The SAG equation

    The SAG equation is a diffusion Fick equation of a specie c:

    \displaystyle \nabla^2 c = 0 with boundary condition of cristal growth on the substrate \frac{\partial c}{\partial y} = bi\; c on y=0 (bi is a kind of Biot number) and no growth \frac {\partial c}{\partial y}=0 on y=0 on the mask and far from the wall, there is a constant arrival of species c(x,top)=1, right and left are periodic conditions (or infinite domain), we solve it in 3D.

    This system can be solved with the poisson solver.

    //#include "grid/multigrid.h"
    #include "grid/multigrid3D.h"
    #include "poisson.h"
    #include "utils.h"

    Concentration source and flux, and some obvious variables

    scalar c[],s[],flux[];
    double bi,dx;

    the exact solution without mask

    double cexact(double y)
    { double ce;
        ce = (bi*y+1)/(bi*L0+1);
        return ce;

    the mask function

    double masque(double x, double z)
        return ((fabs(x)<1 && z<0)? 1:0);

    Boundary conditions: a given concentration at the top, no reaction on the mask (no flux: \partial c(t)/\partial y=0), a complete reaction on the cristal \partial c(t)/\partial y = bi\; c(t) this mixed condition is written (c[0,0]-c[bottom])/Delta = bi (c[0,0] + c[bottom])/2

    c[top]    =  dirichlet(1);
    c[bottom] = ( masque(x,z))?   neumann(0) :c[]*(2.-bi*Delta)/(2.+bi*Delta)  ;
    c[right]  =  neumann(0);
    c[left]   =  neumann(0);
    c[back]   =  neumann(0);
    c[front]  =  neumann(0);
    flux[bottom] = neumann(0);


    The size of the domain L0. it is a cube of reacting surface on the plane -L_0/2<x<L0/2 -L_0/2<z<L0/2, the gaz is 0<y<L0

    int main() {
        L0 = 10.;
        X0 = -L0/2;
        Y0 = 0;
        Z0 = -L0/2;
        N =  256/2;
        dx = L0/N;
        init_grid (N);

    parameter of flux

        bi = 1.2;

    Solve Laplace

    Initialisation of the solution of laplace equation with poisson \displaystyle \nabla^2 c = s with a zero source term s=0

          c[] = s[] = 0.;
        boundary ({c, s});
        poisson (c,s);

    Computation of the flux

          flux[] =  ( c[0,0] - c[0,-1] )/Delta;
        boundary ({flux});
    Save the results
        double c00=1./(1+bi*L0);
        FILE *  fpx = fopen("Fcxz.txt", "w");
        for (double x = -L0/2 ; x < L0/2; x += dx){
            for (double z = -L0/2 ; z < L0/2; z += dx){
            fprintf (fpx, "%g %g %g %g %g \n",
                     x, z, 1-masque(x,z),  interpolate (flux, x, 0, z)/c00/bi, interpolate (c, x, 0, z)/c00);}
            fprintf (fpx,"\n");}
        FILE *  fpc = fopen("Fcut.txt", "w");
        for (double x = -L0/2 ; x < L0/2; x += dx){
            for (double z = -L0/2 ; z < L0/2; z += dx){
                fprintf (fpc, "%g %g %g %g  \n",
                         x,  interpolate (flux, x, 0, 0)/c00/bi, z, interpolate (flux, 0, 0, z)/c00/bi);}
            fprintf (fpc,"\n");}
        fprintf(stdout," end\n");


    Then compile and run:

     rm sag3D; qcc  -g -O2 -DTRASH=1 -Wall  sag3D.c -o sag3D ; ./sag3D

    or better

     make sag3D.tst;make sag3D/plots    
     make sag3D.c.html ; open sag3D.c.html 


    plots of flux

     set hidden3d
     sp 'Fcxz.txt' u 1:2:4 w l
    3D plot of flux (script)

    3D plot of flux (script)

    concentration mask …

     set multiplot;set hidden3d;
     set size 0.6,0.6
     set origin .0,.5 ;;set view 30,30
     sp [-5:5][:][0:]'Fcxz.txt'  u 1:2:4 t'flux' w l
     set origin .5,.5;set view 90,0
     sp [:][][0:]'Fcxz.txt'   u 1:2:4t'flux' w l
     set origin .5,.0
     set view 0,0;set contour;set nosurface;
     sp [:][:][0:]'Fcxz.txt' u 1:2:3 t'mask'w l
     set origin .0,.0 ;
     set view 90,270
     set size 0.4,0.4
     p'Fcut.txt'  u 1:2 t'cut x' w l,''u 3:4 t'cut z'w l
     unset multiplot
    multiplot (script)

    multiplot (script)

     set pm3d; set palette rgbformulae 22,13,-31;unset surface;
     set ticslevel 0;
     unset border;
     unset xtics;
     unset ytics;
     unset ztics;
     unset colorbox;
     #set xrange[-3:3];set yrange[-3:3];
     set view 0,0
    sp'Fcxz.txt' u 1:2:(($4)>1 ? $4:  1) not
    with colors (script)

    with colors (script)


    • N. Dupuis, J. Décobert, P.-Y. Lagrée, N. Lagay, D. Carpentier, F. Alexandre (2008): “Demonstration of planar thick InP layers by selective MOVPE”. Journal of Crystal Growth issue 23, 15 November 2008, Pages 4795-4798

    • N. Dupuis, J. Décobert, P.-Y. Lagrée , N. Lagay, C. Cuisin, F. Poingt, C. Kazmierski, A. Ramdane, A. Ougazzaden (2008): “Mask pattern interference in AlGaInAs MOVPE Selective Area Growth : experimental and modeling analysis”. Journal of Applied Physics 103, 113113 (2008)

    Version 1: april 2015

    ready for new site 09/05/19