R-20

UNIT-5

Superconductors

Syllabus:
Superconductors-Properties: Zero resistance, Critical temperature, Critical magnetic field, Critical
current density, Meissner effect- Type-I and type-II Superconductors -BCS Theory- Josephson effect
(AC & DC)-High TC Superconductors- Applications of superconductors: transformers and electrical
machines, magLev trains, SQUID

Superconductors:

Superconductivity is the phenomenon in which electrical resistance of materials suddenly

disappears below a certain temperature. The materials that exhibit superconductivity and
which are in the superconducting state are called superconductors.

Transition temperature:
The temperature at which a normal material abruptly changes into a superconductor is called
transition temperature, TC. It is also known as the critical temperature.

PROPERTIES OF SUPERCONDUCTORS

1. Zero Electrical Resistance

A super conductor is characterized by zero electrical resistivity. A method devised by Onnes

consists of measuring the decrease of the current in a closed ring of superconducting wire.
The superconducting ring is kept in a magnetic field and it is cooled to below the critical
temperature so that it goes into the superconducting state. When the external magnetic field is
switched off, a current is induced in the ring. If the ring had a finite resistance, R, the current
circulating in the ring would decrease according to the equation
𝑅𝑇
𝐼(𝑡) = 𝐼(0)𝑒 − 𝐿
where L is the inductance of the ring. The decay current is monitored by a change in the
magnetic flux through a test coil held close to the superconducting ring. Any change in the
magnetic flux of the superconducting ring will induce an emf in the test coil. Careful
measurements established that the resistivity of superconductors could be taken as zero.

2. Critical Temperature

When a superconducting material is

cooled below a certain temperature,
it goes into the superconducting
state from normal state. The
temperature at which a material in
normal state goes into
superconducting state is known as
the critical temperature, TC .

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Different materials have different critical temperatures. The transition is reversible. When the
temperature of the material is increased above the critical temperature, it passes into the
normal state. The superconducting transition is sharp for a chemically pure and structurally
perfect specimen while the transition range is broad for specimens which are structurally
imperfect or which contain impurities.

3. Critical Magnetic Field

Superconducting state depends on

the strength of the magnetic field in
which the material is placed.
Superconductivity vanishes if a
sufficiently strong magnetic field is
applied. The minimum magnetic
field, which is necessary to regain the
normal resistivity, is called the
critical magnetic field, HC. When
the applied magnetic field exceeds
the critical value HC, the
superconducting state is destroyed
and the material goes into normal
state. The value of HC varies with
temperature.

The critical field required to destroy the superconducting state decreases progressively with
increasing temperature. The dependence of critical field on temperature is governed by the
following relation.
𝑇 2
𝐻𝐶 (𝑇) = 𝐻𝐶 (0) [1 − ( ) ]
𝑇𝐶

4. Critical Current Density

If a superconducting ring carries a
current I, it gives rise to its own
magnetic field. As the current increases
to a critical value, IC, the associated
magnetic field increases to HC and the
superconductivity disappears. The
maximum current density at which the
superconductivity disappears is called
the critical current density, JC. For
any value of J < JC, the current can
sustain itself whereas for values J > JC,
the current cannot sustain itself. This
effect is known as Silsbee effect. The
variation of critical current density JC
and critical magnetic field HC with
temperature is as shown.
A superconducting ring of radius R ceases to be a superconductor when the current is
𝐼𝐶 = 2𝜋𝑅𝐻𝐶

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Meissner Effect:

When superconductors are cooled below their critical temperature TC in the presence of a
magnetic field, the magnetic flux is expelled from the interior of the specimen and the
superconductor becomes a perfect diamagnetic. This phenomenon is known as Meissner
effect.
The effect is reversible. When the temperature is raised from below TC, the flux suddenly
penetrates the specimen at T = TC and the material returns to the normal state.

The magnetic induction inside the specimen is given by

𝐵 = 𝜇0 (𝐻 + 𝑀) = 𝜇0 (1 + 𝜒)𝐻

where H is the magnetic field applied externally and M is the magnetization produced within
the specimen.

