-
Notifications
You must be signed in to change notification settings - Fork 7
/
kp_14bands_Luttinger_Fishman_f.m
171 lines (129 loc) · 7.24 KB
/
kp_14bands_Luttinger_Fishman_f.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
function[E]=kp_14bands_Luttinger_Fishman_f(k_list, Eg8c, Eg7c, Eg6c, Dso, EP, EP14, 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/
% Moustafa El Kurdi
% PhD thesis: "Dispositifs a ilots de GeSi pour la microphotonique proche infrarouge sur silicium"
% Chapter 1, page 31
% https://www.theses.fr/2003PA112148
% Said Ridene et al., PRB 64, 085329 (2001)
% "Infrared absorption in Si/Si(1-x)Ge(x)/Si quantum wells"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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;
EP = EP *e;
EPx = EP14(1)*e;
EPp = EP14(2)*e;
P = sqrt(EP *hbar^2/(2*m0)) ;
Px= sqrt(EPx*hbar^2/(2*m0)) ;
Pp= sqrt(EPp*hbar^2/(2*m0)) ;
%gc= 1+2*F + EP*(Eg6c-2*Eg7v/3) / (Eg6c*(Eg6c-Eg7v)) ; % =1/mc electron in CB eff mass
% renormalization of all the gamma for the 8bands Hamiltonian
g1=g123(1); g2=g123(2); g3=g123(3);
%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);
%gc123=[0 0 0]; %% flat band
%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);
% if Silicon or Ge
%gc1=-1;
%gc2=0;
%gc3=0;
%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;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% 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 ;
Hdiag = [ EH8c EL8c EL8c EH8c E7c E7c E6c E6c EH8v EL8v EL8v EH8v E7v E7v];
% 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
];
H=[H6x6 H6x8 ; zeros(8,6) H8x8];
H14 = H' + H + diag(Hdiag);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
E(:,i) = eig(H14)/e;
end
end