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% Calculate ZincBlende Bandstructure from Walter Harrison's book with an
% added extra s* state for the Conduction band (labelled as t since * inside
% variable names is not allowed) as in Vogl's paper
%Onsite energies are given for cation then anion (same for elemental)
% e.g. ecs is energy for cation s, eap is energy for anion p.
hm = 7.62; % hbar^2/m in eV.A^2
% Cubic lattice constant
alat = 5.65; % GaAs lattice constant
% GaAs
% Onsite Matrix elements (in eV)
eas = -8.3431;
eap = 1.0414;
eat = 8.5914;
ecs = -2.6569;
ecp = 3.6686;
ect = 6.7386;
%vectors to neighboring cations
d1 = alat/4 * [1 1 1];
d2 = alat/4 * [1 -1 -1];
d3 = alat/4 * [-1 1 -1];
d4 = alat/4 * [-1 -1 1];
d = [d1; d2; d3; d4];
%phase factors
g0 = inline('exp(i*k*transpose(d(1,:)))+exp(i*k*transpose(d(2,:)))+exp(i*k*transpose(d(3,:)))+exp(i*k*transpose(d(4,:)))','k','d');
g1 = inline('exp(i*k*transpose(d(1,:)))+exp(i*k*transpose(d(2,:)))-exp(i*k*transpose(d(3,:)))-exp(i*k*transpose(d(4,:)))','k','d');
g2 = inline('exp(i*k*transpose(d(1,:)))-exp(i*k*transpose(d(2,:)))+exp(i*k*transpose(d(3,:)))-exp(i*k*transpose(d(4,:)))','k','d');
g3 = inline('exp(i*k*transpose(d(1,:)))-exp(i*k*transpose(d(2,:)))-exp(i*k*transpose(d(3,:)))+exp(i*k*transpose(d(4,:)))','k','d');
%Interatomic Matrix Elements (eV) using Harrison's universal parameters
dn = norm(d1);
% Composite Matrix elements
Ess = -1.4054* hm/dn^2;
Esp = 1.0392*(hm/dn^2);
Exx = 0.3111*(hm/dn^2);
Exy = 0.8298*(hm/dn^2);
Etp = 0.9549* hm/dn^2; % Harrison parameter for s*ps from Vtps*dn^2/hm
Ett = 0; % In Dow's model no coupling between s* states on adjacent atoms is allowed
% Zone boundary and number of k points
qbz = 2*pi/alat;
nk=20;
j=1;
E=1;clear E;
kx=1; clear kx;
for q=0:qbz/nk:qbz,
k = [q 0 0];
%LCAO hamiltonian for the zincblende structure
% Calculate Upper Half of the LCAO Hamiltonian Matrix and use its
% Hermiticity to get lower half
Hu = [[ ecs/2 Ess *g0(k,d) 0 0 0 Esp*g1(k,d) Esp * g2(k,d) Esp* g3(k,d) 0 0];
[ 0 eas/2 -Esp * conj(g1(k,d)) -Esp * conj(g2(k,d)) -Esp * conj(g3(k,d)) 0 0 0 0 0];
[ 0 0 ecp/2 0 0 Exx * g0(k,d) Exy * g3(k,d) Exy * g2(k,d) 0 -Etp * g1(k,d)];
[ 0 0 0 ecp/2 0 Exy * g3(k,d) Exx * g0(k,d) Exy * g1(k,d) 0 -Etp * g2(k,d) ];
[ 0 0 0 0 ecp/2 Exy * g2(k,d) Exy * g1(k,d) Exx * g0(k,d) 0 -Etp * g3(k,d)];
[ 0 0 0 0 0 eap/2 0 0 Etp * g1(k,d) 0 ];
[ 0 0 0 0 0 0 eap/2 0 Etp * g2(k,d) 0 ];
[ 0 0 0 0 0 0 0 eap/2 Etp * g3(k,d) 0 ];
[ 0 0 0 0 0 0 0 0 eat/2 Ett *g0(k,d) ];
[ 0 0 0 0 0 0 0 0 0 eat/2];
];
%Its adjoint
Hd = (Hu)';
%The full Hermitian Matrix
H = Hu+Hd;
%Calculate Eigenvalues
E(j,:) = eig(H)';
kx(j)=q;
j=j+1;
end;
plot(kx,E,'k');
ylabel('Energy (eV)');