9.7 Band–Band Transitions
273
(a)
(b)
Fig. 9.14 a Absorption edge of Si at two different temperatures. Adapted from [849]. b Phonon energies in silicon along
[001] obtained from neutron scattering (black: unidentified, green: TA, purple: LA, blue: LO, red: TO). The vertical
grey bar indicates the position of the conduction band minimum, the horizontal grey bars the energies of the phonons
observed at the indirect optical absorption edge. The dark grey overlap areas indicate that TO and TA phonons contribute.
Adapted from [850]
(a)
E
k
8
L
L 6
7
(b)
6
7
.
0
8
7
.
0
0
9
.
0
6
8
.
0
8
8
.
0
4
8
.
0
2
8
.
0
0
8
.
0
10
4
10
10
10
10
10
10
3
2
1
0
-1
-2
-1
Ge
Fig. 9.15 a Scheme of indirect and direct optical transitions starting at the top of the valence band in Ge. Vertical solid
lines represent the involved photon, the horizontal dashed line the involved phonon. b Experimental absorption spectrum
of Ge (T = 20 K). Adapted from [849]
9.7.4 Urbach Tail
Instead of the ideal (E − E g )
1/2 dependence of the direct band-edge absorption, often an exponential
tail is observed (see Fig. 9.17). This tail is called the Urbach tail [853] and follows the functional
dependence (for E < E g )
α(E) ∝ exp
E − E g
E 0
,
(9.48)
where E 0 is the characteristic width of the absorption edge, the so-called Urbach parameter.
The Urbach tail is attributed to transitions between band tails below the band edges. Such tails
can originate from disorder of the perfect crystal, e.g. from defects or doping, and the fluctuation of
electronic energy bands due to lattice vibrations. The temperature dependence of the Urbach parameter
E 0 is thus related to that of the band gap as discussed in [854, 855].
273
(a)
(b)
Fig. 9.14 a Absorption edge of Si at two different temperatures. Adapted from [849]. b Phonon energies in silicon along
[001] obtained from neutron scattering (black: unidentified, green: TA, purple: LA, blue: LO, red: TO). The vertical
grey bar indicates the position of the conduction band minimum, the horizontal grey bars the energies of the phonons
observed at the indirect optical absorption edge. The dark grey overlap areas indicate that TO and TA phonons contribute.
Adapted from [850]
(a)
E
k
8
L
L 6
7
(b)
6
7
.
0
8
7
.
0
0
9
.
0
6
8
.
0
8
8
.
0
4
8
.
0
2
8
.
0
0
8
.
0
10
4
10
10
10
10
10
10
3
2
1
0
-1
-2
-1
Ge
Fig. 9.15 a Scheme of indirect and direct optical transitions starting at the top of the valence band in Ge. Vertical solid
lines represent the involved photon, the horizontal dashed line the involved phonon. b Experimental absorption spectrum
of Ge (T = 20 K). Adapted from [849]
9.7.4 Urbach Tail
Instead of the ideal (E − E g )
1/2 dependence of the direct band-edge absorption, often an exponential
tail is observed (see Fig. 9.17). This tail is called the Urbach tail [853] and follows the functional
dependence (for E < E g )
α(E) ∝ exp
E − E g
E 0
,
(9.48)
where E 0 is the characteristic width of the absorption edge, the so-called Urbach parameter.
The Urbach tail is attributed to transitions between band tails below the band edges. Such tails
can originate from disorder of the perfect crystal, e.g. from defects or doping, and the fluctuation of
electronic energy bands due to lattice vibrations. The temperature dependence of the Urbach parameter
E 0 is thus related to that of the band gap as discussed in [854, 855].