Page 108 - Electrical Properties of Materials

90 The free electron theory of metals

rules of quantum mechanics, it may still suffer reflection. Thus, the probability
of escape is 1 – r(p x ), where r(p x ) is the reflection coefficient. If the number of
electrons having a momentum between p x and p x +dp x is N(p x )dp x , then the
number of electrons arriving at the surface per second per unit area is

p x
N(p x )dp x , (6.28)
m
and the number of those escaping is
p x
{1 – r(p x )} N(p x )d p x . (6.29)
m
Adding the contributions from all electrons that have momenta in excess of
, we can write for the emission current density,
p x 0
e ∞
J = {1 – r(p x )}p x N(p x )dp x . (6.30)
m
p x 0
We may obtain the number of electrons in an infinitesimal momentum range
in the same way as for the infinitesimal energy range. First, it consists of two
factors, the density of states and the probability of occupation. The density of
states, Z(p x ), can be easily obtained by noting from eqns (6.2) and (6.26) that
h h h
p x = n x , p y = n y , p z = n z . (6.31)
2 2 2
The number of states in a cube of side one is exactly one. Therefore, the
number of states in a volume of sides dn x ,dn y ,dn z is equal to dn x dn y dn z ,
which with the aid of eqn (6.31) can be expressed as
3
2

d p x dp y d p z . (6.32)
h
Dividing again by 8 (because only positive integers matter) and multiplying
by two (because of spin) we get
2
Z(p x , p y , p z )= . (6.33)
h 3
Hence, the number of electrons in the momentum range p x , p x +dp x ; p y ,
p y +d p y ; p z , p z +dp z is

N(p x , p y , p z )dp x d p y dp z
2 dp x d p y dp z
= . (6.34)
2
2
2
3
h exp[{(1/(2 m))(p + p + p )– E F }/k B T]+1
x y z
To get the number of electrons in the momentum range p x , p x +dp x ,the
above equation needs to be integrated for all values of p y and p z ,
N(p x )d p x
2 ∞ ∞ d p y dp z
= dp x . (6.35)
2
2
2
h 3 –∞ –∞ exp[{(1/(2 m))(p + p + p )– E F }/k B T]+1
y
x
z
This integral looks rather complicated, but since we are interested only in
those electrons exceeding the threshold (ϕ k B T), we may neglect the unity