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XIV International Conference on Gas Discharges and their Applications (GD2002) in Liverpool
The University of Liverpool, 1-6 September 2002
Developing an Efficient PIC/MC Simulator
for RF Glow Discharge
using a Higher Order Sampling Technique
K.Satoh1, M.Kozaki1, H.Date2, H.Itoh1 and H.Tagashira1
1
2
Department of Electrical and Electronic Engineering,
Muroran Institute of Technology, Muroran 050-8585, Japan
College of Medical Technology, Hokkaido University, Sapporo 060-0812, Japan
[email protected]
Agenda
1. Motivation & Objective
2. Methodology
3. Simulation model & conditions
4. Results
5. Conclusions
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Introduction
Particle-In-Cell/Monte Carlo (PIC/MC) simulator is one of the most
reliable and useful simulators for RF glow discharges.
 The behaviour of charged particles both in equilibrium and in non-equilibrium
regions is calculated accurately and the effects of boundaries are treated properly.
 The velocity distributions of the charged particles, which are important and useful
information to understand the characteristics of RF glow discharges, are obtained.
However, statistical fluctuations are included in the PIC/MC essentially.
 Traced MC particles have weight factor irrespective to the position in a gap, so that
number of the MC particles in the sheaths is smaller than that in the bulk plasma.
 Since the sheath regions are important to sustain the discharge, the statistical
fluctuations due to the shortage of MC particles may cause the error of the PIC/MC
simulation.
In order to remove the instability of the PIC/MC simulation, the
statistical fluctuations must be reduced.
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(Ardehali et al., IEEE Trans. Plasma Sci., Vol.25, pp.1081-5, 1997)
 Sheath regions are split and MC particles are added in the regions.
 Fluctuations are reduced with a small amount (< 5%) of increase
in computation time.
Legendre Polynomial Weighted Sampling (LPWS)
(Ventzek et al., J.Appl.Phys., vol,75, pp.3785-88, 1994)
 The higher order derivatives of the bin of energy distribution are
sampled, and then the density distribution in the bin is described
using Legendre polynomials.
 The distributions are multiplied by the weight of B-spline, and
then superposed to eliminate the discontinuity between successive
bins.
sheath
sheath
bulk
Number of
MC particles
SPLIT-PIC/MC simulator
Ion density
Approaches to reduce fluctuations in MC simulation
This technique has a wider applicability to MC sampling.
In this work we developed a PIC/MC simulator coupled with LPWS to reduce
the statistical fluctuations in the density distributions of charged particles.
The developed simulator is applied to the simulation of a RF glow discharge
in nitrogen.
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METHODOLOGY - Conventional sampling & principle of LPWS •Conventional method
(simple counting)
n(z)
e
Ni=Ni+1 〇
bin
•Legendre Polynomial
Weighted Sampling
n(z)
bin
e
〇
n1()
n0()
N0
z0
N1
z1
ni()
Ni Ni+1 Ni+2
z2
zi
zi+1
Dz
z
 Number of electrons in a bin is added.
If (zi＜z≦zi＋1) then Ni  Ni 1
To obtain detailed distribution n(z)
・fine mesh(bin) is needed.
・statistical fluctuations increase.
・a number of electron must be sampled.
・considerable computing time
z0
z1
z2
zi
ni+2()
ni+1()
zi+1
Dz
z
1  χ  1
 Density distribution ni(x) in a bin is described
using Legendre polynomial.
ni (  )  P0 (  ) f 0  P1 (  ) f1  P2 (  ) f 2  ・・・
1 1

f

0
1 P0 (  )ni (  )d

1

2
2

 P0  1, P1   , P2  2 (3  1)
3 1

 f1   P1 (  )ni (  )d



2 1
   2 z  ( zi 1  zi ) / 2


5 1
zi 1  zi

f

P2 (  )ni (  )d
2

2 1

 To obtain detailed distribution n(z)
・increasing terms of the polynomial.
・accurate n(x) is independent on bin size.
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METHODOLOGY - LPWS : overlap sampling & superposition Problem
ni
zi
ni+1
Solutions
zi+2
zi+1
・To sample enough number of particles.
・Overlap sampling & superposition of ni and ni+1 using B-spline (LPWS).
①
ni(z)
{
n(z)
At boundary, ni and ni+1 are not necessarily continuous.
Superposition (smoothing) by the primary B-Spline
1  χ  1
S(z)
z
Si(z)
z
②
{
n(z)
Si+1(z)
S(z)
z
Si(z)
Si+1(z)



