Page 118 - Electrical Properties of Materials

100 The free electron theory of metals

the Fermi level to be regarded as unoccupied. When a voltage Show (i) that the tunnelling current takes the form
U is applied [Fig. 6.14(b)] electrons may tunnel through the

E F 2 +eU
insulator from left to right. Assume that the tunnelling current I ∼ (E – eU – E 0 ) 1/2 1/2 d E
E
in each energy range d E is proportional to the number of filled E F 2
states from which tunnelling is possible and to the number of and (ii) that Ohm’s law is satisfied for small voltages.
empty states on the other side into which electrons can tunnel.
In the coordinate system of Fig. 6.14(a) the density of states 6.11. Assume that the energy levels in a certain system are in-
as a function of energy may be written as tegral multiples of a basic unit (zero energy being permitted),
and each energy level is doubly degenerate, which means that
Z 1 (E)= C 1 (E – E 0 ) 1/2 for E > E 0 two different states can have the same energies. Assume fur-
ther that there are only five fermions (particles which obey the
rule that only one of them can occupy a state) in the system
and
with a total energy of 12 units.
1/2
Z r (E)= C r E for E > 0,
Find 10 allowed distributions of the particles into the energy
where C 1 , C r ,and E 0 are constants. levels mentioned.