Potential
+–
Sheath
+
+
Bulk
Plas
ma
Sheath
+ –
–
+ –
– + –
+
+– +–
Substrate
Density(cm^-3)
: Electron and Ion
X direction
~ V0 cos(t )
• 1D Planar RF VoltageDriven System
Klystron
3 cm
2 cm
Phase space
uz
Density
Kinetic energy
Plasma Applicatio
Modeling, POSTE
PIC Overview
 Applications of PIC model
• Basic plasma physics: waves and instabilities
• Magnetic fusion
• Gaseous discharges
• Electron and ion optics
• Microwave-beam devices
• Plasma-filled microwave-beam devices
Congratulations! It's nice to get emails like this, isn't it. Cheers,
Mike Lieberman
On Sep 28, 2005, at 5:46 PM, 이재구 wrote:
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Subject: Your article has been downloaded 250 times!
Dear Professor Lee,
I am pleased to tell you that your article, "Particle and fluid
simulations of low-temperature plasma discharges: benchmarks and
kinetic effects", in Journal of Physics D: Applied Physics, Vol 38,
ppR283 (2005), has been downloaded 250 times so far.
> This was achieved in 12 days from the date of publication. To put
this into context, across all IOP journals 10% of articles were
accessed over 250 times this quarter.
You can link directly to your article at:
http://stacks.iop.org/0022-3727/38/R283
I would like to thank you for supporting Journal of Physics D:
Applied Physics and I trust that you have found our publication
process to be friendly and efficient. I hope you will consider
submitting further papers to the journal and encourage your
colleagues to do likewise.
Our publication times are highly competitive and we use a fully
electronic editorial process from submission to peer review to
production. Electronic submission to Institute of Physics journals
can be done very simply at the following web page:
http://www.iop.org/journals/authorsubs
We look forward to working with you again. If you have any queries
about the journal, please don't hesitate to contact me.
If you would prefer not to receive further updates on the number of
times this article has been downloaded, please reply to this email
with the word remove in the subject line.
Kind regards
Sarah Quin , Publisher
Journal of Physics D: Applied Physics
Institute of Physics Publishing
> Dirac House, Temple Back, Bristol BS1 6BE, England
PIC Overview
 PIC Codes Overview
• PIC codes simulate plasma behavior of a large
number of charges particles using a few
representative “super particles”.
• These type of codes solve the Newton-Lorentz
equation of motion to move particles in
conjunction with Maxwell’s equations (or a
subset).
• Boundary conditions are applied to the
particles and the fields to solve the set of
equations.
• PIC codes are quite successful in simulating
kinetic and nonlinear plasma phenomenon like
ECR, stochastic heating, etc.
PIC-MCC Flow Chart
• Particles in continuum space
( x,v)i
• Fields at discrete mesh locations in space (E,B ) j
• Coupling between particles and fields
I
II
V
IV
III
IV
Fig: Flow chart for an explicit PIC-MCC scheme
I. Particle Equations of Motion
 Newton-Lorentz equations of motion
d
mu  F  q (E  v  B)
dt
u  v
d
xv
dt
 In finite difference form, the leapfrog method
Fig: Schematic leapfrog integration
I. Particle Equations of Motion
ut  t / 2  ut t / 2 q t ut  t / 2  ut t / 2
t
 (E 

B
)
t
t
m
2
xt  t  xt ut  t / 2
 t  t / 2
t

• Second order accurate
• Requires minimal storage
• Requires few operations
• Stable for w p t  2
I. Particle Equations of Motion
• Boris algorithm

u u
t  t / 2
qtEt

2m
t
q

t
E
u   u t  t / 2 
2m
u  u
q

 t (u   u  )  Bt
t
2 m
u
u  u

u  u
u

t
q
B
t
ˆ
t  b tan( ) 
2
2 t m
t
I. Particle Equations of Motion
Finally,
u'  u   u   t t
2t t
u  u  u' 
1 tt  tt


2t t
u' 
1 tt  tt
u

u'
u

u  tt
II. Particle Boundary
 Absorption
• Conductor : absorb charge, add to the global σ
• Dielectric : deposit charge, weight q locally to mesh
 Reflection
• Physical reflection
reverse x bc  x  x bc  x
 vx  vx
• Specular reflection
 1st order error
v
t  t / 2*
 v
t  t / 2
qE t 2 | x t  xbc |

(
 t )
t  t / 2
m
|v
|
 Thermionic Emission
 Fowler-Nordheim Field Emission
 Child’s Law Field Emission
 Gauss’s Law Field Emission
II. Particle Boundary
 Secondary electron emission
+ io n
p h o to n , ex
– electro n
– se
• Photoemission
• Ion impact secondary emission
• Electron impact secondary emission
 Important in processes related to high-power
microwave sources
III. Electrostatic Field Model
 Possion’s equation
  (x, t )   (x, t ),
• Finite difference form in 1D planar geometry
 j 1  2 j   j 1
x 2
j

