9.9 Absorption in the Presence of Free Charge Carriers
297
(a)
E
k
X
E
(b)
10
2
10
10
10
10
1
0
-1
-2
-1
0.2 0.4 0.6 0.8 1.0 1.2 1.4
n-InP
Fig. 9.47 a Schematic of inter-valley conduction band transitions involving a photon (solid line arrow) and a phonon
(dashed line arrow). b Optical absorption coefficient of InP with n = 1.65 × 10 18 cm −3 . Experimental data (solid
line) and calculation of inter-valley band contribution (dashed line). The extrapolated free carrier contribution is shown
as dash-dotted line and the difference of experimental absorption and extrapolated free carrier contribution as circles.
Adapted from [927]
9.9.4 Inter-Valley Transitions
Electrons at the conduction band minimum can undergo optical transitions to the same band at a
different point of the Brillouin zone. Such intervalley transition, as sketched in Fig. 9.47a, is phononassisted to fulfill momentum conservation and occurs around the energy difference between the
two valleys (cmp. Table 8.4).
For InP with an electron concentration of n = 1.65 × 10
18 cm
−3 , below the fundamental band
edge at 1.4 eV, an additional contribution starting around 0.8–0.9 eV is found besides the free carrier
absorption (Fig. 9.47b) [927]. Taking into account the filling of the bottom of the conduction band,
an energy separation for the two valleys of = 0.90 ± 0.02 eV was found for various values
of the electron concentrations. This energy corresponds to the energy difference of conduction band
minima at and X in InP. The lineshape of this absorption processes can be modeled and fits well the
difference of measured absorption and extrapolated free-carrier absorption spectra. Transitions to the
lower minimum at L ( = 0.6 eV) are not observed, possibly masked by the free-carrier absorption.
9.9.5 Intra-Band Transitions
Phonon-assisted transitions within the lowest conduction band (not to a different valley), as indicated
schematically in Fig. 9.48a for the SnO 2 band structure [928], can cause absorption at photon energies
below the fundamental absorption edge. Actually in SnO 2 , the optical transition across the fundamental
band gap is only weakly dipole-allowed and leads to small absorption coefficient below 100 cm
−1
directly above the fundamental band gap of about 3.6 eV. The strong dipole-allowed transition with
absorption coefficient around 10
5 cm
−1 begins at about 4.3 eV and stems from electrons in a lower
valence band [929]. The free-carrier absorption due to transitions within the lowest conduction band
are calculated to dominate below 2.8 eV (Fig. 9.48b) and thus can also impact transparency in the
visible spectral range. The calculated slope is close to α ∝ λ
3 (cmp. (9.75)), expected from the linear
dispersion of the conduction band away from the -point [928]. A similar effect with the contribution
297
(a)
E
k
X
E
(b)
10
2
10
10
10
10
1
0
-1
-2
-1
0.2 0.4 0.6 0.8 1.0 1.2 1.4
n-InP
Fig. 9.47 a Schematic of inter-valley conduction band transitions involving a photon (solid line arrow) and a phonon
(dashed line arrow). b Optical absorption coefficient of InP with n = 1.65 × 10 18 cm −3 . Experimental data (solid
line) and calculation of inter-valley band contribution (dashed line). The extrapolated free carrier contribution is shown
as dash-dotted line and the difference of experimental absorption and extrapolated free carrier contribution as circles.
Adapted from [927]
9.9.4 Inter-Valley Transitions
Electrons at the conduction band minimum can undergo optical transitions to the same band at a
different point of the Brillouin zone. Such intervalley transition, as sketched in Fig. 9.47a, is phononassisted to fulfill momentum conservation and occurs around the energy difference between the
two valleys (cmp. Table 8.4).
For InP with an electron concentration of n = 1.65 × 10
18 cm
−3 , below the fundamental band
edge at 1.4 eV, an additional contribution starting around 0.8–0.9 eV is found besides the free carrier
absorption (Fig. 9.47b) [927]. Taking into account the filling of the bottom of the conduction band,
an energy separation for the two valleys of = 0.90 ± 0.02 eV was found for various values
of the electron concentrations. This energy corresponds to the energy difference of conduction band
minima at and X in InP. The lineshape of this absorption processes can be modeled and fits well the
difference of measured absorption and extrapolated free-carrier absorption spectra. Transitions to the
lower minimum at L ( = 0.6 eV) are not observed, possibly masked by the free-carrier absorption.
9.9.5 Intra-Band Transitions
Phonon-assisted transitions within the lowest conduction band (not to a different valley), as indicated
schematically in Fig. 9.48a for the SnO 2 band structure [928], can cause absorption at photon energies
below the fundamental absorption edge. Actually in SnO 2 , the optical transition across the fundamental
band gap is only weakly dipole-allowed and leads to small absorption coefficient below 100 cm
−1
directly above the fundamental band gap of about 3.6 eV. The strong dipole-allowed transition with
absorption coefficient around 10
5 cm
−1 begins at about 4.3 eV and stems from electrons in a lower
valence band [929]. The free-carrier absorption due to transitions within the lowest conduction band
are calculated to dominate below 2.8 eV (Fig. 9.48b) and thus can also impact transparency in the
visible spectral range. The calculated slope is close to α ∝ λ
3 (cmp. (9.75)), expected from the linear
dispersion of the conduction band away from the -point [928]. A similar effect with the contribution