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

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

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.

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