At T < TC, B = 0 and therefore 𝜇0 (𝐻 + 𝑀) = 0 . It follows that M = – H

𝑀
∴ The susceptibility of the material is 𝜒 = 𝐻 = −1
The specimen is therefore diamagnetic and the state in which magnetization cancels the
external magnetic field completely is referred to as perfect diamagnetism. The Meissner
effect contradicts the fundamental principles of electromagnetism. The condition of perfect
diamagnetism cannot be explained from the simple definition that superconductivity is a state
𝑑𝐵
of zero resistivity. Meissner effect shows that in the superconductor not only 𝑑𝑡 = 0 but also
B = 0. Thus, two mutually independent properties, namely zero resistivity and perfect
diamagnetism are the essential properties that characterize the superconducting state.

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TYPE-I AND TYPE-II SUPERCONDUCTORS

Type-I Superconductor Type-II Superconductor

1. They exhibit complete Meissner 1. They do not exhibit complete Meissner effect
Effect
2. They show perfect diamagnetic 2. They do not show perfect diamagnetic
behavior Behavior
3. They have only one critical magnetic 3. They have two critical magnetic fields,
field, HC lower critical magnetic field, HC1 and
upper critical magnetic field, HC2

4.There is no mixed state or intermediate 4. Mixed state or intermediate state is present in

state in case of these materials these materials.

5. The material loses magnetization 5. The material loses magnetization gradually.

abruptly
6. Highest value for HC is about 0.1 Wb/ 6. Upper critical field is of the order of 30
m2 Wb/m2

7. They are known as soft superconductors 7. They are known as hard superconductors
8. Lead, tin, mercury are examples 8. Nb-Sn, Nb-Ti, Nb-Zr, Va-Ga are examples

BCS THEORY

Three American physicists J. Bardeen, L.N. Cooper and J.R. Schrieffer developed the
quantum theory of superconductivity, which came to be known as BCS theory.

The two principal features of BCS theory are

1. Electrons form pairs, called Cooper pairs, which propagate throughout the lattice
2. Such propagation is without resistance because the electrons move in resonance with
phonons.

Each electron experiences an attraction towards its nearest positive ion. When the electrons
get very close to each other in the region between ions, they repel each other due to Coulomb
force. In an equilibrium condition, a balance between attraction and repulsion is established

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and the two electrons combine to form a Cooper pair. At normal temperatures, the attractive
force is too small and pairing of electrons does not take place. However, at lower
temperatures, such pairing is energetically advantageous. In a typical superconductor, the
dense cloud of Cooper pairs form a collective state and the motion of all the Cooper pairs is
correlated. As such the pairs drift cooperatively through the material. Thus, the
superconducting state is an ordered state of the conduction electrons. Since the density of
Cooper pairs is very high, even large currents require only a small velocity. The small
velocity of ordered Cooper pairs minimize collision processes and leads to zero resistivity.

The electrons of a Cooper pair have a lower energy than two unpaired electrons. The
theory predicted the existence of an energy gap between the ground state (superconducting
state) and first excited state .The energy gap represents the energy required to break up a
Cooper pair. Hence, larger energy gaps correspond to more stable superconductors.
According to BCS theory, the energy gap at 0K is given by
𝐸𝑔 (0) = 2∆ ≡ 3.52 𝐾𝑇𝐶

Josephson Effect.
Consider a thin insulating layer sandwiched between two superconductors. This insulating
layer acts as a potential barrier for flow of electrons from one superconductor to another.
Since. Josephson predicted that the barrier is so thin the quantum mechanically electron can
tunnel through it the tunneling can occur without any resistance, giving rise to a direct current
when the voltage applied across the junction is zero and an alternating current when the
applied voltage is a dc voltage

D.C. Josephson Effect.

According to Josephson, when tunnelling occurs through the insulator it introduces a phase
difference ϕ between the two parts of the wave function on opposite sides of the junction.

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The tunnelling current is given by

I = Iosin(ϕo)

where Io is the maximum current that flows through the junction without any potential
difference across the junction. Io depends on the thickness of the junction and the
temperature. With no applied voltage, a D.C. current flows across the junction. The
magnitude of the current varies between Io and -Io according to the value of phase difference
ϕo= (ϕ1- ϕ2). This is called D.C. Josephson Effect.