Generally,
m 1
n( z )   ni ( z ) Si ( x) ,
z
i 1
x
z  zi
zi  z  zi 1 
zi 1  zi
m : the order of B-spline, Si : B-spline
n(z)
n(z)
②

z  zi
n( z )  (1   )  ni    ni 1　 
zi 1  zi

ni+1(z)
1  χ  1
①
0  χ 1
zi-1 zi
zi+1 zi-2
z
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METHODOLOGY - PIC/MC simulation coupled with LPWS ① conventional
Flowchart
Calculating flight, collision &
scattering of MC particles in Dt
Charge
assignment
n(z) between electrodes
is obtained by LPWS,
and then charge is
assigned to grid points.
e
 q
Dz
 1
Dz
②
①
Charge is assigned to
grid points as a
function of its position
between grid points.
n(z)
1q
Dz
q
z
② using LPWS
n(z)
Calculating electric field
using poisson’s eq.
Dz
z
Mesh width
PIC/MC simulator
PIC/MC simulator with LPWS
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Simulation Conditions
Electron-N2 collision cross sections
x=0 (cm)
(Y.Ohmori et al., J. Phys. D: vol.21, pp.724-9, 1987)
-14
10
Cg=2.5 (pF)
qm
-15
10
qi
2
cross section (cm )
Cb=2.0 (pF)
・N2
・0.5 Torr
・20 ℃
・n0=1.0×107 (cm-3)
・MC partical : 30,000
x=3 (cm)
-16
10
qv(10)
qex(20)
-17
10
-18
10
-19
10
V=300sinwt w/2p =13.56MHz
LPWS for ne(z) and np(z)
Number of terms (Legendre Polynomial) : 7 terms
○ B-spline : primary B-Spline
○
1
10
electron energy (eV)
2
3
10
10
(Phelps, J.Phys.Chem. Ref. Data, vol.20, pp.557-73, 1991)
10
-13
Qm
10
-14
Q(Vib)
2
number for field calculation (Dx≦lD)
60 meshes (Dx=0.05cm), 300 meshes (Dx=0.01cm)
○ Simulation Step (Dt≦1/fe )
Dt(=0.123ns)=1/600cycle
0
10
N2+-N2 collision cross sections
cross sections [cm ]
○ Mesh
-1
10
10
-15
QCT
Q(N+)
10
10
-16
-17
Q(N 3+)
Q(391)
Q(300-500)
10
-18
0.1
1
10
100
1000
electron energy [eV]
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Spatiotemporal variations of electron & ion densities 1499-1500 cycle
2.5x10
8
2.0x10
8
1.5x10
8
1.0x10
8
60 grid points for charge assignment
Ion density
2.5x10
8
2.0x10
8
1.5x10
8
1.0x10
8
5.0x10
7
5.0x10
-3
7
0.0
0.0
P
0.5
1.0
1.5
positio
n (cm)
2.0
2.5
1500.0
1499.8
1499.6
)
1499.4
cle
1499.2
cy
e(
1499.0
tim
3.0
0.0
0.0
P
G
2.5x10
8
2.0x10
8
1.5x10
8
1.0x10
8
2.5x10
8
2.0x10
8
1.5x10
8
1.0x10
8
5.0x10
7
0.5
1.0
1.5
positio
n (cm)
2.0
2.5
3.0
1500.0
1499.8
1499.6 )
e
1499.4
cl
cy
1499.2
e(
tim
1499.0
G
5.0x10
7
0.0
0.0
P
0.5
1.0
1.5
positio
n (cm)
2.0
2.5
1500.0
1499.8
1499.6
)
1499.4
cle
1499.2
cy
e(
1499.0
m
i
t
3.0
-3
PIC/MC
with LPWS
electron density (cm )
-3
electron density (cm )
Standard
PIC/MC
electron density (cm )
-3
electron density (cm )
Electron density
0.0
0.0
P
0.5
1.0
1.5
positio
n (cm)
G
Statistical fluctuations are reduced by LPWS.
2.0
2.5
3.0
1500.0
1499.8
1499.6 )
1499.4
cle
cy
1499.2
e(
tim
1499.0
G
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Spatiotemporal variations of electric field
1499-1500 cycle
60 grid points for charge assignment
Standard PIC/MC
PIC/MC with LPWS
20000
0
