J. Plasma Fusion Res. SERIES, Vol.4 (2001) 340-344

Self-consistent Sheaths in RF Dusty Plasmas
CHUTOV Yuriy*, GOEDHEER Wimr, KRAVCHENKO Olexandr,
ZIJZ Yadym and YAN Min2
Faculty of Radio Physics, Taras Shevchenko Kiev University,
Volodymyrs'ka Str. 64, 252017 Kiev, UkrainelFoM-Institutefor Plasma Physics "Rijnhuizen", P.O. Box 1207,
3430 BE Nieuwegein, The Netherlands
2Department of Chemistry, University of Antwerp, Universiteitsplein 1,
8-26 l0 Wilrijk-Antwerp, Belgium(Received: 5 December 2000 / Accepted: 18 August 2001)

AbstractSelf-consistent RF sheaths are investigated using a self-consistent lD PICA4C simulation of a RF
discharge in argon with dust particles distributed uniformly in an interelectrode gap. Spatial distributionsof discharge parameters across the discharge interelectrode gap were simulated at various densities ofdust particles, namely: the electron and ion densities, the electric potential and field, the time-averaged
electron energy distribution function, as well as the charge of dust particles. Obtained results shown that
the RF discharge with dust particles has non-stationary sheaths with a strong electric field separating theelectrodes from a quasi-neutral central part. The dust particles essentially influence the spatial
distributions of discharge parameters, in particular an increase of a dust particle density causes anexpansion of sheaths. The dust particle charge changes non-monotonously across the interelectrode gap
and has maximum close to a sheath edge, due to the spatial distribution of plasma parameters and apeculiarity of the electron energy distribution function in the quasi-neutral central part of the RFdischarge.

Keywords:
dusty plasma, sheath, RF discharge, electron energy distribution function, dust charge

1. Introduction
RF sheaths separate electrodes under a RF voltage

the force of the mean electric field. Besides, the conditions needed for the creation of a plasma crystal canfrom plasma in many cases including RF heating of

be satisfied here. The first requirement is a strong non-fusion plasmas by RF antennas, plasma diagnostics, andplasma processing in the microelectronics industry (e.g.
plasma etching, plasma deposition, and plasma surfacemodification). The sheaths can contain dust particles,

ideality of the considered system, i.e. a large ratio of

which can appear here due to an interaction of theplasma with the electrodes [-3] or can be immersed

now.

potential and kinetic energies of dust particles screened

by background electrons and ions. Plasma crystals
created by dust particles are intensively investigated
Dust particles can essentially influence the properties of RF sheaths due to the action of a self-consistentelectric field created by charged dust particles and duefrom outside [4-7]. Dust particles can be levitated close
to a RF sheath edge because the gravitational force
acting on dust particles can be compensated here by a*02001 by The Japan Society of Plasma

Corresponding author's e-mail: yuch@ mail.univ.kiev.ua

Science and Nuclear Fusion Research340

Chutov Y. et a/., Self-consistent Sheaths in RF Dusty Plasmas

cm filled with Ar at a pressure of 0.1 Torr is simulated.
Immobile dust particles of I pm radius are distributed

to the collection of background electrons and ions by
dust particles, like the influence on DC sheaths [8]. Bothreasons indicated above depend strongly on theinteraction between dust particles. Dust particles interact
in dusty plasmas through a shielded Coulomb potential

usually uniformly in the interelectrode gap with adensity No. The uniform spatial distributions of dust
particles allow us better estimate their role in various
regions of the RF discharge. The dust particles collectand scatter electrons and ions distributed in thedischarge with a (non-uniform) density n" and ni,respectively. A harmonic external voltage V.(t) =V.sin(a;t) at a frequency f = 13,56 MHz with anthat is determined by both the self-consistent charge ofdust particles and the characteristic shielding length.
Both the charge and the shielding length have non-equilibrium values in dusty plasmas, especially insheaths, due to the non-equilibrium distribution of
background electrons and ions caused by their selectivecollection by dust particles [8,9] as well as due to themutual interaction of dust particles [0].

