10. Appendix

642 Appendix B
How to explain a Positive Eg/T?
We note that the above approach based on a second-order perturbation treatment
of electron-phonon interaction tends to predict a decrease in Eg with
increase in T. Clearly a different approach is necessary to explain the increase
in Eg with T observed experimentally in the lead chalcogenides (even after
deduction of the effect of thermal expansion) . This approach has been proposed
by Antončik [E. Antončik, Czech. J. Phys. 5, 449 (1955)]. The main idea
of this approach is that one starts with the pseudopotential method for calculating
the energy gap Eg at T 0 (to be more precise, with the atoms at rest,
so as to avoid quantum-mechanical zero-point vibrations) and then calculates
Eg(T) from the temperature dependent pseudopotential form factors Vg. Let
us assume that we define the T 0 pseudopotential form factors by (2.25):
Vg(0) 1
V(r) exp[ig · r]dr
ø ø
At T 0 we would expect that both V(r) and the unit cell volume ø would
change with T. Again, we neglect the effect of thermal expansion on the lattice
constant so ø remains unchanged. Under this assumption the effect of T
is simply to cause the atoms to vibrate with amplitude ¢R(t) about the T 0
equilibrium position R0. In principle, this vibration will cause V to be a function
of time t and hence Vg also becomes time dependent. Since the period
of atomic vibration is typically much shorter than the time of measurement
of Eg in an experiment, Eg can be assumed to depend on the time-averaged
pseudopotential 〈V〉 only. It is not easy to calculate this average 〈V〉 since we
do not know how V will change as a result of the atomic vibration. Even if
we can assume that the ion cores vibrate as a rigid body there is no reason to
assume that the charge distribution of the valence electrons will rigidly follow
the ion cores. One way to understand this is to think of what happens when
we shake an egg. The egg yolk will not necessarily follow the shell rigidly.
For simplicity, one can assume that the whole atom, including all the valence
electrons, will vibrate as a rigid body. This means that when the atom moves
from R0 to R0 ¢R(t) the pseudopotential changes from V to V ′ where V ′
is related to V simply by a displacement of the coordinate system by ¢R(t).
If we define a new coordinate system so that the origin is displaced by ¢R(t)
and a point r in the old system becomes: r ′ r ¢R in the new system then
V ′ (r ′ ) V(r). The new pseudopotential form factor Vg(T) is given by:
Vg(T) 1
V
ø ø
′ (r) exp[ig · r]dr
1
V(r
ø ø
′ ) exp[ig · (r ′ ¢R)dr ′
[exp ig · ¢R]
V(r
ø
′ ) exp[ig · r ′ ]dr ′
ø