10000
0
electric field (V/m)
10000
electric field (V/m)
20000
-10000
-10000
1500.0
1499.8
1499.6
time 1499.4
(cyc
le) 1499.2
1499.0 0.0
P
0.5
1.0
1.5
2.0
)
n (cm
pisitio
2.5
-20000
3.0
G
1500.0
1499.8
1499.6
time 1499.4
(cyc
le) 1499.2
1499.0 0.0
P
0.5
1.0
1.5
2.0
)
n (cm
pisitio
2.5
-20000
3.0
G
The fluctuations in the charged particle densities do not make large influence on the
electric field in this case.
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Spatial variations of densities & electric field
electron density
300 grid points for charge assignment
ion density
6
6
300x10
300x10
250
Standard PIC
PIC with LPWS
250
100
Standerd PIC
PIC with LPWS
-3
ion density (cm )
-3
electron density (cm )
2999.3 cycle
200
150
100
Electric field
200
150
100
50
50
0
0
0.5
1.0
1.5
2.0
2.5
3.0
0.5
Position [cm]
1.0
1.5
2.0
Position [cm]
Net charge density
6
-3
net charge density (cm )
150x10
2.5
3.0
electric field (V/cm)
50
0
100
-50
50
0
Standerd PIC
PIC with LPWS
-50
Standerd PIC
PIC with LPWS
-100
0.5
1.0
1.5
Position [cm]
2.0
2.5
3.0
-100
0.5
1.0
1.5
2.0
2.5
3.0
Position [cm]
60 meshes → 300 meshes
 Fluctuations in the densities by the standard PIC/MC increase. (both in sheath and in
bulk plasma)
 Fluctuation in the densities by PIC/MC with LPWS do not increase significantly.
 Spatial variation of electric fields by the PIC/MC with agrees well with that by the
standard PIC/MC. However, in the bulk small fluctuations are found.
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Spatiotemporal variations of electron density
100 cycle
6
 At 100 and 200 cycle, the electron density profiles
by these methods do not agree, however, the profiles
have a tendency to agree each other at 500 cycle.
-3
electron density (cm )
250x10
200
150
100
50
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
position (cm)
200 cycle
-3
electron density (cm )
 It is likely that the simulation using the PIC/MC
with LPWS tends to reach convergence faster than
the standard PIC/MC simulation.
6
250x10
200
150
100
50
0
0.0
 It is found that the density profile by the PIC/MC
with LPWS increases faster than that of the
standard PIC/MC.
0.5
1.0
1.5
2.0
2.5
3.0
 Similar tendency is reported by Ardehali et al. for
the comparison between the SPLIT-PIC/MC and
the standard PIC/MC simulation. (IEEE Trans. Plasma
Sic., vol.25, pp.1081-5, 1997)
position (cm)
500 cycle
6
electron density (cm
-3
)
300x10
250
200
Standard PIC/MC
PIC/MC with LPWS
DSMC (Date et al, 1992)
Propagator
150
100
50
0
0.0
0.5
1.0
1.5
position (cm)
2.0
2.5
3.0
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Comparison with measured data
Generation rate of excited
molecules (C3u) density
Experiment
PIC/MC with LPWS
6.0x10
2
4.0x10
2
3
density of C  u (a.u.)
Spatiotemporal variation of excited
molecule (C3u) density
K
2.0x10
0.0
0
K
6.0x10
14
4.0x10
14
2.0x10
14
2
A
A
0
A
A
37
