Computational Study of Interaction Between

Multicomponent Plasma and Solid

T. Ibehej and R. Hrach

Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.

Abstract. Presented computer model simulates an interaction of low-temperatureAr/O2 plasma with conductive solid substrate. Mixtures of argon with elec-tronegative gases are often used in material processing technologies. The resultscan also be used for an analysis of Langmuir probe measurements. Electrostaticpotential, number densities of species contained in plasma near the biased sub-strate and fluxes of the species to the substrate are studied in both static anddynamic regimes. Presented two-dimensional simulation was written in C languagewith additional parallel computing and force calculation libraries. Computationaltechnique is a hybrid Molecular Dynamics / Monte Carlo particle approach withadvanced collisions-computing method and a two-dimensional Particle in Cell forcecalculation.

Introduction

Particle simulations of low-temperature chemically active plasmas interacting with solids are veryuseful in both engineering and scientific applications. Results of the simulations can help with anexplanation of measured probe characteristics as well as they can assist in the improvement of variousmaterial treatment technologies. The main advantage of particle simulations in general is the detailin which the final information is given. They provide local information about fluxes of each kind ofparticles, their energy and angular distributions and number densities. It is also possible to get globalparameters of the system like electrostatic potential or electrical current. On the other hand, extremelyhigh demands on computational time are the biggest disadvantage.

The argon-oxygen mixture which is studied in this paper is a typical medium used in technologiessuch as plasma etching or cleaning. The composition of the undisturbed plasma was obtained from achemical kinetics model with an initial volume ratio [Ar]:[O2] of 9:1. In [Ibehej, Hrach, 2011] we studiedproperties of argon plasma and general electronegative plasma with variable electronegativity, while thiscontribution focuses on concrete Ar/O2 mixture. The mixture consists of five charged species – electrons,O−, O+, Ar+ and O+

2. We also included 29 of the most frequent types of collisions between charged

and neutral species. For the needs of collisions, the neutral background was composed of Ar, O andO2. A separate continuous timescale was applied for collisions, according to the “advanced null collisionmethod”. Besides the Monte Carlo approach, for the remaining parts of the simulation we used moleculardynamics computational technique.

In contrast to some previously published one-dimensional particle simulations [e.g. Diomede et

al., 2005; van der Straaten et al., 1998], our presented simulation is two-dimensional. Therefore, morecomplicated geometries could be studied. The static results were obtained for the plasma near a groovedplanar substrate. The same shape of the solid is also discussed in [Ibehej, Hrach, 2011] for electropositiveand general electronegative plasma. In addition, some dynamical results are presented for a non-groovedplanar substrate.

Description of the simulation

Working area and source of particles

The situation for the static simulation is schematically displayed in Figure 1. The groove dimensionswere 10 × 5 mm, while the working area, including the solid, was square-shaped with the side of 2 cm.The source of particles was a simple model of undisturbed plasma. The velocity distribution in thesource was Maxwell-Boltzmann distribution with the mean energy of 3.6 eV for electrons and 0.039 eVfor all ion species. Periodic boundary conditions (PBC) were applied to the borders of the source. Oneside of the source was permeable, so the particles from the source could enter the working area. Insidethe source, the particles neither underwent any collision nor were affected by the electrical field from thesubstrate.

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IBEHEJ AND HRACH: SIMULATION OF PLASMA-SOLID INTERACTION

Figure 1. Configuration for static simulations. (1) grooved substrate, (2) two-dimensional workingarea, (3) source of particles from undisturbed plasma.

Table 1. Number densities of charged and neutral species in the undisturbed plasma. The plasmapressure was set approximately to 145 Pa. The density ratios were taken from the model of chemicalkinetics.

e 9.8 × 1014 m−3 Ar 2.9 × 1022 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

O+

22 × 1013 m−3

The PBC were also applied to both opposite sides of the working area perpendicular to the substratesurface. Particles exiting the area towards the source were discarded and so were the particles which hitthe substrate – before, they were counted towards the electrical current.

For the dynamic simulations, a simple non-grooved planar substrate was used in order to preservesimplicity of the results, plotted with additional time scale.

For both types of simulations, we used plasma with pressure of 145 Pa. The ratios of number densitiesof particular species were taken from the results of a chemical kinetics simulation which included 129oxygen, 12 argon and 20 argon-oxygen reactions [Cerny, 2011]. The cited model simulated processes in apositive column of a DC glow discharge in an Ar/O2 mixture with reduced electrical field E/N = 60 Td.The densities are summarized in Table 1.

