
of (a) 500, (b) 600, (c) 700, (d) 800, and (e) 900 V. Other conditions are Ar
plasma at 20 mTorr and 400 V 40 MHz.
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FIG. 7. Time-averaged electron density for 800 kHz rectangular wave voltage of
(a) 500, (b) 600, (c) 700, (d) 800, and (e) 900 V. Other conditions are Ar plasma
at 20 mTorr and 400 V 40 MHz.
J. Vac. Sci. Technol. B 40(3) May/Jun 2022; doi: 10.1116/6.0001732
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250 V at 40 MHz and 20 mTorr in Ar. A lower 40 MHz voltage has
been used for sinusoidal wave to attempt to get approximately
similar ne with the two bias waveforms. We discuss later why
40 MHz couples more efficiently with sinusoidal wave bias, thus
requiring a smaller 40 MHz voltage to obtain similar ne. The potential in the plasma bulk is reasonably uniform, and the DC voltage
at the powered electrode increases in magnitude with increasing
VLF. The sheath at the powered electrode gets thicker with increasing VLF. Comparing the results in Figs. 6 and 9, one can observe
that the “time-averaged” sheath is thicker with rectangular wave LF
bias than with sinusoidal waveform, which is primarily because the
sheath is fully expanded for a longer period with rectangular wave.
Corresponding time-averaged ne is shown in Fig. 10. The electron
density peaks at the chamber center when VLF = 500 V. As VLF is
increased, ne decreases and the sheath gets thicker, which is
observed with rectangular wave bias as well. However, the plasma
peak remains at the chamber center until larger VLF with sinusoidal
LF bias. The plasma expands in the radial direction becoming more
uniform at higher VLF. The time-averaged source of electron and
Ar+ ion production is shown in Fig. 11. The plasma is produced
reasonably uniformly between the two electrodes for the range of
VLF considered in this article. Compared to the results in Fig. 8,
one can observe that sources are smaller with sinusoidal wave bias
for comparable plasma densities, which is primarily due to the
lower 40 MHz voltage.
Some of the salient differences between the two LF bias waveforms are that (1) a larger 40 MHz voltage is used with rectangular
wave for comparable ne and (2) ne profile evolves differently at
higher LF voltages. We next present detailed modeling results to
explain these trends. When we reach steady-state, plasma production and loss balance each other. In an Ar plasma, charged species
are primarily lost at the surfaces. We have shown time-averaged ion
flux to the powered electrode for the two LF bias waveforms in
Fig. 12. The ion flux does not change significantly with LF voltage
even though the electron and Ar+ densities decrease by a factor of 2
between VLF = 500 and 900 V. This relative insensitivity is because
the ion drift current increases with an electric field. The effect of
an electric field on ion flux is particularly noticeable at the edge of
the powered electrode where the electric field is larger due to the
adjacent grounded wall. Ion flux is larger at the electrode edge than
at the chamber center even though charged species densities are
comparable or lower near the electrode edge. Between the two LF
bias waveforms, the ion flux is noticeably higher for rectangular
wave than for sinusoidal wave. This difference in ion flux is
because the voltage at the powered electrode remains at the peak
negative value for a longer period for rectangular LF voltage so ions
get extracted more strongly. On a time-averaged basis, an equal
number of electron and Ar+ leave the plasma in steady-state so
electron loss is higher with rectangular wave too. Due to higher
plasma loss with rectangular wave than sinusoidal LF bias, a larger
40 MHz voltage is needed with rectangular LF bias to obtain
plasma of comparable density.
The plasma is not produced similarly throughout the LF cycle.
As explained below, the time variation of plasma production
strongly influences the plasma profile. We have shown a spatially
integrated source of electrons versus time during the last four LF
cycles in Figs. 13(a) and 14(a) for rectangular and sinusoidal LF
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FIG. 8. Time-averaged electron source for 800 kHz rectangular wave voltage of
(a) 500, (b) 600, (c) 700, (d) 800, and (e) 900 V. Other conditions are Ar plasma
at 20 mTorr and 400 V 40 MHz.