amplitude V" = 350 V sustains the RF discharge. The

Therefore many experiments were carried out todischarges without dust particles is modified forcomputer simulations of the RF discharge with dustparticles. The self-consistent electric field E is obtainedby solving Poisson's equation using a computationalgrid, which is introduced by a uniform division of theinterelectrode gap into 128 simulation cells. An electrode collects a "superparticle" if its center reaches theelectrode atx=d is grounded.
The PIC/MC method described in detail in [17] formeasure the dust charge and the dust screening length
using the properties of dust oscillations or dust waves aswell as shocks in RF sheaths (see for example l2,3,ll141). However, the properties of dust oscillations or dust
waves in RF sheaths depend not only on the dust charge
and the dust screening length but also on the spatial
distribution of the sheath parameters, especially theelectrode surface. Each superparticle represents 8 x 107
real electrons or ions. The size of the electrodes, needed
to compute the total current. is 0.04 m2.
The Monte Carlo technique [8] is used to describe
electron and ion collisions. The collisions include elastic
collisions of electrons and ions with atoms, ionizationand excitation of atoms by electrons, charge exchange
between ions and atoms, Coulomb collisions of electrons and ions with dust particles, as well as the electron
and ion collection by dust particles. The electron-argon
collision cross-sections used in the model are the sameas those used in [l9]. The Coulomb cross-section o forelectron and ion scattering by immobile dust particles iselectric potential. Therefore, detailed investigations of
RF sheaths with levitated dust particles, including
computer simulations, are very actual for understanding
the influence of dust particles on the sheaths. The aim of
this work is lD-computer kinetic simulations of RF
discharges in argon with dust particles, which can beconsidered as a physical model of dusty plasmas. The
kinetic simulation of dusty RF discharges was not
carried out earlier.

2. ModelElectrodes or walls disturb plasmas due to thedifference in the amount of electrons and ions absorbed
from plasma; the disturbance penetrates far into plasma.

taken from [20]. The cross-section for collection of anelectron or ion by a dust particle is taken from [21].
The simulation starts with an initial uniform distribution of electrons and ions with densities equal to l0l5
m-3 and is prolonged by iterations up to a moment when
the change of the discharge parameters is less than 0.1
7o. Simulations show that 400 cycles are enough toobtain the periodic steady state.

The disturbed plasma region can be divided into acharged sheath and a quasi-neutral presheath, if these
sub-regions essentially exceed a characteristic Debye
length [15].

Self-consistent sheaths have to be simulated
together with the corresponding presheaths in order tosolve the complex problem of providing boundary

3. Results

conditions for sheaths [15]. Another approach is to use aself-consistent solution for a rarefaction wave in plasma[16]. However in case of sheaths created in a gas
discharge plasma, the best approach is a self-consistent
simulation of the entire gas discharge with corresponding boundary conditions on electrodes or walls.
In our case, a one-dimensional RF discharge between two plane electrodes separated by a gap of d = 2.'7

Obtained spatial distributions of the electric field E
across the interelectrode gap of the RF discharge areshown in Fig. I for various densities N6 of dust particlesof I pm radius for various phases t/? of the sustaining
external voltage where 7 is the RF period. Thedistributions show the existence of the central quasineutral region and non-neutral RF sheaths close to both341

Chutov Y. el al., Self-consistent Sheaths in RF Dusty Plasmas

E*10-5, V/m

I0.41
tlT = 0.25 - - '- 0.32-'- 0.5 -..-.-.0.58
0.6-.-. .

-n.

---- n, tlT= 0.25. 0.32 -.- -.- 0.41

No=00

Na=0-101

n*10-tt, m-'

0-101n*10-1t, m-'20,5xld0,5Fig. 1 Spatial distributions of the electric field E forx/dvarious densities No of dust particles including thecase without dust particles (Nd = 0), and forvarious phases f/f of the sustaining external
voltage where 7. is the period of the externalvoltage.

Fig. 2 Spatial distributions of the electron n" and ion ni

densities for various densities No of dust particles
including the case without dust particles (N, = 0),and for various phases f/f of the sustaining

harmonic external voltage where f is the periodof the external voltage.
electrodes. The quasi-neutral part of the interelectrodegap corresponds here to E = 0 so that the sheath edge isa point between this region and the region with E + 0.
The sheath edge, which corresponds to the boundary

for conditions of Fig.l. As can be seen in Fig.2, there
are stationary spatial distributions of the ion density n1

between the charged sheath and the quasi-neutral

and changed spatial distributions of the electron density

presheath [15], shifts to or from electrodes depending onn" because the sustained voltage frequency by far

the phase of the sustaining external voltage. As can beseen in the Fig. 1, the sheath edge approaches theexceeds the ion plasma frequency and is comparable togrounded electrode and is moving away from theopposite electrode between phases rc/2 and 3n/2 of theelectron density decreases smoothly in the RF sheath,unlike the stepwise drop in the front model [22],or less than the electron plasma frequency. Note that thesustaining external voltage. Note that the size of thebecause the Debye length is not negligable compared tocentral quasi-neutral part is decreased with the increaseof the dust particle density No. This is caused by theinfluence ofdust particles on the RF sheath size. Indeed,
the sheath size is determined by a condition that an
applied voltage is compensated by the total space sheathcharge. As can be seen later (Fig. 3), immobile dust

the sheath size in our case.

particles have a negative charge and therefore compensate partially the ion screening of the sustained

increase of the dust particle density so that the ion
density by far exceeds the electron density at large
densities of dust particles. The indicated difference iscaused by the negative total electric charge Q6 of dust
particles of which spatial distributions are shown in theupper part of Fig. 3 for various values ofN6.

