ECE490O: Special Topics in EM-Plasma Simulations
JK LEE (Spring, 2006)
ECE490O: Special Topics in EM-Plasma Simulations
JK LEE (Spring, 2006)
• ODE Solvers• PIC-MCC• PDE Solvers (FEM and FDM)• Linear & NL Eq. Solvers
ECE490O: PDE Gonsalves’ lecture notes (Fall 2005)
JK LEE (Spring, 2006)
Plasma ApplicationModeling @ POSTECH
Plasma Display PanelPlasma Display Panel Many Pixels
the flat panel display using phosphor luminescence by UV photons produced in plasma gas discharge
DischargeDischargeDischarge
White light emission
(1) Electric input power
(2) Discharge
(3) VUV radiation
(4) Phosphor excitation
(5) Visible light in cell
(6) Display light
bus electrode
dielectricITO electrodeMgO layer
barrier phosphors
addresselectrode
Front panel
Back panel
PDP structure
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Simulation domainy
x
ny
nx
dielectric layer
dielectric and phosphor layer
Sustain 1 Sustain 2
address
Electric field, Density
Potential, Charge
Flux of x and y
i i+1
j
j+1
Light, Luminance, Efficiency, Power, Current and so on
Finite-Difference Method
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Flow chartfl2p.c
initial.c
charge.c
field.c
continuity.c
Calculate efficiencyhistory.c
time_step.c
current.c, radiation.c, dump.c, gaspar.c, mu_n_D.c, gummel.c
diagnostics.c
flux.c
pulse.c
source.c
Plasma ApplicationModeling @ POSTECH
Basic equations• Continuity Equation with Drift-Diffusion Approx.
spspsp St
n
Γ
• Poisson’s Equation
• Boundary conditions on dielectric surfaceiiiii nn v25.0 nEnΓ
nΓnEnΓ seeeeee nn v25.0
i
iisese - ΓΓ , for secondary electron
Mobility-driven drift term
Isotropic thermal flux term
exexex n v25.0 nΓ
for ion
for electron
for excited species
ll nqV )(
: surface charge density on the dielectric surfaces
EΓ ppppp nnD
EΓ eeeee nnD
exexex nD Γ
and
m
TkBth
8
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Partial Differential Eqs.
General form of linear second-order PDEs
with two independent variables0 gfueuducubuau yxyyxyxx
eq.) sPoisson' (ex. Elliptic ,04
eq.) Continuity (ex. Parabolic ,04
eq.) Wave(ex. Hyperbolic ,04
2
2
2
acb
acb
acb
In case of elliptic PDEs,
Jacobi-Iteration methodGauss-Seidel method
Successive over-relaxation (SOR) method
In case of parabolic PDEs,
Alternating direction implicit (ADI) method
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Continuity equation (1)
yS
xt
nn kjiy
kjiyk
jijixjix
kjiji
5.0,,5.0,,,
*,5.0,
*,5.0,,
*,
2/
spspsp St
n
Γ density nsp
Spatially discretized forms are converted to tridiagonal systems of equations which can be easily solved.
jijijijijijiji DnCnBnA ,*
,1,*,,
*,1,
Alternating direction implicit (ADI) method
ADI method uses two time steps in two dimension to update the quantities between t and t+t. During first t/2, the integration sweeps along one direction (x direction) and the other direction (y direction) is fixed. The temporary quantities are updated at t+t/2. With these updated quantities, ADI method integrates the continuity equation along y direction with fixed x direction between t+t/2 and t+t.
Discretized flux can be obtained by Sharfetter-Gummel method.jijixjix,i,j,jx,i nn ,1,,,5.0
1st step
(k means the value at time t)( * means the temporal value at time t+t/2 )
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Tridiagonal matrix (1)
3
2
1
3
2
1
33
222
11
0
0
D
D
D
BA
CBA
CB
3
2
11
2
3
2
1
33
222
11
21
1
2
0
0
D
D
DB
A
BA
CBA
CB
AB
B
A
R2
Based on Gauss elimination
3
2
12
3
2
1
33
222
122
0
0
D
D
DR
BA
CBA
CRA
3
122
12
3
2
1
33
2122
122
0
0
0
D
DRD
DR
BA
CCRB
CRA
2B2D
3
22
3
12
3
2
1
33
22
32
2
3
122
0
0
0
D
DB
ADR
BA
CB
AB
B
ACRA
R3
233
23
12
3
2
1
233
233
122
00
0
0
DRD
DR
DR
CRB
CRA
CRA
3B
3D
3
23
12
3
2
1
3
233
122
00
0
0
D
DR
DR
B
CRA
CRA
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Tridiagonal matrix (2)
3
23
12
3
2
1
3
233
122
00
0
0
D
DR
DR
B
CRA
CRA
3
33 B
D
2
33322
3
322332323 ,)(
B
ARCD
A
RDRCRA
)(1
3222
2 CDB
1
22211
2
211221212 ,)(
B
ARCD
A
RDRCRA
)(1
)(1
2111
2111
1 CDB
CDB
iiiii BCD )( 1
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Tridiagonal matrix (3)
/* Tridiagonal solution */
void trdg(float a[], float b[], float c[], float d[], int n){
int i; float r; for ( i = 2; i <= n; i++ ) { r = a[i]/b[i - 1]; b[i] = b[i] - r*c[i - 1]; d[i] = d[i] - r*d[i - 1]; }
d[n] = d[n]/b[n];
for ( i = n - 1; i >= 1; i-- ) { d[i] = (d[i] - c[i]*d[i + 1])/b[i]; } return; }
Calculate the equations in increasing order of i until i=N is reached.
