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Copy pathkp_6bands_DKK_WZ_strain_f.m
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kp_6bands_DKK_WZ_strain_f.m
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function[E]=kp_6bands_DKK_WZ_strain_f(k_list, Dcr, Dso, AA, DD, exx, ezz)
% Stefan Birner (Nextnano)
% PhD thesis: "Modeling of semiconductor nanostructures and semiconductor-electrolyte interfaces" (2011)
% Chapter3, page 36: "Multi-band k.p envelope function approximation"
% Download:
% https://mediatum.ub.tum.de/doc/1084806/1084806.pdf
% https://www.nextnano.com/downloads/publications/PhD_thesis_Stefan_Birner_TUM_2011_WSIBook.pdf
% S. L. Chuang et al. PRB, 54, 2491 (1996)
% "k.p method for strained wurtzite semiconductors"
% M. Kumagai, S. L. Chuang et al. PRB, 57, 15304 (1998)
% "Analytical solutions of the block-diagonalized Hamiltonian for strained wurtzite semiconductors"
% Seoung-Hwan Park and S. L. Chuang, PRB, 59, 4726 (1999)
% "Crystal-orientation effects on the piezoelectric field and electronic properties of strained wurtzite semiconductors"
% D. J. Dugdale, S. Brand, and R. A. Abram, PRB, 61, 12934 (2000)
% "Direct calculation of k"p parameters for wurtzite AlN, GaN, and InN"
% W. J. Fan et al. J. Crystal Growth, 287, 28 (2006)
% "Electronic structures of wurtzite ZnO and ZnO/MgZnO quantum well"
% P. Rinke, PRB, 77, 075202 (2008)
% "Consistent set of band parameters for the group-III nitrides AlN, GaN, and InN"
% http://cmt.dur.ac.uk/ssterratum/paper/node4.html
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Constants %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
h=6.62606896E-34; %% Planck constant [J.s]
hbar=h/(2*pi);
e=1.602176487E-19; %% electron charge [Coulomb]
m0=9.10938188E-31; %% electron mass [kg]
H0=hbar^2/(2*m0) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Dcr = Dcr*e;
Dso = Dso*e;
A1=AA(1);A2=AA(2);A3=AA(3);A4=AA(4);A5=AA(5);A6=AA(6);A7=AA(7)*e*1e-10;
D1=DD(1)*e;D2=DD(2)*e;D3=DD(3)*e;D4=DD(4)*e;D5=DD(5)*e;D6=DD(6)*e;
% here is the modification of the parameters due to the 6 bands instead of 8 bands model
%L1 = H0*(A2+A4+A5-1)+P1^2/Eg;
%L2 = H0*(A1-1)+P2^2/Eg;
L1 = H0*(A2+A4+A5-1);
L2 = H0*(A1-1);
M1 = H0*(A2+A4-A5-1);
M2 = H0*(A1+A3-1);
M3 = H0*(A2-1);
% here is the modification of the parameters due to the 6 bands instead of 8 bands model
%N1 = H0*2*A5 + P1^2/Eg;
%N2 = H0*sqrt(2)*A6 + P1*P2/Eg;
N1 = H0*2*A5 ;
N2 = H0*sqrt(2)*A6 ;
N3 = 1i*sqrt(2)*A7;
B1=0;
B2=0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
eyy = exx;
exy = 0; eyx=0;
ezx = 0; exz=0;
eyz = 0; ezy=0;
ee = exx+eyy+ezz;
l1 = D2+D4+D5;
l2 = D1;
m1 = D2+D4-D5;
m2 = D1+D3;
m3 = D2;
n1 = 2*D5;
n2 = sqrt(2)*D6;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%% Strained Hamiltonian %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% since it does not depend on k, it can be outside the loop %%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Hst=[
l1*exx+m1*eyy+m2*ezz n1*exy n2*exz
n1*exy m1*exx+l1*eyy+m2*ezz n2*eyz
n2*exz n2*eyz m3*(exx+eyy)+l2*ezz
];
Hst = [Hst zeros(3,3) ; zeros(3,3) Hst];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% Building of the Hamiltonien %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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);
Hdiag = H0*k^2 *ones(1,6);
%%%%%%%%%%%%%%%%%%%%%%%%%%%% DKK Hamiltonian %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
H_DKK=[
L1*kx^2+M1*ky^2+M2*kz^2 N1*kx*ky N2*kx*kz
N1*kx*ky M1*kx^2+L1*ky^2+M2*kz^2 N2*ky*kz
N2*kx*kz N2*ky*kz M3*(kx^2+ky^2)+L2*kz^2
];
H_DKK = [H_DKK zeros(3,3) ; zeros(3,3) H_DKK];
%%%%%%%%%%%%%%%%%%%%%%%%% Spin-orbit Hamiltonian %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Hso=[
0 1 0 0 0 1i
-1 0 0 0 0 1
0 0 0 -1i -1 0
0 0 -1i 0 -1 0
0 0 1 1 0 0
1i -1 0 0 0 0
];
%%%%%%%%%%%%%%%%%%%%%%% Crystal field Hamiltonian %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Hcr=[ Dcr Dcr 0 Dcr Dcr 0 ];
Hcr=diag(Hcr);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
H = diag(Hdiag) + H_DKK + Hso*Dso/(3i) + Hcr + Hst;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E(:,i) = eig(H)/e ;
end
end