九州大学大学院総合理工学報告第 40 巻 第 2 号 1-5 頁 平成 31 年 2 月

Engineering Sciences Reports, Kyushu University
Vol. 40, No. 2, pp. 1-5, FEB, 2019

Two Dimensional PIC Simulation of VHF Capacitively
Coupled Ar Plasma
Li-Wen SU*1†, Kiichiro UCHINO*2 and Yoshinobu KAWAI*3
E-mail of corresponding author: su@kyudai.jp†(Received December 7, 2018, accepted December 8, 2018)
A very high frequency capacitively coupled plasma source (100 MHz) was studied using a
two-dimensional PIC simulation, and spatial distributions of the plasma parameters were examinedat a discharge gap of 25 mm. In the simulation, electromagnetic effects were not taken into account.
Electron energy distribution functions were also calculated. It was found that the density andtemperature of electrons had a peak value at the center and near the edge of the discharge electrodes,
while the plasma potential was fairly uniform.
Key words: Capacitively coupled plasma, Energy distribution, PIC simulation, Etching1.

Introduction

Very high frequency capacitively coupledplasma (VHF CCP) sources are widely used for
applications such as plasma enhanced
chemical vapor deposition (PECVD) andetching because of a high electron density witha low electron temperature. There has been acurrent tendency that a power generator
frequency is increased to obtain high rates ofdeposition and etching. Recently, VHF plasma
sources at excitation frequencies of 100-150
MHz have become popular in etching [1-2]. In
fact, when the frequency is increased, theelectron density increases, leading to a highetching rate. However, as is well known,electromagnetic effects such as standing wave
and skin effects [3] arise at high frequencies,
and as a result, the electron density
distribution along the discharge electrodes
becomes nonuniform [4]. Overzet and Hopkins[5] observed the off-axis radial peak in electron
density in the Ar plasma even at a frequency of
13.56 MHz, which was explained by the fluid
model simulation [6]. On the other hand,
Yamazawa [7] investigated the effect of
harmonics on the electron density andreported a center-peaked electron density
*1
*2
*3

Department of Applied Science for Electronics andMaterials, Graduate Student
Department of Electrical and Material Science
Professor Emeritus

profile that was caused by a nonlinear electron
resonance heating. In addition, Sawada et al[8] observed that the electron density in the
VHF Ar plasma at a frequency of 60 MHz hada center-peaked profile. Thus, the study of a
spatial distribution of VHF Ar CCPs is an
important subject in etching and has been
examined by many researchers [2, 8-9].
The Langmuir probe method is a simple andpowerful diagnostic tool in a low temperatureplasma and is mostly used [5] to measure theplasma parameters such as the density andtemperature of electrons. However, there isfear that a Langmuir probe disturbs VHF
discharges; spatial distributions of the plasma
parameters are deformed by introducing theprobe. Therefore, in order to develop a new
VHF CCP source for etching, we have studied
the characteristics of VHF plasmas bysimulation. As is well known, there are threekinds of simulation models to examine thebehaviors of a low temperature plasma, that is,
fluid model, hybrid model, and particle-in-cell(PIC) model. The fluid model is useful at highpressure while the PIC model is effective at
low pressures. The properties of CCPs havebeen extensively investigated using a
two-dimensional (2D) hybrid model [2, 9]. Yang
and Kushner [2] explained a shift in the peakelectron density toward the center of theplasma source as the power source frequency
increased. Bera et al [10] tried to control

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Two Dimensional PIC Simulation of VHF Capacitively Coupled Ar Plasma

plasma uniformity using two frequency powersources at a combination frequency of 60 and
180 MHz and compared with the simulation by
a 2D fluid model including Maxwell equations.
They also calculated the electron density
profile by a one-dimensional (1D) PIC
simulation, where electromagnetic effects were
not considered, and found that the electron
density became maximum at the center withlarge contribution of the 180 MHz powersource to plasma production and the electronenergy distribution deviated from Maxwellian
one.
Generally, an electron energy distributionfunction in VHF plasmas is not Maxwellian at
low pressure. In fact, Godyak and Piejak [11]observed that electrons had a bi-Maxwellian
distribution. Thus, we need to incorporate akinetic description of electrons and ions in thesimulation for understanding self consistently
theplasmacharacteristics.
The
best
calculation is to use the PIC/Monte Carlo (MC)method [12] because the fundamental
equations can be employed without muchapproximation. In this study, we calculated
two dimensional (2D) profiles of the plasma
parameters using a 2D Particle-In-Cell with
Monte Carlo Collision Module (PIC-MCCM) of
PEGASUS
Software
[13],
where
electromagnetic effects were not included, thatis, we assumed that electrostatic fields were
dominant in the VHF discharge.

