272
9 Optical Properties
Fig. 9.13 Absorption edge
of GaP (
√ α versus E) at
various temperatures. The
index ‘e’ (‘a’) indicates
phonon emission
(absorption) during the
optical absorption process.
The theoretical excitonic
gap (E gX ) at T = 77 K is
indicated. Adapted
from [845]
Energy (eV)
8
6
4
2
0
Absorption coefficient
(cm )
-1/2
2.15 2.18
2.27
2.21 2.24
2.30
2.36
2.33
2.39 2.42
GaP
296K
77K
1.6K
TA e
LA e
TO e
LA a
TA a
E gX
77K
216K
TA e
LA e
LA a
TA a
TO a
(LO+TA) a
Considering the temperature dependent population of the phonon density of states (Bose statistics,
(E.3)) the absorption coefficients for transitions with phonon emission (α e ) and phonon absorption
(α a ) are:
α e (E) ∝
(E − (E g + ω ph ))
2
1 − exp(− ph /kT )
(9.47a)
α a (E) ∝
(E − (E g − ph ))
2
exp( ph /kT ) − 1
.
(9.47b)
The two-particle process is less probable than the direct absorption that only involves one photon. The
strength of indirect absorption close to the band gap is about 10
−3 smaller than for the direct transition.
An 11-parameter formula based on terms like (9.47a) can describe the room temperature absorption
spectrum of silicon in the visible with a precision of a few percent [847].
The absorption spectra close to the absorption edge are shown for GaP (Fig. 9.13) and Si (Fig. 9.14a).
According to (9.47a), the plot of
√ α versus energy (Macfarlane–Roberts plot [848]) yields a straight
line beyond the spectral region of phonon effects. The complicated form close to the (indirect) gap
energy is due to the contribution of different phonons. The phonon energies found to contribute to the
silicon absorption edge [849] agree with the TA and TO energy at the X minimum [850] (Fig. 9.14b).
Also multiple phonons can contribute (Fig. 9.13). The momentum conservation can also be achieved
by impurity scattering or electron-electron scattering [851].
We note also that the indirect semiconductors have an optical transition between valence- and
conduction-band states. However, this transition is at higher energies than the fundamental band gap,
e.g. for Si (E g = 1.12 eV) at 3.4 eV (see Fig. 6.9a). In Fig. 9.15, the absorption scheme for indirect and
direct absorption processes starting with an electron at the top of the valence band is shown together
with an experimental absorption spectrum for Ge with the direct transition ( 8 → 7 ) at 0.89eV,
0.136 eV above the fundamental band gap.
In Fig. 9.16, the absorption edge of BaTiO 3 is shown. An indirect transition with an increase of
(weak) absorption ∝ E
2 and an indirect gap of E i = 2.66 eV and a direct transition with an increase
of (strong) absorption ∝ E
1/2 and a direct gap of E d = 3.05 eV are observed. These transitions could
be due to holes at the M (indirect gap) and (direct gap) points (cf. Sect. 6.3.11), respectively.
9 Optical Properties
Fig. 9.13 Absorption edge
of GaP (
√ α versus E) at
various temperatures. The
index ‘e’ (‘a’) indicates
phonon emission
(absorption) during the
optical absorption process.
The theoretical excitonic
gap (E gX ) at T = 77 K is
indicated. Adapted
from [845]
Energy (eV)
8
6
4
2
0
Absorption coefficient
(cm )
-1/2
2.15 2.18
2.27
2.21 2.24
2.30
2.36
2.33
2.39 2.42
GaP
296K
77K
1.6K
TA e
LA e
TO e
LA a
TA a
E gX
77K
216K
TA e
LA e
LA a
TA a
TO a
(LO+TA) a
Considering the temperature dependent population of the phonon density of states (Bose statistics,
(E.3)) the absorption coefficients for transitions with phonon emission (α e ) and phonon absorption
(α a ) are:
α e (E) ∝
(E − (E g + ω ph ))
2
1 − exp(− ph /kT )
(9.47a)
α a (E) ∝
(E − (E g − ph ))
2
exp( ph /kT ) − 1
.
(9.47b)
The two-particle process is less probable than the direct absorption that only involves one photon. The
strength of indirect absorption close to the band gap is about 10
−3 smaller than for the direct transition.
An 11-parameter formula based on terms like (9.47a) can describe the room temperature absorption
spectrum of silicon in the visible with a precision of a few percent [847].
The absorption spectra close to the absorption edge are shown for GaP (Fig. 9.13) and Si (Fig. 9.14a).
According to (9.47a), the plot of
√ α versus energy (Macfarlane–Roberts plot [848]) yields a straight
line beyond the spectral region of phonon effects. The complicated form close to the (indirect) gap
energy is due to the contribution of different phonons. The phonon energies found to contribute to the
silicon absorption edge [849] agree with the TA and TO energy at the X minimum [850] (Fig. 9.14b).
Also multiple phonons can contribute (Fig. 9.13). The momentum conservation can also be achieved
by impurity scattering or electron-electron scattering [851].
We note also that the indirect semiconductors have an optical transition between valence- and
conduction-band states. However, this transition is at higher energies than the fundamental band gap,
e.g. for Si (E g = 1.12 eV) at 3.4 eV (see Fig. 6.9a). In Fig. 9.15, the absorption scheme for indirect and
direct absorption processes starting with an electron at the top of the valence band is shown together
with an experimental absorption spectrum for Ge with the direct transition ( 8 → 7 ) at 0.89eV,
0.136 eV above the fundamental band gap.
In Fig. 9.16, the absorption edge of BaTiO 3 is shown. An indirect transition with an increase of
(weak) absorption ∝ E
2 and an indirect gap of E i = 2.66 eV and a direct transition with an increase
of (strong) absorption ∝ E
1/2 and a direct gap of E d = 3.05 eV are observed. These transitions could
be due to holes at the M (indirect gap) and (direct gap) points (cf. Sect. 6.3.11), respectively.