Chemistry Reference
In-Depth Information
Band structure of solids
3.1 Introduction
We saw in the last chapter how we can build a good understanding of
molecules and solids by describing the electronic structure using linear
combinations of atomic orbitals. This method gives a very useful picture,
particularly of trends in bonding properties. However, our earlier discuss-
sion gave at best a partial description of the electronic structure of solids.
In particular, we only stated that isolated atomic and molecular energy
levels broaden into bands of allowed energy states in solids, separated
by forbidden energy gaps. In this chapter we consider in more detail the
structure of these allowed energy bands.
There are about 10
23
valence electrons which contribute to the bonding
in each cubic centimetre of a typical solid. This implies that the calculation
of the electronic structure should be a complex many-body problem, as the
exact wavefunction and energy of each electron depend on those of all the
others. However, there are at least two factors which considerably simplify
the calculation of the energy spectrum.
First, it is found that in many cases each electron effectively sees a sim-
ilar average potential as all the others, so that instead of having to solve
something like a 10
23
body problem, we can use an 'independent electron
approximation', and calculate the energy spectrum using the one-electron
Schrödinger equation introduced in Chapter 1. While we may not know
the exact form of this average potential we expect that it should be closely
related to the isolated atomic potentials of the atoms which form the solid.
Second, many interesting solid state materials are crystalline, with a
periodic lattice. Because the ground state electronic structure must also be
periodic, with the same charge distribution in each unit cell, we find that
the potential
V
(
r
)
is periodic, with
V
(
r
+
R
)
=
V
(
r
)
(3.1)
where
R
is a vector joining the same point in two different unit cells,
as illustrated in fig. 3.1. It can be shown that the individual electron