,

 Boundary condition : External circuit
Fig: Schematic one-dimensional bounded plasma
with external circuit
EECE695: Computer Simulation (2005)
Particle-in-Cell Techniques
HC Kim, SJ Kim, and JK Lee
Plasma Application Modeling Group,
POSTECH
 References:
• Minicourse by Dr. J. P. Verboncoeur (PTS Group of
UC Berkeley) in IEEE International Conference on
Plasma Science (2002)
• “Plasma Physics via Computer Simulation” by C.K.
Birdsall and A.B. Langdon (Adam Hilger, 1991)
XPDx1 Flow Chart
I
II
V
IV
III
IV
Fig: Flow chart for an explicit PIC-MCC scheme
Collisions
 Electron-neutral collisions
• Elastic scattering (e + A → e + A)
• Excitation (e + A → e + A*)
• Ionization (e + A → e + A+ + e)
 Ion-neutral collisions
• Elastic scattering (A+ + A → A+ + A)
• Charge exchange (A+ + A → A + A+)
V. Monte-Carlo Collision Model
• The MCC model statistically describes the
collision processes, using cross sections for each
reaction of interest.
• Probability of a collision event
Pi  1  exp[ ng (x) T ( i )i t ]
where  T ( i )   j  j ( i )
• For a pure Monte Carlo method, the timestep is
chosen as the time interval between collisions.
ti  
ln( 1  R)
ng (x) T ( i )i
where 0< R< 1is a uniformly distributed random
number.
However, this method can only be applied when
space charge and self-field effects can be neglected.
V. Monte-Carlo Collision Model
• Computing the collision probability for each particle
each timestep is computationally expensive.
→ Null collision method
1. The fraction of particles undergoing a collision each
time step is given by
PT  1  exp[  max t ].
2. The particles undergoing collisions are chosen at
random from the particle list.
3. The type of collisions for each particle is determined
by choosing a random number, 0  R   max .
Null collision
Collision
type 3
Collision type 2
Collision type 1
Fig: Summed collision frequencies for the null collision
method.
Capacitively Coupled Plasma –
1D PIC-MCC
dx j (t )
~ Vhf cos(hf t )
+–
Sheath
Sheath
+ –
–
+ –
– + –
+
+– +–
j = 1,  , N
dv j (t ) q j
 E(t, x j )
dt
mj
+
+
Bulk
Plas
ma
dt
 v j (t )
1 N
  E(t , x)   q j (x  x j (t ))
 0 j 1
Substrate
~ Vlf cos(lf t )
• 1D Asymmetric
Dual-Freq. VoltageDriven System
e
 (ni  ne )
0
• MCC (Monte-Carlo Collision)
Processes
- Electron-Neutral Collisions
(Ionization, Scattering,
Excitation)
- Ion-Neutral Collisions
(Charge-exchange, Scattering)
Plasma Application
Modeling
POSTECH
Capacitively Coupled Plasma –
1D PIC-MCC
Vx vs. x for electrons
Density vs. x
Vx vs. x for ions
Potential vs. x
Ion Energy Distribution
Function
10
mTorr
20
mTorr
30
mTorr
50
mTorr
 High frequency : 100MHz (100V)
 Low frequency :1MHz (102V)
 Symmetric Discharge
 L = 2.5 cm
Plasma Application
Modeling
POSTECH
Electron Energy Distribution
Function
In the discharge center
Plasma Application
Modeling
POSTECH
Comparison Between PIC and
Experiment (II)
EEPF [A.U ]
Single langmuir probe
measurement result
from KAIST
e
2
e
1
e
0
60mT(Te:3.4eV)
e
-1
e
-2
e
-3
e
e
-5
-Electron temperature
is comparable to PIC
simulation
27
MH
z/2
MH
z
30 30 60 60
0/3 0/9 0/3 0/9
00 00 00 00
Te
2.7
(eV) 3
80mT(Te:2.5eV)
-4
-Electron temperature
is decreased as
pressure is increased
70mT(Te:3eV)
5
10
Electron Energy [eV]
2.7
7
2.7
Table 1 Electron
temperature from
PIC simulation
RF Power : 200Watt/200Watt
0
2.7
15
(pressure=110mT)
Plasma Application
Modeling
POSTECH
PIC-MCC versus Fluid
Simulation (I)
~
4eV
Due to Ramsauer
minimum
• Low-energy
electrons
-3
: 2.40 eV (1.11e15
cm )
-3
• High-energy
electrons
: 1.00 eV (7.82e12
cm )
• EEPF obtained from
PIC/MCC is totally in
disagreement with that of
LFA because of nonlocal
behavior of electrons.
Plasma Application
Modeling
POSTECH
IV. Coupling Fields to Particles
 Particle and force weighting
: connection between grid and particle quantities
• Weighting of charge to grid
• Weighting of fields to particles
grid point
a point charge
IV. Coupling Fields to Particles
• Nearest grid point (NGP) weighting
 fast, simple bc, noisy
• Linear weighting
: particle-in-cell (PIC) or cloud-in-cell (CIC)
 relatively fast, simple bc, less noisy
• Higher order weighting schemes
 slow, complicated bc, low noisy
Quadratic spline
NGP
1.0
Linear spline
Cubic spline
0.5
0.0
xi  2x
Position (x)
xi  x
xi
xi  x
xi  2x
Fig: Density distribution function of a particle at xi
for various weightings in 1D
IV. Coupling Fields to Particles
Areas are assigned to grid
points; i.e., area a to grid
point A, b to B, etc
Fig: Charge assignment for linear weighting in 2D
PDP Structure
AC PDP
Sustain Electrode
Discharge in PDP
Bus Electrode
Visible Light
Sustain Electrode
Bus Electro
Front Glass Substrate
Protection
Layer
Dielectric LayerMgO
Discharge
Barrier Rib
Dielectric lay
Barrier
UV
Address Elec
o
Address
Electrode
Phosphor(R,G,B)
Phosphor
Rear Glass Substrate 90
rotation
Plasma Application
Modeling
POSTECH
Striation Profiles in PDP – 2D
PIC/MCC
Anode
Cathode
100Torr
200Torr
500Torr
• Pressure dependence of striations
: Number of peaks depend on the pressure and el
Plasma Application
Modeling
POSTECH
Input file(II)
The
direction
1.00
1.00, 1.00
The comparison of like-sign and opposite-sign
Likesign
Oppositesign
• Like-sign : electron – electron two stream case
• Opposite-sign : electron – ion two stream case
Plasma Application
Modeling, POSTECH
The comparison of like-sign and opposite-sign
Field
energy
Electron-Ion two stream case
Kinetic
energy
6.5
Electron-Electronon two stream case
6.0
2.0
5.5
Tot_K.E in El-Ion case
El_K.E in El-Ion case
Ion_K.E in El-Ion case
Tot_K.E in El-El case
El_K.E in El-Ion case
El_K.E in El-Ion case
5.0
Kinetic Energy
Field energy
1.5
1.0
4.5
4.0
3.5
3.0
0.5
2.5
2.0
0.0
0
50
100
150
Time
200
250
0
50
100
150
200
250
Time
• Field & Kinetic energy distribution in time is different.
The peak field energy value in like-sign case is higher than
that of oppositesign case and the time to reach to the peak value longer
than that of opposite-
sign case.
Plasma Application
Modeling, POSTECH
Simulation Domain of Klystron (SJ Kim)
RF input port
10.05 cm
RF output port
13.07 cm
9.55 cm
E-beam
7.569 cm
6.66 cm
Cylindrical Axis
37.2 cm
Simulation condition:
Beam emitter: I= 12 kA, ud =2.48e8 m/s
Input port : Rin=2300 , R=20 , f=7.69 GHz
Output port : R=47.124 
Plasma Applicatio
Modeling, POSTE
Example of Klystron Simulation
Phase space
uz
Density
Kinetic energy
Plasma Applicatio
Modeling, POSTE
Simulation Results at 10 ns
8
5.0x10
07
8
06
4.5x10
Velocity (m/s)
05
04
03
8
4.0x10
8
3.5x10
02
8
3.0x10
01
00
0.05
0.10
0.15
0.20
0.25
0.300.000.350.05
0.10
0.15
z (m)
0.20
0.25
0.30
z (m)
10 nsec
5
4.0x10
5
3.8x10
KE (eV)
0.00
8
2.5x10
5
3.6x10
5
3.4x10
5
3.2x10
0.0
-9
2.0x10
-9
4.0x10
6.0x10
-9
8.0x10
-9
1.0x10
-8
Time (sec)
Plasma Applicatio
Modeling, POSTE
Simulation Results at 6 us
5
7.0x10
6 usec
14
0
5
total
density
6.5x10
14
0
5
6.0x10
14
0
KE (eV)
5
13
0
13
5.5x10
5
5.0x10
5
4.5x10
0
5
13
4.0x10
0
13
3.5x10
0.0
3.0x10
0
0.0
5
5
0.0-6 1.0x10-6-6 2.0x10-6 3.0x10-6 4.0x10-6 5.0x10-6 6.0
-6
-6
-6
-6
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10 6.0x10
Time (sec)
Time (sec)
9
9
0
5.0x10
output po
input port
0.0
.0
9
-5.0x10
9
0
Power (W)
10
10
10
-1.0x10
10
-1.5x10
10
-2.0x10
10
-2.5x10
10
10
-3.0x10
10
10
-3.5x10
10
0.0
10
-6
1.0x10
-6
2.0x10
-6
3.0x10
-4.0x10
-6
-6 -6
-6
-6
-6
-6
0.0-6 6.0x10
4.0x10 5.0x10
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10
Time (sec)
Time (sec)
Plasma Applicatio
Modeling, POSTE
KE as a Function of Beam Current
5
7.0x10
5
6.5x10
5
6.0x10
5
KE (eV)
5.5x10
5
5.0x10
5
4.5x10
I= 6kA
I= 9kA
I=12kA
I=15kA
I=18kA
5
4.0x10
5
3.5x10
5
3.0x10
5
2.5x10
-6
-6
-6
-6
-6
-6
0.0
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10 6.0x10
Time (sec)
60
20 ns
6 us
58
KE efficiency (%)
56
54
52
50
48
46
44
6
8
10
12
current (kA)
14
16
18 6
8
10
12
14
16
current (kA)
Plasma Applicatio
Modeling, POSTE
KE as a Function of Beam Energy
5
1.5 M
0.12 MeV
2.2x10
6
0
5
0
6
2.0x10
KE (eV)
4
0
4
0
6
1.8x10
6
1.6x10
4
0
6
1.4x10
4
0
0.0
-6
-6
-6
-6
-6
-6
-6
-6
-6 -6
0.0
1.0x10
2.0x10 3.0x10 4.0x10 5.0x10
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10
6.0x10
Time (sec)
Time (sec)
0
60
6
20 ns
40
0
20
KE efficiency (%)
0
0
0
0
0
0
-20
-40
-60
0
-80
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Beam energy (MeV)
1.4 0.01.6 0.2
0.4
0.6
0.8
1.0
1.2
1.4
Beam energy (MeV)
Plasma Applicatio
Modeling, POSTE
Klystron
3 cm
2 cm
Phase space
uz
Density
Kinetic energy
Plasma Applicatio
Modeling, POSTE
Simulation Results at 0.5 ns and 2.5 ns
5
8
4.0x10
10
0.5
5
8
3.9x10
8
3.8x10
10
KE (eV)
5
10
8
10
5
3.7x10
5
3.6x10
8
10
5
3.5x10
8
10
5
3.4x10
0.00
0.05
0.10
0.15
0.20
0.25
0.30 0.00.35 1.0x10-10 2.0x10-10 3.0x10-10 4.0x10-10 5.
z (m)
Time (sec)
8
10
5
8
4.0x10
10
2.5 n
8
10
5
3.8x10
8
10
5
8
3.6x10
KE (eV)
10
8
10
8
10
5
3.4x10
5
8
3.2x10
10
8
10
5
3.0x10
8
10
5
8
2.8x10
10
0.00
0.05
0.10
0.15
0.20
z (m)
0.25
0.30 0.00.35 5.0x10-10 1.0x10-9 1.5x10-9 2.0x10-9 2.
Time (sec)
Plasma Applicatio
Modeling, POSTE
Simulation Results at 10 ns and 20 ns
8
5
0
4.0x10
8
0
10
5
3.8x10
8
0
5
KE (eV)
3.6x10
8
0
8
0
5
3.4x10
5
3.2x10
8
0
5
3.0x10
8
0
5
2.8x10
8
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30 0.0
0.35 2.0x10-9 4.0x10-9 6.0x10-9 8.0x10-9 1
z (m)
Time (sec)
8
10
5
4.0x10
8
10
8
20 ns
5
10
3.8x10
8
10
5
3.6x10
8
KE (eV)
10
8
10
8
10
5
3.4x10
5
3.2x10
8
10
5
3.0x10
8
10
8
10
5
2.8x10
8
10
0.00
0.05
0.10
0.15
0.20
z (m)
0.25
0.30 0.0
0.35
-9
5.0x10
-8
1.0x10
1.5x10
-8
2.0
Time (sec)
Plasma Applicatio
Modeling, POSTE
Simulation Results at 6 us
6
14
1x10
14
9x10
14
8x10
5
KE (eV)
5
13
13
5
7x10
5
6x10
5
5x10
13
5
4x10
13
5
3x10
5
0
0.0
2x10
-6
-6
-6
-6
-6
-6 -6
-6
-6
-6
-6
0.0
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10
6.0x10
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10
Time (sec)
Time (sec)
9
0
output por
input
0.0port
0
9
0
9
Power (W)
-5.0x10
10
10
10
10
-1.0x10
10
-1.5x10
10
10
10
-2.0x10
10
0.0
-6
-6
-6
-6
-6
-6 -6
-6
-6
-6
0.0
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10
6.0x10
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10
Time (sec)
Time (sec)
Plasma Applicatio
Modeling, POSTE
Simulation Results at 0.5 ns and 2.5 ns
5
8
4.0x10
10
0.5
5
8
3.9x10
8
3.8x10
10
KE (eV)
5
10
8
10
5
3.7x10
5
3.6x10
8
10
5
3.5x10
8
10
5
3.4x10
0.00
0.05
0.10
0.15
0.20
0.25
0.30 0.00.35 1.0x10-10 2.0x10-10 3.0x10-10 4.0x10-10 5.
z (m)
Time (sec)
8
10
5
8
4.0x10
10
2.5 n
8
10
5
3.8x10
8
10
5
8
3.6x10
KE (eV)
10
8
10
8
10
5
3.4x10
5
8
3.2x10
10
8
10
5
3.0x10
8
10
5
8
2.8x10
10
0.00
0.05
0.10
0.15
0.20
z (m)
0.25
0.30 0.00.35 5.0x10-10 1.0x10-9 1.5x10-9 2.0x10-9 2.
Time (sec)
Plasma Applicatio
Modeling, POSTE
Simulation Results at 10 ns and 20 ns
8
5
0
4.0x10
8
0
10
5
3.8x10
8
0
5
KE (eV)
3.6x10
8
0
8
0
5
3.4x10
5
3.2x10
8
0
5
3.0x10
8
0
5
2.8x10
8
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30 0.0
0.35 2.0x10-9 4.0x10-9 6.0x10-9 8.0x10-9 1
z (m)
Time (sec)
8
10
5
4.0x10
8
10
8
20 ns
5
10
3.8x10
8
10
5
3.6x10
8
KE (eV)
10
8
10
8
10
5
3.4x10
5
3.2x10
8
10
5
3.0x10
8
10
8
10
5
2.8x10
8
10
0.00
0.05
0.10
0.15
0.20
z (m)
0.25
0.30 0.0
0.35
-9
5.0x10
-8
1.0x10
1.5x10
-8
2.0
Time (sec)
Plasma Applicatio
Modeling, POSTE
Simulation Results at 6 us
6
14
1x10
14
9x10
14
8x10
5
KE (eV)
5
13
13
5
7x10
5
6x10
5
5x10
13
5
4x10
13
5
3x10
5
0
0.0
2x10
-6
-6
-6
-6
-6
-6 -6
-6
-6
-6
-6
0.0
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10
6.0x10
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10
Time (sec)
Time (sec)
9
0
output por
input
0.0port
0
9
0
9
Power (W)
-5.0x10
10
10
10
10
-1.0x10
10
-1.5x10
10
10
10
-2.0x10
10
0.0
-6
-6
-6
-6
-6
-6 -6
-6
-6
-6
0.0
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10
6.0x10
1.0x10 2.0x10 3.0x10 4.0x10 5.0x10
Time (sec)
Time (sec)
Plasma Applicatio
Modeling, POSTE
Charge Conservation Scheme
Boris-DADI correction (OOPIC) Langdon-Marder correction (MA
ECE586: Advanced E&M Simulation (2004)
On PDX1 Program
HyunChul Kim and J.K. Lee
Plasma Application Modeling, POSTECH
2004. 9. 16
 References:
• Minicourse by Dr. J. P. Verboncoeur (PTS Group of
UC Berkeley) in IEEE International Conference on
Plasma Science (2002)
• “Plasma Physics via Computer Simulation” by C.K.
Birdsall and A.B. Langdon (Adam Hilger, 1991)
A Series of XPDX1*
 XPDx1: X window (using xgrafix library),
Plasma Device, 1 Dimensional (1d3v), Bounded
(with external circuit drive), Electrostatic Code
• XPDP1 (x=P) : Planar Configuration
• XPDC1 (x=C) : Cylindrical Configuration
• XPDS1 (x=S) : Spherical Configuration
r
~
LRC
Computation Space
* Developed by PTS group, UC Berkeley
All are available at http://ptsg.eecs.berkeley.edu
PIC Overview
 PIC Codes Overview
• Plasma behavior of a large number of charges
particles are simulated by using a few
representative “super particles”.
• PIC codes solve fundamental equations, the
Newton-Lorentz equation of motion to move
particles in conjunction with Maxwell’s
equations (or a subset) with few approximations.
• PIC codes are quite successful in simulating
kinetic and nonlinear plasma phenomenon like
ECR, stochastic heating, etc.
Computer Simulation of Plasma
Kinetic
Description
Vlasov,
Fokker-Planck
Codes
Particle
Codes
Fluid
Description
Hybrid
Codes
 Particle codes
• The particle-particle model
• The particle-mesh model
• The particle-particleparticle-mesh model
Fluid
Codes
XPDx1 Flow Chart
• Particles in continuum
space ( x,v)i
• Fields at discrete mesh
locations in space (E,B ) j
• Coupling between
particles and fields
I
II
Subcycling : ti  slow  kt fast , k  1
IV
III
IV
Fig: Flow chart for an explicit PIC-MCC scheme
V
II. Particle Boundary
 Absorption
• Conductor : absorb charge, add to the global σ
 Secondary electron emission
+
–
io n
electron
–
se
• Ion impact secondary emission
• Electron impact
secondary emission
III. Electrostatic Field Model
 Maxwell’s equation in vacuum
 E  
B
,   D   , D  0 E
t
D
 H  J 
,   B  0 , B  0 H
t
• In electrostatics,
  E  0  E  
  2  

