156
6 Band Structure
Fig. 6.28 LED chain with part at room temperature (left) and a part in a dewar filled with liquid nitrogen (right)
(a)
Si
0
-50
-100
-150
-200
E (meV)
g
0 100 200 300 400 500 600 700 800
(b)
Fig. 6.29 Temperature dependence of the band gap of a Si (data from [498]) and b ZnO (experimental data from
photoluminescence (triangles) and ellipsometry (circles)). The solid lines are fits with (6.34) and the parameters given
in Table 6.4
∂ E g
∂ T
p
=
∂ E g
∂ T
V
−
α
β
∂ E g
∂ p
T
,
(6.31)
where α is the volume coefficient of thermal expansion and β is the volume compressibility. A recommendable discussion of the thermodynamic role of the band gap as chemical potential for the mass
action law (7.12), entropy contributions and its temperature dependence can be found in [497].
An anomaly is present for the lead salts (PbS, PbSe, PbTe) for which the temperature coefficient is
positive (Fig. 6.30a). Theoretical calculations [499] show that both terms in (6.31) are positive for the
lead salts. The L
+
6 and L
−
6 levels (see Fig. 6.12) shift as a function of temperature in such a way that
their separation increases (Fig. 6.30b).
Also in copper and silver halides [500, 501] (Fig. 6.31a) and chalcopyrites [502] (Fig. 6.31b) the
increase of band gap with increasing temperature has been found, sometimes only for a certain temperature range. This effect is attributed to the p-d electron hybridization in the valence band with Cu
3d electrons and to even stronger effect with Ag 4d electrons.
For many semiconductors the temperature dependence can be described with the empirical, threeparameter Varshni formula [503],
6 Band Structure
Fig. 6.28 LED chain with part at room temperature (left) and a part in a dewar filled with liquid nitrogen (right)
(a)
Si
0
-50
-100
-150
-200
E (meV)
g
0 100 200 300 400 500 600 700 800
(b)
Fig. 6.29 Temperature dependence of the band gap of a Si (data from [498]) and b ZnO (experimental data from
photoluminescence (triangles) and ellipsometry (circles)). The solid lines are fits with (6.34) and the parameters given
in Table 6.4
∂ E g
∂ T
p
=
∂ E g
∂ T
V
−
α
β
∂ E g
∂ p
T
,
(6.31)
where α is the volume coefficient of thermal expansion and β is the volume compressibility. A recommendable discussion of the thermodynamic role of the band gap as chemical potential for the mass
action law (7.12), entropy contributions and its temperature dependence can be found in [497].
An anomaly is present for the lead salts (PbS, PbSe, PbTe) for which the temperature coefficient is
positive (Fig. 6.30a). Theoretical calculations [499] show that both terms in (6.31) are positive for the
lead salts. The L
+
6 and L
−
6 levels (see Fig. 6.12) shift as a function of temperature in such a way that
their separation increases (Fig. 6.30b).
Also in copper and silver halides [500, 501] (Fig. 6.31a) and chalcopyrites [502] (Fig. 6.31b) the
increase of band gap with increasing temperature has been found, sometimes only for a certain temperature range. This effect is attributed to the p-d electron hybridization in the valence band with Cu
3d electrons and to even stronger effect with Ag 4d electrons.
For many semiconductors the temperature dependence can be described with the empirical, threeparameter Varshni formula [503],