Page 102 - Electrical Properties of Materials

The free electron

6 theory of metals

Struggling to be free, art more engaged
Hamlet
Much have I travelled in the realms of gold,
And many goodly states and kingdoms seen.
Keats On First Looking in Chapman’s Homer

6.1 Free electrons
The electrical and magnetic properties of solids are mainly determined by the
properties of electrons in them. Protons can usually be relegated to subordin-
ate roles, like ensuring charge neutrality. Neutrons may sometimes need to
be considered, as for example in some superconducting materials, in which the
critical temperature depends on the total mass of the nucleus, but on the whole,
the energy levels of electrons hold the key to the properties of solids.
The mathematical problem is not unlike the one we met in the case of indi-
vidual atoms. How can we determine the energy levels of electrons in a solid?
Take a wave function depending on the coordinates of 10 25 electrons; write
down the Coulomb potential between each pair of electrons, between electrons
and protons; and solve Schrödinger’s equation. This is an approach which, as
you have probably guessed, we are not going to try. But what can we do in-
stead? We can take a much simpler model, which is mathematically soluble,
and hope that the solution will make sense.
Let us start our search for a simple model by taking a piece of metal and
noting the empirical fact (true at room temperature) that there are no elec-
trons beyond the boundaries of the metal. So there is some mechanism keeping
the electrons inside. What is it? It might be an infinite potential barrier at
the boundaries. And what about inside? How will the potential energy of an
electron vary in the presence of that enormous number of nuclei and other
electrons? Let us say it will be uniform. You may regard this as a sweeping
assumption (and, of course, you are absolutely right), but it works. It was in-
troduced by Sommerfeld in 1928, and has been known as the ‘free electron’
The electrons inside the metal
model of a metal.
(more correctly, the valence elec-
You may recognize that the model is nothing else but the potential well we
trons, which occupy the outer ring)
met before. There we obtained the solution for the one-dimensional case in the
are entirely free to roam around,
following form,
but they are not allowed to leave
the metal. k h n 2
2 2
2
E = = . (6.1)
2 m 8 m L 2