(Poisson’s equation)
0
Or Gauss’ law
 D  d s 
S
V
 dV Qenclosed
III. Electrostatic Field Model
 Possion’s equation
  (x, t )   (x, t ),
• Finite difference form in 1D planar geometry
 j 1  2 j   j 1
x 2
j

,

 Boundary condition : External circuit
0
E0 

J
EJ  

From Gauss’s law,
 t0  1t  0t   0t x 2
E1/ 2 

x

t
Q t  Q t  t
t
t  t
 0   0   J plasmadt 
t  t
A
• Short circuit
 0 (t ) is specified,  J (t )  0
• Open circuit
 
t
0
t  t
0

t
t  t
J plasmadt
III. Electrostatic Field Model
• Voltage driven series RLC circuit
From Kirchhoff’s voltage law,
d 2Q (t )
dQ (t ) Q (t )
L

R

2
dt
dt
C
 V (t )   J (t )   0 (t )
― One second order difference equation
where
IV. Coupling Fields to Particles
 Weighting in 1D
• For linear weighting in cylindrical coordinates,
(0<j<N)
XPDx1 Flow Chart
I
II
V
IV
III
IV
Fig: Flow chart for an explicit PIC-MCC scheme
V. Monte-Carlo Collision Model
• There is a finite probability that the i-th particle will
undergo more than one collision in the timestep.
Thus, the total number of missed collisions
(error in single-event codes)

2
P
r   Pi k  i
1  Pi
k
Hence, traditional PIC-MCC codes are constrained by
vmax t  1 for accuracy.
where vmax  max (ng ( x )) max ( T ( ) )
x

