Page 127 - Electrical Properties of Materials

The Feynman model 109

7.4 The Feynman model
This is the one I like best, because it combines mathematical simplicity with
an eloquent physical picture. It is essentially a generalization of the model we
used before to understand the covalent bond—another use of the coupled mode
approach.
Remember, the energy levels of two interacting atoms are split; one is
slightly above, the other slightly below the original (uncoupled) energy. What
happens when n atoms are brought close together? It is not unreasonable to ex-
pect that there will be an n-fold split in energy. So if the n atoms are far away
from each other, each one has its original energy levels denoted by E 1 and E 2
in Fig. 7.9(a), but when there is interaction they split into n separate energy
levels. Now looking at this cluster of energy levels displayed in Fig. 7.9(b), we
are perfectly entitled to refer to allowed energy bands and to forbidden gaps
between them.
To make the relationship a little more quantitative, let us consider the one-
dimensional array of atoms shown in Fig. 7.10. We shall now put a single
electron on atom j into an energy level E 1 and define by this the state ( j). Just
as we discussed before in connection with the hydrogen molecular ion, there is
a finite probability that the electron will jump from atom j to atom j + 1, that is
from state ( j) into state ( j + 1). There is of course no reason why the electron
should jump only in one direction; it has a chance of jumping the other way
too. So the transition from state ( j) into state ( j–1) must have equal probability.
It is quite obvious that a direct jump to an atom farther away is also possible
but much less likely; we shall therefore disregard that possibility.

) a ( ) b (
n
n –1
n/2+2
E n/2+1 Allowed
2 n/2
n/2–1
1
0
Forbidden
n
n –1
n/2+2
n/2+1 Fig. 7.9
E n/2 Allowed
1
n/2–1 There is an n-fold split in energy
when n atoms are brought close to
2 each other, resulting in a band of
1 allowed energies, when n is large.

j–2 j–1 j j+1 j+2
Fig. 7.10
a
A one-dimensional array of atoms.