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kp_16bands_Luttinger_Fishman_f.m
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kp_16bands_Luttinger_Fishman_f.m
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function[E]=kp_16bands_Luttinger_Fishman_f(k_list, Eg8c, Eg7c, Eg6c, Dso, Eg6v, EP, EP14, EPs, F, gc123, g123)
% Guy Fishman
% "Semi-Conducteurs: les Bases de la Theorie k.p " (2010)
% Structure de bande: II, 4.B Appendice : La matrice 14x14 ou hamiltonien H14, page 274
% https://www.amazon.fr/Semi-Conducteurs-Bases-Theorie-K-P-Fishman/dp/2730214976/ref=sr_1_fkmr1_1?ie=UTF8&qid=1548234034&sr=8-1-fkmr1&keywords=guy+fishman+kp
% https://www.abebooks.fr/semi-conducteurs-bases-th%C3%A9orie-k.p-Fishman-ECOLE/30091636895/bd
% https://www.decitre.fr/livres/semi-conducteurs-les-bases-de-la-theorie-k-p-9782730214971.html
% https://www.unitheque.com/Livre/ecole_polytechnique/Semi_conducteurs_les_bases_de_la_theorie_K.p-35055.html
% https://www.eyrolles.com/Sciences/Livre/semi-conducteurs-les-bases-de-la-theorie-k-p-9782730214971/
% Matrix H16 is based on the H14
% To get the H16, I had the valence band Eg6v~-12.5eV with the Eps parameters
% I find the way to write the 2 matrix lines in S. Ben Radhia, JAP 92, 4422, (2002)
% The results are not nuch different from the 14x14 bands
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Constants %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
h=6.62606896E-34; %% Planck constant [J.s]
hbar=h/(2*pi);
e=1.602176487E-19; %% charge de l electron [Coulomb]
m0=9.10938188E-31; %% electron mass [kg]
H0=hbar^2/(2*m0) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Eg8c = Eg8c*e;
Eg7c = Eg7c*e;
Eg6c = Eg6c*e;
Eg8v = 0;
Eg7v = -Dso*e;
%Dso = Dso*e;
Eg6v = Eg6v*e;
EP = EP *e;
EPx = EP14(1)*e;
EPp = EP14(2)*e;
EPs = EPs *e;
P = sqrt(EP *hbar^2/(2*m0)) ;
Px = sqrt(EPx *hbar^2/(2*m0)) ;
Pp = sqrt(EPp *hbar^2/(2*m0)) ;
Ps = sqrt(EPs(1)*hbar^2/(2*m0)) ;
Psp= sqrt(EPs(2)*hbar^2/(2*m0)) ;
%gc= 1+2*F + EP*(Eg6c-2*Eg7v/3) / (Eg6c*(Eg6c-Eg7v)) ; % =1/mc electron in CB eff mass
gc1=gc123(1); gc2=gc123(2); gc3=gc123(3);
g1=g123(1); g2=g123(2); g3=g123(3);
% renormalization of all the gamma for the 8bands Hamiltonian
%gc = gc - EP/3*( 2/Eg6c + 1/(Eg6c+Dso) ) + EPp/3*( 2/(Eg8c-Eg6c) + 1/(Eg7c-Eg6c) );
gc = 1+2*F + EPp/3*( 2/(Eg8c-Eg6c) + 1/(Eg7c-Eg6c) );
g1 = g1 - EP/(3*Eg6c) - EPx/3*(1/(Eg7c-Eg8v) + 1/(Eg8c-Eg8v) ) ;
g2 = g2 - EP/(6*Eg6c) + EPx/6/(Eg7c-Eg8v);
g3 = g3 - EP/(6*Eg6c) - EPx/6/(Eg7c-Eg8v);
gD1=g1;gD2=g2;gD3=g3;
% gD1 = g1 - EP/3/Eg6c - EPx/3*(1/Eg7c + 1/Eg8c - 2/(Eg8c-Eg7v)) ;