Numerical Parameters
 Choose Δx and Δt to resolve the smallest important
physical feature
 Require Δx < Debye length, sheath length, wave
length, Larmor radius, boundary feature, etc.
 Require t  x / max for all species (“particle
Courant”) for accurate sampling of fields.
 Require t  0.2 / w p for accuracy of explicit leap
frog mover or for accuracy when space charge forces are
important.
 Require t  1/ when collisions are important.
 Require # of superparticles per cell > 10. It should
be larger in simulations where particles remain trapped
for long times.
Example of XPDP1 Input File
RF DISCHARGE(IN MKS UNITS)
Voltage-driven with electron-neutral collisions (Argon atom)
-nsp---nc---nc2p---dt[s]---length[m]--area[m^2]--epsilonr---B[Tesla]---PSI[D]-2 400 1.8e6 8e-12 0.03
0.01
1.0
0 .0
0.0
-rhoback[C/m^3]---backj[Amp/m^2]---dde--extR[Ohm]--extL[H]---extC[F]---q0[C]0.0
0.0
0.0 0.0
0.0
1.0
0.0
-dcramped--source--dc[V|Amp]--ramp[(V|Amp)/s]---ac[V|Amp]---f0[Hz]--theta0[D]0
v
0.0
0.0
100
13.56e6
0
--secondary--e_collisional---i_collisional---reflux---nfft--n_ave--nsmoothing--ntimestep-1
1
2
0
0
276549 6
0
--seec(electrons)---seec(ions)---ion_species----Gpressure[Torr]---GTemp[eV]---imp-0.0
0.2
2
100e-3
0.026
0
---GAS----psource--nstrt-1
0
0
SPECIES 1
----q[C]-------m[Kg]---j0L[Amp/m^2]---j0R[Amp/m^2]----initn[m^-3]----k--1.602e-19 9.11e-31
0.0
0.0
5e14
1
--vx0L[m/s]---vxtL[m/s]--vxcL[m/s]---vxLloader(0=RNDM,1=QS)-0.0
4.19e5
0.0
1
--vx0R[m/s]---vxtR[m/s]--vxcR[m/s]---vxRloader
0.0
4.19e5
0.0
1
--v0y[m/s]---vty[m/s]---vyloader---v0z[m/s]---vtz[m/s]--vzloader-0.0
4.19e5
1
0.0
4.19e5
1
--nbin----Emin[eV]----Emax[ev]---maxnp—
200
0.0
20.0
300000
-For-Mid-Diagnostic---nbin----Emin[eV]---Emax[eV]----XStart--XFinish—
300
0.0
20.0
0.0
0.03
SPECIES 2
----q[C] ------m[Kg]---j0L[Amp/m^2]---j0R[Amp/m^2]----initn[m^-3]----k1.602e-19 6.68e-26
0.0
0.0
5e14
1
-vx0L[m/s]---vxtL[m/s]--vxcL[m/s]---vxLloader(0=RNDM,1=QS)-0.0
97.8
0.0
1
--vx0R[m/s]---vxtR[m/s]--vxcR[m/s]---vxRloader
0.0
97.8
0.0
1
--v0y[m/s]---vty[m/s]---vyloader---v0z[m/s]---vtz[m/s]--vzloader-0.0
97.8
0
0.0
97.8
1
--nbin----Emin[eV]----Emax[ev]---maxnp-100
0.0
100.0
300000
-For-Mid-Diagnostic---nbin----Emin[eV]---Emax[eV]----XStart--XFinish-200
0.0
0.3
0.0
0.03
Some Input Parameters
 nsp : Number of species.
 nc: The number of spatial cells. Δx=length/nc
 nc2p: Superparticle to actual particle weight. The initial
number of superparticles is N=initn·area·length/nc2p.
 dt: Timestep for simulation in seconds.
 length: The length of the system (distance between
electrodes) in meters.
 B: Applied homogeneous magnetic field in Tesla
 PSI: Angle of the B-field in degrees
 extC: The external circuit capacitance in Farads. Used in
conjuction with extL, extR and the driving source.
 source: Either a voltage (v) or current (i) source
 f0: AC frequency of the source.
 GAS: The type of gas, important when collisions are
turned on. Helium = 1, Argon = 2, Neon = 3, Oxygen = 4.
 Gpressure : Background gas pressure in Torr.
 q: Charge of the particle in Coulombs.
 m: Mass of the particle in Kgs.
 initn: Initial particle number density
 For details, refer the source code itself or the manual
inside the package of source file.
Example of Result (driven by RF)
Vx vs. x for electrons
Density vs. x
Ion flux vs. Ion Energy
Vx vs. x for ions
Potential vs. x
Electron Temperature vs. x
Semi-conductor
Plasma
Processing
( Comparison of Plasma Kinetic Properties of
Various Equipments )
H.S. Ko and J.K. Lee
Department of Electronic and Electrical
Engineering,
POSTECH
CCP (Capacitively
Coupled Plasma)
Objective of
research
Recently,
dual-frequency
capacitively coupled plasma (CCP)
has been of much interest as the
mainstream plasma source for
dielectric etchers. Through the
numerical
modeling,
we
understand the characteristics of
plasma discharge and optimize the
etching process.
Silicon Electrodes
For this work, one-dimensional
electrostatic particle in cell/Monte
Carlo (PIC/MC) code is used.
Although the particle simulation is
rather computationally intensive,
it has the advantage of being a
self-consistent and fully kinetic
method where the electron and ion
velocity distribution functions can
be obtained.
+–
Sheath
+
+
E
+ –
–+ –
– + –
+
Sheath
E
+ – + –SEE
Exelan DFC
Etcher of LAM
Corp.
M/
F
27
MHz
M/
F
Cext
2
MHz
~ V cos(t)
Outline of Charge-up Simulation
Plasma Etching Reactor
(CCP, ICP, etc)
Ions are accelerated in sheath region
Charge-up & etch
Plasma source
Simulation space
simulation space
Ion
Electron
Plasma
wafer
Plasma Application
Modeling Group
POSTECH
Single Frequency CCP: Effect of pressure (20, 45, 100mTorr)
17
200
100 mTorr
Small electrode (LHS)
100 mTorr
17
100
1,0x10
0,012
-3
Concentration (m )
20 mTorr
-100
-200
16
8,0x10
16
6,0x10
IEDF
0
45 mTorr
20 mTorr
0,008
16
4,0x10
0,004
20 mTorr
16
2,0x10
-300
0,0
0,030
0,035
0,040
0,000
0,045
0,030
0,035
z (m)
0,040
0
0,045
200
400
Ion energy (eV)
z (m)
600
Plasma potential profiles, plasma density and corresponding ion energy distributions at the smaller
(LHS) electrode. The IEDF spread corresponds to the mean potential drop at the left electrode.
2.88 eV
100 mT
2 eV
3
1E14
20 mTorr
1E13
1E12
1.73 eV
1
Power density
45 mT
3
0,6
Electrons, 20 mTorr
Ions, 20 mTorr
Electons, 100 mTorr
Ions, 100 mTorr
6
EEPF in the bulk
45 mTorr
1E15
4
2
100 mTorr
1E16
j E (10
20 mT
5
W/m )
1E17
6
Te, eV
Potential (V)
0,016
1,2x10
100 mTorr
0,4
0,2
0,0
1E11
0,030
0,035
0,040
z (m)
0,045
0
5
10
15
20
Energy (eV)
25
30
35
0,030
0,035
0,040
0,045
z (m)
Electron temperature profiles, electron energy probability function and power absorbed by
electrons and ions for different values of pressure
Plasma Applica
ModelingPOSTE
EEDF Comparison for a SmallGap CCP
 Langmuir
Probe by Godyak
J = 2.65
Thomson
Scattering by
Elsabbagh, Muraoka
J=
3.8
 Using PIC-MCC
Simulation
J = 3.8
mA/cm2
Comparison with
Experimental Result
Our PIC-MCCN ~ 1.2e5
New combined PIC-MCC
N ~ 5000
I.V. Schweigert et al,
PSST (2004)
PIC-MCC, Vahedi et al, PSST (19
Experiment, Godyak et
al.(1992)
 argon gas
 current density :
2
2.65
mA
/
cm
 gap distance : 2 cm
Plasma Application
Modeling, POSTEC
Low-Freq. Current Effect on EEDF
(Summary)
• W/O
SEEC :
from
collisional
(Ohmic) to
 Depending on SEEC,
collisionles
the main mechanism for EEDF
s
(stochastic) change differs.
• With
SEEC : γ mode
transition
HC Kim & JK Lee, PRL (to appear)
Plasma Applicatio
Modeling POSTE
SEEC: ION-ENERGY and ANGLE DEPENDENT SECONDARY EMIS
New SEEC for argon is incorporat
in XPDC1 code
For argon:
0.006  i
4  i  80
 i i  
 1.05 10
1.5
1  i /10
1  i / 8000
1.2
For each variant
below
SS is reached
Electron yield per ion
1
1
 i ( i , )   i ( i ) cos 
For oxygen:
 i  i   7 10
4
 i 15
1.5
1   i / 5000
1.3
Old SEEC = 0.2
0,1
New SEEC for argon
Low voltage
High voltage
0,01
10
Energy
100
1000
Plasma Applica
Modeling POST
IEDF (RHS) for Xe/Ar mixtures, 27 MHz, 800 V, 50 mTorr Compariso
(Natalia B.)
Xe 50 W
Xe 50 W
1400
1400
CF
Ar
COF
CF2
CHF2
CF3
C3F3
C2F4
C3F5
Xe
C3F6
C4F7
1400
1200
1000
800
600
400
200
0
20
40
60
80
600
600
400
200
200
0
0
0
20
40
Ion energy60 (eV)
80
0
100
20
40
60
80
100
Ion energy (eV)
0,06
100
0,06
Ion energy (eV)
800
400
0,05
0,05
Experimental results for IEDF
0,04
The curves for Xe+ and Ar+ ions
are clipped from here and shown
0,03
in the same scale
a. u.
0
Ar+
800
Ar+
Simulation
Xe+
Simulation
0,04
a. u.
1600
Xe+
1000
Y Axis Title
Y Axis Title
1800
Experiment
1200
1000
2000
Y Axis Title
Experiment
1200
Xe 50 W
2200
0,03
0,02
0,02
0,01
0,01
0,00
0,00
0
Ion energy (eV)
200
0
200
Ion energy (eV)
Simulations: IEDF for Xe:Ar=20:80 mixture, 27 MHz, 800 V, 50 m
Qualitative comparison with experiment
Plasma Applica
Modeling Grou
POSTECH
Electron density and temperature: PIC
cathode
anode
160m
140m


675ns
700ns
810ns

Electron Temperature (eV)
1260m
675ns
700ns
2.0
810ns
1.6
1.2
0.8
0.4
1
2
3
4
5
6
Position number
Striation density and temperature: out of phase Plasma Applicati
Modeling @ POST
EECE 586
How to calculate potential
and electric field
in 3D charge-up simulation
S.J. Kim, H.J. Lee, and J.K. Lee
Plasma Application Modeling Lab.
Department of Electronic and Electrical Engineering
Pohang University of Science and Technology
What is charge-up effect?
Ion: anisotropic
Electron: isotropic
High charge-up potential
Because of the electron shading effect in high aspect ratio etching,
most of the ions reach the bottom of trench.
High potential is generated at the bottom of trench.
Trajectory of ions is changed.
Plasma Application
Modeling, POSTECH
Simulation routine and basic assumptions
Ion
Electron
Move particles
(by E-field)
All particles
arrive at
boundary
Solve Laplace
Equation
Update Potential
0.2um
Due to the small size of simulation domain( ~ 1um)
• Particle flight time is much shorter than the time interval of each
entering particle.
Number of space charge particle is small in the simulation
domain.
Ignore space charge effect in the E-field calculation.
• Collisions in the simulation domain are neglected.
mean free path of ions or electrons(~mm) is much longer than
the simulation domain size.
• The potential at the top and bottom boundary is the same.
Surface current is neglected.
Plasma Application
Modeling, POSTECH
Flow chart
Plasma Application
Modeling, POSTECH
Simulation domain and boundary conditions
Plasma Application
Modeling, POSTECH
Solving electrostatic potential in 3D charge-up simulation(1)
Space gradient of dielectric constant being considered,
Poisson’s equation is as follows:
     
Surface charges in the dielectric surfaces is only considered in right-hand.
For solving PDE numerically, Poisson’s equation is represented as follows:


 1 i 1 jk    1   1 ijk   1 i 1 jk
i  jk
i  jk
i  jk
2
2 
2
 i  2 jk

2
x



 1 ij 1k    1   1 ijk   1 ij 1k
ij  k
ij  k
ij  k
2
2 
2
 ij  2 k

2
y



 1ijk1    1   1 ijk   1ijk1
ijk 
ijk 
ijk 
2
2 
2
 ijk  2
   ijk
2
z
Plasma Application
Modeling, POSTECH
Solving electrostatic potential in 3D charge-up simulation(2)
Poisson’s equation is described as Matrix equation
aijki 1 jk  bijki 1 jk  cijkij1k  dijkij1k 
eijkijk1  f ijkijk1  gijkijk   ijk

aijk 
x

cijk 
g ijk
1
ij  k
2
2
y

eijk 
1
i  jk
2
2
1
2
2

, bijk 
x

, d ijk 
,
f ijk 
1
ij  k
2
2
y

ijk 
1
i  jk
2
2
1
2
2
ijk 
z
z
 aijk  bijk  cijk  d ijk  eijk  f ijk 
Plasma Application
Modeling, POSTECH
Successive OverRelaxation (SOR)
• Elliptic PDE
aCi , j  a Li 1, j  a Ri 1, j  a Bi , j 1  aTi , j 1  Si , j
• SOR method for solving PDEs
i(,tj)  Si , j  a Li(t1), j  a Ri(t1,1j)  a Bi(,tj)1  aTi(,tj11) / aC  (1  )i(,tj1)
where,  = 1 : Gauss-seidel (GS) method
0<  < 1 : underrelaxation
1<  <2 : overrelaxation
How to choose optimal 
If Jacobi is the spectral radius of the Jacobi iteration,
 optimal
2
2
1  1   Jacobi

