Page 151 - Electrical Properties of Materials

Scattering 133

e 2
σ = τN e . (8.37)
m
An obvious modification is to put the effective mass in place of the actual
mass of the electron. But there is still τ, the mean free time between collisions.
What will τ depend on and how?
We have now asked one of the most difficult questions in the theory of
solids. As far as I know no one has managed to derive an expression for τ
starting from first principles (i.e. without the help of experimental results).
Let us first see what happens at absolute zero temperature. Then all the
∗
atoms are at rest; so the problem seems to be: how long can an electron travel ∗ For the purpose of the above discus-
in a straight line without colliding with a stationary atom? Well, why would sion we can assume that the atoms are
at rest, but that can never happen in an
it collide with an atom at all? Classically a lattice looks to an electron like
actual crystal. If the atoms were at rest
a dense forest through which there is no chance of passing without bumping then we would know both their posi-
into a tree here and there. But the quantum-mechanical picture is quite differ- tions and velocities at the same time,
ent. As we know from the Feynman model, the electron does something quite which contradicts the uncertainty prin-
ciple. Therefore, even at absolute zero
different. It sits in an energy level of a certain atom, then tunnels through the
temperature, the atoms must be in some
adverse potential barrier and takes a seat at the next atom, and again at the next motion.
atom—so it just walks across the crystal without any collision whatsoever. The
mean free path is the length of the crystal—so it is not the presence of the
atoms that causes the collisions. What then? The imperfections? If the crystal
were perfect, we should have nice periodic solutions [as in eqn (7.33)] for the
wave function, and there would be equal probability for an electron being at
any atom. It could thus start at any atom and could wriggle through the crystal
to appear at any other atom. But the crystal is not perfect. The ideal periodic
structure of the atoms is upset, partly by the thermal motion of the atoms, and
partly by the presence of impurities, to mention only the two most important
effects. So, strictly speaking, the concept of collisions, as visualized for gas
molecules, makes little sense for electrons. Strictly speaking, there is no jus-
tification at all for clinging to the classical picture. Nevertheless, as so often
before, we shall be able to advance some rough classical arguments, which lead
us into the right ballpark.
First notice [eqns (1.11) and (1.13)] that the mean free path may be written
with good approximation as proportional to
l ∼ τT 1/2 . (8.38)

Arguing now that the mean free path is inversely proportional to the scattering
probability, and the scattering probability may be taken to be proportional to
the energy of the lattice wave (i.e. to T), we obtain for the collision time

τ thermal ∼ lT –1/2 ∼ T –3/2 . (8.39)

The argument for ionized impurities is a little more involved. We could say
that no scattering will occur unless the electron is so close to the ion that the
2
2
electrostatic energy [given by eqn (4.2) if e is replaced by Ze ] is comparable
with the thermal energy
Ze 2 3
= k B T, (8.40)
4π r s 2