9.4 Absorption
263
Fig. 9.5 Schematic
absorption spectrum of a
typical semiconductor.
From [837]
closest band edge (donor to conduction and acceptor to valence band). A continuous background is
due to free-carrier absorption.
If absorption is considered, the reflectance (9.15) needs to be modified. Using the complex index
of refraction n
∗
= n r + iκ, it is given as
R =
n
∗
− 1
n ∗ + 1
2
=
(n r − 1)
2
+ κ
2
(n r + 1) 2 + κ 2 .
(9.17)
9.5 Dielectric Function due to Optical Phonons
In this section, the dielectric function around the resonance energy of optical phonons is developed.
Adjacent atoms oscillate with opposite phase in an optical phonon. If the bond has (partial) ionic
character, this leads to a time-dependent polarization and subsequently to a macroscopic electric
field. This additional field will influence the phonon frequencies obtained from a purely mechanical
approach. We consider in the following the case k ≈ 0. The phonon frequency for TO and LO vibrations
is given by
ω 0 =
2 C
M r
,
(9.18)
where M r is the reduced mass of the two different atoms (cf. Sect. 5.2.2). u is the relative displacement
u 1 − u 2 of the two atoms in a diatomic base. When the interaction with the electric field E (which will
be calculated self-consistently in the following) is considered, the Hamiltonian for the long-wavelength
limit is given by [838]:
ˆ
H (p, u) =
1
2
1
M r
p
2
+ b 11 u
2
+ 2b 12 u · E + b 22 E
2
.
(9.19)
263
Fig. 9.5 Schematic
absorption spectrum of a
typical semiconductor.
From [837]
closest band edge (donor to conduction and acceptor to valence band). A continuous background is
due to free-carrier absorption.
If absorption is considered, the reflectance (9.15) needs to be modified. Using the complex index
of refraction n
∗
= n r + iκ, it is given as
R =
n
∗
− 1
n ∗ + 1
2
=
(n r − 1)
2
+ κ
2
(n r + 1) 2 + κ 2 .
(9.17)
9.5 Dielectric Function due to Optical Phonons
In this section, the dielectric function around the resonance energy of optical phonons is developed.
Adjacent atoms oscillate with opposite phase in an optical phonon. If the bond has (partial) ionic
character, this leads to a time-dependent polarization and subsequently to a macroscopic electric
field. This additional field will influence the phonon frequencies obtained from a purely mechanical
approach. We consider in the following the case k ≈ 0. The phonon frequency for TO and LO vibrations
is given by
ω 0 =
2 C
M r
,
(9.18)
where M r is the reduced mass of the two different atoms (cf. Sect. 5.2.2). u is the relative displacement
u 1 − u 2 of the two atoms in a diatomic base. When the interaction with the electric field E (which will
be calculated self-consistently in the following) is considered, the Hamiltonian for the long-wavelength
limit is given by [838]:
ˆ
H (p, u) =
1
2
1
M r
p
2
+ b 11 u
2
+ 2b 12 u · E + b 22 E
2
.
(9.19)