% gD2 = g2 - EP/6/Eg6c - EPx/12*(1/Eg8c + 1/(Eg8c-Eg7v) -2/Eg7c) ;
% gD3 = g3 - EP/6/Eg6c + EPx/12*(1/Eg8c + 1/(Eg8c-Eg7v) -2/Eg7c) ;
gc1=gc123(1); gc2=gc123(2); gc3=gc123(3);
%gc1 = gc123(1) + EPp/(3*(Eg8c-Eg6c)) + EPx/3*(1/(Eg8c-Eg8v) + 1/(Eg8c-Eg7v) ) ;
%gc2 = gc123(2) + EPp/(6*(Eg8c-Eg6c)) - EPx/6/(Eg8c-Eg7v);
%gc3 = gc123(3) + EPp/(6*(Eg8c-Eg6c)) + EPx/6/(Eg8c-Eg7v);
%gcD1 = gc1 + EPx/3 *(1/(Eg8c-Eg8v) + 1/(Eg8c-Eg7v) - 2/Eg7c ) ;
%gcD2 = gc2 + EPx/12*(1/(Eg8c-Eg8v) + 1/Eg7c - 2/(Eg8c-Eg7v)) ;
%gcD3 = gc3 - EPx/12*(1/(Eg8c-Eg8v) + 1/Eg7c - 2/(Eg8c-Eg7v)) ;
gcD1=gc1;gcD2=gc2;gcD3=gc3;
gv=-1; % unknown parameter... It gives the curvature of the lowest BV: Eg6v
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% Building of the Hamiltonien %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%k+ = kx + 1i*ky
%k- = kx - 1i*ky
for i=1:length(k_list(:,1))
kx = k_list(i,1);
ky = k_list(i,2);
kz = k_list(i,3);
k=sqrt(kx.^2 + ky.^2 + kz.^2);
kpp = kx + 1i*ky;
kmm = kx - 1i*ky;
AA = H0*g2 *( 2*kz^2 - kx.^2 - ky.^2 );
AAc = H0*gc2*( 2*kz^2 - kx.^2 - ky.^2 );
BB = H0*2*sqrt(3)*g3 *kz*(kx - 1i*ky) ;
BBc = H0*2*sqrt(3)*gc3*kz*(kx - 1i*ky) ;
CC = H0*sqrt(3)*(g2 *(kx^2-ky^2)-2i*g3 *kx*ky);
CCc = H0*sqrt(3)*(gc2*(kx^2-ky^2)-2i*gc3*kx*ky);
AA_D = H0*gD2 *( 2*kz^2 - kx.^2 - ky.^2 );
AAc_D = H0*gcD2*( 2*kz^2 - kx.^2 - ky.^2 );
BB_D = H0*2*sqrt(3)*gD3 *kz*(kx - 1i*ky) ;
BBc_D = H0*2*sqrt(3)*gcD3*kz*(kx - 1i*ky) ;
CC_D = H0*sqrt(3)*(gD2 *(kx^2-ky^2)-2i*gD3 *kx*ky);
CCc_D = H0*sqrt(3)*(gcD2*(kx^2-ky^2)-2i*gcD3*kx*ky);
EH8c = Eg8c - gc1 *H0*k^2 + AAc;
EL8c = Eg8c - gc1 *H0*k^2 - AAc;
E7c = Eg7c - gcD1*H0*k^2 ;
E6c = Eg6c + gc *H0*k^2 ;
EH8v = 0 - g1 *H0*k^2 + AA ;
EL8v = 0 - g1 *H0*k^2 - AA ;
E7v = Eg7v- gD1 *H0*k^2 ;
E6v = Eg6v- gv *H0*k^2 ;
Hdiag = [ EH8c EL8c EL8c EH8c E7c E7c E6c E6c EH8v EL8v EL8v EH8v E7v E7v E6v E6v];
% Ec Ec HH LH LH HH SO SO
H8x8=[
0 0 -sqrt(1/2)*P*kpp sqrt(2/3)*P*kz sqrt(1/6)*P*kmm 0 sqrt(1/3)*P*kz sqrt(1/3)*P*kmm % Ec
0 0 0 -sqrt(1/6)*P*kpp sqrt(2/3)*P*kz sqrt(1/2)*P*kmm sqrt(1/3)*P*kpp -sqrt(1/3)*P*kz % Ec
0 0 0 BB CC 0 sqrt(1/2)*BB_D sqrt(2) *CC_D % HH
0 0 0 0 0 CC -sqrt(2) *AA_D -sqrt(3/2)*BB_D % LH
0 0 0 0 0 -BB -sqrt(3/2)*BB_D' sqrt(2) *AA_D % LH
0 0 0 0 0 0 -sqrt(2) *CC_D' sqrt(1/2)*BB_D' % HH
0 0 0 0 0 0 0 0 % SO
0 0 0 0 0 0 0 0 % SO
];
% HHc LHc LHc HHc SOc SOc
H6x6=[
0 BBc CCc 0 sqrt(1/2)*BBc_D sqrt(2) *CCc_D % HHc
0 0 0 CCc -sqrt(2) *AAc_D -sqrt(3/2)*BBc_D % LHc
0 0 0 -BBc -sqrt(3/2)*BBc_D' sqrt(2) *AAc_D % LHc
0 0 0 0 -sqrt(2) *CCc_D' sqrt(1/2)*BBc_D' % HHc
0 0 0 0 0 0 % SOc
0 0 0 0 0 0 % SOc
];
Deltap=0;%Dso; % unknown parameter...