A.C. Josephson Effect.

Assume that’s a static potential Vo is applied across the junction. This results in additional
phase difference introduced by the cooper pair during the tunnelling across the junction. This
additional phase difference ∆ϕ at any time t can be calculated using quantum mechanics.
𝐸𝑡
∆ϕ =
ħ

Where E is the total energy of the system.

In this case E = (2e)Vo. Since cooper pairs contain 2 electrons, the factor 2 appears in the
above equation.
2𝑒𝑉0 𝑡
∆ϕ =
ħ

The tunnelling current

2𝑒𝑉0 𝑡
I= Io sin (ϕo+ Δϕ) = Io sin(ϕo+ )
ħ

This is of the form, I= Io sin (ϕo+ ω t)

ω = 2eVo/ ħ

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This represents an alternating current with angular frequency ω. This is the a.c. Josephson
Effect.

Current Voltage characteristic of a Josephson junction.

1) When Vo=0 there is a current flow of dc current Ic through the junction. This current
is called super conducting current and the effect is the d.c. Josephson effect.
2) So long Vo < Vc, a constant d.c current Ic flows.
3) When Vo > Vc, the junction has a finite resistance and the current oscillates with a
frequency ω = 2eVo/ ħ. This effect is called a.c. Josephson Effect.

High TC Superconductors:

Superconductors are divided into low TC and high TC superconductors based on their
transition temperature. Broadly, materials having TC below 24 K are regarded as low TC
superconductors and those having TC above 27 K are regarded as high TC superconductors.
However, in practice, materials for which liquid nitrogen cooling can cause transition to
superconducting state may be regarded as high TC superconductors.

Examples:
1. It is found that mixed metallic oxide of lanthanum-barium-copper (La1Ba2Cu3O7)
exhibited superconductivity at about 30 K. The superconductivity of the oxide was
linked with the deficiency of oxygen ions in the oxide compound.
2. YBa2Cu3O7 showed a transition temperature of about 95 K.
3. Bi2Ca2Sr2Cu3O10+x has the transition temperature 110 K.
4. A high TC of 133 K was achieved in mercury based copper oxide HgBa2Ca2Cu3O1+ x
Properties:
1. The high TC superconductors are brittle in nature.
2. The properties of the normal state of these materials are highly anisotropic.

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3. The Hall coefficient is positive indicating that the charge carriers are holes.
4. Their behaviour cannot be explained by BCS theory.
5. The isotope effect is almost absent in these materials.
6. The magnetic properties of these materials are highly anisotropic.
7. The effect of pressure is different on different materials.

Applications of Superconductors:

1. Superconducting coils in transformers and electrical machines generate much stronger

magnetic fields than magnetic circuits employing ferromagnetic materials produce.
The normal eddy current losses and hysteresis losses will not be present in
superconducting devices and hence the size of motors and generators will be
drastically reduced.
2. The mag-Lev train has superconducting magnets built into the base of its carriages.
An aluminium guideway is laid on the ground and carries electric current. The
repulsion between the two powerful magnetic fields, namely the field produced by the
superconductor magnet and the field produced by the electric current in the
aluminium guideway causes magnetic levitation of the train. Once the train is
levitated in air, it glides forward on the air cushion.

3. A superconducting quantum interference device (SQUID) is a device used to measure

extremely weak magnetic flux. Thus, it is basically a sensitive magnetometer. The
heart of a SQUID is a superconducting ring, which contains one or more Josephson
junctions.

When the currents are tunneling through the junctions, a single wave function
describes all the Cooper pairs. The wave function experiences a phase shift at the
junctions. In the absence of magnetic field the phase differences at the two junctions
are equal. When a magnetic field B is applied perpendicular to the loop, the flux
passes through the loop, and changes the quantum mechanical phase difference across
each of the two junctions. The wave functions at the two Josephson junctions interfere
with each other. According to Josephson’s theory, the phase difference between the
reunited currents is directly proportional to the magnetic flux. SQUID is a very
sensitive magnetometer, which can measure very weak magnetic fields of the order of
10–13 Wb/m2.