 Jacobi
2
 x 

cos
   cos
xgrid  y 
ygrid

2
 x 
1   
 y 
where,  =  : Dirichlet or Neumann boundary conditions.
 = 2 : Periodic boundary conditions
Plasma Application
Modeling, POSTECH
SOR algorithm for 2nd-order elliptic PDE
From 2D finite difference model,
ai , j ui 1, j  bi , j ui 1, j  ci , j ui , j 1  d i , j ui , j 1  ei , j ui , j  si , j
Iterative procedure:
1
u 
( si , j  ai , j ui 1, j  bi , j ui 1, j  ci , j ui , j 1  di , j ui , j 1 )
ei , j
*
*
old
uinew


u

(
1


)
u
,j
i, j
i, j
Residual is calculated as follows:
 i , j  ai , j ui 1, j  bi , j ui 1, j  ci , j ui , j 1  d i , j ui , j 1  ei , j ui , j  si , j
u
new
i, j
u
old
i, j

i, j
ei , j
Finding optimal  in SOR with Chebyshev acceleration
 ( 0)  1
2
 (1/ 2)  1 /(1   Jacobi
/ 2)
2
 ( n 1/ 2)  1 /(1   Jacobi
 ( n ) / 4),
n  1 / 2,1,, 
 (  )  optimal
Plasma Application
Modeling, POSTECH
Solving electric field in 3D charge-up simulation
Electric field is calculated by using Gauss’s law.
i) Cases without changes in 
Ex ,ijk 
E z ,ijk 
i 1 jk  i 1 jk
2x
E y ,ijk 
ij1k  ij1k
2y
ijk1  ijk1
2z
ii) Cases with changes in 
x-directional electric field in the plane of yz1 :
E x ,ijk 
1

1
i  jk
2


 x 
 
x 
E 1  E 1 
E


E
 ijk   1  E 1 
1
1  



i  jk
x ,ij  k
y
,
ij

k
y
,
ij

k
z
,
ijk

z
,
ijk

2y 
2z 

2 
2
2
2 
2
2  

where,
E
E
1
x ,i  jk
2
z ,ijk 
1
2


i 1 jk  ijk
x
E
1
y ,ij  k
2

ij1k  ijk
y
ijk1  ijk
z
Plasma Application
Modeling, POSTECH
Main.c
/***********************************************************/
/* The main physics loop
*/
void XGMainLoop()
{
int
isp, phi_flag;
it++;
phi_flag=0; /* to determine if it is necessary to solve Laplace's Equaton */
for(isp=0; isp<nsp; isp++) {
(*move_ptr)(isp);
boundary(isp);
phi_flag+=sp[isp].np;
}
if (theRunWithXFlag) history2();
if (phi_flag==0) {
/* if no particle is left */
t+=DT;
fields();
load();
if (theRunWithXFlag) history1();
}
}
/***************************************************************/
void display_title()
{
puts("\nEtch-3d Version 1.0");
puts("Made by Hae June Lee");
puts("Plasma Application Modeling Group");
puts("Pohang University of Science and Technology\n");
}
/***************************************************************/
Plasma Application
Modeling, POSTECH
Motions of electrons and ions
Electron
Initial
Saturated
Ion
Initial
Saturated
Plasma Application
Modeling, POSTECH
Potential profiles
BC
A
Aspect ratio = 7
0.00
0.00
A
0.00
0.00
B
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.000.00
0.00
C
0.00
0.00
0.00
0.00
AR=3
0.00
0.00
0.000.00
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
-20
0.00
0.00
AR=3, long z-direction
Plasma Application
Modeling, POSTECH
3D Radiation Transport
Simulation for Plasma
Display Panels
HyunChul Kim and J. K. Lee
Plasma Application Modeling,
POSTECH
 References:
• A.F. Molisch and B.P. Oehry, Radiation
Trapping in Atomic Vapours (Oxford,
New York, 1998).
• H.C. Kim, S.S. Yang, and J.K. Lee, J.
Appl. Phys. 93, 9516 (2003).
Plasma Application
Modeling, POSTEC
1
History of FL3P Simulator
2000/12 : Initial cut of FL3P (FLuid simulator for 3-d Plasma
Display Panels)
as a generalization of FL2P
2001/09 : Implementation of self-consistent radiation model in
FL2P
2001/10 : Its generalization to FL3P
The first journal publication on 3D fluid PDP simulation
2002/02 : H.C. Kim et al., IEEE Trans. Plasma Sci. 30, 188.
2002/06 : H.C. Kim et al., J. Appl. Phys. 91, 9513.
The journal publication on RT simulation in PDP
2002/06 : 2D self-consistent radiation transport model for Plasma
Display Panels
H.J. Lee et al., Phys. Plasmas 9, 2822.
2003/06 : 3D self-consistent radiation transport model …
H.C. Kim et al., J. Appl. Phys. 93, 9516.
Plasma Application
Modeling, POSTEC
2
Resonance Radiation Trapping (I)
i) Emission of Independent
Excited-State Atoms
Level
2
 *
n (r , t)  n0 (r ) exp( t /  )
A*  A  h
Level
1
Atom
1
*
Atom
2
ii) Interaction between ExcitedState Atoms (Radiation trapping:
absorption/reemission processes)
h
A*  A  h
Imprisonment of resonant
radiation
A  A* h
A*  A  h
 *
n (r , t)  n0 (r ) exp( t / g0 )
*
Plasma Application
Modeling, POSTEC
3
Resonance Radiation Trapping (II)
 Particle diffusion vs. radiation trapping
Particle: Flight time >> Scattering time
Photon: Scattering time >> Flight time
• Spectral Line
Broadening
i) Natural
broadening
• Caused by the finite lifetime of the atomic levels
and uncertainty principle
• Lorentz line shape with FWHM
1 1 1
 n    
2  1  2 
ii) Doppler
broadening
• Caused by the thermal motion of the atoms
and Doppler effect
D 2 2 ln( 2)k BT
• Doppler line shape with FWHM
 
0
c
M
iii) Pressure
broadening
• Caused by the collisions with other atoms
• Lorentz line shape with FWHM
Resonance Radiation Trapping
(III)
4
k ( )
[Ref] H.J. Lee and J.P. Verboncoeur, Phys. Plasmas 8, 3077 (2001)
• Frequency redistribution at
absorption/reemission process


n* (r , t )   n*j (r ) exp( t / g j )
j
Plasma Application
Modeling, POSTEC
5
Holstein Equation (I)
•
Holstein equation (Continuity eqn. with
radiation transport)
•
Kernel function (Prob. that photon emitted
•
Transmission factor (Freq-avg. prob. of
at r’ is reabsorbed at r.)
traversing a distance R)
where R=|r-r’|.
Plasma Application
Modeling, POSTEC
6
Holstein Equation (II)
• Holstein
Equation
  1 * 1 *  
 *
n (r , t )      S (r , t )  n (r , t )   n (r , t )G(r , r )dr 
t


i) When
n* is spatially uniform,
it is reduced to
 1 *
 *
n (r , t )      S (r , t )   n (r , t )
t
 (r )
  : Effective
 (r )  
(r ) lifetime
Escape factor (escape prob.
of a photon) :

(r )  1   G(R)dV
V
eˆR
1
  T (R) 2  eˆr dS
4 S R
ii) When
n* distribution is identical to the
lowest-order
eigenmode,
*
*
n (r , t)  n0 (r ) exp( t / g0 )
 t n* (r,t )  g0
: Average time that a photon spends in the
vapor before it escapes.
g0 : Trapping factor (mean number
   g0 : Effective
of absorption (or emission)
lifetime
processes that a photon suffers)
Plasma Application
Modeling, POSTEC
7
Numerical Methods for
Holstein Equation
  1 * 1 *  
 *
n (r , t )      S (r , t )  n (r , t )   n (r , t )G(r , r )dr 
t


: Fredholm integral equation of the 2nd kind
• Piecewise-Constant
Approximation (PCA)
 Piecewise-constant approximation of
the eigenfunctions
 Superior to PFM only when subcell #
low (i.e.Carlo
< 300)Simulation (MC)
•isMonte
 Easy to program but slower than PFM
• Propagator Function Method
(PFM)
 Numerical method used for our
research
Plasma Application
Modeling, POSTEC
Calculation of Resonant State
Density (I)
8
• Escape factor method cannot treat the redistribution of
excited state density profile.
• In this study, we use Propagator Function Method with
full calculation of Holstein equation.
  1 * 1 *  
 *
n (r , t )      S (r , t )  n (r , t )   n (r , t )G(r , r )dr 
t


Propagator Function
Method
1 Cells *
  Ak,m nm (t )
m
eˆR 
Ak,m   T (R) 2  ds
Sm
4R
 
From cell
rm or( r 
(
)
 
