292
9 Optical Properties
Taking the square of this equation yields
n
2
r − κ
2
= r + i
σ i
0 ω
= r −
n e
2
0 m ∗
τ
2
1 + ω 2 τ 2
(9.74a)
2 n r κ =
σ r
0 ω
=
n e
2
0 ω m ∗
τ
1 + ω 2 τ 2 .
(9.74b)
The absorption coefficient is related to κ by (9.9). For the case of higher frequencies, i.e. ωτ 1, the
absorption is
α =
n e
2
0 c n r m ∗ τ
1
ω 2 ∝ λ
2
.
(9.75)
The absorption decreases with increasing frequency like ω
−r . The classical Drude treatment as followed
here results in an exponent of r = 2. This is the case for neutral impurity scattering and also for
small frequencies ω E F . A more detailed discussion of the energy dependence of free-carrier
absorption can be found in [913]. Other exponents have been derived for scattering by acoustical
phonons (r = 3/2), LO phonons (r = 5/2) and ionized impurities (r = 7/2). More detailed quantum
mechanical treatments of free-carrier absorption in the presence of impurities and phonons can be
found in [914–916].
For semiconductors free-carrier absorption is particularly important in the mid- and far-infrared
regions when carriers are present due to doping or thermal excitation. In Fig. 9.39a absorption spectra
of n-type Ge for various doping concentrations are shown. The absorption coefficient in the transparency
regime varies proportionally to λ
2 as predicted in (9.75). In Fig. 9.39a, the absorption can be seen to
rise for photon energy above 0.7 eV due to absorption in the band structure. Electrons are excited from
the valence band across the fundamental band gap into the conduction band (cmp. Sect. 9.7.3), which
is an indirect transition in Ge.
In Fig. 9.39b the absorption coefficient due to free carrier absorption at fixed wavelength is shown
as a function of dopant concentration.
11 The slope is slightly overlinear, indicating a weak dependence
τ (n). A sub-linear relation has been found for heavily p-doped GaAs [917].
The index of refraction is given by (also for ωτ 1)
n
2
r = r −
ne
2
0 m ∗ ω 2 + κ
2
= r
1 −
ω p
ω
2
+
2
r
4n 2
r
ω p
ω
4 1
ω 2 τ 2
(9.76)
≈ r
1 −
ω p
ω
2
,
where
ω p =
n e 2
r 0 m ∗
(9.77)
is the plasma frequency. The approximation is valid for small absorption and when (ωτ )
−2 can be
neglected. A graphical representation is given in Fig. 9.40a. For coupling to electromagnetic waves
(still ωτ 1)
(ω) = r
1 −
ω p
ω
2
=
c
2 k
2
ω 2
(9.78)
must be fulfilled. It follows that the dispersion relation in the presence of free carriers (Fig. 9.40b) is
11 Even at low temperature, n ≈ N D since N D N c (cf. [594] and Sect. 7.5.7).
9 Optical Properties
Taking the square of this equation yields
n
2
r − κ
2
= r + i
σ i
0 ω
= r −
n e
2
0 m ∗
τ
2
1 + ω 2 τ 2
(9.74a)
2 n r κ =
σ r
0 ω
=
n e
2
0 ω m ∗
τ
1 + ω 2 τ 2 .
(9.74b)
The absorption coefficient is related to κ by (9.9). For the case of higher frequencies, i.e. ωτ 1, the
absorption is
α =
n e
2
0 c n r m ∗ τ
1
ω 2 ∝ λ
2
.
(9.75)
The absorption decreases with increasing frequency like ω
−r . The classical Drude treatment as followed
here results in an exponent of r = 2. This is the case for neutral impurity scattering and also for
small frequencies ω E F . A more detailed discussion of the energy dependence of free-carrier
absorption can be found in [913]. Other exponents have been derived for scattering by acoustical
phonons (r = 3/2), LO phonons (r = 5/2) and ionized impurities (r = 7/2). More detailed quantum
mechanical treatments of free-carrier absorption in the presence of impurities and phonons can be
found in [914–916].
For semiconductors free-carrier absorption is particularly important in the mid- and far-infrared
regions when carriers are present due to doping or thermal excitation. In Fig. 9.39a absorption spectra
of n-type Ge for various doping concentrations are shown. The absorption coefficient in the transparency
regime varies proportionally to λ
2 as predicted in (9.75). In Fig. 9.39a, the absorption can be seen to
rise for photon energy above 0.7 eV due to absorption in the band structure. Electrons are excited from
the valence band across the fundamental band gap into the conduction band (cmp. Sect. 9.7.3), which
is an indirect transition in Ge.
In Fig. 9.39b the absorption coefficient due to free carrier absorption at fixed wavelength is shown
as a function of dopant concentration.
11 The slope is slightly overlinear, indicating a weak dependence
τ (n). A sub-linear relation has been found for heavily p-doped GaAs [917].
The index of refraction is given by (also for ωτ 1)
n
2
r = r −
ne
2
0 m ∗ ω 2 + κ
2
= r
1 −
ω p
ω
2
+
2
r
4n 2
r
ω p
ω
4 1
ω 2 τ 2
(9.76)
≈ r
1 −
ω p
ω
2
,
where
ω p =
n e 2
r 0 m ∗
(9.77)
is the plasma frequency. The approximation is valid for small absorption and when (ωτ )
−2 can be
neglected. A graphical representation is given in Fig. 9.40a. For coupling to electromagnetic waves
(still ωτ 1)
(ω) = r
1 −
ω p
ω
2
=
c
2 k
2
ω 2
(9.78)
must be fulfilled. It follows that the dispersion relation in the presence of free carriers (Fig. 9.40b) is
11 Even at low temperature, n ≈ N D since N D N c (cf. [594] and Sect. 7.5.7).