As can be seen inFig. 2, dust particles essentiallychange the spatial distributions of the electron and iondensities. First of all, a difference between these
densities appears in the central part of the RF discharge
with dust particles. The difference increases with anvoltage in the sheath and consequently increase the RFsheath size at the same voltage.
Spatial distributions of the electron (n") and ion (4)density across the interelectrode gap are shown in Fig. 2J+L

Chutov Y. et al., Self-consistent Sheaths in RF Dusty Plasmas

Na=lo,tmo - ---- NoJYlorr^'
16,00-iN a=l0t2 m" .''- - - N

o=2*1 fz

0.4
0.3

-6,0

^a3,0Eq"a_e)x/d-_0.5

F"*1 otu, moev'

0.2
0.1

0,1|,'

9'96

14,00

6,0

No=3!'1gtr, a3,00,1o'oo'
0,1-'-0 1

0,5,/d10e, gV20Fig. 4 Averaged electron energy distribution functions F"

Fig. 3 Spatial distributions of the total electric charge

in various points x/d of the interelectrode gap forvarious densities No of dust particles including thecase without dust particles (Nd = 0).

density Oo of dust particles and the dust particle
charge gd across the interelectrode gap forvarious densities N6 of dust particles.

region of the interelectrode gap, due to a free mixing ofNote the non-monotonic spatial distributions of thedust particle chatge q6 shown in the bottom part of Fig.
3. As can be seen here, the charge qa in the quasi-neutral
central part of the RF discharge is equal to about 2.5 x
103 e, corresponding to a surface potential of 3.6 eV.
The charge 4d increases towards the sheaths and thendecreases in the sheaths. As a result, the charge 46 has afast electrons in this almost equipotential region.
Therefore, the electron charging current onto a dust
particle is almost the same for various points in thecentral region. The constant electron charging current
and the decrease of the ion charging current cause theincrease of the dust particle charge 1461, when moving
outward from the centre (Fig. 4).
Another situation prevails in the sheaths where afree mixing of electrons is not possible due to the strong
voltage drop in sheaths (Fig. 1). Therefore the numberof fast electrons is not constant in various sheath points
Gig. a) decreasing towards electrodes so that the electron charging current decreases also. Obtained results
show that the ratio of the time-averaged electron density<ne> to the constant ion density n; decreases in thesheaths towards electrodes. This results in a decreasing

maximum close to the sheath edge. This maximum
shifts together with the sheath edge when the dust particle density N6 changes.
The complex spatial distribution of the dust particle
charge is caused by the balance of the electron and ion
charging currents onto a dust particle, as usually [7]. In

our case of a non-uniform plasma with negatively
charged dust particles, the ion charging current is proportional to the ion density and therefore has to decrease
monotonically from the center of the interelectrode gap
towards the electrodes due to the monotonic decrease of
the ion density shown in Fig. 2. The electron charging
current depends on the surface potential of the dust

dust particle charge q6 towards the electrodes, creating amaximum close to the sheath edge (Fig. 4).

4. Conclusion

particles and the time-averaged energy distributionfunction .F'" shown in Fig. 4 for the cases of Fig. 1. Note

Self-consistent lD PIC/MC simulations of a RF
discharge in argon with dust particles distributed
uniformly in the interelectrode gap show that the dustthat the distribution functions F" practically coincide in
the energy region e > 3 eV in the quasi-neutral central343

Chutov Y. et al., Self-consistent Sheaths in RF Dusty plasmas

particles essentially influence the spatial distribution of(ELSEVTER, 2000).

the discharge parameters. An increase of the dust

t6l M. Schabel et al., J. Appl. Phys. 86, 1834 (1999).
t7l G.E. Morfill et al., in Advances in Dusty Plasmas

particle density causes an expansion of sheaths.
The dust particle charge varies non-monotonously(World Scientific, 1997).
Yu.I.
Chutov et al., PhysicaB. 262, 415 (1999).
[8][9] Yu.I. Chutov et al.,PbysicaB. l2E, l1 (1996).
[10] J. Goree, Plasma Sources Sci. Technol. 3, 400

across the interelectrode gap and has maximum close toa sheath edge, due to the spatial distribution of theplasma parameters and free maxing of the energetic
electrons in the quasi-neutral central part of the RF(1994\.

llll A. Homann et al.,Phys. Rev. E 59, R3836 (1999).
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tl3l D. Samsonov et al.,Phys. Rev. E 61,5557 (2000).
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Acknowledgments
This work was partially supported by INTAS(Contract No 96-0617) and by a granr from the
Ukrainian Committee of Science and Technoloev.

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