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Fermi3D_np.m
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Fermi3D_np.m
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function[Ef]=Fermi3D_np(Eg,T,meffn,meffp,Dop3D)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Constants %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
h = 6.62606896E-34; %% Planck constant [J.s]
hbar = h/(2*pi);
e = 1.602176487E-19; %% electron charge [C]
m0 = 9.10938188E-31; %% electron mass [kg]
kB = 1.3806488E-23; %% Boltzmann's constant [J/K]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% patch to avoid problem...
if abs(Dop3D)<1e13
Ef=-Eg/2;
return
end
%%%%%%%%%%%%%%%%%%%%%%%%%% Fermi level at T=0K %%%%%%%%%%%%%%%%%%%%%%%%%%%%
if Dop3D>0
Ef0 = hbar^2 / (2*e*meffn*m0) * (3*pi^2* Dop3D )^(2/3);
elseif Dop3D<0
Ef0= -Eg - hbar^2 / (2*e*meffp*m0) * (-3*(pi^2)* Dop3D )^(2/3);
elseif Dop3D==0
Ef0=0;
end
if T==0
Ef=Ef0;
return
end
%%%%%%%%%%%%%%%%%%%%%%%%% 3D density of states %%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Here, I try to optimize the meshing
if Dop3D>0
Emax=Ef0+0.3;
Emin=-Eg;
else
Emin=Ef0-0.3;
Emax=0;
end
En1=linspace( 0 , Emax ,1e4);
En2=linspace( Emax , 3 ,50);
En=sort([En1 En2]);
Ep1=linspace( -Eg-3 , Emin , 50 );
Ep2=linspace( Emin , -Eg , 1e4 );
Ep=sort([Ep1 Ep2]);
ro3Dn = (1/(2*pi^2))*( (2*e*meffn*m0/(hbar^2))^(3/2) ) * sqrt(En) ;
ro3Dp = (1/(2*pi^2))*( (2*e*meffp*m0/(hbar^2))^(3/2) ) * sqrt( -(Ep+Eg) ) ;
%%%%%%%%%%%%%%%%%%%%% Fermi level at any temperature %%%%%%%%%%%%%%%%%%%%%%
Ef=Ef0;
FEc = 1./(1+exp(+(En-Ef)/(kB*T/e))) ;
FEv = 1./(1+exp(-(Ep-Ef)/(kB*T/e))) ;
ro3DEfn=ro3Dn.*FEc;
NtotX=trapz(En,ro3DEfn);
ro3DEfp=ro3Dp.*FEv;
PtotX=trapz(Ep,ro3DEfp);
NPtotX=NtotX-PtotX;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% the Fermi level is obiously at T=0K the max possible for n doped
% so now, it scans down with big step (ddE)
% to find the Fermi level when T is not zero
ddE=0.005; % step of the scan in eV to pass below the Fermi level
while ( ((NPtotX - Dop3D)) > 0)
Ef = Ef - ddE;
FEc = 1./(1+exp(+(En-Ef)/(kB*T/e))) ;
FEv = 1./(1+exp(-(Ep-Ef)/(kB*T/e))) ;
ro3DEfn = ro3Dn.*FEc;
NtotX = trapz(En,ro3DEfn);
ro3DEfp = ro3Dp.*FEv;
PtotX = trapz(Ep,ro3DEfp);
NPtotX = NtotX-PtotX;
end
% the Fermi level is obiously at T=0K the min possible for p doped
% so now, it scans up with big step (ddE)
% to found the Fermi level when T is not zero
% step of the scan in eV to pass above the Fermi level
while ( ((NPtotX - Dop3D)) < 0)
Ef = Ef + ddE;
FEc = 1./(1+exp(+(En-Ef)/(kB*T/e))) ;
FEv = 1./(1+exp(-(Ep-Ef)/(kB*T/e))) ;
ro3DEfn = ro3Dn.*FEc;
NtotX = trapz(En,ro3DEfn);
ro3DEfp = ro3Dp.*FEv;
PtotX = trapz(Ep,ro3DEfp);
NPtotX = NtotX-PtotX;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Now, it will try to get as close as posible to the real Ef with an
% error of 1% by dichotomy
Ef1=Ef;
Ef2=Ef1+ddE;
Epsilon = 1e-10;
while abs((NPtotX - Dop3D)/Dop3D) > Epsilon % find the Fermi level at any temperature
if NPtotX > Dop3D
Ef= Ef - abs(Ef1-Ef2)/2 ;
Ef1=Ef ;
FEc= 1./(1+exp((En-Ef)/(kB*T/e))) ;
FEv = 1./(1+exp(-(Ep-Ef)/(kB*T/e))) ;
else
Ef= Ef + abs(Ef1-Ef2)/2 ;
Ef2=Ef ;
FEc= 1./(1+exp((En-Ef)/(kB*T/e))) ;
FEv = 1./(1+exp(-(Ep-Ef)/(kB*T/e))) ;
end
ro3DEfn = ro3Dn.*FEc;
NtotX = trapz(En,ro3DEfn);
ro3DEfp = ro3Dp.*FEv;
PtotX = trapz(Ep,ro3DEfp);
NPtotX = NtotX-PtotX;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% figure('position',[150 30 1000 650]);
% plot(ro3Dn,En,'b')
% hold on;grid on;box on;
% plot(ro3Dp,Ep,'b')
%
% plot(ro3DEfn,En,'r.')
% plot(ro3DEfp,Ep,'g.')
%
% plot([0 max(max(ro3Dp),max(ro3Dn))],Ef*[1 1],'m')
%
% xlabel('Densité d etat Bulk (eV-1.m-3)');
% ylabel('Energy (eV)');
end