sandbox/M1EMN/BASIC/sag3D.c
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);
Parameters
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
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
Computation of the 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");}
fclose(fpx);
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");}
fclose(fpc);
fprintf(stdout," end\n");
}
Run
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
Results
plots of flux
set hidden3d
sp 'Fcxz.txt' u 1:2:4 w l
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
reset
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
Bibliography
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