Phonon: A phonon is the description of the elementary vibration of lattice or atoms in solid state

material, which oscillates with single frequencies. In classical mechanics it is referred to as normal
modes of vibration. Any arbitrary vibration of lattice or atoms is the super position of these normal
modes of vibration.

Classical free electron theory: This theory was proposed by Drude-Lorentz. The assumption of classical
free electron theory as follows
1. All metals contain a large number of free electrons which move freely through the positive ionic
core of the metals. Since these free electrons cause conduction under a applied electric field
they are also called conduction electrons.
2. The free electrons are treated as equivalent to gas molecule (electron gas), which have three
degrees of freedom. They move in random motion, continuously collides with positive ion core
with a mean free path and a mean collision time. In the absence of an electric field, the kinetic
energy associated with an electron at a temperature T is given by,
1 2 3
m v th = kT
2 2
Where vth is the thermal velocity.

3. The electric field (or potential) due to the positive ionic cores is considered to be constant.
4. The repulsion between the free electrons is considered to be negligible.
5. The electric current flows in a metal due to an externally applied electric field is s consequency
of the drift velocity of the electrons in a direction opposite to the direction of the field
6. The motion of free electrons obeys the classical Maxwell-Boltzmann velocity distribution law
and the law of kinetic theory of gases.
7. According to classical free electron theory, the conductivity of electrons is given by
2
ne τ
σ=
m
Where n is the free electron concentration, e is the charge of electron,  is the relaxation time of
free electrons, m is the mass of the free electron.

Failure of classical free electron theory:

1. Temperature dependence of electrical conductivity: The conductivity of a metal is proportional to
inverse of temperature, i.e,
1
σ∝
T
But following the Drude-Lorentz theory one arrives at the condition,
1
σ∝
√T
Which is incorrect.

2. Electrical conductivity and electron concentration: As per the equation of conductivity obtained in
this theory,  is proportional to n the free electron concentration in metal. The values of n for Zinc
and Aluminium are 13.10 x 1028/m3 and 18 x 1028/m3 respectively. But these metals are
comparatively less conducting than coppoer and silver whih have values 8.45 x 1028/m3, and 5.85 x
1028/m3for n. This is also an other significant failure for the theory.
Quantum free electron theory:

Quantum free electron theory was proposed by Sommerfield. The main assumption of quantum
free electron theory are,

1. The energy values of the free electrons are quantized

2. The free electrons obey the Pauli’s exclusion principle.
3. The distribution of electrons in various allowed energy levels obey the Fermi-Dirac quantum
statistics
4. Free electrons have the same potential energy everywhere within the metal because the
potential field due to the ion cores is uniform throughout the metal.
5. Both the attraction between the electrons and the lattice points and the repulsion between the
electrons themselves are neglected and therefore electrons are treated free.
6. Electrons are treated as wave-like particles.

According to classical electron theory all the valence electrons participate in the electrical
conductivity phenomenon. But according to quantum free electron theory only electrons near to the
Fermi surface participate in the electrical conductivity phenomenon. Before the application of electric
field, the velocity vectors cancel each other pair wise at equilibrium and no net velocity of the electron
exist. The velocity vectors are plotted in three dimensional k-space. The velocity is given by,

hk
v=
m

Fermi velocity (vf) is the maximum velocity that an electron can assume i.e., vf is the actual velocity value
of the electron at the Fermi level. The shape having k in three dimensional space is given in Fig. Before
the application of external electric field, the velocity vectors cancel each other pair wise at equilibrium
and not net velocity of the electron exists. According to quantum theory the relaxation time of only
those electrons which are at Fermi level occurs conductivity.

The equation of motion of each electron in the Fermi surface under the influence of a static
electric field intensity, E is

h ( dkdt )=−eE
This means that in the absence of collisions, the Fermi sphere will be at a constant rate in k-space. If 
be the relaxation time or collision time, integrating the above equation of motion, one gets

−eEτ
k ( τ )−k ( 0 )=
h

−eEτ
∆ k=
h
Since,

p=m v f =h k f

The change in velocity is given by

h
∆ v= ∆k
m

∆ v= ( mh )(−eEτ
h )=
−eEτ
m

Since the current density is expressed as,

J=n(−e )∆ v

Substituting the value of v in the above equation we get,

( )
2
−eEτ n e Eτ
J=n (−e ) =
m m
'
J=σE(Oh m s law )

Where the electrical conductivity,  is given by

2
ne τ
σ=
m

The relaxation time can be written in terms of mean free path and Fermi velocity as,

