Superconductivity

Superconductivity:
We know that the resistance of a material depends upon the length of the conductor, area
and the temperature of the conductor. It can be represented by
𝜌𝑙𝑇
𝑅=
𝐴
where ρ is the resistivity of the material.
If a graph is plotted between resistance and temperature,
it is seen that in case of a normal conductor, as the
resistance decreases, the temperature also decreases.
But, in case of a superconductor, at a particular
temperature 𝑇𝐶 known as the critical temperature, the
resistance becomes zero, which implies that the current
will flow infinitely without any resistance. This
phenomenon is known as superconductivity.
H. K. Onnes discovered this phenomena in the
year 1911.
Superconductivity: The phenomenon in which the electrical resistance of a given material
suddenly falls to zero on cooling it to extremely low temperature is called
superconductivity.
Superconductors: The materials which show superconductivity at such low temperatures
are called superconductors. Eg. Aluminium, Copper Oxide, Niobium, Magnesium diboride.
Transition or Critical Temperature (𝑇𝐶 ): The temperature at which the resistance of a
given material becomes zero and it enters a new state i.e. the superconducting state, is
called critical or transition temperature. For eg. The value of 𝑇𝐶 for Hg (mercury) is 4.2 K.
Meissner Effect: Meissner effect states that if a superconductor
is cooled in a magnetic field down to the critical temperature,
then the magnetic lines of force will be expelled out of the body
of a superconductor. This is known as the Meissner effect.

Superconductors show Meissner effect if the temperature is less

than that of the critical temperature 𝑇𝐶
Properties of Superconductor:

1. Zero electrical resistance or Infinite Conductivity

A superconductor is a material which shows zero resistance below a certain temperature
called the critical temperature. Above the critical temperature, the material is in normal
state. For eg. Mercury shows zero resistance below 4.2 K.
2. Effect of impurity
When some amount of impurities are added to a superconducting material, the value of
its critical temperature decreases.
3. Effect of pressure and stress
On increasing the pressure and stress on a certain material like Caesium, it starts showing
superconducting behaviour at a critical temperature of 1.5 K, when a pressure of 110 kilo
bar is applied.
4. Persistent Current

Let us consider a superconducting ring

whose electrical resistance is zero
below a critical temperature.

When current flows through a superconducting ring below critical temperature, then the
loss in current in the form of heat is negligible which means there is no loss in energy
conduction due to extremely low resistance. Hence the current flows through the ring
for an infinite time.

The constant flow of current in a superconducting ring without any potential

driving it, is known as the persistent current.
Types of Superconductors:
Depending upon the behaviour of superconductors in the presence of an external
magnetic field, they have been classified into 2 categories.
(a) Type-I Superconductors/ Soft Superconductors
(b) Type-II Superconductors/ Hard Superconductors
Now, let us discuss these two superconductors based on their magnetization curve.
(a) Type-I Superconductors:
Let us draw a graph between magnetic field strength (𝐻) and intensity of magnetization
(𝑀). As 𝐻 is increased, 𝑀 also increases. Till here the superconductivity behaviour exists.
But as soon as the value of 𝐻 reaches a particular value 𝐻𝐶 (known as the critical magnetic
field), the intensity of magnetization (𝑀) becomes zero. It implies that till 𝐻𝐶 , the
superconductor will show Meissner effect. Thus, in Type-I superconductors, at a particular
value of 𝐻𝐶 , it looses its superconducting state and changes to the normal state. The
following graphs both for Type-I and Type-II superconductors are shown so as to
differentiate between these two.
(b) Type-II Superconductors:
In case of Type-II Superconductors, as we increase
𝐻, 𝑀 increases to a particular value 𝐻𝐶1 (lower
critical magnetic field), then the intensity of
magnetization goes on decreasing till it reaches
another value of 𝐻 i.e. 𝐻𝐶2 (upper critical
magnetic field) at which 𝑀 becomes zero.
Unlike Type-I superconductors, here 𝑀
does not abruptly falls down to zero or it can be
said that the superconductor does not change its
state abruptly. In the present case, there are
three regions namely the superconducting state,
the mixed state and the normal state. Thus, the
transition first takes place from superconducting
state to mixed state and then finally to the normal
state.
Differences between Type-I and Type-II Superconductors:
Type-I Superconductors Type-II Superconductors
1. They follow Meissner effect strictly. 1. They do not follow Meissner effect strictly.
2. They are perfectly diamagnetic. 2. They show perfect diamagnetism for fields
less than 𝐻𝐶1 .
3. They directly jump from the superconducting 3. They do not jump abruptly to the normal
state to the normal state. state as the field is increased.
4. The value of 𝐻𝐶 for these superconductors is 4. Since large magnetic field is required to
very low (0.1 T), because of which they make a destroy the superconductivity properties,
very quick transition from superconducting state therefore they are also known as Hard
to the normal state and hence known as Soft Superconductors.
Superconductors.
5. These superconductors find very limited 5. These have many practical applications (eg.
applications because of their lower 𝐻𝐶 . Fusion reactors, Particle accelerators, etc.)
Critical Magnetic Field: When a sufficiently strong magnetic field is applied to a
superconductor then it looses its superconductivity. The value of the magnetic field at
which the superconductor looses its superconductivity is called critical magnetic field (𝐻𝐶 ).
In other words, a particular value of the magnetic field at which the intensity of
magnetization becomes zero as the magnetic field is increased is known as critical magnetic
field (𝐻𝐶 ).
The relation between critical magnetic field (𝐻𝐶 ) and critical temperature (𝑇𝐶 ) is
given by
𝐻𝐶 = 𝐻0 1 − 𝑇ൗ𝑇𝐶 (1)
where 𝐻0 is the magnetic field strength at T=0.
When there is no magnetic field i.e. 𝐻𝐶 = 0 , then
𝐻0 1 − 𝑇ൗ𝑇 = 0
𝐶
֜ 1 − 𝑇ൗ𝑇𝐶 = 0 ֜ 𝑇 = 𝑇𝐶 (2)
which means if magnetic field is not applied, then the temperature of a material is equal to
that of the critical temperature.
BCS Theory: There were many theories for the explanation of the phenomena of
superconductivity. But the most widely accepted theory is the BCS theory developed in
1957 named after three scientists Bardeen, Cooper and Schrieffer for which they got the
Nobel Prize in the year 1972.

