Computational study of Ar/O 2 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. Introduction
Chemically 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/O 2
mixture. A significant improvement was achieved
especially in modeling of charged-neutral particles
interactions.
2. Computational description
Presented 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 2×2 cm 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
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.
Input 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.
OO
+
2×1013 m −3
14
−3
8.2×10 m
Ar+
1.6×1014 m −3
O2+
2×1013 m −3
Ar
2.9×10 22 m−3
O
5.6×1021 m −3
O2
Species Interaction
20
3.5×10 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
References
e
Ar
elastic, excitation 11.5 eV,
ionization 15.8 eV
[10]
e
O
elastic, excitations 1.97 and 4.19 eV
ionization 13.62 eV
[11]
[12]
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]
e
Ar electron loss
[13]
O
-
O
[14]
O
-
O2 elastic
[15]
O
+
O
[14]
charge transfer
charge transfer
Ar+ Ar elastic, charge transfer
[15]
Ar+ O
charge transfer
[16]
Ar+ O2 charge transfer
[17]
+
[18]
+
[15]
O2 Ar charge transfer
Neutral species
9.8×1014 m−3
e
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.
O-
3. Parameters of simulation
Charged species
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.
O2 O2 charge transfer
Table 2. List of interactions included in the simulation with
corresponding references.
4. Results
4.1 Static regime
At 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
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.
4.2 Dynamic regime
The 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.
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.
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.
Acknowledgement
The 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).
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.
References
[1] T. Ibehej and R. Hrach, “Computational study of sheath
structure for plasma-assisted technologies in the presence
of electronegative plasma,” Vacuum (2010), in print.
[2] R. Hrach and M. Vicher, Czech. J. Phys. 51, 557 (2001).
[3] R. Hrach, V. Hrachová and M. Vicher, Comput. Phys.
Commun. 147 , 505 (2002).
[4] R. Hrach, D. Sedlák, M. Vicher and J. Šimek, Thin Solid
Films 459, 137 (2004).
[5] P. Černý, S. Novák, R. Hrach and P. Bruna, Contrib.
[6]
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. Conclusion
Presented 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.
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