Simulation of capacitively coupled RF discharge in argon
In this work, the axial profiles of the density of electrons and positive ions, the mean electron energy, the electric field strength, and the potential were obtained, both on average over the period and in dynamics. It was shown that argon discharges are dominated by ionization by electrons that ga...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2023
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| Cite this: | Simulation of capacitively coupled RF discharge in argon / V. Lisovskiy, S. Dudin, A. Shakhnazarian, P. Platonov, V. Yegorenkov // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 129-133. — Бібліогр.: 21 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859648956966895616 |
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| author | Lisovskiy, V. Dudin, S. Shakhnazarian, A. Platonov, P. Yegorenkov, V. |
| author_facet | Lisovskiy, V. Dudin, S. Shakhnazarian, A. Platonov, P. Yegorenkov, V. |
| citation_txt | Simulation of capacitively coupled RF discharge in argon / V. Lisovskiy, S. Dudin, A. Shakhnazarian, P. Platonov, V. Yegorenkov // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 129-133. — Бібліогр.: 21 назв. — англ. |
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| description | In this work, the axial profiles of the density of electrons and positive ions, the mean electron energy, the electric field strength, and the potential were obtained, both on average over the period and in dynamics. It was shown that argon discharges are dominated by ionization by electrons that gained energy by stochastic heating during the expansion of near-electrode sheaths. This ionization occurs in two pulses during one RF period. At low RF voltage between the electrodes, the role of Ohmic heating of electrons in the electric field in a quasi-neutral plasma increases, but the contribution of stochastic heating remains dominant. The time-averaged plasma potential was found to increase non-linearly with the RF voltage between the electrodes Urf. It is shown that at low Urf values (when the RF voltage approaches the discharge extinction curve), the average potential Φ can reach Urf due to the axial redistribution of the instantaneous potential in the gap between
Отримано осьові профілі густини електронів та позитивних іонів, середньої енергії електронів, напруженості електричного поля та потенціалу як у середньому за період, так й їхню динаміку. Показано, що в розряді в аргоні переважає іонізація електронами, які набули енергії під час стохастичного нагріву при розши ренні приелектродних шарів. Ця іонізація відбувається двома імпульсами протягом одного ВЧ-періоду. При низькій ВЧ-напрузі між електродами зростає роль омічного нагріву електронів у електричному полі в квазінейтральній плазмі, але внесок стохастичного нагріву залишається домінуючим. З’ясовано, що середній за часом потенціал плазми Φ нелінійно зростає з ВЧ-напругою Urf між електродами. Показано, що при низьких значеннях Urf (при наближенні ВЧ-напругою до кривої згасання розряду) середній потенціал Φ може досягати Urf завдяки осьовому перерозподілу миттєвого потенціалу в проміжку між електродами.
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| first_indexed | 2025-12-07T13:31:03Z |
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ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 129
https://doi.org/10.46813/2023-146-129
SIMULATION OF CAPACITIVELY COUPLED RF DISCHARGE
IN ARGON
V. Lisovskiy, S. Dudin, A. Shakhnazarian, P. Platonov, V. Yegorenkov
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
E-mail: lisovskiy@yahoo.com
In this work, the axial profiles of the density of electrons and positive ions, the mean electron energy, the electric
field strength, and the potential were obtained, both on average over the period and in dynamics. It was shown that
argon discharges are dominated by ionization by electrons that gained energy by stochastic heating during the ex-
pansion of near-electrode sheaths. This ionization occurs in two pulses during one RF period. At low RF voltage
between the electrodes, the role of Ohmic heating of electrons in the electric field in a quasi-neutral plasma increas-
es, but the contribution of stochastic heating remains dominant. The time-averaged plasma potential was found to
increase non-linearly with the RF voltage between the electrodes Urf. It is shown that at low Urf values (when the RF
voltage approaches the discharge extinction curve), the average potential F can reach Urf due to the axial redistribu-
tion of the instantaneous potential in the gap between the electrodes.
PACS: 52.80.Hc
INTRODUCTION
Radio-frequency capacitive discharge in gases is
widely used in many plasma technologies. With its help,
various materials are sputtered, various films are depos-
ited (both metal and dielectric), semiconductor materials
are etched during the production of microchips, gas dis-
charge lasers are pumped, and the surfaces of techno-
logical and research chambers are cleaned (from centi-
meter size to the scale of large tokamaks and stellara-
tors), etc. Therefore, considerable attention is paid to
both experimental and theoretical research of its proper-
ties. Usually, either the external parameters of the RF
capacitive discharge (for example, the current-voltage
characteristics) or the time-averaged plasma parameters
are measured experimentally [1 - 7]. But in order to
understand exactly how the processes of birth and loss
of charged particles take place, and what the dynamics
of the internal parameters of the plasma look like, nu-
merical modeling is frequently used [2, 7 - 9].