2.

Description of the model

The PIC/MC method is based on a kinetic
description of charge particle motions in phasespace. Charged particles move in the
self-consistent electric field that is obtained bysolving the Poisson equation. The Monte Carlo
method is used to calculate collisions including
excitation and ionization. The PIC method has
been described in detail by Birdsall [14],Turner [15], and Verboncoeur [12]. In the
PIC-MCCM of PEGASUS Software, electrons
and ions are modeled as a large number of
20 mm

Quart

representative particles that are referred to as
simulated particles (superparticles). A motionof the individual superparticle is subject to the
Newton’s law in the electric field. Various
kinds of collisions are also taken into account
using
Monte
Carlo
collision
method.
Macroscopic quantities of the electron and ion
densities, mean energy, particle flux are
obtained by statistical calculations using
velocities
andpositions
of
all
the
superparticles. Here, we assume that there are
two kinds of charged particles in the plasma
for saving the computation time: electrons andAr+ ions. Figure 1 is a schematic of our CCP
source simulated (380 × 40 mm2), where a
Cartesian coordinate is used and a symmetricplane at the center between two discharge
electrodes is assumed. A power source
frequency was 100 MHz, where external
circuit was not included. The gas used was Ar
and the pressure was varied from 20 to 100
mTorr. The initial conditions for superparticles
were: the density=3 × 105 m-3 for electrons (1015m-3) and ions (1015 m-3).

3.

Simulation Results and Discussion

Since a VHF CCP source provides a highelectron density plasma with a low electron
temperature, it has been popular for etching.
On the other hand, the self-bias voltage formedon the power electrode that is indispensable for
etching is much lower than that at a powersource frequency of 13.56 MHz. Therefore,
recently dual-frequency power sources havebeen employed for etching. In this study, we
performed the simulations at a single powersource frequency as a first step for developinga large-area VHF CCP source. At first, we
calculated the time history of the self-bias
voltage Vdc at a pressure of 40 mTorr. Figure 2
shows that Vdc saturates to – 3.3 V that is very
low for etching, as expected.
We calculated a two-dimensional (2D) mapof the plasma parameters. The results at 30
mTorr are shown in Figs. 3-5, where thePowered electrode

z
25 mm

PlasmaCounter electrode
380 mmFig.1

Schematic of the VHF CCP source.

平成 30 年度

九州大学大学院総合理工学報告

amplitude is 30 V. Figure 3 shows that theelectron density ne amounts to 3.6×1016 m-3.
Note that ne peaks at x= 75 and 190 mm, thatis, the electron density peaks both near theedge of the discharge electrodes and thechamber center. Bera et al [16] simulated a
VHF-CCP at a frequency of 180 MHz using the2D fluid model coupled with Maxwell
equations and found the two-peak profile at a
narrow gap of 17.8 mm at a pressure of 100
mTorr. They also reported that as the gap
distance was increased, the electron density
profile became to a center-peaked profile. In
addition, Sawada et al [8] carried out theexperiments on a VHF-CCP at power source
frequencies of 60 and 106 MHz and comparedwith the simulations taking into account of
Maxwell equations. They pointed out thathigher harmonics of a power source frequency
affected the electron density profile. As alreadydescribed, here we did not solve Maxwell
equations, so we can conclude that thetwo-peak profile may be due to other effects.
Figure 4 shows that the electron
temperature Te also has a peak value at x=75
and 190 mm and Te is around 2 eV. Note thatTe near the discharge electrodes is around 3 eVthat is a little higher than Te at the center. The
CCP is produced by electron impact ionization,
so Te near the electrodes becomes higher. In
this study, the power source frequency is 100
MHz, so VHF effect will be higher, leading to a
low electron temperature. There have been
many reports on spatial distributions of theelectron density of VHF-CCPs, that is,researchers have mostly focused on theelectron density profile. Spatial distributionsof the electron temperature, especially thosenear the discharge electrodes, are also
important because it is closely related to thesheath formation on the substrate.
Figure 5 shows that the plasma potential Vs
is around 32 V and fairy uniform over thedischarge electrodes. Looking carefully at Vs
close to the discharge electrodes, it is not
uniform, and the thickness of the sheath
formed on the grounded electrode is not
uniform, suggesting that it will not easy tocontrol energy distribution functions ofcharged particles by dual-frequency technology[2].
Energy distribution functions of charged
particles play an important role in etching aswell as plasma production. We calculated theenergy distribution functions of electrons(EEDF) and ions (IEDF) at different

