CHO ET AL.
403
Fig. 7 Time of the probe peak signals for the step-bias experiment,
T
pr4, measured for different step-bias voltages.
III. Simulation
A. Simulation Code
As the simulationscheme,we use theMonteCarlo particle-in-cell
5;6
(
)
PTC method that follows the motion of charged particles self-
consistentlywith their own space charge and collisionswith neutral
particles. The simulation domain is axisymmetric, and we consider
r
z
spatialvariationsonly in the and directions,while all of the three
componentsof particlevelocitiesare considered.Figure8 showsthe
computationaldomain. The computationaldomain simulatesa slice
¡8
Fig. 8 Schematic of computationaldomain.
£
of cylinder with an angle of 7 10 rad whose radius is 1.24 m
and length is 3.3 m.
11
10 m¡3
7.5 ¹m , the computationalgrids are clusteredin directionsusing
(
)
(
)
n D n D
£
z
Initially, a plasma of uniform density
T D T D
5
e
i
(
)
z D
with the Maxwellian distribution at ·
·
2:4 eV is loaded
Eq. 5.220 of Ref. 7. The smallest grid size is 1
2 ¹m and the
e
i
z D
r
inside the computational domain, and once the simulation starts,
no particle is added from the boundary. The plasma source is not
included in the simulation. The plasma parameters simulate the ex-
perimental condition in Ref. 2. In a PIC code, a particle followed
in the simulation represents a group of real charged particles or a
fractionof a real particle,which is often called a superparticle.Each
superparticlehas its weight as an indicator of how many real parti-
clesit represents.If we loadthe superparticleswith equalweighting,
only few particles are loaded near the axis because the cell volume
becomes increasinglysmall as the radius approaches zero.
largest grid size is 1
r D
3 cm. In the direction, the grid size is
r
equal to 1
0:02 m. The total number of grid points is 63 in the
z
direction and 651 in the direction. Initially, the insulator surface,
polyimide and acrylic, has a potential of zero. The initial surface
charge to have the zero potential is calculated from the capacitance
matrix method.8 We give the same potential pro le to the electrode
as the experiment, such as the one shown in Fig. 2. The potential
data are taken from the step-bias experiment and used as an input
le for the potentialpro le in the simulation. Because in the exper-
iment the oscilloscope was triggered at the threshold of 52 V and
recorded that time as time zero, we have shifted the oscilloscope
data by 1.6 ¹s as shown in Fig. 2 so that time zero corresponds to
the time when the step pulse begins. The time 1.6 ¹s was chosen
because the potentialnever went to negative after that time until the
end of the record.
To avoid this problem, we assign smaller weight to particles
loaded near the axis, and load the superparticles so that an equal
number of superparticles, 18, is loaded in each cell. Then the ini-
tial number of particles in the domain is 730,000 for each species.
The ion to electron mass ratio is set to 73,400 to simulate the
real mass ratio of an argon ion to an electron. The computational
boundary is a xed wall except for the center axis, to simulate the
chamber wall, where the potential is set to zero. Once a particle
reaches the wall, it is removed from the domain. The simulation
To validatethesimulationcode,we placea langmuirprobeshaped
like a fat disk, whose radius is 2 cm and the thickness is 2 cm.
z
The planarsurface is perpendicularto the axis. In the experiment,
the probe is a disk of 3 mm radius. The thickness is about 1 mm.
time step is 1 7:5 10¡10 s, and the simulation is run typically
up to 15 32 ¹s. Typicalcomputationaltime for one caseis 2 4 days
on a UNIX workstation with an Alpha 21264 500-MHz CPU.
The computational domain is lled with argon gas of uniform
The planar probe surface is parallel to the axis in the experiment.
t D
£
z
–
–
The probe current consists of displacement current and the con-
ductioncurrent.Thedisplacementcurrentisdue tothe time variation
of the surface charge induced on the probe surface as the electric
eld on the surface varies. The conduction current is mainly due
to ions entering the probe surface for the present case because the
plasma potential is mostly positive with respect to the probe. As it
was explained in Ref. 2, the probe signal in the experiment corre-
sponds to the the time integral of the current because of the large
resistance, 100 kÄ, compared to the parallel capacitance. There-
fore, as a comparison, we calculate the surface charge induced on
the probe surface by integrating the electric eld perpendicular to
the probe surface. By dividing the surface charge by the probe ca-
pacitancewith respecttothe ground,600pF, we canobtaintheprobe
signal in the voltage as if it were observed in the experiment. Un-
fortunately,however, the probe signal calculatedthis way cannotbe
compared quantitatively because the electric eld and the surface
charge on the probe surface depend on the size and shape of the
probe. Nevertheless,the probe signal in the experimentshows very
similartemporalpro le to the computationallycalculatedprobesig-
nal. In the simulation, the probe potential is xed to zero, though
the probe actually has a small potential equal to the probe signal
output voltage, less than 1V. Because the probe is surrounded by
the plasma,whose space potentialis 20 V or higher, this assumption
on the probe potentialcauses no signicant error.
n
density n . Collisionbetween charged particlesand neutralargon is
taken into account, though no motion of neutral argon is taken into
account. Quantitative accuracy of the collision cross section was
already checked against the the transport coef cient data, such as
(
ionizationand driftvelocity,taken from the literature see Ref. 6 for
)
details . Once ionization collision occurs, a pair of secondary elec-
trons and ions is placed with an appropriate energy at the collision
point.
To simulate the insulator and electrode used in the experiment,
we placean electrodeinsulatedby acrylicand polyimide.The cylin-
drical coordinate of the computational domain has its center at the
polyimide surface for comparison with the experiment. In the ex-
periment, the insulatorsurfacewas 40 cm off axis from the chamber
r
axis in the direction. The effect caused by this slight difference
is insignicant as long as the the sheath from the insulator does
not extend over 85 cm, where the closest chamber wall existed
in the experiment.The geometryreproducesthe real geometry with
the same thickness of the electrode, the polyimide, and the acrylic.
The area of the polyimide lm is also the same so that it has the
same capacitance as the experiment. The dielectric constants are
3:5 for the polyimide and 2:7 for the acrylic. To model the thin lm