430
15 External Fields
(a)
E g
h
E C
E V (b)
e2
e1
h1
h3
h2
h4
h5
e2
e1
h1
h3
h2
h4
h5
h
h
E 0
E=0
Fig. 15.1 Impact of an electric field on a bulk material (tilt of bands) and b a quantum well (quantum confined Stark
effect, QCSE)
(a) -10 -8 -6 -4 -2 0 2 4
-0.4
-0.2
0
0.2
0.4
(b) -10 -8 -6 -4 -2 0 2 4
-1
-0.5
0
0.5
Fig. 15.2 a Airy function Ai(x), b Ai’(x)
with ξ =
E z
− z
2μ
2 e E
1/3 and the optoelectronic energy =
e
2 E
2
2
2μ
1/3
. The solution of (15.3) is
given by the Airy function Ai (cf. Fig. 15.2):
φ E z (ξ ) =
√
e E
Ai(ξ ) .
(15.4)
The pre-factor guaranties the orthonormality (with regard to the E z ). The absorption spectrum is then
given by
α(ω, E) ∝
1
ω
√
π
Ai
2
(η) − η Ai
2
(η)
,
(15.5)
with η = (E g − E)// and Ai
(x) = d Ai(x)/dx.
Optical transitions below the band gap become possible that are photon-assisted tunneling processes.
The below-bandgap transitions have the form of an exponential tail. Additionally, oscillations develop
above the band gap, the so-called Franz-Keldysh oscillations (FKO) [1417, 1418] (Fig. 15.3a).
The absorption spectrum scales with the optoelectronic energy . The energy position of the FKO
peaks E n is periodic with (ν ∼ 0.5)
E n − E g
3/2 ∝ (n − ν) E
√
μ .
(15.6)
15 External Fields
(a)
E g
h
E C
E V (b)
e2
e1
h1
h3
h2
h4
h5
e2
e1
h1
h3
h2
h4
h5
h
h
E 0
E=0
Fig. 15.1 Impact of an electric field on a bulk material (tilt of bands) and b a quantum well (quantum confined Stark
effect, QCSE)
(a) -10 -8 -6 -4 -2 0 2 4
-0.4
-0.2
0
0.2
0.4
(b) -10 -8 -6 -4 -2 0 2 4
-1
-0.5
0
0.5
Fig. 15.2 a Airy function Ai(x), b Ai’(x)
with ξ =
E z
− z
2μ
2 e E
1/3 and the optoelectronic energy =
e
2 E
2
2
2μ
1/3
. The solution of (15.3) is
given by the Airy function Ai (cf. Fig. 15.2):
φ E z (ξ ) =
√
e E
Ai(ξ ) .
(15.4)
The pre-factor guaranties the orthonormality (with regard to the E z ). The absorption spectrum is then
given by
α(ω, E) ∝
1
ω
√
π
Ai
2
(η) − η Ai
2
(η)
,
(15.5)
with η = (E g − E)// and Ai
(x) = d Ai(x)/dx.
Optical transitions below the band gap become possible that are photon-assisted tunneling processes.
The below-bandgap transitions have the form of an exponential tail. Additionally, oscillations develop
above the band gap, the so-called Franz-Keldysh oscillations (FKO) [1417, 1418] (Fig. 15.3a).
The absorption spectrum scales with the optoelectronic energy . The energy position of the FKO
peaks E n is periodic with (ν ∼ 0.5)
E n − E g
3/2 ∝ (n − ν) E
√
μ .
(15.6)