第 40 巻第2号

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monitoring positions denoted by Pi. As seen in
Fig. 6(a), electrons have a Maxwellian
distribution function in the low energy region.
In fact, we confirmed that semi-log plots of the
EEDFs fit to a straight line in the energy
region of (0-15) eV and high energetic electrons
are truncated, that is, the obtained EEDF
looks a Druyvesteyn-like distribution function.
Experimentally, Godyak and Piejak [11]observed the bi-Maxwellian and Druyvesteyn
distribution functions at low and highpressures, respectively, at a frequency of 13.56
MHz. Abdel-Fattah and Sugai [17] alsomeasured EEDFs for different power source
frequencies ranging from 13.56 to 60 MHz andreported that the EEDFs became a
Druyvesteyn distribution function below 30
MHz and a bi-Maxwellian distributionfunction above 30 MHz. In this simulation, theVHF plasma was generated at 40 mTorr at 100
MHz, but the bi-Maxwellian distributionfunction was not obtained, so we can conclude
that the effect of stochastic heating [3] on theplasma generation is negligible small. Figure 6
also indicates that the electron density is lower
and electron temperature is higher at amonitoring position of P1 (inside the ion
sheath) which agrees with the results of Figs. 3
and 4. On the other hand, ions were cold except
at P1 where ions are accelerated by the sheathpotential.
As aforementioned, we have found a
two-peak profile of the electron density in the
VHF electrostatic discharge by the PIC
simulation. As is well known, the PIC
simulation is more accurate than the fluid
model
simulation
becauseoffewer
assumptions for modeling. To understand thetwo-peak profile, we need to examine theparametric dependences of the 2D map such as
the pressure and gap dependences. This will be
discussed in future.

Fig.2 Time history of the self-bias voltage, Vdc, on the

powered electrode at a pressure of 40 mTorr,
where the amplitude is 30 V

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Two Dimensional PIC Simulation of VHF Capacitively Coupled Ar Plasma

P1
P2
P3

(mm)

40

P4

20
0

50

100

150 (mm)Electron Density (m-3)(a)
Fig.3 2D map of the electron density ne at a pressureof 40 mTorr, where the amplitude is 30 V.

Y
P1~P4

0

X
P1 (51.75, 24.25)
P2 (52.25, 20.25)

(mm)

40
P3 (51.75, 16.25)

20

P4 (51.75, 13.25)

0

50

100

150

(mm)

Electron Temperature (eV)Fig.4

2D map of the electron temperature Te at apressure of 40 mTorr, where the amplitude is 30
V.

.

(mm)

40
20
0

50

100

150 (mm)

Plasma potential (Vs)

Fig.5 2D map of the plasma potential Vs at a pressure of
40 mTorr, where the amplitude is 30 V.

(b)Fig. 6 Energy distributions: (a) EEDFs for different
monitoring positions at a pressure of 30 mTorr,
where the amplitude is 30 V and (b) the monitoring
positions of electron energy distributions

4.

Conclusion

We calculated the characteristics of the
VHF Ar plasma (100 MHz) at a pressure of 40
mTorr using the PIC-MCCM of PEGASUS
Software, where we assumed that the plasma
is produced by electrostatic fields. We found
the following results:(1)
The self-bias voltage is as low as a
few volts.
(2)
The 2D maps of the electron densityand electron temperature show that they peak
both near the center and the edge of thedischarge electrodes, while the plasma
potential is fairly uniform.
(3)
The electron energy distributionfunction is similar to the Druyvesteyn-like one.
We conclude that the above-mentioned
results will be useful for designing a VHF
plasma source for large-area etching. The
simulation using the dual-frequency powersources will be our future study.
Acknowledgment

平成 30 年度

九州大学大学院総合理工学報告We thank Dr. M. Matsukuma at Tokyo
Electron Ltd. for fruitful discussion.
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