9.7 Band–Band Transitions
271
Fig. 9.12 Optical selection
rules for band–band
transitions in bulk material
for a single photon
transitions and b
two-photon transitions
(with photon energy equal
to half the transition
energy)
1/2
hh
m j
1/2
3/2
1/2
3/2
1/2
electrons
h
h
h
l
l h
holes
m j
1/2
hh
1/2
1/2
3/2
3/2
1/2
h
h
h
l
l h
)
b
(
)
a
(
When bands run in parallel, i.e. with the same separation, in the E(k) diagram, the absorption processes
accumulate at the same transition energy. In this way peaks at higher energy in the complex part of the
dielectric function and in the absorption spectrum due to the E 1 or E
0 transitions originate as shown
in Fig. 9.1.
The selection rules for transitions from valence to conduction band must take into account the angular
momentum and spin states of the wavefunctions. The optical transitions for circularly polarized light
are shown in Fig. 9.12a, fulfilling the selection rule j = ±1. A lifting of the energetic degeneracies
of these states occurs, e.g. by magnetic fields (cmp. Fig.15.12) or spatial confinement (cmp. Fig. 12.30).
For two-photon absorption (Chap.9.7.14), the selection rule is j = ±2 as shown in Fig. 9.12b [844].
We note that in some materials the direct transition between certain bands is forbidden. An example
is SnO 2 where the direct transition from the topmost valence band into the lowest conduction band (at
) is forbidden (cmp. Fig. 9.48). If the matrix element increases linearly with E − E g , the absorption
coefficient varies like
α(E) ∝ (E − E g )
3/2
.
(9.46)
9.7.3 Indirect Transitions
In an indirect band structure the missing k difference (across the Brillouin zone) between valence- and
conduction-band state needs to be provided by a second quantum. A phonon can provide the necessary
momentum and additionally contributes a small amount of energy ph . There are several steps in the
absorption spectrum due to various involved phonons (or combinations of them). At low temperature
(T = 1.6 K, Fig. 9.13) phonons can only be generated and the absorption starts at energies above the
band gap. At higher temperatures (typically above 40 K [845], Fig. 9.13), acoustical phonons assisting
the optical absorption transition can also be absorbed from the crystal; in this case due to energy
conservation the absorption starts already at an energy E g − ph below the band gap. At even higher
temperatures (> 200 K, Fig. 9.13), also optical phonons can be absorbed.
The perturbation calculation yields an absorption coefficient with a quadratic dependence on energy
(9.47a) [846]. Essentially, for the absorption into a specific (empty) conduction band state (with squareroot density of states) various initial (filled) valence band states (also with square-root density of states)
are possible, making the probability depend on the product of the DOS and thus on the energy to the first
power. Integrating over all energy states with energy separation E ± ph , yields an E
2 -dependence.
5
5 A flat optical phonon dispersion is assumed.
271
Fig. 9.12 Optical selection
rules for band–band
transitions in bulk material
for a single photon
transitions and b
two-photon transitions
(with photon energy equal
to half the transition
energy)
1/2
hh
m j
1/2
3/2
1/2
3/2
1/2
electrons
h
h
h
l
l h
holes
m j
1/2
hh
1/2
1/2
3/2
3/2
1/2
h
h
h
l
l h
)
b
(
)
a
(
When bands run in parallel, i.e. with the same separation, in the E(k) diagram, the absorption processes
accumulate at the same transition energy. In this way peaks at higher energy in the complex part of the
dielectric function and in the absorption spectrum due to the E 1 or E
0 transitions originate as shown
in Fig. 9.1.
The selection rules for transitions from valence to conduction band must take into account the angular
momentum and spin states of the wavefunctions. The optical transitions for circularly polarized light
are shown in Fig. 9.12a, fulfilling the selection rule j = ±1. A lifting of the energetic degeneracies
of these states occurs, e.g. by magnetic fields (cmp. Fig.15.12) or spatial confinement (cmp. Fig. 12.30).
For two-photon absorption (Chap.9.7.14), the selection rule is j = ±2 as shown in Fig. 9.12b [844].
We note that in some materials the direct transition between certain bands is forbidden. An example
is SnO 2 where the direct transition from the topmost valence band into the lowest conduction band (at
) is forbidden (cmp. Fig. 9.48). If the matrix element increases linearly with E − E g , the absorption
coefficient varies like
α(E) ∝ (E − E g )
3/2
.
(9.46)
9.7.3 Indirect Transitions
In an indirect band structure the missing k difference (across the Brillouin zone) between valence- and
conduction-band state needs to be provided by a second quantum. A phonon can provide the necessary
momentum and additionally contributes a small amount of energy ph . There are several steps in the
absorption spectrum due to various involved phonons (or combinations of them). At low temperature
(T = 1.6 K, Fig. 9.13) phonons can only be generated and the absorption starts at energies above the
band gap. At higher temperatures (typically above 40 K [845], Fig. 9.13), acoustical phonons assisting
the optical absorption transition can also be absorbed from the crystal; in this case due to energy
conservation the absorption starts already at an energy E g − ph below the band gap. At even higher
temperatures (> 200 K, Fig. 9.13), also optical phonons can be absorbed.
The perturbation calculation yields an absorption coefficient with a quadratic dependence on energy
(9.47a) [846]. Essentially, for the absorption into a specific (empty) conduction band state (with squareroot density of states) various initial (filled) valence band states (also with square-root density of states)
are possible, making the probability depend on the product of the DOS and thus on the energy to the first
power. Integrating over all energy states with energy separation E ± ph , yields an E
2 -dependence.
5
5 A flat optical phonon dispersion is assumed.