9.9 Absorption in the Presence of Free Charge Carriers
291
9.9 Absorption in the Presence of Free Charge Carriers
In the presence of charge carriers, various absorption processes can occur. First, the dissipative motion
of carriers leads to infrared absorption, termed the free carrier absorption (Sect. 9.9.1). Filling of a band
with carriers leads to a shift of the band-band absorption edge, the Burstein-Moss shift (Sect. 9.9.2).
Besides the free-carrier absorption, free carriers present in the semiconductor can lead to further
absorption processes with transition energies below the band gap. These processes are due to transitions
within the band structure and can be
• inter-valence band transitions of holes (Sect. 9.9.3),
• phonon-assisted inter-valley transitions of electrons (Sect. 9.9.4),
• phonon-assisted intra-band transitions of electrons (Sect. 9.9.5).
9.9.1 Absorption Coefficient, Plasma Frequency
The absorption due to free carriers in the infrared spectral range (away from phonon resonances) can
be described with the Drude model [911].
A time-dependent electric field accelerates the charge carriers within a band. The excess energy
is subsequently transferred to the lattice via scattering with phonons. A review of the effect of free
carriers on optical properties can be found in [912]. In the relaxation-time approximation energy is
relaxed with a time constant τ . Thus energy is absorbed from the electromagnetic wave and dissipated.
Effectively, this process represents an intra-band excitation.
The complex conductivity (8.37) is given by
σ
∗
= σ r + iσ i =
n e
2
τ
m ∗
1
1 + ω 2 τ 2 + i
ωτ
1 + ω 2 τ 2
.
(9.69)
We note that a static magnetic field introduces birefringence as discussed in more detail in Sect. 15.2.2.
The wave equation for the electric field is
∇
2 E = r 0 μ 0 ¨
E + σ
∗
μ 0 ˙
E .
(9.70)
For a plane wave ∝ exp[i(kr − ωt)] the wavevector obeys
k =
ω
c
r + i
σ ∗
0 ω
,
(9.71)
where c = ( 0 μ 0 )
−1/2 is the velocity of light in vacuum, r is the background dielectric constant (for
large ω).
The part FC of the dielectric function due to free carriers is
FC =
i
0 ω
σ
∗
.
(9.72)
The complex index of refraction is
n
∗
= n r + i κ =
r + i
σ ∗
0 ω
.
(9.73)
291
9.9 Absorption in the Presence of Free Charge Carriers
In the presence of charge carriers, various absorption processes can occur. First, the dissipative motion
of carriers leads to infrared absorption, termed the free carrier absorption (Sect. 9.9.1). Filling of a band
with carriers leads to a shift of the band-band absorption edge, the Burstein-Moss shift (Sect. 9.9.2).
Besides the free-carrier absorption, free carriers present in the semiconductor can lead to further
absorption processes with transition energies below the band gap. These processes are due to transitions
within the band structure and can be
• inter-valence band transitions of holes (Sect. 9.9.3),
• phonon-assisted inter-valley transitions of electrons (Sect. 9.9.4),
• phonon-assisted intra-band transitions of electrons (Sect. 9.9.5).
9.9.1 Absorption Coefficient, Plasma Frequency
The absorption due to free carriers in the infrared spectral range (away from phonon resonances) can
be described with the Drude model [911].
A time-dependent electric field accelerates the charge carriers within a band. The excess energy
is subsequently transferred to the lattice via scattering with phonons. A review of the effect of free
carriers on optical properties can be found in [912]. In the relaxation-time approximation energy is
relaxed with a time constant τ . Thus energy is absorbed from the electromagnetic wave and dissipated.
Effectively, this process represents an intra-band excitation.
The complex conductivity (8.37) is given by
σ
∗
= σ r + iσ i =
n e
2
τ
m ∗
1
1 + ω 2 τ 2 + i
ωτ
1 + ω 2 τ 2
.
(9.69)
We note that a static magnetic field introduces birefringence as discussed in more detail in Sect. 15.2.2.
The wave equation for the electric field is
∇
2 E = r 0 μ 0 ¨
E + σ
∗
μ 0 ˙
E .
(9.70)
For a plane wave ∝ exp[i(kr − ωt)] the wavevector obeys
k =
ω
c
r + i
σ ∗
0 ω
,
(9.71)
where c = ( 0 μ 0 )
−1/2 is the velocity of light in vacuum, r is the background dielectric constant (for
large ω).
The part FC of the dielectric function due to free carriers is
FC =
i
0 ω
σ
∗
.
(9.72)
The complex index of refraction is
n
∗
= n r + i κ =
r + i
σ ∗
0 ω
.
(9.73)