J. Vac. Sci. Technol. B 40(3) May/Jun 2022; doi: 10.1116/6.0001732
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FIG. 9. Time-averaged plasma potential for 800 kHz sinusoidal wave voltage of
(a) 500, (b) 600, (c) 700, (d) 800, and (e) 900 V. Other conditions are Ar plasma
at 20 mTorr and 250 V 40 MHz.
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FIG. 10. Time-averaged electron density for 800 kHz sinusoidal wave voltage of
(a) 500, (b) 600, (c) 700, (d) 800, and (e) 900 V. Other conditions are Ar plasma
at 20 mTorr and 250 V 40 MHz.
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FIG. 11. Time-averaged electron source for 800 kHz sinusoidal wave voltage of
(a) 500, (b) 600, (c) 700, (d) 800, and (e) 900 V. Other conditions are Ar plasma
at 20 mTorr and 250 V 40 MHz.
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FIG. 12. Ar+ flux at the powered electrode for (a) rectangular wave LF voltage
and 400 V 40 MHz and (b) sinusoidal wave LF voltage and 250 V 40 MHz.
Simulations have been done for Ar plasma at 20 mTorr.
bias waveforms, respectively. These results are for VLF = 900 V.
Note that the vertical scales are different in these two graphs. With
rectangular LF bias, the plasma is produced more intensely when
the low frequency voltage at the powered electrode is positive and
the sheath collapses there. During this stage, the sheath can oscillate strongly and stochastic heating due to 40 MHz is more effective. Once the LF voltage at the electrode becomes negative, the
sheath expands there and plasma production plummets. With
sinusoidal LF voltage, even though plasma production varies
during the LF cycle, plasma production occurs over a longer period
and the changes are more gradual. This time variation of plasma
production in dual-frequency CCPs has been extensively
studied,41,42 and it is referred to as frequency coupling in the
literature.
Plasma production location changes during the LF cycle.
We have shown the electron source at different times in
Figs. 13(b)–13(d) and 14(b) and 14(c) for rectangular and sinusoidal LF waveforms, respectively. Average of the source in discrete
time windows is plotted to reduce the statistical noise and more
easily explains the trends using static images. Note that different
J. Vac. Sci. Technol. B 40(3) May/Jun 2022; doi: 10.1116/6.0001732
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FIG. 13. (a) Spatially averaged electron source during the final four low frequency cycles of the simulation. Time-averaged electron source for (b) 0.0–0.1
fraction of the LF cycle (c) 0.1–0.2 fraction of the LF cycle and (d) 0.3–0.9 fraction of the LF cycle. These results are for Ar plasma at 20 mTorr generated
using 400 V at 40 MHz and 900 V at 800 kHz rectangular wave.
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FIG. 14. (a) Spatially averaged electron source during the final four low frequency cycles of the simulation. Time-averaged electron source for (b) 0.1–0.3
fraction of the LF cycle and (c) 0.3–0.9 fraction of the LF cycle. These results
are for Ar plasma at 20 mTorr generated using 250 V at 40 MHz and 900 V at
800 kHz sinusoidal wave.
scales have been used for the different time fraction ranges. During
the sheath collapse phase with rectangular LF wave, plasma production occurs in the whole interelectrode region with slightly higher
production at the chamber center. Plasma production profile
during this time mimics the plasma density profile. This intense
production, however, occurs for the short time. As the LF voltage
at the powered electrode becomes negative, the sheath expands at
the powered electrode, making it difficult for plasma to diffuse
between the chamber center and edge. For a large fraction of the
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FIG. 15. (a) Radial flux, (b) vertical flux, and (c) energy of electrons at 0.0412
fraction of the low frequency 800 kHz cycle. Vectors representing the electron
flux are also superimposed on the flux plots. These results are for Ar plasma at
20 mTorr excited by 400 V at 40 MHz and 900 V at 800 kHz rectangular wave.
LF cycle, the plasma is produced at a slow rate and production
strongly occurs beyond the electrode edge where the sheath is thin.
This plasma is not able to diffuse to the chamber center due to the
thin plasma region, effectively splitting the plasma into two
regions. With sinusoidal LF wave, plasma production does vary
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between the sheath collapse and expansion phases. However, the
sheath fully expands for only a short period, so production occurs
more uniformly during both these phases.
