Page 119 - Electrical Properties of Materials

The band theory of solids

7

Band of all evils, cradle of causeless care.
Sir Philip Sidney

7.1 Introduction
Most properties of metals can be well explained with the aid of the
free-electron model, but when we come to insulators and semiconductors the
theory fails. This is not very surprising because the term ‘free electron’, by
definition, means an electron free to roam around and conduct electricity; and
we know that the main job of an insulator is to insulate; that is, not to conduct
electricity. It is not particularly difficult to find a model explaining the absence
of electrical conductivity. We only need to imagine that the valence electrons
cling desperately to their respective lattice ions and are unwilling to move away.
So we are all right at the two extremes; free electrons mean high conductivity,
tightly bound electrons mean no conductivity. Now what about semiconduct-
ors? They are neither good conductors nor insulators; so neither model is
applicable. What can we do? Well, we have touched upon this problem be-
fore. Silicon and germanium are semiconductors in spite of the covalent bonds
between the atoms. The bonding process uses up all the available electrons, so
at absolute zero temperature there are no electrons available for conduction.
At finite temperatures, however, some of the electrons may escape. The lat-
tice atoms vibrate randomly, having occasionally much more than the average
thermal energy. Thus, at certain instants at certain atoms there is enough energy
to break the covalent bond and liberate an electron. This is a possible descrip- The higher the temperature the
tion of the electrical properties of semiconductors and, physically, it seems more likely it is that some elec-
quite plausible. It involves no more than developing our physical picture of the trons escape.
covalent bond a little further by taking account of thermal vibrations as well.
All we need to do is to put these arguments into some quantitative form, and
we shall have a theory of semiconductors. It can be done, but somehow the
ensuing theory never caught the engineers’ imagination.
The theory that did gain wide popularity is the one based on the concept of
energy bands. This theory is more difficult to comprehend initially, but once
digested and understood it can provide a solid foundation for the engineers’
flights of fancy.
The job of engineers is to invent. Physicists discover the laws of nature, and
engineers exploit the phenomena for some useful (sometimes not so useful)
end. But in order to exploit them, the engineer needs to combine the phe-
nomena, to regroup them, to modify them, to interfere with them; that is, to
create something new from existing components. Invention has never been an
easy task, but at least in the good old days the basic mechanism was simple
to understand. It was not very difficult to be wise after the event. It was,