9.7 Band–Band Transitions
269
3D
2D
1D
M 0
M 1
M 0
M 1
M 2
M 3
M 0
M 1
M 2
M 0
M 1
M 2
M 2
M 3
M 0
M 1
M 0
M 1
r
i
Fig. 9.8 Shape of the real (left panel) and imaginary (right panel) parts of the dielectric function in the vicinity of
critical points in 3, 2 and 1 dimensions (for labels see Table 9.3). The dashed line in each graph indicates the energy
position of the critical point E 0 . Adapted from [841]
Fig. 9.9 a Direct optical
transition and b indirect
optical transitions between
valence and conduction
bands. The indirect
transition involves a
phonon with energy ph
(index a: phonon
absorption, e: phonon
emission) and wavevector
k ph
(a)
E
+E
C
g
=E V
k
E
E V
h
(b)
E
k
h ph,a
k ph
E V
E =E +E
C
V
g
h ph,e
When the energy dependence of the matrix element is neglected, the absorption coefficient is determined
by the corresponding square-root joint density of states (M 0 critical point):
α(E) ∝
E − E g
E
≈ ∝
E − E g .
(9.45)
The approximation is valid if the considered energy interval, e.g. around a band edge, is small.
Absorption spectra of (In x Ga 1−x ) 2 O 3 alloy thin films at room temperature are shown in Fig. 9.10a.
The α
2 versus photon energy so-called Tauc plot shows a linear dependence with broadening and
additional states at the band edge due to disorder effects. The extrapolation of the linear part yields the
absorption edge (Fig. 9.10b).
Absorption spectra of GaAs are shown in Fig. 9.11a for photon energies close to the band gap at
various temperatures. The rapid increase, typical for direct semiconductors, is obvious. In particular at
low temperatures, however, the absorption lineshape close to the band gap is dominated by an excitonic
feature, discussed in Sect. 9.7.6.
Due to the increasing density of states, the absorption increases with the photon energy (Fig. 9.11c).
At 1.85 eV there is a step in the absorption spectrum of GaAs due to the beginning of the contribution
of transitions between the s-o hole band and the conduction band (see E 0 + 0 transition in Fig. 9.1b).
269
3D
2D
1D
M 0
M 1
M 0
M 1
M 2
M 3
M 0
M 1
M 2
M 0
M 1
M 2
M 2
M 3
M 0
M 1
M 0
M 1
r
i
Fig. 9.8 Shape of the real (left panel) and imaginary (right panel) parts of the dielectric function in the vicinity of
critical points in 3, 2 and 1 dimensions (for labels see Table 9.3). The dashed line in each graph indicates the energy
position of the critical point E 0 . Adapted from [841]
Fig. 9.9 a Direct optical
transition and b indirect
optical transitions between
valence and conduction
bands. The indirect
transition involves a
phonon with energy ph
(index a: phonon
absorption, e: phonon
emission) and wavevector
k ph
(a)
E
+E
C
g
=E V
k
E
E V
h
(b)
E
k
h ph,a
k ph
E V
E =E +E
C
V
g
h ph,e
When the energy dependence of the matrix element is neglected, the absorption coefficient is determined
by the corresponding square-root joint density of states (M 0 critical point):
α(E) ∝
E − E g
E
≈ ∝
E − E g .
(9.45)
The approximation is valid if the considered energy interval, e.g. around a band edge, is small.
Absorption spectra of (In x Ga 1−x ) 2 O 3 alloy thin films at room temperature are shown in Fig. 9.10a.
The α
2 versus photon energy so-called Tauc plot shows a linear dependence with broadening and
additional states at the band edge due to disorder effects. The extrapolation of the linear part yields the
absorption edge (Fig. 9.10b).
Absorption spectra of GaAs are shown in Fig. 9.11a for photon energies close to the band gap at
various temperatures. The rapid increase, typical for direct semiconductors, is obvious. In particular at
low temperatures, however, the absorption lineshape close to the band gap is dominated by an excitonic
feature, discussed in Sect. 9.7.6.
Due to the increasing density of states, the absorption increases with the photon energy (Fig. 9.11c).
At 1.85 eV there is a step in the absorption spectrum of GaAs due to the beginning of the contribution
of transitions between the s-o hole band and the conduction band (see E 0 + 0 transition in Fig. 9.1b).