MODULE -4
Chapter -1
ELECTRICAL
CONDUCTIVITY IN METALS
CLASSICAL FREE ELECTRON THEORY
Assumptions of Classical free electron theory
1. Atoms in a metal are considered to be made up of
ion cores
2. There are many free electrons in metal which are
free to move anywhere inside the metal
3. The free electrons are treated equivalent to gas
molecules and hence they obey the laws of kinetic
theory of gas.
4. The effect of electric potential due to positive ions
on the electrons is considered to be constant and
hence neglected.
5. The repulsion between the free electrons and
attraction between ion core and electrons are
considered to be constant negligible and hence
neglected.
Drift velocity- The average velocity with which free
electrons moves, in a steady state, opposite to the
direction of the applied electric filed in a metal is
called drift velocity (vd).
The drift velocity is given by,
Where, E = charge, T = relaxation time, m = mass of
electron, E = electric field
Mean collision time and mean free path
Mean collision time (τ ) is the average time that
elapses between two successive collisions of an electron
with lattice ions.
Average distance traveled by the conduction electrons
between two successive collisions is called mean free
path (λ).
If ‘v’ is the total velocity of an electron, then mean free
path, λ = v τ
FAILURE OF CLASSICAL FREE ELECTRON
THEORY
1. Specific heat
According to the CFT the electrons in metal is
considered equivalent to gas molecule,
therefore, the value of specific heat (Cv) of
electrons is 3 /2 R .
But experimentally it was found that,
Thus, the value of specific heat as per the
theory is far higher than the experimentally
observed value.
Also, the theory predicts that the specific heat
does not depend on temperature. But
experimentally the specific heat is proportional
to temperature.
2. Temperature dependence of electrical resistivity.
According to kinetic theory of gases, kinetic energy of electron is given by
or, we can write v ᾳ √T
But, electrical conductivity,
But experimentally, it has been observed that for metals the electrical
conductivity is inversely proportional to the temperature T.
Thus, it clearly indicates that the theory of classical free electron is not agreeing
with experimental observations.
3. Dependence of electrical conductivity (σ) on electron
concentration:
According to classical free electron theory, the electrical
conductivity σ is given by ,
Where ‘n’ is the electron concentration.
Therefore, σ ᾳ n
But practically it is observed that is not σ strictly proportional to
the electron concentration, indicating the classical free electron
theory does not holds good.
Ex:
QUANTUM FREE ELECTRON THEORY
This theory was developed by Somerfield.
The main assumptions of quantum free electron theory.
1. There are many free electrons in metal which are free to
move anywhere inside the metal
2. The energy values of the conduction electrons are
quantized.
3. The distribution of electrons in the various allowed
energy levels occur as per Pauli’s exclusion principle (they
obey F-D statistics).
4. The effect of electric potential due to positive ions on the
electrons is considered to be constant andhence neglected.
5. The repulsion between the free electrons and attraction
between ion core and electrons areconsidered to be
negligible.
FERMI ENERGY:
The energy of the electron corresponding to the
highest occupied energy levels at zero kelvin is
called as Fermi energy. The corresponding
energy level is referred to as the Fermi level.
The Fermi energy is denoted as EF .
Thus at T = 0 K all the energy levels lying above
the Fermi level are empty, and those lying below
are completely filled as shown in figure below.
FERMI FACTOR :
Fermi factor is the probability of occupation of
energy state in a metal at thermal equilibrium.
The probability of occupation f(E) at a given
energy state with energy E at a temperature T, is
given by,
Where EF is the Fermi energy and k is Boltzmann
constant.
EFFECT OF TEMPERATURE ON FERMI
FACTOR
(i)probability of occupation for E < EF at T = 0K,
Then,
At T = 0, all the energy levels below the Fermi level
are occupied.
ii) Probability of occupation for E > EF at T = 0 :
At T = 0, all the energy levels above Fermi levels are unoccupied.
iii) Probability of occupation at ordinary temperature
(T > 0K):
At ordinary temperatures the f(E) remains one for E<<< EF , it
starts decreasing from 1 as the value of E becomes closer to EF.