The purpose of this work was to carry out numerical
modeling of the parameters of a radio-frequency capaci-
tive discharge in argon using the SIGLO-rf hydrody-
namic code. Plasma parameters averaged over the RF
period were determined, and their changes with time
were investigated. The mechanisms of ionization, the
time and place within the discharge, where and when
they occur, have been clarified.
1. DESCRIPTION OF SIGLO-RF CODE
SIGLO-rf is fluid code, which is the 1D user-
friendly simulation software of capacitively coupled RF
discharges. It was developed by JP Boeuf and LC Pitch-
ford at the University of Toulouse and is based on the
same physics as in [10, 11].
SIGLO-rf numerically solves the continuity, momen-
tum, and energy equations for electrons, the discontinui-
ty and momentum equations for positive ions, and Pois-
son's equation for the electric field.
In the axial profiles shown in the next section, the left
electrode is grounded, and RF voltage Urf oscillating
with a frequency of 13.56 MHz is applied to the right
electrode. The distance between the electrodes is fixed
and is 2 cm. The gas pressure is 1 Torr.
Calculations were made for argon. Accordingly, the
necessary information was entered into the file with gas
parameters. The dependences of electron mobility, aver-
age electron energy, and ionization coefficient (the first
Townsend coefficient a/p) on the reduced electric field
E/p for gas temperature T = 300 K were preliminary cal-
culated using the Bolsig+ code [12]. At the same time,
cross-sections of electron collisions with argon atoms
were used, given in the LXCat database (www.lxcat.net)
[13], where cross-sections from the Phelps set [14] were
selected.
In addition, for argon, the mobilities of positive Ar+
ions in Ar, given in the review [15], were used. Ion-
electron recombination was neglected due to the very
small value of the coefficient for this process [16].
2. EXPERIMENTAL RESULTS
RF capacitive discharge consists of three regions, a
quasi-neutral plasma and two sheaths of space charge,
which are actually transition parts between the plasma
and the electrode walls limiting it. Quasi-neutral plasma
and near-electrode sheaths differ greatly in terms of
parameters. At the same time, it is necessary to consider
not only the average parameters of the discharge over
the RF period but also to take into account that these
parameters also change over time due to the harmonic
change of the RF voltage on the right (potential) elec-
trode. Therefore, below we will present the average pro-
files of plasma parameters for the RF period, and for
some of them, we will find out their dynamics.
The time-averaged axial profiles of electron density
Ne and positive ion density Np are shown in Fig. 1. It can
be seen from the figure that the quasi-neutrality condi-
tion is fulfilled in the plasma volume, that is, the densi-
ties of electrons and positive ions are equal to each oth-
er, and the corresponding parts of the curves in the fig-
ure coincide.
The near-electrode sheaths are dominated by posi-
tive ions, and the electron density here is much lower
than the ion density. That is, here we observe a violation
of the condition of quasi-neutrality. From the plasma to
the boundary of the sheaths, positive ions and electrons
move in the mode of ambipolar diffusion, so their densi-
ties in the plasma are the same.
130 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146)
Fig. 1. Time-averaged axial profiles of the density
of electrons and positive ions
But after reaching the limit of the plasma region,
positive ions are accelerated to the electrodes by a
strong electric field averaged over time, the voltage of
which increases approximately according to a linear law
when approaching the electrodes (Fig. 2). In a quasi-
neutral plasma, this field is small (approximately units
V/cm and increases to 20 V/cm at the boundary of the
sheaths) but it is zero only in the center of the discharge
gap. In the considered case, the average electric field
strength near the electrode surface approaches
1400 V/cm.
As a result of the oscillations of the electrons in the
instantaneous radio-frequency electric field, some of
them enter the electrodes. Therefore, there are more
positive ions than electrons in the discharge gap. Thanks
to this, the plasma has a positive potential in relation to
the electrodes. Let us recall that the left electrode is
grounded, and the potential of the right electrode chang-
es according to the harmonic law and is also zero on
average during the RF period. Therefore, positive ions
are accelerated by this electric field averaged over the
RF period from the plasma to the electrodes, due to
which the RF capacitive discharge has found wide ap-
plication in many plasma technologies.
Fig. 2. Time-averaged axial profiles of potential
and electric field strength
The value of the average plasma potential depends on
the RF voltage applied to the electrodes, the pressure, and
the type of gas. For argon gas at 1 Torr pressure and
500 V RF voltage, the average plasma potential (for the
center of the discharge) is about 230 V (see Fig. 2).
Now let's consider the time-averaged axial profiles
of the mean electron energy, which are shown in Fig. 3.
