Page 106 - Electrical Properties of Materials

88 The free electron theory of metals

4. In the range E ≈ E F the distribution function changes rather abruptly from
nearly unity to nearly zero. The rate of change depends on k B T. For absolute
zero temperature the change is infinitely fast, for higher temperatures (as
can be seen in Fig. 6.1) it is more gradual. We may take this central region
(quite arbitrarily) as between F(E) = 0.9 and F(E) = 0.1. The width of the
region comes out then [by solving eqn (6.11)] to about 4.4 k B T.
Summarizing, we may distinguish three regions for finite temperatures:
from E =0 to E = E F –2.2 k B T, where the probability of occupation is
close to unity, and the probability of non-occupation varies exponentially; from
E = E F –2.2 k B T to E = E F +2.2 k B T, where the distribution function changes
over from nearly unity to nearly zero; and from E = E F +2.2 k B T to E = ∞,
where the probability of occupation varies exponentially.

6.3 The specific heat of electrons

Classical theory, as we have mentioned before (Section 1.8), failed to predict
the specific heat of electrons. Now we can see the reason. The real culprit
is not the wave nature of the electron nor Schrödinger’s equation but Pauli’s
principle. Since only one electron can occupy a state, electrons of lower energy
are not in a position to accept the small amount of energy offered to them
occasionally. The states above them are occupied, so they stay where they are.
Only the electrons in the vicinity of the Fermi level have any reasonable chance
of getting into states of higher energy; so they are the only ones capable of
contributing to the specific heat.
The specific heat at constant volume per electron is defined as
d E
c v = , (6.21)
dT
where E is the average energy of electrons.
A classical electron would have an average energy 3/2 k B T, which tends to
zero as T → 0. Quantum-mechanically, if an electron satisfies the Fermi–Dirac
statistics, then the average energy of the electrons is finite and can quite easily
be determined (see Exercise 6.6). For the purpose of estimating the specific
heat, we may make up a simple argument and claim that only the electrons in
the region E = E F –2.2 k B T to E = E F need to be considered as being able
to respond to heat, and they can be regarded as if they were classical electrons
possessing an energy (3/2) k B T. Hence the average energy of these electrons
is
3 2.2 k B T
E = k B T , (6.22)
2 E F
which gives for the specific heat,
2
k B
c v =6.6 T. (6.23)
E F
A proper derivation of the specific heat would run into mathematical dif-
ficulties, but it is very simple in principle. The average energy of an electron
following a distribution F(E)isgiven by
1 ∞
E = F(E)Z(E)E d E, (6.24)
N 0