When E= EF ,
For E>EF , f(E) falls off to zero very rapidly.
Fermi temperature (TF ) is the temperature at
which the average thermal energy of the free
electrons in a solid becomes equal to the Fermi
energy at 0 k.
Fermi velocity (VF): The velocity of the
electrons which occupy the Fermi level is called
Fermi velocity.
MERITS OF QUANTUM FREE ELECTRON
THEORY
1. Temperature dependence of electrical
resistivity/conductivity.
According to Quantum free electron theory,
Here vF is independent of temperature but λ is temperature
dependent. As temperature increases, the lattice ions start
vibrating with larger amplitudes, which results in the
Reduction in the value of mean free path of the electrons.
2 Specific heat :As per quantum free electron theory ,
electrons that are occupying energy levels close to Fermi
level are capable of absorbing heat energy to get excited to
higher energy levels. Lower energy level electrons will not
absorb the energy. Thus, only a small percentage of the
conduction electrons are capable of receiving the thermal
energy and hence the specific heat value becomes very
small for the metal.
3.Electrical conductivity and electron concentration:
According to quantum free electron theory,
It indicates σ depends on both n and ratio of
In the case of Aluminum, free electrons (n) value is
high compared to copper but the value of ratio is
low for Aluminum and very high for copper.
Therefore, copper shows higher electrical
conductivity.
EFFECT OF TEMPERATURE AND IMPURITY ON
ELECTRICAL RESISTIVITY OF METALS
(MATTHIESSN’S RULE)
The electrical conductivity of metal varies with the temperature.
The resistivity in metals is mainly due to scattering of electrons .
The two scatterings are scattering of electrons by lattice vibrations
and ) scattering of electrons due to the presence of impurities.
At low temperature collision of electron with impurities is more
and this gives a resistance called Residual resistivity ρi. it exists
even at 0K.
As temperature increases the amplitude of lattice vibration
increases hence rate of collision also increases. Resultant
resistivity is called as ideal resistivity ρph .
“The total resistivity of a metal is the sum of the resistivity due to
thermal agitation of the metal ions and the resistivity due to the
presence of impurity in the crystal”
ρ = ρph + ρi.
This is called Matthiessen’s rule.
At very low temperatures, ρ = ρi.
At high temperature ρ = ρph since value of is ρi very small.
MODULE -4
Chapter -2
SUPERCONDUCTIVITY
INTRODUCTION TO SUPERCONDUCTOR
Super conductivity was first observed by
H Kammerlingh Onnes in 1911 while
measuring the resistivity of mercury at low
temperatures.
In certain materials, may be pure or impure, it is
observed that the resistivity becomes zero
suddenly at particular low temperature.
This phenomenon of disappearance of resistivity
in the material at low temperature is called as
superconductivity.
Materials which are showing zero resistivity or
superconducting property is called as
superconductor.
TEMPERATURE DEPENDENCE OF RESISTIVITY IN
SUPERCONDUCTING MATERIALS
The temperature at which the resistivity (ρ) becomes zero, known as
critical temperature (Tc) or transition temperature.
Above the transition temperature, the substance is in the normal state
and below it will be in superconducting state. Tc value is different for
different materials.
MEISSNER EFFECT:
A superconductor kept in a weak magnetic field
expels the magnetic flux out of its body and thus
becomes perfect diamagnet.
This expulsion of magnetic flux lines from the
superconductor when MF is applied is called as
Meissner effect.
Therefore, within superconductor, magnetic
induction B = 0 as shown in figure.
We know that B = µ0(H + M)
(where M is the magnetization and µ0 permeability)
0= µ0(H + M)
On simplification we get, M/H = -1
But M/H is magnetic susceptibility (χ)
Therefore, χ = -1
It indicates the superconductor becomes a perfect diamagnetic
material when MF is applied to it.
(since for diamagnetic materials χ = -1 )
TEMPERATURE DEPENDENCE OF CRITICAL
FIELD:
If the superconducting materials are subjected to a strong magnetic
field (MF), then superconductor will lose its superconducting property
(they return to the normal state).
The minimum MF required to destroy the superconducting property of
the superconductor is called the critical field (Hc).