) rtoorcell
k r
Plasma Application
Modeling, POSTEC
9
PDP Structure and Discharge
AC PDP
Sustain Electrode
Discharge in PDP
Visible Light
Bus Electrode
Sustain Electrode
Bus Electrode
Front Glass Substrate
Protection
Layer
Dielectric Layer
MgO
Discharge
Dielectric laye
Barrier Rib
Barrier
Address Electr
UV
o
Address
Electrode
Phosphor(R,G,B)
Phosphor
90
Rear Glass Substrate
rotation
• Color AC PDPs operate on the same
principle as fluorescent lamps converting UV
to visible light by the use of phosphors.
Plasma Application
Modeling Group
POSTECH
2
1
Xe*(3P1) Density Profiles (I)
Solution of Holstein
Equation
Catho
de
28
8
u
m
Escape Factor
Method
Top
View
Anod
e
Barr
ier
Ribs
0.4
us
10
30
86
4
60
u
m
1.
0
2.
0
20
10
0
14
0
1.0
us
0.
2
(Unit:
1010
#/cm )
5
2
0
40
2
Plasma Application
Modeling, POSTEC
2
6
Energy Diagram
Stand
ard
VUV Emission
(10.9% of
total input
power)
VUV on
Phosphor
(5.47
%)
Visible on
Phosphor
(1.43
%)
Visible Photon
(0.433
%)
147nm 150nm 173nm
(50.1
%)
(45.9
%)
(4.0
%)
(35.1
%)
(31.7
%)
(31.8
%)
(5.1
%)
(4.7
%)
(4.7
%)
(59.8
%)
(63.6
%)
(63.5
%)
Plasma Application
Modeling, POSTEC
Main.c
/******************************************/
/* def.h */
#define EPS0
8.8542e-12 /* (F/m) */
#define NperTORR
8.3221e20
#define HISTMAX
2048
#define HISTMAX2
4096
#define NSMAX
2
#define QUIET_START
#define RANDOM
0
1
#define GROUNDED
#define VOLTAGE_D
#define CURRENT_D
0
1
2
#define DIRICHLET
#define NEUMANN
#define PERIODIC
#define SYMMETRIC
0
1
2
1
#define LEFT
#define RIGHT
#define UP
#define DOWN
0
1
2
3
#define SIDES
2
/* ---- PNO means positive, negative, and ordinary
in each directions, and C and V means concave and convex. --*/
#define NO_FACE
#define FACE_X_UP
#define FACE_X_DOWN
#define FACE_Y_UP
#define FACE_Y_DOWN
#define FACE_Z_UP
#define FACE_Z_DOWN
#define SIDE_PPO_C
#define SIDE_PNO_C
#define SIDE_NPO_C
#define SIDE_NNO_C
#define SIDE_OPP_C
#define SIDE_OPN_C
#define SIDE_ONP_C
0
1
2
3
4
5
6
7
8
9
10
11
12
13
#define SIDE_ONN_C
14
#define SIDE_POP_C
15
#define SIDE_PON_C
16
#define SIDE_NOP_C
17
#define SIDE_NON_C
18
#define CORN_PPP_C
19
#define CORN_PNP_C
20
#define CORN_NPP_C
21
#define CORN_NNP_C
22
#define CORN_PPN_C
23
#define CORN_PNN_C
24
#define CORN_NPN_C
25
#define CORN_NNN_C
26
#define SIDE_PPO_V
27
#define SIDE_PNO_V
28
#define SIDE_NPO_V
29
#define SIDE_NNO_V
30
#define SIDE_OPP_V
31
#define SIDE_OPN_V
32
#define SIDE_ONP_V
33
#define SIDE_ONN_V
34
#define SIDE_POP_V
35
#define SIDE_PON_V
36
#define SIDE_NOP_V
37
#define SIDE_NON_V
38
#define CORN_PPP_V
39
#define CORN_PNP_V
40
#define CORN_NPP_V
41
#define CORN_NNP_V
42
#define CORN_PPN_V
43
#define CORN_PNN_V
44
#define CORN_NPN_V
45
#define CORN_NNN_V
46
// F means flat in xz surface
#define CORN_PPP_F
47
#define CORN_NPP_F
48
#define CORN_PPN_F
49
#define CORN_NPN_F
50
// U means there is a side in up-direction
#define CORN_PPP_U
51
#define CORN_NPP_U
52
#define CORN_PPN_U
53
#define CORN_NPN_U
54
/**************************************/
/* defining internal structures
*/
#define ACCUMULATION
0
#define INJECTION
1
#define OPEN
3
field.c (1)
#include "et3d.h"
#define my_div(x, y)
((fabs(y) <= 1e-25) ? 0 : x/y)
void
set_voltage();
float hdx, hdy, hdz;
/***************************************************************/
void fields()
{
register int isp, i, j, k, n, ip1, im1, jp1, jm1, kp1, km1;
float ex_mag, ey_mag, ez_mag, e_mag, temp, epsi1, epsi2;
static int init_field_flag=1, stat_counter=0;
static float ***epsx, ***epsy, ***epsz;
if(init_field_flag) {
hdx = 0.5*idx;
hdy = 0.5*idy;
hdz = 0.5*idz;
epsx = tensor3(ngx, ngy, ngz);
epsy = tensor3(ngx, ngy, ngz);
epsz = tensor3(ngx, ngy, ngz);
for(i=0; i<ngx; i++)
for(j=0; j<ngy; j++)
for (k=0; k<ngz; k++)
eps_array[i][j][k] *= EPS0;
for(i=0; i<ngx; i++)
for(j=0; j<ngy; j++)
for (k=0; k<ngz; k++) {
if (bc_x==PERIODIC) im1= (i)?i-1:ncx-1;
else
im1= (i)?i-1:0;
Symmetric
jm1= (j)?j-1:0;
Symmetric (y-direction)
km1= (k)?k-1:ncz-1;
Periodic (z-direction)
epsx[i][j][k]=0.25*(eps_array[i][j][k] + eps_array[i][jm1][k] +
eps_array[i][j][km1] + eps_array[i][jm1][km1]);
epsy[i][j][k]=0.25*(eps_array[i][j][k] + eps_array[im1][j][k] +
eps_array[i][j][km1] + eps_array[im1][j][km1]);
epsz[i][j][k]=0.25*(eps_array[i][j][k] + eps_array[im1][j][k] +
eps_array[i][jm1][k] + eps_array[im1][jm1][k]);
}
set_poisson_coefficient();
init_field_flag=0;
}
Setting
initial
values
Call function which sets coefficients
for solving poisson’s eq.
Plasma Application
Modeling, POSTECH
field.c (2)
/**************************************************/
/*** Get the charge densities..
****/
for (i=0; i<= ncx; i++)
for (j=0; j<= ncy; j++)
for (k=0; k<ngz; k++) {
sigma[i][j][k] = 0.0;
source[i][j][k]= 0;
}
for (isp=0; isp< nsp; isp++) {
for (i=0; i< ngx; i++)
for (j=0; j< ngy; j++)
for (k=0; k<ngz; k++) {
source[i][j][k]-= sp[isp].q_per_cell*sp[isp].sigma[i][j][k];
sigma[i][j][k]+= sp[isp].q*sp[isp].sigma[i][j][k];
/* 아직 area로 나눠지지 않았음. */
}
}
/*******************************************/
/*** Solve Poisson's eqn with SOR method */
set_voltage();
Set =0 at the xzs plane.
sor_num=sor_cheb(aa, bb, cc, dd, ee, ff, gg, source, phi, 1.0);
Plasma Application
Modeling, POSTECH
field.c (3)
/**********************************************/
/* Calculation of surface charge of conductor */
/* Divergence theorem 을 이용해서 (i,j,k) grid를 둘러싼 Cell에서의
Note
Q = epsilon* \int E ds - \int rho dV 로 여분의 charge를 구할 수
Q   E ds   dv
있다. 이것을 area로 나누면 균일한 surface charge를 갖게 된다. */
for (i=0; i<ngx; i++)
Note
for (j=1; j<ngy; j++)
ngx=ncx+1
for (k=0; k<ngz; k++) {
ngy=ncy+1
if(grid_mask[i][j][k]==3 && face[i][j][k]) {
ngz=ncz+1
if (bc_x==PERIODIC) {
Note
ip1= (i==ncx)?1:i+1;
ip1(i+1)=1, 2, 3, …, ncx, 1
Grid mask =
im1= (i)?i-1:ncx-1;
im1(i-1)=ncx-1, 0, 1, …, ncx-1
}
0 : empty
else {
(symmetic boundary)
1 : inside dielectric
ip1(i+1)=1, 2, 3, …, ncx, ncx
ip1= (i==ncx)?ncx:i+1;
2 : dielectric surface
im1=
(i)?i-1:0;
im1(i-1)=0, 0, 1, …, ncx-1
3 : conductor
}
4 : simulation boundary
jp1= (j==ncy)?ncy:j+1;
kp1= (k==ncz)?1:k+1;
jm1= (j)?j-1:0;
km1= (k)?k-1:ncz-1;

sigma[i][j][k] = dz*dy*(phi[i][j][k]-phi[ip1][j][k])*idx*
0.25*(eps_array[i][j][k]+eps_array[i][jm1][k]+
eps_array[i][j][km1]+eps_array[i][jm1][km1]);
sigma[i][j][k]+= dz*dy*(phi[i][j][k]-phi[im1][j][k])*idx*
0.25*(eps_array[im1][j][k]+eps_array[im1][jm1][k]+
eps_array[im1][j][km1]+eps_array[im1][jm1][km1]);
sigma[i][j][k]+= dz*dx*(phi[i][j][k]-phi[i][jp1][k])*idy*
0.25*(eps_array[i][j][k]+eps_array[im1][j][k]+
eps_array[i][j][km1]+eps_array[im1][j][km1]);
sigma[i][j][k]+= dz*dx*(phi[i][j][k]-phi[i][jm1][k])*idy*
0.25*(eps_array[i][jm1][k]+eps_array[im1][jm1][k]+
eps_array[i][jm1][km1]+eps_array[im1][jm1][km1]);
sigma[i][j][k]+= dx*dy*(phi[i][j][k]-phi[i][j][kp1])*idz*
0.25*(eps_array[i][j][k]+eps_array[i][jm1][k]+
eps_array[im1][j][k]+eps_array[im1][jm1][k]);
sigma[i][j][k]+= dx*dy*(phi[i][j][k]-phi[i][j][km1])*idz*
0.25*(eps_array[i][j][km1]+eps_array[i][jm1][km1]+
eps_array[im1][j][km1]+eps_array[im1][jm1][km1]);
}
if (sigma[i][j][k])
sigma[i][j][k] = my_div(sigma[i][j][k], area[i][j][k]);

