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

Plasma ApplicationModeling @ POSTECH

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

Plasma ApplicationModeling @ POSTECH

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

Plasma ApplicationModeling @ POSTECH

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

Plasma ApplicationModeling @ POSTECH

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 )

Plasma ApplicationModeling @ POSTECH

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

Plasma ApplicationModeling @ POSTECH

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

Plasma ApplicationModeling @ POSTECH

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,

Plasma ApplicationModeling @ POSTECH

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)