From the figure, one can see that the mean electron en-
ergy in the plasma reaches up to approximately 6 eV,
however, in the space charge sheaths, it quickly de-
creases down to 1 eV near the electrodes. But here, it
should be taken into account that electrons alternately
fill the near-electrode sheaths, and during certain parts
of the RF period they are not present near the electrodes
at all, and, accordingly, their average energy is consid-
ered zero in this case.
Fig. 3 also shows the time-averaged axial profile of
the ionization rate. From its unit of measurement,
cm
-3
s
-1
, it is clear that this value is equal to the product
of the electron density and the ionization frequency
ni·Ne and is the term of the electron and ion balance
equation, which describes the rate of ionization birth of
charged particles in 1 cm
3
per 1 s at a certain point in
the discharge gap. The maximum values of the ioniza-
tion rate appear near the boundaries of the near-
electrode sheaths and indicate the presence of enhanced
electron heating in these areas. A much lower ionization
rate is observed in the central part of the discharge gap.
We will consider the processes of electron heating be-
low.
Now let's pay attention to the time dependences of
the axial profiles of the plasma parameters. First, let's
consider the profiles of the plasma potential for several
moments of time during one-half of the RF period
(Fig. 4). At the initial moment, when the high-amplitude
RF voltage is applied to the right electrode, almost the
entire potential drop is concentrated on the left near-
electrode sheaths, because the left electrode plays the
role of an instantaneous cathode. In the left sheaths, the
electric field strength is maximum, and the right sheaths
has practically collapsed. A flow of electrons enters the
right electrode, which is limited by a small voltage drop
near the electrode.
Fig. 3. Time-averaged axial profiles of average electron
energy and ionization rate
Fig. 4. Axial profiles of potential for different parts
of RF period for voltage between the electrodes of 500 V
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. № 4(146) 131
Then, over time, the voltage drop on the left sheaths
decreases with the simultaneous narrowing of the thick-
ness of the left sheaths. At the right electrode, the volt-
age decreases first to zero, and then further to a negative
amplitude value. Throughout this half-period, the thick-
ness of the right sheaths increases. This half-period ends
with the reverse situation in relation to the initially con-
sidered zero time moment, namely, the left near-
electrode sheaths has collapsed, and the left electrode
receives a flow of electrons (which is regulated by a
small voltage drop near its surface).
Fig. 5. Axial density profiles of positive ions and
electrons for different parts of the period for RF voltage
between electrodes of 500 V
The dynamics of the electron density profile for the
same moments of time are shown in Fig. 5. The positive
ion density profile practically does not change with
time, because the frequency of the RF field of
13.56 MHz applied to the electrodes is too high for
them. Ions are heavy inert particles, in a discharge they
react only to constant (or time-averaged) and low-
frequency (less than 1 MHz) electric fields. Light elec-
trons, in the presence of sufficiently frequent collisions
with gas molecules, are able to sense changes in the
electric field almost without inertia. At the initial mo-
ment (t/T = 0), electrons are pushed out of the left near-
electrode sheaths and, having filled the right sheaths,
enter the right electrode (instantaneous anode). At the
same time, we note that the electron density profile in
the right sheaths practically coincides with the profile of
the density of positive ions, with the exception of a nar-
row region near the very surface of the right electrode.
That is, it can be said that quasi-neutral plasma almost
completely fills the right sheaths at the initial moment
of time. Later, as shown in Fig. 4, the voltage on the
right electrode decreases. The plasma potential also de-
creases, but more slowly than the voltage on the right
electrode, so the voltage drop on the right sheaths in-
creases. This leads to the fact that the thickness of the
right sheaths increases. Electrons, which initially almost
completely filled the right sheaths, are now pushed from
the right electrode back into the plasma because the
potential of this electrode is now more negative in rela-
tion to the plasma. This ejection increases the electron
energy and is called stochastic heating [17 - 20]. This
stochastic heating led to an increase in the ionization
rate near the boundaries of the near-electrode sheaths on
the time-averaged axial profiles (see Fig. 3).
Fig. 6 shows in which places and when exactly sto-
chastic heating occurs in the RF capacitive discharge.
The first peak of the ionization rate (it is located on the
right in the figure) occurs when electrons are pushed out
of the right near-electrode sheaths. The stochastic heat-
ing itself affects the ionization rate both in the right
sheaths and outside it in the region of quasi-neutral
plasma near the sheaths boundary. At the same time,
electrons fill the left sheaths. After the potential on the
right electrode reaches the maximum negative value, the
voltage on the left sheaths begins to increase. This in-
crease in voltage pushes electrons out of the left sheaths,
and as a result of stochastic heating, we observe a spike
in the ionization rate, which is on the left in Fig. 6.
