Page 104 - Electrical Properties of Materials

86 The free electron theory of metals

So the number of states having energies between E and E +dE is equal to
4π 3/2 3/2 3/2
Z(E)d E = K (E +dE) – E
3
~ 3/2 1/2 d E. (6.9)
= 2πK
E
This is not the end yet. We have to note that only positive values of n x , n y , n z
are permissible; therefore we have to divide by a factor 8. Allowing further for
the two values of spin, we have to divide by 4 only. We get finally
3
4πL (2 m) 3/2
1/2
Z(E)d E = CE d E with C = . (6.10)
h 3
Equation (6.10) gives us the number of states, but we would also like to
know the number of occupied states, that is, the number of states that contain
electrons. For that we need to know the probability of occupation, F(E). This
function can be obtained by a not-too-laborious exercise in statistical mechan-
ics. One starts with the Pauli principle (that no state can be occupied by more
than one electron) and works out the most probable distribution on the condi-
Enrico Fermi (Nobel Prize, tion that the total energy and the total number of particles are given. The result
1938) and Paul Dirac (Nobel is the so-called Fermi–Dirac distribution ∗
Prize, 1933) both made funda-
1
mental contributions to quantum F(E)= , (6.11)
mechanics. exp{(E – E F )/k B T}+1
∗ If we use the assumption that a state where E F is a parameter called the Fermi level. It has the easily memorized
may contain any number of particles, the property that at
so-called Bose–Einstein distribution is
obtained. It turns out that all particles 1
belong to one or the other of these dis- E = E F , F(E)= , (6.12)
tributions and are correspondingly called 2
fermions or bosons. For this book, it is of that is, at the Fermi level the probability of occupation is .
1
great importance that electrons are fer- 2
mions and they obey the Fermi–Dirac As may be seen in Fig. 6.1, F(E) looks very different from the classical
distribution. The properties of bosons distribution exp(–E/kT). Let us analyse its properties in the following cases:
(e.g. quantized electromagnetic waves
and lattice waves) are of somewhat less 1. At T =0.
significance. We occasionally need to
refer to them as photons and phonons but F(E)=1 for E < E F (6.13)
their statistics is usually irrelevant for F(E)=0 E > E F .
our purpose. The Bose–Einstein distri-
bution, and the so-called boson condens- Thus, at absolute zero temperature, all the available states are occupied
ation do come into the argument in Sec- up to E F , and all the states above E F are empty. But remember, Z(E)d E
tion 12.14, where we talk briefly about
atom lasers and in Chapter 14 concerned is the number of states between E and E +d E. Thus, the total number of
with superconductivity, but we shall not states is
need any mathematical formulation of
E F

the distribution function.
Z(E)d E, (6.14)
0
3
which must equal the total number of electrons NL , where N is the number
of electrons per unit volume. Thus, substituting eqn (6.10) into eqn (6.14),
the following equation must be satisfied:

E F
3 3/2 3 1/2 3
4πL (2 m) /h E d E = NL . (6.15)
0