166
6 Band Structure
Table 6.6 Valence-band parameters (for (6.48)) A and B in units of ( 2 /2m 0 ), C 2 in units of ( 2 /2m 0 ) 2 , and 0 in eV.
From [164, 523, 524]
Material
A
B
C 2
0
C
−4.24
−1.64
9.5
0.006
Si
−4.28
−0.68
24
0.044
Ge
−13.38
−8.5
173
0.295
GaAs
−6.9
−4.4
43
0.341
InP
−5.15
−1.9
21
0.11
InAs
−20.4
−16.6
167
0.38
ZnSe
−2.75
−1.0
7.5
0.43
Values for A, B, C
2 and 0 for a number of semiconductors are given in Table 6.6. The valence-band
structure is often described with the Luttinger parameters γ 1 , γ 2 , and γ 3 that can be represented through
A, B, and C via
2
2m 0
γ 1 = − A
(6.50a)
2
2m 0
γ 2 = −
B
2
(6.50b)
2
2m 0
γ 3 =
B 2 + C 2 /3
2
.
(6.50c)
The mass of holes in various directions can be derived from (6.48). The mass along the [001] direction,
i.e.
2
/(∂
2 E(k)/∂k
2
x ) for k y = 0 and k z = 0, is
1
m
100
hh
=
2
2 (A + B)
(6.51a)
1
m
100
lh
=
2
2 (A − B) .
(6.51b)
The anisotropy of hole masses has been investigated with cyclotron resonance experiments (Fig. 6.42).
For θ being the angle between the magnetic field and the [001] direction, the effective heavy hole
(upper sign) and light hole (lower sign) mass in cubic semiconductors is given by [515]
m
∗
=
2
2
1
A ±
B 2 + C 2 /4
(6.52)
×
⎧
⎨
⎩
C
2
(1 − 3 cos
2
θ)
2
64
B 2 + C 2 /4
A ±
B 2 + C 2 /4
+ . . .
⎫
⎬
⎭
.
For C
2
= 0 the hole bands are isotropic, as is obvious from (6.48). In this case γ 2 = γ 3 , the so-called
spherical approximation. The average of the hole masses over all directions is
1
m
av
hh
=
2
2
A + B
1 +
2 C
2
15 B 2
(6.53a)