In Fig. 13(a), we can observe a spike in plasma production at
the beginning of the positive voltage phase for rectangular voltage
waveform. Although only a few percent of the plasma is produced
during this period, the electrons in the sheath have been shown to
have significant energy then.18,20 It is, therefore, useful to examine
the electrons during this phase from the uniformity perspective.
We have shown the electron flux in the plasma and the electron
energy just after the sheath has collapsed in Fig. 15. On a timeaveraged basis, an equal number of electrons and ions leave the
plasma at the surfaces during steady-state. However, with rectangular LF wave, ions exit from the plasma for a long fraction of the LF
cycle. For the short period when the sheath collapses, the electrons
are compelled to rush out and the powered electrode is bombarded
by energetic electrons. Electric field has been shown to reverse for a
short time to assist with the production of the large electron
flux.18,20 These energetic electrons have sufficient energy to ionize
and excite with electron energy higher near the electrode edge due
to a higher electric field there. However, due to the low gas pressure, most of these energetic electrons exit at the electrode. We can
notice that the electron flux is higher and electrons are more energetic near the electrode edge. Both the electron flux and energy are
higher at the electrode edge due to a higher electric field in that
region.
IV. CONCLUSIONS
2D PIC modeling has been used to examine the effect of LF
voltage waveform on plasma characteristics in a dual-frequency
(40 + 0.8 MHz) CCP. In such a plasma with disparate frequencies,
the low frequency voltage usually has a strong influence on the
IEDF but contributes less to plasma generation. Past studies have
also shown that the LF voltage waveform can be used to control the
IEDF. In particular, a rectangular voltage waveform with a small
positive period is known to yield a narrow nearly monoenergetic
IEDF. This paper focuses on the effect of the LF voltage waveform
on plasma uniformity. A two-dimensional PIC model is used for
this investigation to adequately treat electron kinetic effects at low
pressure, especially during sharp transitions for the rectangular LF
waveform. The effect of LF voltage amplitude on plasma uniformity
is compared for sinusoidal and rectangular waveforms. When the
LF voltage is low, the charged species densities peak at the chamber
center at the low pressure considered (20 mTorr). This centerpeaked density profile is due to ample diffusion at low pressure and
higher losses at the chamber periphery. As the LF voltage is
increased, the sheath gets thicker at the powered electrode and
charged species densities decrease for a constant 40 MHz voltage.
The plasma profile, however, evolves differently depending on the
LF voltage waveform. With sinusoidal LF voltage, the plasma
spreads out between the electrodes as the LF voltage is increased.
However, the plasma splits into two regions with rectangular LF
voltage waveform: a peak in ne at the chamber center and another
peak near the electrode edge. This double-peaked density profile
with rectangular wave can be attributed to the time (during the LF
cycle) and location where the plasma is generated. Stochastic
J. Vac. Sci. Technol. B 40(3) May/Jun 2022; doi: 10.1116/6.0001732
Published under an exclusive license by the AVS
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heating due to 40 MHz is the main source of plasma production
under the conditions considered. 40 MHz produces plasma most
efficiently when the rectangular wave is positive and the sheath is
thin at the powered electrode. Plasma production during this phase
occurs mostly uniformly between the electrodes, but this period of
uniform plasma production is short. When the LF voltage becomes
negative, the sheath expands over the electrodes and the sheath is
only thin near the electrode edge. Furthermore, the electric field is
larger near the electrode edge due to the adjacent grounded wall.
Plasma is produced near the electrode edge during the longer negative voltage phase of the rectangular LF voltage, resulting in a
double-peaked plasma density profile.
Many different designs for capacitively coupled plasma
reactors are used in practice. Although the principles governing
plasma generation and transport discussed in this article are
expected to be generally applicable, the reported results are specific to the reactor geometry considered. Overall plasma uniformity can, for example, be optimized by modifying the chamber
design at the electrode edges and by changing the volume
beyond the electrodes.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
APPENDIX: PARTICLE MERGING
Particle merging is used in our PIC plasma model to
limit unrestricted growth in the total number of PIC particles
due to particle splitting. In the particle merging scheme, four
particles (Nos. 1–4) of the same type in the same plasma cell
are merged into two particles (a, b) using the following
methodology:
1. Referring to Fig. 16, particles are only merged in cells that are
surrounded by “plasma nodes.” In these cells, we subdivide particles of each type into eight groups based on the direction of
their velocity (+/−vx, +/−vy, +/−vz). Candidate particles for
merging are chosen from the same velocity group.
