Page 135 - Electrical Properties of Materials

Holes 117

where N a is the number of atoms per unit length. Since r may have negative
values as well, the total number of permissible values of k is

2(r max – r min )= N a L. (7.60)

Now to each value of k belongs a wave function; so the total number of
wave functions is N a L, and thus, including spin, the total number of available
states is 2N a L.

7.8 Metals and insulators
At absolute zero some materials conduct well, some others are insulators.
Why? The answer can be obtained from the formulae we have derived.
If each atom in our one-dimensional crystal contains one electron, then the
total number of electrons is N a L, and the band is half-filled. Since dE/dk is
large in the middle of the band, this means that there is a high effective number
of electrons; that is, high conductivity.
If each atom contains two electrons, the total number of electrons is 2N a L;
that is, each available state is filled. There is no conductivity: the solid is an
insulator.
If each atom contains three electrons, the total number of electrons is 3N a L;
that is, the first band is filled and the second band is half-filled. The value of
dE/dk is large in the middle of the second band; therefore a solid containing
atoms with three electrons each (it happens to be lithium) is a good conductor.
It is not difficult to see the general trend. Atoms with even numbers of
electrons make up the insulators, whereas atoms with odd numbers of electrons
turn out to be metals. This is true in general, but it is not true in every case.
All we need to know is the number of electrons, even or odd, and the electric
behaviour of the solid is determined. Diamond, with six electrons, must be
an insulator and aluminium, with thirteen electrons, must be a metal. Simple,
isn’t it?
It is a genuine triumph of the one-dimensional model that the electric prop-
erties of a large number of elements may be promptly predicted. Unfortunately,
it does not work always. Beryllium with four electrons and magnesium with
twelve electrons should be insulators. They are not. They are metals; though
metals of an unusual type in which electric conduction, evidenced by Hall-
effect measurements, takes place both by holes and electrons. What is the
mechanism responsible? For that we need a more rigorous definition of holes.

7.9 Holes
We first met holes as positively charged particles that enjoy a carefree existence
quite separately from electrons. The truth is that they are not separate entities
but merely by-products of the electrons’ motion in a periodic potential. There
is no such thing as a free hole that can be fired from a hole gun. Holes are
artifices but quite lively ones. The justification for their existence is as follows.