Q    E ds    dv
}
Plasma Application
Modeling, POSTECH
field.c (4)
/*******************************************/
/*** calculate the E field...
***/
for(i=0; i<=ncx; i++)
for(j=0; j<=ncy; j++)
for(k=0; k<=ncz; k++) {
if (bc_x==PERIODIC) {
ip1=(i==ncx) ? 1:i+1;
im1=(i) ? i-1:ncx-1;
}
else {
ip1=(i==ncx) ? ncx:i+1;
im1=(i) ? i-1:0;
}
jp1= (j==ncy)?ncy:j+1;
kp1= (k==ncz)?1:k+1;
jm1= (j)?j-1:0;
km1= (k)?k-1:ncz-1;
if (j==0) {
ex[i][j][k]=ez[i][j][k]=0;
ey[i][j][k] = (phi[i][0][k]-phi[i][1][k])*idy ;
}
Note
Ex ,ijk 
E y ,ijk 
E z ,ijk 
i 1 jk  i 1 jk
2x
ij1k  ij1k
2y
ijk1  ijk1
2z
else if (j==ncy) {
ex[i][j][k]=ez[i][j][k]=0;
ey[i][j][k] = (phi[i][jm1][k]-phi[i][j][k])*idy;
}
else if (grid_mask[i][j][k]==3) {
/* biased conductor의 경우, surface charge가 외부 회로에 의해
영향을 받기 때문에 이 방법을 쓸 수 없다. */
ex[i][j][k] = ey[i][j][k] = ez[i][j][k] = 0;
if (face[i][j][k] ==FACE_Z_UP)
ez[i][j][k] = sigma[i][j][k] /epsz[i][j][k];
else if (face[i][j][k]==FACE_Z_DOWN)
ez[i][j][k] = -sigma[i][j][k] /epsz[i][j][km1];
Plasma Application
Modeling, POSTECH
field.c (5)
else if (face[i][j][k]==FACE_Y_UP)
ey[i][j][k] = sigma[i][j][k] /epsy[i][j][k];
else if (face[i][j][k]==FACE_Y_DOWN)
ey[i][j][k] = -sigma[i][j][k] /epsy[i][jm1][k];
else if (face[i][j][k]==FACE_X_UP)
ex[i][j][k] = sigma[i][j][k] /epsx[i][j][k];
else if (face[i][j][k]==FACE_X_DOWN)
ex[i][j][k] = -sigma[i][j][k] /epsx[im1][j][k];
else if (face[i][j][k]==CORN_PPP_F || face[i][j][k]==CORN_NPP_F ||
face[i][j][k]==CORN_PPN_F || face[i][j][k]==CORN_NPN_F)
ey[i][j][k] = sigma[i][j][k] /epsy[i][j][k];
else if (face[i][j][k]==SIDE_PPO_V) {
epsi2=(eps_array[im1][j][km1]+eps_array[im1][j][k]+
eps_array[i][jm1][km1]+eps_array[i][jm1][k]+
eps_array[i][j][km1]+eps_array[i][j][k])/6.0;
ex_mag = (phi[i][j][k] -phi[ip1][j][k])*idx;
ey_mag = (phi[i][j][k] -phi[i][jp1][k])*idy;
e_mag = sqrt(ex_mag*ex_mag + ey_mag*ey_mag);
ex[i][j][k] = fabs(sigma[i][j][k])*my_div(ex_mag, e_mag)/epsi2;
ey[i][j][k] = fabs(sigma[i][j][k])*my_div(ey_mag, e_mag)/epsi2;
}
else if (face[i][j][k]==SIDE_NPO_V) {
epsi2=(eps_array[im1][jm1][km1]+eps_array[im1][jm1][k]+
eps_array[im1][j][km1]+eps_array[im1][j][k]+
eps_array[i][j][km1]+eps_array[i][j][k])/6.0;
ex_mag = (phi[im1][j][k] -phi[i][j][k])*idx;
ey_mag = (phi[i][j][k] -phi[i][jp1][k])*idy;
e_mag = sqrt(ex_mag*ex_mag + ey_mag*ey_mag);
ex[i][j][k] = fabs(sigma[i][j][k])*my_div(ex_mag, e_mag)/epsi2;
ey[i][j][k] = fabs(sigma[i][j][k])*my_div(ey_mag, e_mag)/epsi2;
}
else if (face[i][j][k]==SIDE_OPP_V) {
epsi2=(eps_array[i][j][km1]+eps_array[im1][j][km1]+
eps_array[i][j][k]+eps_array[im1][j][k]+
eps_array[i][jm1][k]+eps_array[im1][jm1][k])/6.0;
ez_mag = (phi[i][j][k] -phi[i][j][kp1])*idz;
ey_mag = (phi[i][j][k] -phi[i][jp1][k])*idy;
e_mag = sqrt(ez_mag*ez_mag + ey_mag*ey_mag);
ez[i][j][k] = fabs(sigma[i][j][k])*my_div(ez_mag, e_mag)/epsi2;
ey[i][j][k] = fabs(sigma[i][j][k])*my_div(ey_mag, e_mag)/epsi2;
}
Plasma Application
Modeling, POSTECH
field.c (6)
else if (face[i][j][k]==SIDE_OPN_V) {
epsi2=(eps_array[i][j][km1]+eps_array[im1][j][km1]+
eps_array[i][j][k]+eps_array[im1][j][k]+
eps_array[i][jm1][km1]+eps_array[im1][jm1][km1])/6.0;
ez_mag = (phi[i][j][km1] -phi[i][j][k])*idz;
ey_mag = (phi[i][j][k] -phi[i][jp1][k])*idy;
e_mag = sqrt(ez_mag*ez_mag + ey_mag*ey_mag);
ez[i][j][k] = fabs(sigma[i][j][k])*my_div(ez_mag, e_mag)/epsi2;
ey[i][j][k] = fabs(sigma[i][j][k])*my_div(ey_mag, e_mag)/epsi2;
}
}
/* end of conductor surface */
else if (grid_mask[i][j][k]==2) {
ex[i][j][k] = hdx*(phi[im1][j][k] - phi[ip1][j][k]);
ey[i][j][k] = hdy*(phi[i][jm1][k] - phi[i][jp1][k]);
ez[i][j][k] = hdz*(phi[i][j][km1] - phi[i][j][kp1]);
if (face[i][j][k]==FACE_X_UP) {
/* For the case that epsilon doesn't change in y,z directions */
epsi1=epsx[im1][j][k];
epsi2=epsx[i][j][k];
ex[i][j][k]=sigma[i][j][k] +
epsi1*( (phi[im1][j][k]-phi[i][j][k])*idx - 0.5*dx*
( idy*(2*phi[i][j][k]-phi[i][jp1][k]-phi[i][jm1][k])*idy
+idz*(2*phi[i][j][k]-phi[i][j][kp1]-phi[i][j][km1])*idz) );
ex[i][j][k]/=epsi2;
}
Note
else if (face[i][j][k]==FACE_X_DOWN) {
epsi2=epsx[im1][j][k];
 ijk   1 

i  jk
epsi1=epsx[i][j][k];


2
ex[i][j][k]=sigma[i][j][k] +





1  E 1  x  E 1  E 1  
epsi1*( (phi[ip1][j][k]-phi[i][j][k])*idx - 0.5*dx*
E x ,ijk 
 x ,ij  2 k 2y  y ,ij  2 k
y ,ij  k  
2 
( idy*(2*phi[i][j][k]-phi[i][jp1][k]-phi[i][jm1][k])*idy
 1 