2. The merged particles need to satisfy the following conservation
equations:
ωx ¼ ωa þ ωb ¼
X4
ωx v x ¼ ωa v a þ ωb v b ¼
ωx εx ¼ ωa kv a k2 þ ωb kv b k2 ¼
ii¼1
ωi ,
X4
ωi v i ,
(A2)
ωi kv i k2 ,
(A3)
ii¼1
X4
ii¼1
(A1)
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ωa α ωb β ¼ 0,
(A9)
ωx ðϵx kv x k2 Þ ¼ ωa α2 þ ωb β 2 ,
(A10)
ωx kqx k ¼ ωa α3 ωb β 3 :
(A11)
E2 ¼ ϵx kv x k2 ,
(A12)
If we define
θ¼
α
,
E
(A13)
Eqs. (A8)–(A11) can be combined to obtain
θ2 1 θkqx k
¼ 0:
E3
(A14)
Equation (A14) can be solved to determine θ, which can then
be used in Eqs. (A5)–(A13) to obtain ωa, ωb, va, and vb.
FIG. 16. Definitions used for describing the particle merging algorithm. nij are
the densities at the adjacent plasma nodes, and x and z are normalized in the
range [0,1].
ωx qx ¼ ωa (va v x )kva v x k2 þ ωb (v b vx )kv b v x k2
X4
¼
ω (v v x )kv i vx k2 ,
(A4)
ii¼1 i i
where some simplifications have been made as all particles have
identical mass. For a vector A, we have defined
kAk2 ¼ A A:
Following the methodology proposed in Ref. 43, we define new
variables α, β, and e, where
v a ¼ vx þ αe,
(A5)
v b ¼ v x βe,
(A6)
q
e¼ x :
kqx k
J. Vac. Sci. Technol. B 40(3) May/Jun 2022; doi: 10.1116/6.0001732
Published under an exclusive license by the AVS
ωa (1 xa )(1 za ) þ ωb (1 xb )(1 zb ) ¼ n11 ,
(A15)
ωa xa (1 za ) þ ωb xb (1 zb ) ¼ n12 ,
(A16)
ωa (1 xa )za þ ωb (1 xb )zb ¼ n21 ,
(A17)
ωa xa za þ ωb xb zb ¼ n22 :
(A18)
Combining these equations, we obtain
xb ¼
n12 þ n22 ωa
xa ,
ωb
ωb
(A19)
zb ¼
n21 þ n22 ωa
za :
ωb
ωb
(A20)
(A7)
Inserting Eqs. (A5)–(A7) into Eqs. (A1)–(A4), we obtain the
following equations that need to be solved for ωa, ωb, α, and β:
ωx ¼ ωa þ ωb ,
(3) Knowing the velocities and weights of the merged particles, the
next goal is to determine their locations. To find the merged
particle locations in 2D, we enforce the condition that the
species density at plasma nodes does not change due to
merging. Therefore, knowing the density at the surrounding
plasma nodes (n11, n12, n21, n22) in Fig. 16, the merged particle
locations (xa, za) and (xb, zb) need to satisfy the following constraints, where the positions are defined in a normalized cell
with dimensions (1,1):
(A8)
The values of xa, xb, za, and zb need to be in the range [0,1].
However, for given densities at the nodes, ωa, and ωb, Eqs. (A19)
and (A20) can be used to ascertain that the valid ranges for xa and
za are even more restricted. Equations (A18)–(A20) can be
40, 032202-12
ARTICLE
19
combined to obtain
za ¼
xa ωa (n21 þ n22 ) þ n22 ωb (n21 þ n22 )(n12 þ n22 )
:
xa ωa (ωa þ ωb ) (n12 þ n22 )ωa
(A21)
We use Eq. (A21) to find a pair (xa, za) that is within the valid
ranges as determined by Eqs. (A19) and (A20). Equations (A19)
and (A20) can then be used to compute xb and zb. If we cannot
find values of xa and za within the valid range, we skip merging of
the four candidate particles.
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