9.2 Complex Dielectric Function
259
lattice vibrations (Sect. 9.5) and transitions within the electronic band structure (Sect. 9.6). In some
cases also its k-dependence is important, known as ’spatial dispersion’ (cmp. Sect. 9.7.8).
An optic axis in the transparency regime (all tensor elements of ∈ R) is the direction in which
the speed of light or the index of refraction is independent of polarization. Uniaxial (biaxial) materials
have one (two) of such axes. The anisotropy of the index of refraction and its polarization dependence
must be taken into account when light propagation is considered in birefringent semiconductors, e.g.
for Raman spectroscopy [827], unless the propagation is along an optic axis.
The dielectric function is generally complex and written as (scalar)
=
+ i
= 1 + i 2 .
(9.3)
The real (
or 1 ) and imaginary (
or 2 ) part of the dielectric function are related to each other via
the Kramers-Kronig relations (Appendix C).
The complex index of refraction n
∗ is
n
∗
=
√
= n r + i κ .
(9.4)
From n
∗2
= follows
= n
2
r − κ
2
(9.5)
= 2 n r κ .
(9.6)
From ¯
= (n
2
r + κ
2
)
2 and (9.5) follows
n
2
r =
+
√
2 + 2
2
(9.7)
κ =
2 n r
.
(9.8)
The real part of the complex index of refraction n r is responsible for the dispersion, the imaginary part
κ is named extinction coefficient and is related to the absorption coefficient for a plane wave (damping
of the intensity ∝ E
2 ) by
α = 2
ω
c
κ =
4π
λ
κ = 2 k κ .
(9.9)
Here, k and λ denote the respective values in vacuum. Through the Kramers-Kronig relations
(Appendix C), birefringence, i.e. the orientational dependence of the index of refraction, is thus automatically related to dichroism, i.e. the orientational dependence of the absorption coefficient.
As an example, in Fig. 9.1 the dielectric function of GaAs is shown in the vicinity of the band edge
and above. Since GaAs is cubic, the dielectric function at each photon energy can represented by a single
complex number. The tensor character of the dielectric function is demonstrated in Fig. 9.2a where
the four independent tensor elements for (monoclinic) β-Ga 2 O 3 are depicted [828]. The contributions
of various dipole oscillators (strength and orientation) to the dielectric function can be analyzed from
these data [829].
In the absorption regime, for biaxial crystals the two optic axes split into four singular optic axes
[830] as visualized for β-Ga 2 O 3 in Fig. 9.2b [831]. It should be noted that optical activity [832] is not
considered in the following.
259
lattice vibrations (Sect. 9.5) and transitions within the electronic band structure (Sect. 9.6). In some
cases also its k-dependence is important, known as ’spatial dispersion’ (cmp. Sect. 9.7.8).
An optic axis in the transparency regime (all tensor elements of ∈ R) is the direction in which
the speed of light or the index of refraction is independent of polarization. Uniaxial (biaxial) materials
have one (two) of such axes. The anisotropy of the index of refraction and its polarization dependence
must be taken into account when light propagation is considered in birefringent semiconductors, e.g.
for Raman spectroscopy [827], unless the propagation is along an optic axis.
The dielectric function is generally complex and written as (scalar)
=
+ i
= 1 + i 2 .
(9.3)
The real (
or 1 ) and imaginary (
or 2 ) part of the dielectric function are related to each other via
the Kramers-Kronig relations (Appendix C).
The complex index of refraction n
∗ is
n
∗
=
√
= n r + i κ .
(9.4)
From n
∗2
= follows
= n
2
r − κ
2
(9.5)
= 2 n r κ .
(9.6)
From ¯
= (n
2
r + κ
2
)
2 and (9.5) follows
n
2
r =
+
√
2 + 2
2
(9.7)
κ =
2 n r
.
(9.8)
The real part of the complex index of refraction n r is responsible for the dispersion, the imaginary part
κ is named extinction coefficient and is related to the absorption coefficient for a plane wave (damping
of the intensity ∝ E
2 ) by
α = 2
ω
c
κ =
4π
λ
κ = 2 k κ .
(9.9)
Here, k and λ denote the respective values in vacuum. Through the Kramers-Kronig relations
(Appendix C), birefringence, i.e. the orientational dependence of the index of refraction, is thus automatically related to dichroism, i.e. the orientational dependence of the absorption coefficient.
As an example, in Fig. 9.1 the dielectric function of GaAs is shown in the vicinity of the band edge
and above. Since GaAs is cubic, the dielectric function at each photon energy can represented by a single
complex number. The tensor character of the dielectric function is demonstrated in Fig. 9.2a where
the four independent tensor elements for (monoclinic) β-Ga 2 O 3 are depicted [828]. The contributions
of various dipole oscillators (strength and orientation) to the dielectric function can be analyzed from
these data [829].
In the absorption regime, for biaxial crystals the two optic axes split into four singular optic axes
[830] as visualized for β-Ga 2 O 3 in Fig. 9.2b [831]. It should be noted that optical activity [832] is not
considered in the following.