ECE490O: Special Topics in EM-Plasma Simulations JK LEE (Spring, 2006)

1. ECE490O: Special Topics in EM-Plasma SimulationsJK LEE (Spring, 2006)

2. ECE490O: Special Topics in EM-Plasma SimulationsJK LEE (Spring, 2006) • ODE Solvers • PIC-MCC • PDE Solvers (FEM and FDM) • Linear & NL Eq. Solvers

3. ECE490O: PDEGonsalves’ lecture notes (Fall 2005)JK LEE (Spring, 2006)

4. Electric input power • Discharge • VUV radiation • Phosphor excitation • Visible light in cell • Display light White light emission Discharge Discharge Discharge Front panel bus electrode dielectric Plasma Application Modeling@ POSTECH MgO layer ITO electrode phosphors barrier address electrode Back panel Plasma Display Panel Plasma Display Panel Many Pixels the flat panel display using phosphor luminescence by UV photons produced in plasma gas discharge PDP structure

5. y x ny Plasma Application Modeling@ POSTECH nx Simulation domain Sustain 1 Sustain 2 dielectric layer dielectric and phosphor layer address Finite-Difference Method    j  Electric field, Density     Potential, Charge     Flux of x and y j+1 i i+1  Light, Luminance, Efficiency, Power, Current and so on

6. Plasma Application Modeling@ POSTECH Flow chart fl2p.c initial.c pulse.c charge.c field.c flux.c continuity.c source.c history.c diagnostics.c Calculate efficiency time_step.c current.c, radiation.c, dump.c, gaspar.c, mu_n_D.c, gummel.c

10. Plasma Application Modeling@ POSTECH Basic equations • Continuity Equation with Drift-Diffusion Approx. and • Poisson’s Equation  : surface charge density on the dielectric surfaces • Boundary conditions on dielectric surface for ion for electron Mobility-driven drift term Isotropic thermal flux term for secondary electron for excited species

11. Plasma Application Modeling@ POSTECH Partial Differential Eqs. General form of linear second-order PDEs with two independent variables In case of elliptic PDEs,  Jacobi-Iteration method  Gauss-Seidel method  Successive over-relaxation (SOR) method In case of parabolic PDEs,  Alternating direction implicit (ADI) method

12. Plasma Application Modeling@ POSTECH Continuity equation (1) density nsp 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. 1st step (k means the value at time t) (* means the temporal value at time t+t/2 ) Discretized flux can be obtained by Sharfetter-Gummel method. Spatially discretized forms are converted to tridiagonal systems of equations which can be easily solved.

13. Plasma Application Modeling@ POSTECH Tridiagonal matrix (1) Based on Gauss elimination R2 R3

14. Plasma Application Modeling@ POSTECH Tridiagonal matrix (2)

15. Plasma Application Modeling@ 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. Ri Calculate the solution for the last unknown by Calculate the following equation in decreasing order of i

16. Plasma Application Modeling@ POSTECH 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.

17. Plasma Application Modeling@ POSTECH Poisson’s eq. (1) 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 Density correction by electric field change between t and t+t The continuity eq. and flux are coupled with Poission’s eq. 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.

18. Plasma Application Modeling@ POSTECH Poisson’s eq. (2) j+1 ci, j bi, j ai, j j is the surface charge density accumulating on intersection between plasma region and dielectric. di, j j-1 i-1 i i+1 Solved using SOR method

22. Plasma Application Modeling@ POSTECH Scharfetter-Gummel method Tridiagonal matrix • 2D discretized continuity eqn. integrated by the alternative direction implicit (ADI) method Scharfetter-Gummel method

29. Gonsalves’ lecture notes (Fall 2005)

31. Gonsalves’ lecture notes (Fall 2005)

36. Gonsalves’ lecture notes (Fall 2005)

38. Gonsalves’ lecture notes (Fall 2005)

45. Gonsalves’ lecture notes (Fall 2005)

46. ECE490O: NL Eq. SolversGonsalves’ lecture notes (Fall 2005)JK LEE (Spring, 2006)