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Pot_GaN_AlN.m
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Pot_GaN_AlN.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Vb=2; %% Potential barrier height[eV]
Mass = 0.23; %% effective mass, constant over all the structure...
Mx=15e-9; %% map X [m]
My=15e-9; %% map Y [m]
Mz=4e-9; %% map Z [m]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
WLt = 0.5e-9; %% Wetting Layer thickness [m]
QDh = 1.8e-9; %% Quantum dot height [m]
QDd = 10e-9; %% Quantum dot diameter [m]
E = 10e8; %% Internal Electrical Field (10MV/cm in GaN/AlN interface)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x=linspace(-Mx/2,Mx/2,Nx);
y=linspace(-My/2,My/2,Ny);
z=linspace(-Mz,Mz,Nz)+0.5e-9;
[X,Y,Z]=meshgrid(x,y,z);
a=QDd/2; %% side length of the hexagon [m]
h=tan(pi/6) * a*sqrt(3)/2; %% height of the pyramid with 30deg angle bw side face and base
% Hexagonal base
idx1= abs(X)<a*sqrt(3)/2;
idx2=(tan(pi/6)*X+a>Y) .* (tan(pi/6)*X-a<Y) .* (-tan(pi/6)*X-a<Y) .* (-tan(pi/6)*X+a>Y);
idx_QD=idx1.*idx2;
idx_QD(Z < 0) = 0;
idx_QD(Z > QDh) = 0;
% Pyramid facet
idx_QD(Z > -2*h/a/sqrt(3)*X + h) = 0;
idx_QD(Z > +2*h/a/sqrt(3)*X + h) = 0;
idx_QD(Z > +h/a/sqrt(3)*X -h/a*Y + h) = 0;
idx_QD(Z > -h/a/sqrt(3)*X -h/a*Y + h) = 0;
idx_QD(Z > +h/a/sqrt(3)*X +h/a*Y + h) = 0;
idx_QD(Z > -h/a/sqrt(3)*X +h/a*Y + h) = 0;
% Wetting Layer index
idx_WL = (Z < 0) & (Z > -WLt);
% Potential index
idx = idx_QD | idx_WL;
V0=(idx)*0 + (1-idx)*Vb ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i=1:length(x)
for j=1:length(y)
idz=find(idx(j,i,:)==1);
z1=z(idz(1));
z2=z(idz(end));
Fbz= +E* (z2-z1)/(z2-z(1)); %% Field in the barrier
Fwz= -E*(z2-z(1)-(z2-z1))/(z2-z(1)); %% Field in the well
V0(j,i,:) = (Fwz*Z(j,i,:)+Fwz*WLt).* idx(j,i,:) + V0(j,i,:) ; % adding the electric field Fwell-z to the potential in the z-direction
V0(j,i,:) = (Fbz*Z(j,i,:)+Fbz*WLt) .* (Z(j,i,:)<-WLt) +V0(j,i,:); % adding the electric field Fbarrier-z to the potential in the z-direction
V0(j,i,:) = (Fbz*Z(j,i,:)-Fbz*z2+V0(j,i,1)-V0(j,i,:) ).* (Z(j,i,:)>0).*(1-idx(j,i,:)) +V0(j,i,:); % adding the electric field Fbarrier-z to the potential in the z-direction
end
end
V0 = Fx*X+V0; % adding the electric field Fx to the potential in the x-direction
V0 = Fy*Y+V0; % adding the electric field Fy to the potential in the y-direction
V0 = Fz*Z+V0; % adding the electric field Fz to the potential in the z-direction
V0=V0-min(min(min(V0)));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure('Name','Potential','position',[100 100 1600 800])
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(2,3,1,'fontsize',15)
hold on;grid on;
pcolor(x*1e9,z*1e9,squeeze(V0(round(end/2),:,:))')
plot([-1 1]*10,[0 0],'w--','linewidth',2)
idz=find(z>QDh);
idz=idz(1)-1;
plot([-1 1]*10,[1 1]*z(idz)*1e9,'w--','linewidth',2)
plot([1 1]*0,[-1 1]*10,'b--','linewidth',2)
