6.7 Temperature Dependence of the Band Gap
157
(a)
E (eV)
g
0.35
0.34
0.33
0.32
0.31
0.30
0.29
0.28
0
50
100
150
PbS
(b)
8.0
7.5
7.0
6.5
6.0
0
100
300
200
PbTe
L 6
L 6
Fig. 6.30 a Band gap versus temperature for PbS. b Theoretical position of L
+
6 and L
−
6 as a function of temperature for
PbTe. Adapted from [468]
(a)
T
0
100
200
300
3.05
3.00
2.95
2.90
CuI Br
1-x
x
x=0
x=1
x=0.1
x=0.2
x=0.42
x=0.8
x=0.64
(b)
T
0
100
200
300
1.82
1.81
1.80
1.79
AgGaSe 2
Fig. 6.31 a Band gap versus temperature for zincblende CuI 1−x Br x alloys with various compositions x (including
binary CuI and CuBr) as labeled. Dashed lines are guide to the eye. Adapted from [500]. b Band gap vs. temperature
for chalcopyrite AgGaSe 2 . Solid line is fit with two-oscillator Bose-Einstein model. Adapted from [502]
E g (T ) = E g (0) −
α T
2
T + β
,
(6.32)
where E g (0) is the band gap at zero temperature. A more precise and physically motivated formula
(based on a Bose-Einstein phonon model [504]) has been given in [505]
E g (T ) = E g (0) −
α B B
2
coth
B
2T
− 1
= E g (0) −
α B B
exp(( B /T ) − 1
,
(6.33)
where α B is a coupling constant and k B is a typical phonon energy; typical values are given in
Table 6.4. This model reaches a better description of the fairly flat dependence at low temperatures.
The more elaborate model of [506] takes into account a more variable phonon dispersion, including
optical phonons, and proposes the four-parameter formula
E g (T ) = E g (0) − α α
1 − 3
2
exp (2/γ ) − 1
+
3
2
2
6
1 + β − 1
(6.34)