❑f
τ=
vf

Substituting the above in the equation for the conductivity, we will get,
2
n e ❑f
σ=
mvf

Density of states:

The density of energy levels in a band is the number of energy levels/unit energy range of in the
band. Each energy level corresponds to one energy value. Each energy value is applicable to two energy
states, one for spin-up, and the other one for an electron with spin-down. The exact dependence of
density of energy states on the energy is realized through a function denoted as g(E) which is known as
density of states. It can be defined as:- it is the number of available states per unit energy range entered
at a given energy E in the valence band of a material of unit volume. It is a mathematically continuous
function and the product g(E)dE gives the number of states in the energy interval dE at E. It can be
shown on the basis of quantum mechanical calculation that,

( 8 √2 π m
)
3/ 2
1/ 2
g ( E ) dE= 3
E dE
h

As per the above equation it is clear that, the number of energy states in an energy interval dE is
proportional to E. A plot of g(E) versus E s shown in Fig. The shape of the curve is a parabola.

Fermi Factor:

The probability f(E) that a given energy state with energy E is occupied at a steady temperature
T, in thermal equilibrium, is given by,

1
f ( E )= (E −E F )/ kT
e +1

Where EF is the Fermi energy, k be the boltzman constant.

Dependence of Fermi factor on temperature and Effect on occupancy of energy levels:

(i) Probability of occupation for E<EF at T=0K

When T=0K and E<EF, we have for the probability,
1 1
f ( E )= = =1
−∞
e +1 0+1
f ( E )=1. for E< E F

The variation of f(E) with temperature is shown in Fig.

Here f(E) = 1 means the energy level is certainly occupied, and E<EF applies to all the energy levels
below EF. Therefore at T=0K, all the energy levels below the Fermi level are occupied.

(ii) Probability of occupation for E>EF at T=0.

When T=0, and E>EF

1 1
f ( E )= =
e +1 ∞
∞

There fore
f ( E )=0 for E> E F
At T=0, all the energy levels above Fermi level are unoccupied. In view of the above two
cases, at T=0K, the variation of f(E) for different energy values, becomes a step fuction as
show in Fig.

(iii) Probability of occupation at ordinary temperature:

At ordinary temperatures, though the value of probability remains for E<<Ef it starts
decreasing from 1 as the values of E become closer to Ef.
The value of f(E) becomes ½ at E=Ef. This is because, for E=Ef,
(E−E F )/kT 0
e =e =1
 Therefore,
1 1 1
f ( E )= = =
(E −E F )/ kT
e +1 1+1 2
Further, for E>Ef, the probability value falls off to zero rapidly.

Merits of Quantum free electron theory (Success of Quantum free electron theory:

The quantum free electron theory has been successful in accounting for many experimental
facts which the classical free electron theory failed to account for. The details of how the explanation is
provided is dealt in the following 2 cases.

1. Temperature dependence of electrical conductivity: The experimentally observed fact that the
electrical conductivity  has dependence on (1/T), but not on (1/T). can be explained as follows. As per
quantum free electron theory, the electrical conductivity for metals is given by,
2
n e ❑f
σ=
mvf

The waves associated with the electrons are subjected to scattering by the vibrating ions of the lattice. If
r is the amplitude of vibrations, then the ions can be considered to present effectively a circular corss-
section of area r2 that blocks the path of the electron waves irrespective of the direction of approach.
The value of mean free path of the electrons is inversely proportional to the cross section area as

1
∝ 2
πr

Now considering the fact that,

a) The energy of a vibrating body is proportional to the square of the amplitude

b) The energy of ions is due to thermal energy which is proportional to the temperature (T), we can
write,
2
r ∝T
1
∝
T

Comparing the above equation with the equation of , we have

1
σ∝
T

Thus the dependence of  on T is correctly explained by the quantum free electron theory.

2. Electrical conductivity and electron concentration:

It was not possible to understand why metals such as Aluminium and Gallium, which have three free
electrons/atom have lower electrical conductivity than metals such as copper and silver which posses
only one free electron/atom. As per quantum free electron theory, we have equation for electrical
coductity as,
2
n e ❑f
σ=
mvf

From the above equation it is clear that, the value of  depends on n, the ratio (/vf) and m*. If we
compare the cases of copper and aluminium, the value of n for aluminium is 2.13 times higher than that
of copper. But the value of (/vf) for copper is about 3.73 times higher than that of aluminium. Thus the
conductivity of copper exceeds that of aluminium. Further the value of m * for Aluminium is 1.08 times
that of copper. Because of the inverse dependence of  on m* , this also serves as a contributing factor
for the higher value of  for copper.