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Computational study of Ar/O2 plasma near uneven substrates
T. Ibehej and R. Hrach
Charles University, Faculty of Mathematics and Physics,
Department of Surface and Plasma Science, V Holešovičkách 2, 180 00
Prague 8, Czech Republic
Abstract: An interaction of low-temperature plasma with solid is
studied in both static and dynamic regimes. Investigated plasma is
a mixture of argon and 10% of oxygen with parameters typical for
positive column of DC glow discharge. In presented self-consistent
particle simulation we used Particle in Cell computational
technique with Monte Carlo collisions. An interaction with a
grooved positively biased substrate is studied in static regime and
the results are spatial distribution of electrostatic potential and
fluxes of negatively charged particles to the substrate. Dynamic
simulation describes time development of plasma properties after a
step change of substrate bias. Studied properties are electrostatic
potential and electron and O+ number densities.
Keywords: particle simulation, low-temperature plasma, surface
treatment
1. IntroductionChemically active low-temperature plasmas are
widely used in many plasma-assisted treatment technologies.
Mixtures of argon and electronegative gases like O2, CF4 or SF6 are
important in material processing such as ashing, etching or
cleaning. The presence of chemically active molecular gas makes
theoretical description of the system more difficult. Therefore,
computer simulations of such systems are very useful.
Simulations of plasma-solid interactions can be divided into
three basic categories – fluid, hybrid and particle simulations.
Particle simulations are highly time-consuming, but can provide
detailed information about the system on both macroscopic and
microscopic level and in both static and dynamic regimes. These
simulations can provide us with spatial distributions of particle
densities, fluxes to the substrate, energy and angular
distributions of particles, electric field etc.
Our previous publication [1] discussed plasma properties near
uneven stepped electrode. The investigated plasma was
electropositive Ar plasma and simplified electronegative plasma
with variable electronegativity. One-dimensional simulations of
dynamic plasma properties near planar or cylindrical probe were
also published previously. In these papers, argon plasma [2],
simplified electronegative
plasma [3] and argon-oxygen mixture [4-5] were studied. In
contrast with the papers [2-5], presented simulations are
two-dimensional which allows us to study geometrically more complex
problems. The simulated plasma is a realistic model of Ar/O2
mixture. A significant improvement was achieved especially in
modeling of charged-neutral particles interactions.
2. Computational descriptionPresented simulations were performed
on microcomputers with following configurations: Intel Xeon W3680
(6 CPU @ 3.33GHz, 12 threads, 16 GB RAM) and Intel Core i7 940 (4
CPU @ 2.93 GHz, 8 threads, 8 GB RAM).
Particle trajectories were determined by Newton's equations of
motion which were integrated by the Verlet velocity algorithm. For
calculations in static regime, different time steps for electrons
1×10−11s and for ions 1×10−8s were used. The common time step of
1×10−12s was used for both electrons and ions in dynamic
simulations.
Working area with size of 2×2cm was divided into 500×500 cells.
Using these cells, electrostatic potential and electric field were
calculated. The method of obtaining electrostatic forces acting on
the particles is called Particle In Cell (PIC), Nearest Grid Point
variation [6]. Numerical solution of the
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Poisson's equation is provided by C library Umfpack [7]. No
external magnetic field is applied and magnetic fields generated by
moving charged particles are neglected.
A planar substrate with defined potential and width of 5 mm was
located at the border of the working area. For the static
calculations, a rectangular groove with depth of 5 mm and width of
1 cm was located on the substrate. Because of a lucidity of the
plots, a flat substrate without the groove was used for the dynamic
calculations.
Behind the opposite boundary of the working area, we assume
undisturbed plasma with Maxwell distributions. Through that
boundary, the particles from the simulation can leave the area and
the particles from undisturbed plasma can enter. At two remaining
sides, periodic boundary conditions are applied.
Coulombian interactions between charged particles are provided
by PIC algorithm. Non-Coulombian interactions between charged and
neutral particles are also very important, due to low ionization
degree of investigated plasma. These interactions are simulated by
the Monte Carlo method. A modified implementation of null collision
method [8] is used.
3. Parameters of simulationInput parameters of the simulation
are plasma composition – density and mean energy of each charged
and neutral species considered in the simulation, dimensions and
bias of the substrate and cross sections of the most important
charged-neutral interactions.
Charged species Neutral speciese 9.8×1014 m−3 Ar 2.9×10 22
m−3
O- 2×1013 m−3 O 5.6×1021 m−3
O+ 8.2×1014 m−3 O2 3.5×1020 m−3
Ar+ 1.6×1014 m−3
O2+ 2×1013 m−3
Table 1. Plasma composition – number densities of charged and
neutral species.
Table 1 shows the list of all charged and neutral species and
their number densities. The ratio of number densities was obtained
from a chemical kinetics simulation in O2/Ar mixture with E/n
ratio
of 60 Td. In this simulation, the neutral gas density ratio
[O2]:[Ar] was assumed 1:9 which is typical for engineering
applications. The electron density approximately corresponds to
experiment [9]. The temperature of electrons in the undisturbed
plasma was approximately 28,000 K and the ion temperature was 300
K.
