154
6 Band Structure
In a perfectly crystalline semiconductor the eigenenergies of the states in the bands are real. An
amorphous semiconductor can be modeled using a spectrum of complex energies [495]. In Fig. 6.27
the band structure of crystalline silicon is shown next to that calculated for amorphous silicon with
α = 0.05.
6.7 Temperature Dependence of the Band Gap
The band gap of a semiconductor typically decreases with increasing temperature. A direct visual
impression can be obtained from the same LED chain at room temperature and dipped into liquid
nitrogen (Fig. 6.28). Experimental data of band gap versus temperature are shown in Fig. 6.29 for bulk
Si and ZnO.
The reasons for the temperature variation of the band gap are the change of electron–phonon
interaction and the expansion of the lattice. The temperature coefficient may be written as
Fig. 6.24 a Band gap of
Si x Ge 1−x alloy
(T = 296 K) with a change
from the conduction-band
minimum at L (Ge-rich) to
X. The inset depicts the
transition energy of the
indirect (–L) and direct
( absorption edge for
low Si content. Adapted
from [489]. b Band gap (at
room temperature) of
In x Ga 1−x As. The solid line
is an interpolation with
bowing (b = 0.6 eV) and
the dashed line is the linear
interpolation. Data from
[486]. c Band gap (at room
temperature) in the ternary
system Al x Ga 1−x As. For
x < 0.4 the alloy is a
direct, for x > 0.4 an
indirect, semiconductor.
E dd denotes the energy
position of a deep donor
(cf. Sect. 7.7.6). Adapted
from [490]. d Band gap (at
room temperature) in the
ternary system
Mg x Zn 1−x O. Data (from
spectroscopic ellipsometry
[491, 492]) are for
hexagonal wurtzite phase
(circles), and Mg-rich
cubic rocksalt phase
(squares). Dashed lines are
fits to data with a different
bowing parameter for each
phase
1.5
1.3
1.1
0.9
0.7
0.5
0.3
0.2
4
.
0
6
.
0
8
.
0
0
.
1
x
0.0
E (eV)
g
In Ga
x
1-x As
(a)
(b)
0.0 0.2
0.4
0.6
0.8 1.0
2.6
2.4
2.2
2.0
1.8
1.6
1.4
0.0
x
E (eV)
g
X 1
Al
As
x Ga 1-x
E dd
1
L 1
(c)
(d)
6 Band Structure
In a perfectly crystalline semiconductor the eigenenergies of the states in the bands are real. An
amorphous semiconductor can be modeled using a spectrum of complex energies [495]. In Fig. 6.27
the band structure of crystalline silicon is shown next to that calculated for amorphous silicon with
α = 0.05.
6.7 Temperature Dependence of the Band Gap
The band gap of a semiconductor typically decreases with increasing temperature. A direct visual
impression can be obtained from the same LED chain at room temperature and dipped into liquid
nitrogen (Fig. 6.28). Experimental data of band gap versus temperature are shown in Fig. 6.29 for bulk
Si and ZnO.
The reasons for the temperature variation of the band gap are the change of electron–phonon
interaction and the expansion of the lattice. The temperature coefficient may be written as
Fig. 6.24 a Band gap of
Si x Ge 1−x alloy
(T = 296 K) with a change
from the conduction-band
minimum at L (Ge-rich) to
X. The inset depicts the
transition energy of the
indirect (–L) and direct
( absorption edge for
low Si content. Adapted
from [489]. b Band gap (at
room temperature) of
In x Ga 1−x As. The solid line
is an interpolation with
bowing (b = 0.6 eV) and
the dashed line is the linear
interpolation. Data from
[486]. c Band gap (at room
temperature) in the ternary
system Al x Ga 1−x As. For
x < 0.4 the alloy is a
direct, for x > 0.4 an
indirect, semiconductor.
E dd denotes the energy
position of a deep donor
(cf. Sect. 7.7.6). Adapted
from [490]. d Band gap (at
room temperature) in the
ternary system
Mg x Zn 1−x O. Data (from
spectroscopic ellipsometry
[491, 492]) are for
hexagonal wurtzite phase
(circles), and Mg-rich
cubic rocksalt phase
(squares). Dashed lines are
fits to data with a different
bowing parameter for each
phase
1.5
1.3
1.1
0.9
0.7
0.5
0.3
0.2
4
.
0
6
.
0
8
.
0
0
.
1
x
0.0
E (eV)
g
In Ga
x
1-x As
(a)
(b)
0.0 0.2
0.4
0.6
0.8 1.0
2.6
2.4
2.2
2.0
1.8
1.6
1.4
0.0
x
E (eV)
g
X 1
Al
As
x Ga 1-x
E dd
1
L 1
(c)
(d)