438
15 External Fields
(a)
-10
2
-10
3
-10
4
Mobility (cm /Vs)
2
10
5
4
10
3
10
2
10
Conductivity (arb. units)
electrons
AlAs/GaAs
X
(b)
10
2
-10
2
-10
3
-10
4
-10
5
10
3
10
4
Mobility (cm /Vs)
2
10
5
4
10
3
10
2
10
Conductivity (arb. units)
InAs/GaSb
electrons
holes
3D
2D
Fig. 15.10 a Mobility spectrum of GaAs/(Al,Ga)As/InAs double quantum well structure. Adapted from [1441]. b
Mobility spectrum of InAs/GaSb quantum well structure. Adapted from [1442]
15.2.2 Free-Carrier Absorption
The absorption of free carriers was treated in Sect. 9.9.1 without the presence of a static magnetic field.
Solving (15.8) for a static magnetic field B = μ 0 H with H = H (h x , h y , h z ) and a harmonic electric
field E ∝ exp(−iωt) yields for the dielectric tensor (cf. (9.72))
=
i
0 ω
σ ,
(15.25)
and by comparing to j = σ E = q n v,
(ω) = −ω
∗2
p
⎡
⎣ (ω
2
+ i ω γ ) 1 − i ω
2
c
⎛
⎝
0 −h z h y
h z 0 −h x
−h y h x 0
⎞
⎠
⎤
⎦
−1
,
(15.26)
where 1 denotes the (3 by 3) unity matrix and γ = 1/τ = q/(m
∗
μ) is the damping parameter with μ
representing the optical carrier mobility (in the non-isotropic case a tensor γ needs to be used). The
(unscreened) plasma frequency is given by (cmp. (9.77))
ω
∗
p =
n
e 2
0 m ∗ .
(15.27)
The free-carrier cyclotron frequency is
ω c = e
μ 0 H
m ∗ .
(15.28)
If the effective mass is treated as a tensor, 1/m
∗ is replaced by m
∗−1 in (15.27) and (15.28). For zero
magnetic field the classical Drude theory for one carrier species is recovered (cf. (9.74a))
(ω) = −
ω
∗2
p
ω (ω + iγ )
.
(15.29)
15 External Fields
(a)
-10
2
-10
3
-10
4
Mobility (cm /Vs)
2
10
5
4
10
3
10
2
10
Conductivity (arb. units)
electrons
AlAs/GaAs
X
(b)
10
2
-10
2
-10
3
-10
4
-10
5
10
3
10
4
Mobility (cm /Vs)
2
10
5
4
10
3
10
2
10
Conductivity (arb. units)
InAs/GaSb
electrons
holes
3D
2D
Fig. 15.10 a Mobility spectrum of GaAs/(Al,Ga)As/InAs double quantum well structure. Adapted from [1441]. b
Mobility spectrum of InAs/GaSb quantum well structure. Adapted from [1442]
15.2.2 Free-Carrier Absorption
The absorption of free carriers was treated in Sect. 9.9.1 without the presence of a static magnetic field.
Solving (15.8) for a static magnetic field B = μ 0 H with H = H (h x , h y , h z ) and a harmonic electric
field E ∝ exp(−iωt) yields for the dielectric tensor (cf. (9.72))
=
i
0 ω
σ ,
(15.25)
and by comparing to j = σ E = q n v,
(ω) = −ω
∗2
p
⎡
⎣ (ω
2
+ i ω γ ) 1 − i ω
2
c
⎛
⎝
0 −h z h y
h z 0 −h x
−h y h x 0
⎞
⎠
⎤
⎦
−1
,
(15.26)
where 1 denotes the (3 by 3) unity matrix and γ = 1/τ = q/(m
∗
μ) is the damping parameter with μ
representing the optical carrier mobility (in the non-isotropic case a tensor γ needs to be used). The
(unscreened) plasma frequency is given by (cmp. (9.77))
ω
∗
p =
n
e 2
0 m ∗ .
(15.27)
The free-carrier cyclotron frequency is
ω c = e
μ 0 H
m ∗ .
(15.28)
If the effective mass is treated as a tensor, 1/m
∗ is replaced by m
∗−1 in (15.27) and (15.28). For zero
magnetic field the classical Drude theory for one carrier species is recovered (cf. (9.74a))
(ω) = −
ω
∗2
p
ω (ω + iγ )
.
(15.29)