Need help on difficulty 3D Plot. - Problem resolved

I got below question. Need help on coding for below issue.

Answer 1

[Alan Stevens]

Here's some code to get you started

% Data
T=300;
k=1.38064852e-23;
Ea=7.2*10^(-21);
Ed=7.2*10^(-21);
Nv=2.5*10^(25)*((0.59*T/300)^(3/2))*10^(-6);
Nc=2.5*10^(25)*((1.08*T/300)^(3/2))*10^(-6);
Ec=0.5*(1.166-0.000473*T*T/(636+T))*1.6*10^(-19);
Ev=0;
% Collect data to be passed to function
data = [T, k , Ea, Ed, Nv, Nc, Ec, Ev];

% Dopant concentrations
Nd = 10^17;
Na = 10^5;

 

Ef0 = 10^-21;   % Initial guess at fermi energy
% Use fzero to find fermi energy, i.e. the value of Ef that makes 
% function Efn return zero
Ef = fzero(@Efn, Ef0,[],data,Nd,Na);  

disp(Ef)


function F = Efn(Ef,data, Nd, Na)
T = data(1);
k = data(2);
Ea = data(3);
Ed = data(4);
Nv = data(5);
Nc = data(6);
Ec = data(7);
Ev = data(8);
kT = k*T;

F = Nc*exp(-(Ec-Ef)/kT) + Na/(1+4*exp(-(Ef-Ea)/kT)) - Nv*exp(-(Ef-Ev)/kT) - Nd/(1+2*exp(-(Ed-Ef)/kT));

end

The above will calculate the Fermi level for one pair of dopant concentrations. See if you can take it from here.

[ZHAO YANG]

Thanks for the detalis explanation.

I have tried below function, but it's not work for me. Could you share more advise.

for j=1:length(Na)

for i=1:length(Nd)

F = Nc.*exp(-(Ec-Ef)./kT) + Na./(1+4.*exp(-(Ef-Ea)./kT)) - Nv.*exp(-(Ef-Ev)./kT) - Nd./(1+2.*exp(-(Ed-Ef)./kT));

end

end

[Alan Stevens]

The following shows how to structure the calculation for several pairs of Nd and Na. In doing this I noticed that the results were sensitive to the initial guess. This was a numerical problem related to the size of the energy levels, so in the listing below the energies are scaled (divided by kT) before being called by fzero. The resulting fermi levels are then rescaled at the end.

% Data
T=300;
k=1.38064852e-23;
Ea=7.2*10^(-21);
Ed=7.2*10^(-21);
Nv=2.5*10^(25)*((0.59*T/300)^(3/2))*10^(-6);
Nc=2.5*10^(25)*((1.08*T/300)^(3/2))*10^(-6);
Ec=0.5*(1.166-0.000473*T*T/(636+T))*1.6*10^(-19);
Ev=0;

% Scale energy levels
kT = k*T;
Ea = Ea/kT; Ed = Ed/kT; Ec = Ec/kT; Ev = Ev/kT;

% Collect data to be passed to function
data = [Ea, Ed, Nv, Nc, Ec, Ev];

% Dopant concentrations
Nd = [10^17, 10^15, 10^15, 10^3, 10^5, 10^5];
Na = [10^5, 10^3, 10^15, 10^15, 10^17, 10^5];

Ef = zeros(numel(Nd),1); 

Ef0 = 10^-21/kT;   % Initial guess at scaled fermi energy
% Use fzero to find fermi energy, i.e. the value of Ef that makes 
% function Efn return zero
for i = 1:numel(Nd)
    Ef(i) = fzero(@Efn, Ef0,[],data,Nd(i),Na(i));  
end

Ef = Ef*kT;  % Rescale  
fprintf('%g\n',Ef)


function F = Efn(Ef,data, Nd, Na)
Ea = data(1);
Ed = data(2);
Nv = data(3);
Nc = data(4);
Ec = data(5);
Ev = data(6);

F = Nc*exp(-(Ec-Ef)) + Na*exp(-(Ef-Ea)) - Nv*exp(-(Ef-Ev)) - Nd./(1+2*exp(-(Ed-Ef)));
end

[ZHAO YANG]

Thank you very much!