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 Application POSTECH Plasma Display Panel Many Pixels the flat panel display using phosphor luminescence by UV photons produced in plasma gas discharge Discharge 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 dielectric ITO electrodeMgO layer barrier phosphors address electrode Front panel Back panel PDP structure

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

Plasma Application POSTECH Flow chart fl2p.c initial.c charge.c field.c continuity.c Calculate efficiency history.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 Application POSTECH Basic equations Continuity Equation with Drift-Diffusion Approx. Poisson’s Equation Boundary conditions on dielectric surface for secondary electron Mobility-driven drift term Isotropic thermal flux term for ion for electron for excited species  : surface charge density on the dielectric surfaces and

Plasma Application 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    

Plasma Application POSTECH Continuity equation (1) density n sp Spatially discretized forms are converted to tridiagonal systems of equations which can be easily solved. 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. 1 st step (k means the value at time t) ( * means the temporal value at time t+  t/2 )

Plasma Application POSTECH Tridiagonal matrix (1) R2R2 Based on Gauss elimination R3R3

Plasma Application POSTECH Tridiagonal matrix (2)

Plasma Application 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 Calculate the following equation in decreasing order of i RiRi

Plasma Application POSTECH Continuity equation (2) 2 nd 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.

Plasma Application 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 The continuity eq. and flux are coupled with Poission’s eq. 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.

Plasma Application POSTECH is the surface charge density accumulating on intersection between plasma region and dielectric. Solved using SOR method Poisson’s eq. (2) i-1ii+1 j-1 j j+1 a i, j b i, j c i, j d i, j

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

Gonsalves ’ lecture notes (Fall 2005)

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