i  jk
+idz*(2*phi[i][j][k]-phi[i][j][kp1]-phi[i][j][km1])*idz) );
2
 x 


E

 
1 E
1 
ex[i][j][k]/=-epsi2;


z ,ijk 
 2z  z ,ijk 2
2

}
else if (face[i][j][k]==FACE_Y_UP) {
i 1 jk  ijk
E 1 
epsi1=epsy[i][jm1][k];
x ,i  jk
x
2
epsi2=epsy[i][j][k];
ij1k  ijk
E 1 
ey[i][j][k]=sigma[i][j][k] +
y ,ij  k
y
2
epsi1*( (phi[i][jm1][k]-phi[i][j][k])*idy - 0.5*dy*
ijk1  ijk
( idx*(2*phi[i][j][k]-phi[ip1][j][k]-phi[im1][j][k])*idx
E
1 
+idz*(2*phi[i][j][k]-phi[i][j][kp1]-phi[i][j][km1])*idz) );
z ,ijk 
z
2
ey[i][j][k]/=epsi2;
}
else if (face[i][j][k]==FACE_Y_DOWN) {
Plasma Application
epsi2=epsy[i][jm1][k];
Modeling, POSTECH
field.c (7)
epsi1=epsy[i][j][k];
ey[i][j][k]=sigma[i][j][k] +
epsi1*( (phi[i][jp1][k]-phi[i][j][k])*idy - 0.5*dy*
( idx*(2*phi[i][j][k]-phi[ip1][j][k]-phi[im1][j][k])*idx
+idz*(2*phi[i][j][k]-phi[i][j][kp1]-phi[i][j][km1])*idz) );
ey[i][j][k]/=-epsi2;
}
else if (face[i][j][k]==FACE_Z_UP) {
epsi1=epsz[i][j][km1];
epsi2=epsz[i][j][k];
ez[i][j][k]=sigma[i][j][k] +
epsi1*( (phi[i][j][km1]-phi[i][j][k])*idz - 0.5*dz*
( idx*(2*phi[i][j][k]-phi[ip1][j][k]-phi[im1][j][k])*idx
+idy*(2*phi[i][j][k]-phi[i][jp1][k]-phi[i][jm1][k])*idy) );
ez[i][j][k]/=epsi2;
}
else if (face[i][j][k]==FACE_Z_DOWN) {
epsi2=epsz[i][j][km1];
epsi1=epsz[i][j][k];
ez[i][j][k]=sigma[i][j][k] +
epsi1*( (phi[i][j][kp1]-phi[i][j][k])*idz - 0.5*dz*
( idx*(2*phi[i][j][k]-phi[ip1][j][k]-phi[im1][j][k])*idx
+idy*(2*phi[i][j][k]-phi[i][jp1][k]-phi[i][jm1][k])*idy) );
ez[i][j][k]/=-epsi2;
}
}
else {
ex[i][j][k] = hdx*(phi[im1][j][k] - phi[ip1][j][k]);
ey[i][j][k] = hdy*(phi[i][jm1][k] - phi[i][jp1][k]);
ez[i][j][k] = hdz*(phi[i][j][km1] - phi[i][j][kp1]);
}
}
}
/***************************************************************/
void set_voltage()
{
register int i, k;
for (i=0;i<ngx;i++)
for (k=0;k<ngz;k++) {
phi[i][0][k]= 0;
phi[i][ncy][k]=0;
}
}
Plasma Application
Modeling, POSTECH
SOR3d.c(1)
#include "et3d.h"
/* Successive Overrelaxation (SOR) with Chebyshev acceleration */
int sor_cheb(float ***a, float ***b, float ***c, float ***d, float ***e,
float ***f, float ***g, float ***s, float ***u, double omega)
{
register int i,j,k;
int im1,ip1,jm1,jp1,km1,kp1;
int iterations;
float residue, change, max_change;
int i_order,j_order,k_order;
int i_max2, j_max2, k_max2;
float anorm, old_anorm=0.0;
/* rjac is input as the spectral radius of the Jacobi iteration, or an
estimate of it */
if (bc_x == SYMMETRIC) i_max2 = ncx;
else if (bc_x == PERIODIC) i_max2 = ncx-1;
if (bc_y == SYMMETRIC) j_max2 = ncy;
else if (bc_y == PERIODIC) j_max2 = ncy-1;
if (bc_z == SYMMETRIC) k_max2 = ncz;
else if (bc_z == PERIODIC) k_max2 = ncz-1;
for(iterations=1;iterations<=SOR_MAX;iterations++)
{
anorm = 0.0;
max_change = 0;
for (i_order = 0; i_order < 2; i_order++)
{
k_order = i_order;
for (i=0;i<=i_max2;i++)
{
j_order = k_order;
for(k=0;k<=k_max2;k++)
{
/* if i_order = 0,
i is even => j_order = 0, even j is swept
i is odd => j_order = 1, odd j is swept
if i_order = 1,
i is even => j_order = 1, odd j is swept
i is odd => j_order = 0, even j is swept
*/
Plasma Application
Modeling, POSTECH
SOR3d.c(2)
for(j=j_order;j<=j_max2;j+=2)
{
if(grid_mask[i][j][k] !=3 && j>0 && j<ncy)
{
im1 = i - 1;
ip1 = i + 1;
jm1 = j - 1;
jp1 = j + 1;
km1 = k - 1;
kp1 = k + 1;
if (bc_x == SYMMETRIC) {
if(im1 == -1) im1 = 1;
if(ip1 == ncx+1) ip1 = ncx-1;
}
else if (bc_x == PERIODIC) {
if(im1 == -1) im1 = ncx-1;
if(ip1 == ncx) ip1 = 0;
}
if (bc_y == SYMMETRIC) {
if(jm1 == -1) jm1 = 1;
if(jp1 == ncy+1) jp1 =ncy-1;
}
else if (bc_y == PERIODIC) {
if(jm1 == -1) jm1 = ncy-1;
if(jp1 == ncy) jp1 = 0;
}
if (bc_z == SYMMETRIC) {
if(km1 == -1) km1 = 1;
if(kp1 == ncz+1) kp1 =ncz-1;
}
else if (bc_z == PERIODIC) {
if(km1 == -1) km1 = ncz- 1;
if(kp1 == ncz) kp1 = 0;
}
Note
Residual (2D):
i , j  ai , j ui 1, j  bi , j ui 1, j  ci , j ui , j 1
 di , j ui , j 1  ei , j ui , j  si , j
/* residual */
residue = a[i][j][k]*u[im1][j][k] + b[i][j][k]*u[ip1][j][k]
+ c[i][j][k]*u[i][jm1][k] + d[i][j][k]*u[i][jp1][k]
+ e[i][j][k]*u[i][j][km1] + f[i][j][k]*u[i][j][kp1]
+ g[i][j][k]*u[i][j][k] - s[i][j][k];
Note
change = omega*residue/g[i][j][k];
if (i_order == 0)
anorm += change*change;
u[i][j][k] -= change;
In 2D case,
old
uinew
, j  ui , j  
i, j
ei , j
Plasma Application
Modeling, POSTECH
SOR3d.c(3)
if (bc_x == PERIODIC && i == 0) u[ncx][j][k] = u[0][j][k];
if (bc_y == PERIODIC && j == 0) u[i][ncy][k] = u[i][0][k];
if (bc_z == PERIODIC && k == 0) u[i][j][ncz] = u[i][j][0];
max_change = max(max_change, fabs(change));
}
}
/* 0 , 1 swap */
j_order = 1 - j_order;
}
/* 0 , 1 swap */
k_order = 1 - k_order;
}
if (iterations==1 && k_order==0)
omega=1.0/(1.0-0.5*rjac_square);
else
omega=1.0/(1.0-0.25*rjac_square*omega);
Note
Finding optimal  in SOR
with Chebyshev acceleration
 ( 0)  1
2
 (1/ 2)  1 /(1   Jacobi
/ 2)
2
 ( n 1/ 2)  1 /(1   Jacobi
 ( n ) / 4),
n  1 / 2,1,, 
 (  )  optimal
}
if (bc_x == PERIODIC && bc_z == PERIODIC)
for (j=0;j<ngy;j++) u[ncx][j][ncz]=u[0][j][0];
if (max_change < tol_pois) return iterations;
#if 0
if (!cheb_flag) {
if (iterations > 1)
{
opt_omega=obtain_opt_omega(anorm, old_anorm, omega);
}
old_anorm = anorm;
}
#endif
} /* End of Iteration Loop */
fprintf(stderr, " Max. # of Iteration exceeded in SOR\n");
return -1;
}
Plasma Application
Modeling, POSTECH
SOR3d.c(4)
/* rjac is input as the spectral radius of the Jacobi iteration, or an
* estimate of it */
void set_rjac_square()
{
double rjac,factor;
factor = dx*dx/dy/dy;
Note
2

 x 

cos
   cos
xgrid  y 
ygrid
 Jacobi 
2
 x 
1   
 y 
where,  =  : Dirichlet or Neumann
boundary conditions.
 = 2 : Periodic boundary
conditions
if (bc_x == SYMMETRIC && bc_y == SYMMETRIC)
rjac = (cos(M_PI/ncx)+factor*cos(M_PI/ncy));
else if (bc_x == PERIODIC && bc_y == PERIODIC)
rjac = (cos(2.*M_PI/ncx)+factor*cos(2.*M_PI/ncy));
else if (bc_x == PERIODIC && bc_y == SYMMETRIC)
rjac = (cos(2.*M_PI/ncx)+factor*cos(M_PI/ncy));
else if (bc_x == SYMMETRIC && bc_y == PERIODIC)
rjac = (cos(M_PI/ncx)+factor*cos(2.*M_PI/ncy));
rjac /= (1.+factor);
rjac_square = rjac*rjac;
}
double obtain_opt_omega(double anorm, double old_anorm, double omega)
{
double mu_W, mu_J, opt_omega;
mu_W = sqrt(anorm/old_anorm);
mu_J = (mu_W+omega-1.0)/(omega*sqrt(mu_W));
Note
opt_omega = 2.0/(1.0+sqrt(1.0-mu_J*mu_J));
return opt_omega;
aijki 1 jk  bijki 1 jk
}
void set_poisson_coefficient()
{
register int i, j, k;
int ii, im1, jm1, jj, km1, kk, isp;
for (i=0;i<ngx;i++)
for (j=0;j<ngy;j++)
for (k=0;k<ngz;k++) {
if (grid_mask[i][j][k] !=3) {
ii = adjust_x(i);
jj = adjust_y(j);
kk = adjust_z(k);
im1 = adjust_x(i-1);
jm1 = adjust_y(j-1);
km1 = adjust_z(k-1);
 cijkij1k  dijkij1k 
eijkijk1  f ijkijk1  gijkijk   ijk

aijk 
x

cijk 
g ijk
1
ij  k
2
2
y

eijk 
1
i  jk
2
2
1
2
2
ijk 

, bijk 
x

, d ijk 
1
i  jk
2
2
1
ij  k
2
2
y

1
2
2
ijk 
, f ijk 
z
z
 aijk  bijk  cijk  d ijk  eijk  f ijk 
Plasma Application
Modeling, POSTECH
SOR3d.c(5)
aa[i][j][k]= eps_array[im1][jj][kk] + eps_array[im1][jm1][kk] +
eps_array[im1][jj][km1]+ eps_array[im1][jm1][km1];
bb[i][j][k]= eps_array[ii][jj][kk] + eps_array[ii][jm1][kk] +
eps_array[ii][jj][km1]+ eps_array[ii][jm1][km1];
cc[i][j][k]= eps_array[ii][jm1][kk] + eps_array[im1][jm1][kk] +
eps_array[ii][jm1][km1]+ eps_array[im1][jm1][km1];
dd[i][j][k]= eps_array[ii][jj][kk] + eps_array[im1][jj][kk] +
eps_array[ii][jj][km1]+ eps_array[im1][jj][km1];
ee[i][j][k]= eps_array[ii][jj][km1] + eps_array[im1][jj][km1] +
eps_array[ii][jm1][km1]+ eps_array[im1][jm1][km1];
ff[i][j][k]= eps_array[ii][jj][kk] + eps_array[im1][jj][kk] +
eps_array[ii][jm1][kk]+ eps_array[im1][jm1][kk];
}
Note
aa[i][j][k]*=idx2;
bb[i][j][k]*=idx2;
idx2=0.25*idx*idx
cc[i][j][k]*=idy2;
dd[i][j][k]*=idy2;
ee[i][j][k]*=idz2;
ff[i][j][k]*=idz2;
gg[i][j][k]= -(aa[i][j][k] + bb[i][j][k] + cc[i][j][k] +
dd[i][j][k] + ee[i][j][k] + ff[i][j][k]);
}
}
int adjust_x(int i)
{
if (bc_x == SYMMETRIC)
{
/* not valid for all integer range */
if (i < 0) i = (-i-1) % ncx;
else if (i >= ncx) i = ncx - 1 - i % ncx;
}
else if (bc_x == PERIODIC)
{
if (i < 0) i = ncx - 1 - (-i-1) % ncx;
else if (i >= ncx) i = i % ncx;
}
return i;
}
Plasma Application
Modeling, POSTECH
SOR3d.c(6)
int adjust_y(int j)
{
if (bc_y == SYMMETRIC)
{
if (j < 0) j = (-j-1) % ncy;
else if (j >= ncy) j = ncy - 1 - j % ncy;
}
else if (bc_y == PERIODIC)
{
if (j < 0) j = ncy - 1 - (-j-1) % ncy;
else if (j >= ncy) j = j % ncy;
}
return j;
}
int adjust_z(int k)
{
if (bc_z == SYMMETRIC)
{
if (k < 0) k = (-k-1) % ncz;
else if (k >= ncz) k = ncz - 1 - k % ncz;
}
else if (bc_z == PERIODIC)
{
if (k < 0) k = ncz - 1 - (-k-1) % ncz;
else if (k >= ncz) k = k % ncz;
}
return k;
}
Plasma Application
Modeling, POSTECH