List of interactions included in the model is presented in table
2. These interactions are responsible for energy loss of charged
particles which are also accelerated by the substrate bias.
Unfortunately, it is quite difficult to find the cross sections in
the literature, especially for the heavy ions. As shown in table 2,
some basic interaction data were found nearly for every combination
of charged and neutral species. In three remaining combinations,
data for similar ions were used.
Species Interaction References
e Ar elastic, excitation 11.5 eV,ionization 15.8 eV
[10]
e Oelastic, excitations 1.97 and 4.19 eVionization 13.62 eV
[11][12]
e O2
elastic, excitations 0.02, 0.19, 0.38, 0.57, 0.75, 0.98, 1.63
and 4.5 eV, dissociation 6.0, 8.4 and 10 eV, ionization 12 eV
[10]
O- Ar electron loss [13]
O- O charge transfer [14]
O- O2 elastic [15]
O+ O charge transfer [14]
Ar+ Ar elastic, charge transfer [15]
Ar+ O charge transfer [16]
Ar+ O2 charge transfer [17]
O2+ Ar charge transfer [18]
O2+ O2 charge transfer [15]
Table 2. List of interactions included in the simulation with
corresponding references.
4. Results
4.1 Static regimeAt first, we studied the interaction of plasma
described above with an uneven substrate in the static regime. Near
the substrate with bias of 5 V, a negative charge density is formed
to shield the electric field. Figure 1 shows the electrostatic
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potential near the substrate when the stationary state has been
reached. On the bottom of the groove, thickness of the sheath is
larger, while close to the edges it is thinner.
Figure 1. Electrostatic potential near the uneven substrate with
bias of +5 V.
Thinner sheath and higher electric field at the groove edge
cause higher fluxes of negatively charged particles to the
substrate.
Figure 2. Fluxes of negatively charged particles to the uneven
substrate along the substrate border.
Figure 2 shows the fluxes of negatively charged particles to the
substrate along the substrate border. The fluxes of both electrons
and O- ions significantly increase near the groove edge. The O-
flux is lower by five orders of magnitude. This also confirms the
results in [1] which show that the sheath formed by the plasma with
lower electronegativity is almost completely composed of
electrons.
After relaxation, the system was held in stationary state for a
time long enough to obtain smooth data. Total time requirements
were about nine days.
4.2 Dynamic regimeThe study in dynamic regime shows the response
of plasma after the substrate bias was changed from +5 V to +10 V.
Undisturbed plasma properties are the same as in the static regime.
In this case, a planar substrate, without the groove, was used.
This symmetrical configuration allows us to integrate the plasma
properties in the direction parallel to the substrate. Therefore,
time dependencies can be added to the plots.
Figure 3 shows the time development of electrostatic potential.
The step change of bias from +5 V to +10 V occurred at time t=0.
Most of the additional positive potential was shielded after a few
tens of nanoseconds. It then took several microseconds to shield
the remaining potential.
Figure 3. Time development of electrostatic potential after a
step change of substrate bias from +5 V to +10 V.
Figures 4 and 5 show the responses of electron and O+ number
densities. Gray-scale represents number densities and contour lines
connect places with the same density, thus actually representing
flow lines of plasma. Data in both figures were computed with time
step of 1×10−12. The figures show that the electron response is
much faster than the response of O+ ions which is due to their
different masses. The time development of densities corresponds to
the development of electrostatic potential. Most of the potential
is shielded by electrons during first several nanoseconds.
Remaining potential is then shielded by the heavy ions – positive
ions are moving away and negative ions are heading towards the
substrate.
Total time requirements for computing in dynamic regime were
approximately sixteen days.
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Figure 4. Time development of electron number density after a
step bias change. Contours represent densities from 2×1013 m−3 to
2.6×1014 m−3 with step of 2×1013 m−3.
Figure 5. Time development of O+ density after a step bias
change. Contours represent densities from 2×1013 m−3 to 2.2×1014
m−3 with step of 2×1013 m−3.
5. ConclusionPresented results showed that particle simulations
are able to provide detailed information about studied systems and
their time requirements are getting feasible even on common
microcomputers. To demonstrate the results we chose as an example a
multicomponent Ar/O2 plasma mixture interacting with grooved and
planar substrate.
The use of grooved substrate showed the benefits of using
two-dimensional simulations instead of widely used one-dimensional
ones. We described the shape of the sheath near the groove and the
fluxes of electrons and ions to the substrate.
The dynamic simulation showed the difference between reaction
time of electrons and heavy ions. The sheath formation was studied
after a change of substrate bias. Our simulations explained the
different behavior of particles with different masses using the
example of electrons and O+ ions in the Ar/O2 plasma mixture.
AcknowledgementThe work is a part of the research plan
MSM0021620834 financed by the Ministry of Education of Czech
Republic. The authors acknowledge support of Charles University
(project SV263302), of the Grant Agency of Czech Republic (project
P205/10/0979) and of the Grant Agency of Charles University Prague
(project 46310/2010).
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1. Introduction2. Computational description3. Parameters of
simulation4. Results4.1 Static regime4.2 Dynamic regime
5. ConclusionAcknowledgementReferences