Fig. 6. Spatio-temporal dependence of ionization rate
for RF voltage between electrodes of 500 V
Let's note one more important feature of RF capaci-
tive discharge, which occurs in any gas, namely, the
balance of positive and negative charges arriving on the
surface of each of the electrodes. Fig. 7 shows the cur-
rent densities of positive ions and electrons that reach
the surface of the left electrode during one RF period. It
can be seen from the figure that the flow of positive ions
is maximal at the beginning and at the end of the RF
period (corresponding to the amplitude of positive RF
voltage on the right electrode).
Fig. 7. Time dependence of the current density
of positive ions and electrons on the left electrode
for RF voltage between the electrodes of 500 V
In the middle of the period, the ion flow decreases,
but not to zero. That is, during the entire RF period,
positive ions arrive on the surface of the left electrode.
Electrons also arrive at the same electrode but in the
form of a short burst with a maximum when the RF
voltage on the right electrode reaches the negative am-
plitude value. Therefore, for the stable burning of the
discharge, it is necessary that the balance of positive and
negative charges be fulfilled, that is, the number of elec-
trons that entered the left electrode during the burst
|
| id | nasplib_isofts_kiev_ua-123456789-196190 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T13:31:03Z |
| publishDate | 2023 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Lisovskiy, V. Dudin, S. Shakhnazarian, A. Platonov, P. Yegorenkov, V. 2023-12-11T12:35:45Z 2023-12-11T12:35:45Z 2023 Simulation of capacitively coupled RF discharge in argon / V. Lisovskiy, S. Dudin, A. Shakhnazarian, P. Platonov, V. Yegorenkov // Problems of Atomic Science and Technology. — 2023. — № 4. — С. 129-133. — Бібліогр.: 21 назв. — англ. 1562-6016 PACS: 52.80.Hc DOI: https://doi.org/10.46813/2023-146-129 https://nasplib.isofts.kiev.ua/handle/123456789/196190 In this work, the axial profiles of the density of electrons and positive ions, the mean electron energy, the electric field strength, and the potential were obtained, both on average over the period and in dynamics. It was shown that argon discharges are dominated by ionization by electrons that gained energy by stochastic heating during the expansion of near-electrode sheaths. This ionization occurs in two pulses during one RF period. At low RF voltage between the electrodes, the role of Ohmic heating of electrons in the electric field in a quasi-neutral plasma increases, but the contribution of stochastic heating remains dominant. The time-averaged plasma potential was found to increase non-linearly with the RF voltage between the electrodes Urf. It is shown that at low Urf values (when the RF voltage approaches the discharge extinction curve), the average potential Φ can reach Urf due to the axial redistribution of the instantaneous potential in the gap between Отримано осьові профілі густини електронів та позитивних іонів, середньої енергії електронів, напруженості електричного поля та потенціалу як у середньому за період, так й їхню динаміку. Показано, що в розряді в аргоні переважає іонізація електронами, які набули енергії під час стохастичного нагріву при розши ренні приелектродних шарів. Ця іонізація відбувається двома імпульсами протягом одного ВЧ-періоду. При низькій ВЧ-напрузі між електродами зростає роль омічного нагріву електронів у електричному полі в квазінейтральній плазмі, але внесок стохастичного нагріву залишається домінуючим. З’ясовано, що середній за часом потенціал плазми Φ нелінійно зростає з ВЧ-напругою Urf між електродами. Показано, що при низьких значеннях Urf (при наближенні ВЧ-напругою до кривої згасання розряду) середній потенціал Φ може досягати Urf завдяки осьовому перерозподілу миттєвого потенціалу в проміжку між електродами. This work was supported by the National Research Foundation of Ukraine in the framework of the project 2021.01/0204. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Problems of Atomic Science and Technology Gas discharge, plasma-beam discharge and their applications Simulation of capacitively coupled RF discharge in argon Моделювання ВЧ ємнісного розряду в аргоні Article published earlier |
| spellingShingle | Simulation of capacitively coupled RF discharge in argon Lisovskiy, V. Dudin, S. Shakhnazarian, A. Platonov, P. Yegorenkov, V. Gas discharge, plasma-beam discharge and their applications |
| title | Simulation of capacitively coupled RF discharge in argon |
| title_alt | Моделювання ВЧ ємнісного розряду в аргоні |
| title_full | Simulation of capacitively coupled RF discharge in argon |
| title_fullStr | Simulation of capacitively coupled RF discharge in argon |
| title_full_unstemmed | Simulation of capacitively coupled RF discharge in argon |
| title_short | Simulation of capacitively coupled RF discharge in argon |
| title_sort | simulation of capacitively coupled rf discharge in argon |
| topic | Gas discharge, plasma-beam discharge and their applications |
| topic_facet | Gas discharge, plasma-beam discharge and their applications |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/196190 |
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