276
9 Optical Properties
Fig. 9.19 a Exciton
binding energy versus band
gap for various
semiconductors. b
Schematic dispersion of
excitonic levels. The
K -vector refers to the
center-of-mass motion
(a)
(b)
2.14
2.035
2.15
2.16
2.17
4
(b)
Cu O
2
2.030
1PA (a.u.)
(a)
green
Fig. 9.20 One-photon (top) and two-photon (bottom) absorption spectra of Cu 2 O at T = 4.2 K. Arrows denote theoretical
peak positions. Adapted from [864]
the crystal. The complete dispersion is (see Fig. 9.19b)
E = E g + E
n
X +
2
2M
K
2
.
(9.52)
The oscillator strength of the exciton states decays ∝ n
−3 . The absorption due to excitons is visible
in Fig. 9.11a for GaAs at low temperatures. If inhomogeneities are present, typically only the n = 1
transition is seen. However, under special conditions also higher transitions of the exciton Rydberg
series are seen (e.g. n = 2 and 3 in Fig. 9.11b).
The exciton concept was introduced first for absorption in Cu 2 O [862]. The J = 1/2 absorption spectrum (‘yellow series’) is shown in Fig. 9.20. In this particular material both the valence and
conduction bands have s character, thus the 1s transition of the exciton is forbidden and the np transitions are observed in normal (one-photon) absorption. With two-photon absorption also the s (and d)
transitions can be excited. On a piece of natural Cu 2 O, the Rydberg series has been measured up to
n = 25 [863] (Fig. 9.21a). The peak energy and the oscillator strength follow the n
−2 (E
b
X = 92 meV,
E g = 2.17208 eV) and n
−3 laws, respectively, expected from a hydrogen model (Fig. 9.21b). The
deviation from the n
−3 -dependence for the oscillator strength at large n is due to interaction effects of
excitons with large radius at finite exciton density.
9 Optical Properties
Fig. 9.19 a Exciton
binding energy versus band
gap for various
semiconductors. b
Schematic dispersion of
excitonic levels. The
K -vector refers to the
center-of-mass motion
(a)
(b)
2.14
2.035
2.15
2.16
2.17
4
(b)
Cu O
2
2.030
1PA (a.u.)
(a)
green
Fig. 9.20 One-photon (top) and two-photon (bottom) absorption spectra of Cu 2 O at T = 4.2 K. Arrows denote theoretical
peak positions. Adapted from [864]
the crystal. The complete dispersion is (see Fig. 9.19b)
E = E g + E
n
X +
2
2M
K
2
.
(9.52)
The oscillator strength of the exciton states decays ∝ n
−3 . The absorption due to excitons is visible
in Fig. 9.11a for GaAs at low temperatures. If inhomogeneities are present, typically only the n = 1
transition is seen. However, under special conditions also higher transitions of the exciton Rydberg
series are seen (e.g. n = 2 and 3 in Fig. 9.11b).
The exciton concept was introduced first for absorption in Cu 2 O [862]. The J = 1/2 absorption spectrum (‘yellow series’) is shown in Fig. 9.20. In this particular material both the valence and
conduction bands have s character, thus the 1s transition of the exciton is forbidden and the np transitions are observed in normal (one-photon) absorption. With two-photon absorption also the s (and d)
transitions can be excited. On a piece of natural Cu 2 O, the Rydberg series has been measured up to
n = 25 [863] (Fig. 9.21a). The peak energy and the oscillator strength follow the n
−2 (E
b
X = 92 meV,
E g = 2.17208 eV) and n
−3 laws, respectively, expected from a hydrogen model (Fig. 9.21b). The
deviation from the n
−3 -dependence for the oscillator strength at large n is due to interaction effects of
excitons with large radius at finite exciton density.