Calculate the solution for the last unknown by
NNN BD
Calculate the following equation in decreasing order of i
iiiii BCD )( 1
jijijijijijiji DnCnBnA ,*
,1,*,,
*,1,
Ri
iDiB
Plasma ApplicationModeling @ POSTECH
xS
yt
nn jixjixkji
kjiy
kjiyji
kji
*
,5.0,*
,5.0,,
15.0,,
15.0,,
*,
1,
2/
',
11,
',
1,
',
11,
', ji
kjiji
kjiji
kjiji DnCnBnA
Continuity equation (2)
2nd step
From the temporally updated density calculated in the 1st step, we can calculated flux in x-direction (*) at time t+t/2. Using these values, we calculate final updated density with integration of continuity equation in y-direction.
(k+1 means the final value at time t+t)( * means the temporal value at time t+t/2 )
(tridiagonal matrix)
From the final updated density calculated in the 2nd step, we can calculated flux in y-direction (k+1) at time t.
ji
ji
z
z
jiji
ji e
e
Dz
x
tA
,2
1
,2
1
12
,1,
2
1
2,
112
1,
2
1
,2
1
,2
1
,2
1,
2
1
,2,ji
ji
jiz
jiz
z
ji
jiji
e
z
e
e
z
Dx
tB
12 ,
2
1
,1,
2
1
2,
jiz
jiji
ji
e
Dz
x
tC
k
ji
k
ji
kji
kjiji y
tS
tnD
2
1,
2
1,
,,, 2
jijijijijijiji DnCnBnA ,*
,1,*,,
*,1,
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Poisson’s eq. (1) ep nneV )( 0
Poisson equation is solved with a successive over relation (SOR) method. The electric field is taken at time t when the continuity equations are integrated between t and t+t. Time is integrated by semi-implicit method in our code. The electric field in the integration of the continuity equation between t and t+t is not the field at time t, but rather a prediction of the electric field at time t+t. The semi-implicit integration of Poisson equation is followed as
t
nntnn
eV ppk
ekp
k
0
1 )(
The continuity eq. and flux are coupled with Poission’s eq.
l
kl
k
l
kl
kl nsign
eVnt
e)(
0
1
0
Density correction by electric field change between t and t+t
This Poisson’s eq can be discriminated to x and y directions, and written in matrix form using the five-point formula in two dimensions.
jijijijijijijijijijiji fVeVdVcVbVa ,,,1,,1,,,1,,1,
Plasma ApplicationModeling @ POSTECH
l
jijijijijijiji nnte
xa 1,1,,,
01,,2,
2
1
l
jijijijijijiji nnte
xb 1,11,1,1,1
01,1,12, 2
1
l
jijijijijijiji nnte
yc ,1,1,,
0,1,2,
2
1
l
jijijijijijiji nnte
yd 1,11,11,1,
01,11,2, 2
1
)( ,,,,, jijijijiji dcbae
)( .1 0
, diek
N
l
lji
qf
diek is the surface charge density accumulating on intersection between plasma region and dielectric.
Solved using SOR method
Poisson’s eq. (2)jijijijijijijijijijiji fVeVdVcVbVa ,,,1,,1,,,1,,1,
i-1 i i+1
j-1
j
j+1
ai, jbi, j
ci, j
di, j
Plasma ApplicationModeling @ POSTECH
Scharfetter-Gummel method
St
n
Γ
• 2D discretized continuity eqn. integrated by the alternative direction implicit (ADI) method
yS
xt
nn kjiy
kjiyk
jijixjix
kjiji
5.0,,5.0,,,
*,5.0,
*,5.0,,
*,
2/
xS
yt
nn jixjixkji
kjiy
kjiyji
kji
*
,5.0,*
,5.0,,
15.0,,
15.0,,
*,
1,
2/
EΓ nqnD )sgn(
jijijijijijiji DnCnBnA ,*
,1,*,,
*,1,
',
11,
',
1,
',
11,
', ji
kjiji
kjiji
kjiji DnCnBnA
Tridiagonal matrix
jijixjix,i,j,jx,i nn ,1,,,5.0 Scharfetter-Gummel method
Gonsalves’ lecture notes (Fall 2005)
Gonsalves’ lecture notes (Fall 2005)
Gonsalves’ lecture notes (Fall 2005)
Gonsalves’ lecture notes (Fall 2005)
Gonsalves’ lecture notes (Fall 2005)
ECE490O: NL Eq. SolversGonsalves’ lecture notes (Fall 2005)
JK LEE (Spring, 2006)