BCS Theory is based on the following two interactions:

(a) electron-phonon-electron interaction &

(b) Cooper pair.

(a) electron-phonon-electron interaction:

According to BCS theory, the lattice vibrations play an important role in superconductivity
phenomena. Let us consider a lattice consisting of positive ions. The positive ions are
equally spaced and are identical with each other. These ions are vibrating about their mean
position and are not at rest. The vibrations are normally small because of which the
electrons can easily travel from one end of the lattice towards the other end.
When an electron passes through the lattice, it interacts with the
positively charged ions electrostatically i.e. Coulomb’s attraction
take place between an electron and positive ions. As a result, the
lattice gets distorted and a region of high positive charge density is
created which is called phonon. Now, when another electron enters
the lattice, it will be attracted by the region of high positive charge
density (phonon) and the two electrons will form a weak bond
between them instead of Coulomb’s repulsion between the
electrons. In this way, the second electron interacts with the first
electron through phonon. This type of interaction is called electron-
phonon-electron interaction.
(b) Cooper Pair:
A cooper pair is formed when two electrons will interact with each
other through phonon attractively and by overcoming the
Coulomb’s repulsive forces. The binding energy of cooper pair is of
the order of 10−3 eV which is very small.
Cooper Pair (contd.):
As the superconductivity phenomena is due to these cooper pairs, thus superconductivity
is low temperature phenomena. The electrons in cooper pair have opposite spin and
hence the total spin of cooper pair is zero and so they behave as Bosons following Bose
Einstein Statistics. If the temperature is greater than the critical temperature (𝑇𝐶 ), cooper
pairs are broken resulting in transition from superconducting state to the normal state.

Isotopic Effect:
According to this effect, the transition temperature of a superconductor is inversely
proportional to the square root of isotopic mass.
1
i.e. 𝑇𝐶 ∝ or 𝑇𝐶 𝑀 = 𝑘 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑀

In other words, the isotopic effect states that the product of the transition temperature
and the square root of the isotopic mass is always constant.
From the above relation, it is seen that for different isotopic mass, the transition
temperature will be different
Magnetic Levitation:
When a piece of superconductor starts floating above a set of
magnets which are placed in a table in the presence of liquid
nitrogen, is known as magnetic levitation or simply maglev.
Reason of magnetic levitation:

The magnets that are placed on the table are oriented in such a way that their North
poles are pointing in an upward direction. Now, if the superconductor tries to drop
because of gravity towards the magnet, there will be a increase in the flux in upward
direction. As this happens, a current will be set up to generate its own downward flux so
that the net flux is constant.
This will turn the superconductor itself into a magnet with the
North Pole facing the North poles of the magnet that are
underneath, as a result there will be a repulsion between the
North poles and hence the superconductor will float.
On the other hand, if the superconductor tries to fly away from
the magnets, there will be a decrease in the upward flux which
means there will be less amount of magnetic flux passing
through the superconductor. Again, a current will be set up in
opposite direction with respect to the previous one. This will
turn the superconductor into a magnet with its South Pole
facing the magnets that are underneath it. Thus, there will be
attraction between the superconductor and the magnets below,
which implies that if the superconductor tries to rise then the
South Pole of the superconductor will attract the N-Pole of the
magnets thereby making the superconductor float.