% Ec Ec HH LH LH HH SO SO
H6x8=[
-sqrt(1/2)*Pp*kmm 0 1/3*Deltap sqrt(1/3)*Px*kpp sqrt(1/3)*Px*kz 0 sqrt(1/6)*Px*kpp sqrt(2/3)*Px*kz % HHc
sqrt(2/3)*Pp*kz -sqrt(1/6)*Pp*kmm -sqrt(1/3)*Px*kmm 1/3*Deltap 0 sqrt(1/3)*Px*kz 0 -sqrt(1/2)*Px*kpp % LHc
sqrt(1/6)*Pp*kpp sqrt(2/3)*Pp*kz -sqrt(1/3)*Px*kz 0 1/3*Deltap -sqrt(1/3)*Px*kpp sqrt(1/2)*Px*kmm 0 % LHc
0 sqrt(1/2)*Pp*kpp 0 -sqrt(1/3)*Px*kz sqrt(1/3)*Px*kmm 1/3*Deltap sqrt(2/3)*Px*kz -sqrt(1/6)*Px*kmm % HHc
sqrt(1/3)*Pp*kz sqrt(1/3)*Pp*kmm -sqrt(1/6)*Px*kmm 0 -sqrt(1/2)*Px*kpp -sqrt(2/3)*Px*kz -2/3*Deltap 0 % SOc
sqrt(1/3)*Pp*kpp -sqrt(1/3)*Pp*kz -sqrt(2/3)*Px*kz sqrt(1/2)*Px*kmm 0 sqrt(1/6)*Px*kpp 0 -2/3*Deltap % SOc
];
H14=[H6x6 H6x8 ; zeros(8,6) H8x8];
% Here are the 2 additionnal bands @~-12.5eV for Gamma6v
% Gamma6v Gamma6v
H16x2=[
-sqrt(1/2)*Ps *kmm 0 % HHc
sqrt(2/3)*Ps *kz -sqrt(1/6)*Ps *kmm % LHc
sqrt(1/6)*Ps *kpp sqrt(2/3)*Ps *kz % LHc
0 sqrt(1/2)*Ps *kpp % HHc
sqrt(1/3)*Ps *kz sqrt(1/3)*Ps *kmm % SOc
sqrt(1/3)*Ps *kpp -sqrt(1/3)*Ps *kz % SOc
0 0 % Ec
0 0 % Ec
-sqrt(1/2)*Psp*kmm 0 % HH
sqrt(2/3)*Psp*kz -sqrt(1/6)*Psp*kmm % LH
sqrt(1/6)*Psp*kpp sqrt(2/3)*Psp*kz % LH
0 sqrt(1/2)*Psp*kpp % HH
sqrt(1/3)*Psp*kz sqrt(1/3)*Psp*kmm % SO
sqrt(1/3)*Psp*kpp -sqrt(1/3)*Psp*kz % SO
0 0 % Gamma6v
0 0 % Gamma6v
];
H16=[ [H14;zeros(2,14)] H16x2 ];
H = H16' + H16 + diag(Hdiag);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E(:,i) = eig(H)/e;
end
end