284
9 Optical Properties
(a)
-5
0
-10
0
1
2
3
E
xc (Ry)
r s
Ge
Si
(b)
CdS
CdSe
CdTe
ZnO
1
2
0
4
5
3
0
-1
-2
-3
-4
-5
r s
Fig. 9.29 a Theoretical exchange and correlation energies in units of the exciton Rydberg energy as a function of the
dimensionless variable r s for Ge, Si and a model system (with one isotropic conduction and valence band each). The
solid line is a fit according to (9.64). Adapted from [886]. b Band gap renormalization in terms of the excitonic Rydberg
for various II–VI semiconductors. Solid line is the relation according to (9.64), dashed line is the dependence predicted
in [887] for T = 30 K. Data are compiled in [888]
9.7.12 Band Gap Renormalization
The band structure theory has been developed so far for small carrier densities. If the carrier density
is large the interaction of free carriers has to be considered. The first step was exciton formation.
However, at high temperatures (ionization) and at large carrier density (screening) the exciton is not
stable. Exchange and correlation energy leads to a decrease of the optical absorption edge that is called
band gap renormalization (BGR).
An effect due to significant carrier density is to be expected when the density is of the order of the
exciton volume, i.e. n ∼ a
−3
B . For a B ∼ 15 nm (GaAs) this means n ∼ 3×10
17 cm
−3 . The dimensionless
radius r s is defined via
4π
3
r
3
s =
1
n a
3
B
.
(9.63)
The sum of exchange and correlation energies E xc is found to be mostly independent of material
parameters [886] (Fig. 9.29a) and follows the form
E xc =
a + b r s
c + d r s + r 2
s
,
(9.64)
with a = −4.8316, b = −5.0879, c = 0.0152 and d = 3.0426. Thus the density dependence of the
band gap at small carrier density is ∝ n
1/3 . Experimental data for a number of II–VI semiconductors
roughly follow such a dependence (Fig. 9.29b).
In Fig. 9.30, a theoretical calculation of the absorption spectrum of bulk GaAs for various carrier
densities (n=p) [889] is shown. With increasing density, the excitonic resonance broadens and vanishes.
The shape approaches the electron–hole plasma shape. The absorption edge shifts to smaller energies.
At high carrier density, the absorption becomes negative in a spectral range before absorption sets
in. In this spectral region, the material exhibits gain and an incoming light wave is amplified (cmp.
Sect. 10.2.6).
9 Optical Properties
(a)
-5
0
-10
0
1
2
3
E
xc (Ry)
r s
Ge
Si
(b)
CdS
CdSe
CdTe
ZnO
1
2
0
4
5
3
0
-1
-2
-3
-4
-5
r s
Fig. 9.29 a Theoretical exchange and correlation energies in units of the exciton Rydberg energy as a function of the
dimensionless variable r s for Ge, Si and a model system (with one isotropic conduction and valence band each). The
solid line is a fit according to (9.64). Adapted from [886]. b Band gap renormalization in terms of the excitonic Rydberg
for various II–VI semiconductors. Solid line is the relation according to (9.64), dashed line is the dependence predicted
in [887] for T = 30 K. Data are compiled in [888]
9.7.12 Band Gap Renormalization
The band structure theory has been developed so far for small carrier densities. If the carrier density
is large the interaction of free carriers has to be considered. The first step was exciton formation.
However, at high temperatures (ionization) and at large carrier density (screening) the exciton is not
stable. Exchange and correlation energy leads to a decrease of the optical absorption edge that is called
band gap renormalization (BGR).
An effect due to significant carrier density is to be expected when the density is of the order of the
exciton volume, i.e. n ∼ a
−3
B . For a B ∼ 15 nm (GaAs) this means n ∼ 3×10
17 cm
−3 . The dimensionless
radius r s is defined via
4π
3
r
3
s =
1
n a
3
B
.
(9.63)
The sum of exchange and correlation energies E xc is found to be mostly independent of material
parameters [886] (Fig. 9.29a) and follows the form
E xc =
a + b r s
c + d r s + r 2
s
,
(9.64)
with a = −4.8316, b = −5.0879, c = 0.0152 and d = 3.0426. Thus the density dependence of the
band gap at small carrier density is ∝ n
1/3 . Experimental data for a number of II–VI semiconductors
roughly follow such a dependence (Fig. 9.29b).
In Fig. 9.30, a theoretical calculation of the absorption spectrum of bulk GaAs for various carrier
densities (n=p) [889] is shown. With increasing density, the excitonic resonance broadens and vanishes.
The shape approaches the electron–hole plasma shape. The absorption edge shifts to smaller energies.
At high carrier density, the absorption becomes negative in a spectral range before absorption sets
in. In this spectral region, the material exhibits gain and an incoming light wave is amplified (cmp.
Sect. 10.2.6).