The variation of Hc with temperature is shown in figure.
From figure it can be seen that when temperature decreases less than
Tc, then Hc increases and reaches maximum when temperature
becomes equal to 0K
Where Hc(0) is the critical field at 0 K.
DEMONSTRATION OF MEISSNER EFFECT
(EXPERIMENTAL PROOF)
Consider a superconducting material is wound with
primary and secondary coils.
The primary coil is connected to the battery through a key
and the secondary coil is connected to ballistic
galvanometer (BG) as shown in figure.
When the key is pressed, the current will flow through the
primary coil and this will set up a MF around it and this
MF is linked with secondary coil, as secondary coil is
around the primary coil.
If input current in primary changes then MF also changes.
This change in MF induces current in the secondary and
can be observed by deflection in the BG.
If there is no further changes in the primary current then
no changes in flux and so no current in the secondary coil.
During experiment, primary current was kept constant and
temperature of the superconductor was decreased
gradually.
When temperature reached critical temperature and
material became superconductor, suddenly deflection in BG
was observed.
It indicates current flow in secondary even though primary
current was kept constant.
BCS THEORY
This theory is developed by Bardeen, Cooper and Schreiffer.
This theory explains superconductivity phenomenon based on electron-lattice-electron interaction.
Consider an electron is moving between lattice points within a superconductor.
There is an attractive force between electron and positive lattice points.
This attraction makes the lattice points to move towards electron.
As a result, electron will be surrounded by lattice points and it creates high concentration of positive
charge around the electron.
Suppose, another electron is moving near this positive charge, then it will be attracted and move towards
this positive charge. This represents the motion of the second electron towards the first electron.
This represents the attraction between two electrons in the presence of lattice point.
This interaction is called as electron-lattice-electron interaction.
Due to this attractive force, two electrons will be couple together and forms electron pairs called as ‘Cooper
pairs’.
As a result, the attraction force becomes greater than the repulsive force between two electrons.
At temperature below TC, the electron-lattice-electron interaction continues and all the electrons form a
cloud of cooper pairs.
The resistance/scattering encountered by one electron will be exactly nullified by another electron, hence
no resistance to the flow. This forms the state of superconductivity.
As long as the superconductor is cooled to very low temperatures, the Cooper pairs stay together.
As the superconductor gains heat energy the vibrations in the lattice become more violent and break the
pairs.
It results in disappearance of superconducting property.
TYPES OF SUPERCONDUCTORS
There are two types of superconductors based on
differences in the magnetization exhibited by
them.
1. Type I (soft) superconductor
2. Type II (hard) superconductors
TYPE I (SOFT) SUPERCONDUCTOR
It has Low critical temperature
It has Low Critical magnetic field
It Perfectly obey the Meissner effect.
These are completely diamagnetic.
It Exhibits single critical magnetic field.
The transition from a superconducting state to a normal state due to the
external magnetic field is sharp and abrupt for type-I
No mixed state exists in typeI Superconductors.
Easily lose the superconducting state by low intensity magnetic field.
BCS theory can be used to explain the super conductivity of type-I
superconductors.
These are also called as Soft Superconductors.
These are also called as Low temperature Superconductors.
Due to the low critical magnetic field, type I superconductors have limited
technical applications.
Examples: Hg, Pb, Zn
TYPE – II SUPERCONDUCTORS
It has High critical temperature
It has High Critical magnetic field
It Partly obey the Meissner effect.
These are not completely diamagnetic.
Exhibits two critical magnetic field
The transition from a superconducting state to a normal state due to the
external magnetic field is gradually but not shape and abrupt. The state
between lower critical magnetic field and upper magnetic field is known
as an inter mediate state or mixed.
Mixed state exists in type II Superconductors.
Does not easily lose the superconducting state by external magnetic field.
BCS theory cannot be used to explain the super conductivity of type-II
superconductors.
These are also called as Hard Superconductors.
These are also called as High temperature Superconductors.
Due to the high critical magnetic field, type II superconductors have
wider technical applications.
Examples: Nb Ti, Nb3 Sn, etc
