Analysis of Wurtzite GaN/AlGaN Quantum Well Lasers from First-Principles Calculations
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near the F point of wurtzite structure is given by,
H(k)
H, HC HVV/
(1)
Hcv indicates the direct interaction between the conduction band minimum (CBM) and the valence band maximum (VBM). Hc, and Hv, indicate the conduction band and the valence band without the interaction between CBM and VBM, respectively. The interactions with the other bands were treated as the second order perturbation. Hc, is given by,
R 0 Q' 0 R 0 Q-
(2)
He=Q0 Q 0 0 where
Q = -R =
P._(k. + iky),
ho(Slp.jX)(k. + ik5 )
h(SjpzIZ)k; = Pi1k,.
Note that Hc, includes (Slp i 1i)(i = x,y, z), which is related to the dipole matrix element. Owing to the large energy-gaps of GaN and AIN, the 8 x 8 Hamiltonian can be splitted into the 2 x 2 Hamiltonian Hcc for the conduction bands and the 6 x 6 Hamiltonian H',, for the valence bands. H,, was renormalized into the matrix elements of H'CC and H',,. They are given by,
Hc=
(Ec 0
S)
0 Ec
F
0
-H*
(3)
0 -H*
0 K*
0
G
-H
A
A A
0
-H
0
A
A
K
0
I
A
G
0
0
I
0
F I
0
K
0
0 I*
0
where 2
Ek2
S= E+ h-k + h
'
F = Al+A2+A+0,
G=A,-A 926
2 +A+0,
(4)
0.47
1-l0.46 r40--
a•
JI_ •r
I
00o
7
A-
"-'
/8
0.45
•
.
-'
•
2 0%07
7,
---- kop Method 0 FLAPW
S0.44 Pb
0"0
0.43 0.06
0.04 0.02 k x (2nt/c)
0.00
0.03 0.06 k z (27t/c)
0.09
Fig. 1. Valence band structures of GaN by k. p method and FLAPW.
H = iA 6 kzk+ - A7 k+, K
= A5 k+,
0 =
A =V
A 3 k + AZ 4 kI,
I = iA 6 kzk+ + A7 k+, A3 ,
A= E
ak
k± = k +±iky, k=
+ A 2kI, k,,k .
A1 and A 2 ,3 represent the crystal-field and spin-orbit splitting energies, respectively. Ai correspond to the Luttinger parameters in the zincblende crystals. Then, the parameters in H' and H', were determined independently, by reproducing the band structures near the band edge. The valence band structure, fitted by the k. p method (dashed lines) is shown in Fig. 1. The open circles are result from the FLAPW method. The parameters in the k - p calculation are indicated in Table 1. 11and I mean the directions parallel and perpendicular to the c-axis, respectively. The effective hole masses of GaN from the Ai are m1 9 = 1.1, m' = 1.65 and very heavy. There are three bands, labeled F9 , F', and F2 at the r point. These eigenstates can be approximately expressed by,
Ir, +)
= IX+ iY,-
lrl±) -IX±iy,+Fl Ir2 1) - IZ, )I due to A,,o 0. Therefore, in the k, direction, the conduction band is strongly coupled with the only F2 state through kp 2 perturbation and it causes the only r' hole mass to be light. On the other hand, in the k, direction, the conduction band is strongly coupled with the mixed state (Ir9 ±-) + IrN + 1) - IX)). Then, only one mixed band at the finite wavenumber has small effective hole mass.
927
50
"0
0
, -50
1-111
1
LH
LH2 HH1
-50
LH2
H CHI
-100
=-100 HJH2 -150 -200
-150
0
5
10
-200 20 0
15
Wave Number ( x10 6 cni)
5
10
15
Wave Number (x 106
20
Crd1 )
Fig. 2. Subband structures in k.-ky plane.
When the heterojunction is perpendicular to the c-axis, k, becomes an operator in H', and H',,. T
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