plot([1 1]*x(end)*1e9,[-1 1]*10,'r--','linewidth',2)
colormap(jet)
colorbar
shading flat
caxis([floor(min(V0(:))) ceil(max(V0(:)))])
xlim([-1 1]*Mx/2*1e9)
ylim([-1 1]*My/2*1e9)
xlabel('x (nm)')
ylabel('z (nm)')
title(strcat('Potential (eV): Vxz @y=0nm'))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(2,3,4,'fontsize',15)
hold on;grid on;
pcolor(y*1e9,z*1e9,squeeze(V0(:,round(end/2),:))')
plot([-1 1]*10,[0 0],'w--','linewidth',2)
idz=find(z>QDh);
idz=idz(1)-1;
plot([-1 1]*10,[1 1]*z(idz)*1e9,'w--','linewidth',2)
plot([1 1]*0,[-1 1]*10,'b--','linewidth',2)
plot([1 1]*x(end)*1e9,[-1 1]*10,'r--','linewidth',2)
colormap(jet)
colorbar
shading flat
caxis([floor(min(V0(:))) ceil(max(V0(:)))])
xlim([-1 1]*Mx/2*1e9)
ylim([-1 1]*My/2*1e9)
xlabel('y (nm)')
ylabel('z (nm)')
title(strcat('Potential (eV): Vyz @x=0nm'))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(2,3,2,'fontsize',15)
hold on;grid on;
idz=find(z>0);
idz=idz(1);
pcolor(x*1e9,y*1e9,squeeze(V0(:,:,idz)))
colormap(jet)
colorbar
shading flat
caxis([floor(min(V0(:))) ceil(max(V0(:)))])
xlim([-1 1]*My/2*1e9)
ylim([-1 1]*My/2*1e9)
xlabel('x (nm)')
ylabel('y (nm)')
title(strcat('Potential (eV): Vxy @z=',num2str(z(idz)*1e9,'%.2f'),'nm'))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(2,3,5,'fontsize',15)
hold on;grid on;
idz=find(z>QDh);
idz=idz(1)-1;
pcolor(x*1e9,y*1e9,squeeze(V0(:,:,idz)))
colormap(jet)
colorbar
shading flat
caxis([floor(min(V0(:))) ceil(max(V0(:)))])
xlim([-1 1]*My/2*1e9)
ylim([-1 1]*My/2*1e9)
xlabel('x (nm)')
ylabel('y (nm)')
title(strcat('Potential (eV): Vxy @z=',num2str(z(idz)*1e9,'%.2f'),'nm'))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(2,3,3,'fontsize',15)
hold on;grid on;
plot(z*1e9,squeeze(V0(round(end/2),round(end/2),:)) ,'b.-')
plot(z*1e9,squeeze(V0(round(end/2),end,:)) ,'r.-')
ylim([floor(min(V0(:))) ceil(max(V0(:)))])
xlabel('z (nm)')
ylabel('Energy (eV)')
title(strcat('Potential (eV): Vz @y=0nm and \color{blue}x=0nm ; \color{red}x=',num2str(x(end)*1e9),'nm'))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot(2,3,6,'fontsize',15)
hold on;grid on;
view (-38, 30);
aa=[
-sqrt(3)/2*a -a/2 0
-sqrt(3)/2*a +a/2 0
0 +a 0
+sqrt(3)/2*a +a/2 0
+sqrt(3)/2*a -a/2 0
0 -a 0
-sqrt(3)/2*a -a/2 0
]*1e9;
plot3(aa(:,1),aa(:,2),aa(:,3),'k','linewidth',2)
bb=[
+(QDh-h)/h*sqrt(3)/2*a -(h-QDh)*a/h/2 QDh
+(QDh-h)/h*sqrt(3)/2*a +(h-QDh)*a/h/2 QDh
0 +(h-QDh)*a/h QDh
-(QDh-h)/h*sqrt(3)/2*a +(h-QDh)*a/h/2 QDh
-(QDh-h)/h*sqrt(3)/2*a -(h-QDh)*a/h/2 QDh
0 -(h-QDh)*a/h QDh
+(QDh-h)/h*sqrt(3)/2*a -(h-QDh)*a/h/2 QDh
]*1e9;
plot3(bb(:,1),bb(:,2),bb(:,3),'k','linewidth',2)
for j=1:length(aa(:,1))
plot3([aa(j,1) bb(j,1)],[aa(j,2) bb(j,2)],[aa(j,3) bb(j,3)],'k','linewidth',2)
end
M=max([Mx My]);
xlim([-1 1]*M/3*1e9)
ylim([-1 1]*M/3*1e9)
zlim([-0.8 0.8]*M/3*1e9)
xlabel('x (nm)')
ylabel('y (nm)')
zlabel('z (nm)')