Motion of particles

The motion of particles can be divided into two processes. The first one is a smooth motion causedby the electrical field from the substrate and from other particles. For the numerical integration ofNewton’s second law of motion we used the Verlet’s velocity algorithm [Verlet, 1967]. Neutral particleswere not included in the molecular dynamics scheme, because they did not directly affect the formationof sheath and presheath and their density was too high compared to the density of charged particles.However, they could influence the motion of charged particles through collisions. The collisions weresimulated as a random process by the “advanced null collision method” presented in [Hrach et al. 2009].This approach allowed each particle to scatter more than once during one time step and, therefore,enabled us to use longer time steps even for higher pressures.

Force calculation

For the Verlet’s velocity scheme, total force acting on each particle was needed. For the force calcu-lation, we used Particle In Cell algorithm in the basic Nearest Grid Point variation [Birdsall, Langdon,1991]. The force is obtained from the electrostatic potential which is calculated at each time step. ThePoisson’s equation

∆U (~r) = −ρ (~r)

ǫ0(1)

is solved numerically using the sparse LU factorization C library Umfpack [Davis, 2004]. The force actingon i-th particle located in ~ri is then

~F (~ri) = −qi∇U (~ri) . (2)

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IBEHEJ AND HRACH: SIMULATION OF PLASMA-SOLID INTERACTION

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25 30

flux

[1018

m-2

s-1

]position along substrate border [mm]

x105

electronsO- (x105)

Figure 2. Electron and O− fluxes to the substrate. The horizontal axis runs along the substrate border.

Figure 3. Spatial distribution of electron number density. Contours: minimum 1 × 1013 m−3, step2 × 1013 m−3.

Figure 4. Charge density. Contours: minimum −20 µC m−3, step 2 µC m−3, maximum −2 µC m−3.

ResultsStatic results

In the static regime, sheath and presheath properties after relaxation were studied. The bias ofthe groove was set to +5 volts. Therefore, the negatively charged species, i.e. electrons and O− ions,were attracted to the substrate. The sheath inside the groove got thicker (approximately 4 mm), whileoutside the groove we observed sheath with thickness of approximately 2.5 mm. At the outer cornersof the groove, a higher electrical field was observed, which resulted in higher fluxes in this region. Thedistribution of electron and O− fluxes is displayed in Figure 2.

Close to the inner corners of the groove, we noticed very low fluxes of negatively charged particles.As it is shown in Figure 3, a small number of electrons could penetrate to those two regions. In Figure 3near the outer corners of the groove, a slight increase of electron density can be seen. This effect is typicalfor sheath formed at a substrate with curved surface. Therefore, it cannot be observed at a non-groovedplanar probe.

Figure 4 shows the charge density near the substrate. The minimum of the charge density is locatedat the boundary of the sheath. The sheath thickness of approximately 2.5 mm corresponds to thethickness estimated from Figure 3.

Dynamic results

In dynamic regime the sheath development near a non-grooved planar substrate was investigated.A periodical voltage with frequency of 1 MHz was applied to the substrate and the reactions of electrons

and ions were studied separately. The electron plasma frequency ωp,e =√

nee2

meǫ0was approximately 1.8

GHz. The O+ ion plasma frequency was about 13 MHz, thus approaching the voltage frequency. As

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IBEHEJ AND HRACH: SIMULATION OF PLASMA-SOLID INTERACTION

Figure 5. A time development of electron (left) and O+ (right) number density in Ar/O2 mixture neara planar substrate. To the substrate a harmonic (top) and rectangular (bottom) voltage was appliedwith frequency of 1 MHz.

shown in Figure 5, two different voltage shapes were applied to the substrate.Electrons, due to their higher plasma frequency, were able to react very quickly to the step change

of voltage. The O+ ions could also register the change, but their reaction time was much longer andduring the period, they did not reach the balance for the actual value of voltage.

Conclusion

A multicomponent Ar/O2 plasma in interaction with solids was studied in both, static and dynamicregimes. In the static regime, the advantage of a two-dimensional simulation was taken by studyingmore complicated geometry. The sheath near a grooved substrate was described. We observed changesof fluxes to the surface of the groove and effects away from the surface which are typical for curvedgeometries. Results of the dynamic regime proved a great difference between ion and electron plasmafrequency. The time development of electron and O+ number density was studied near the non-groovedplanar surface while a periodical voltage of two different shapes was applied to the substrate.

Acknowledgments. The work is a part of the research plan MSM0021620834 financed by the Ministry of Ed-

ucation of Czech Republic. The authors acknowledge support of the Grant Agency of Charles University Prague (project

46310/2010).

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