Ti
m
Gap
)
(mm
)
s)
(n
74
0
s
(n
K
30
e
e
m
Ti
37
excitation rate of C3u
p=0.5Torr, P=60W, f=13.56MHz
0.0
3
K
2
74
1
0
tio
pos i
n (cm
)
H.Itoh et al., Trans. of IEE Japan, Vol.121-A, pp.465-70, 2001
Large peaks are seen near the instantaneous cathode (K), and small peaks are also
seen near the instantaneous anode (A).
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Computation time
method
computation time
(for 100 cycle)
accuracy
PIC/MC simulator
14.35 hours
〇
PIC/MC with LPWS simulator
14.70 hours
◎
DEC Alpha station 533MHz
The increase of computation time ・・・ 2.4%
It is entirely fair to say that accuracy of PIC/MC simulation is improved by
LPWS without adding MC particles.
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Conclusions
A PIC/MC simulator coupled with LPWS is developed in this work.
 It is found that statistical fluctuations are reduced substantially
by this simulator with the slight increase (2.4%) of computation
time.
 It is also found that accuracy of particle simulation can be
improved without adding MC particles by the developed
simulator.
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Direct Simulation Monte Carlo (DSMC)
(Date et al, J.Phys.D, vol.25, pp.442-53, 1992)
before
after
E
Calculating electric field using
Poisson’s eq.
z
n(z)
n(z)
z
E
slab
slab
z
E
e
e
e
e
e
Powered
electrode
4.5
4
f (vz,vr)
Grounded
electrode
Powered
electrode
4.5
3.5
4
3
3.5
2.5
3
2
2.5
1.5
f (vz,vr)
1
0.5
0
Grounded
electrode
2
1.5
1
0.5
0
Free flight, collision and
scattering in Dt are
calculated by MC method.
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Spatial variations of densities & electric field
60 grid points for charge assignment
Net charge density
electron density
1499.7 cycle
6
6
150x10
250x10
net charge density (cm )
100
-3
-3
electron density (cm )
Standard PIC
PIC with LPWS
200
150
100
50
0
0.5
1.0
1.5
2.0
2.5
3.0
50
0
-50
Standerd PIC
PIC with LPWS
Position [cm]
-100
0.5
ion density
1.0
Standerd PIC
PIC with LPWS
2.0
2.5
3.0
Position [cm]
Electric field
6
250x10
1.5
100
-3
ion density (cm )
200
electric field (V/cm)
150
100
50
0
0.5
1.0
1.5
2.0
2.5
3.0
50
0
-50
Position [cm]
Standerd PIC
PIC with LPWS
-100
0.5
1.0
1.5
2.0
2.5
3.0
Position [cm]
1499.7 cycle
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Comparison with other method (propagator)
Propagator method
p=0.5Torr, f=13.56MHz
PIC/MC with LPWS
300
400
150
200
A
A
100
75
0
0
-100
-75
A
A
3
-200
3000.00
K
3000.25
3000.50
3000.75
time (cy
3001.00 0
cle)
2
)
(cm
1
n
o
i
t
i
pos
0.0
18.5
37.0
time (n
s)
3
K
55.5
electric field (V/cm)
electric field (V/cm)
225
K
K
300
-150
2
1
74.0
0
m)
n (c
o
i
t
i
pos
Double layer is formed in the nitrogen rf glow discharge.
Spatiotemporal variation of electric field by PIC/MC with LPWS agrees well with
that by propagator method.
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