9.7 Band–Band Transitions
275
Fig. 9.18 Imaginary part
of the dielectric function of
amorphous (solid line) and
crystalline (trigonal)
selenium (dash-dotted lines
for two different
polarization directions).
From [857]
2
4
6
8
1 0
2nk
5
0
10
15
20
Se
E||c
trigonal
amorphous
Table 9.4 Exciton (E b
X ) and biexciton (E b
XX , see Sect. 9.7.10) binding energies in various bulk semiconductors. Values
for 10 nm GaAs/15 nm Al 0.3 Ga 0.7 As quantum well (QW) are taken from [861]
Material
E b
X (meV)
E b
XX (meV)
E b
XX /E b
X
GaAs
4.2
GaAs QW
9.2
2.0
0.22
ZnSe
17
3.5
0.21
GaN
25
5.6
0.22
CdS
27
5.4
0.20
ZnS
37
8.0
0.22
ZnO
59
15
0.25
momentum
K = k e + k h .
(9.49)
The relative motion yields hydrogen-like quantized states E n ∝ n
−2 (n ≥ 1):
E
n
X = −
m
∗
r
m 0
1
2
r
m 0 e
4
2(4ππ 0 ) 2
1
n 2 ,
(9.50)
where m
∗
r denotes the reduced effective mass m
∗−1
r
= m
∗−1
e
+ m
∗−1
h . The third factor is the atomic
Rydberg energy (13.6 eV). The exciton binding energy E
b
X = −E
1
X is scaled by (m
∗
/m 0 )
−2
r ≈ 10
−3 .
A more detailed theory of excitons beyond the simple hydrogen model presented here, taking into
account the valence-band structure, can be found in [858] for direct and [859] for indirect cubic and in
[860] for wurtzite semiconductors. The exciton binding energies for various semiconductors are listed
in Table 9.4 and shown in Fig. 9.19a versus the band gap.
The radius of the exciton is
r
n
X = n
2 m 0
m ∗
r
r a B ,
(9.51)
where a B = 0.053 nm denotes the hydrogen Bohr radius.
6 The Bohr radius of the exciton is a X = r
1
X
(14.6 nm for GaAs, ∼ 2 nm for ZnO). The exciton moves with the center-of-mass K-vector through
6 Cf. (7.22); an electron bound to a donor can be considered as an exciton with an infinite hole mass.