Department of Physics

Module - 4
Superconductivity
Syllabus: Introduction to Superconductors- temperature dependence of resistivity of metals and
superconductors-critical temperature- Meissner‟s effect- critical field- temperature dependence of
critical field-types of superconductors- BCS theory (Qualitatively) - flux quantization, DC
Josephson Junction (Qualitatively). Application of superconductivity in DC SQUIDs. Numerical
problems.
Introduction:
The phenomenon of superconductivity was discovered by Kemerling Onnes in 1911, when he
was measuring the resistivity of mercury at low temperatures. He observed that the electrical
resistivity of pure mercury drops to zero at 4.2K. The material has passed into a new state called
the superconducting state.
Superconductors are those materials whose electrical resistance is zero below critical
temperature. Superconductivity is the phenomenon of obtaining superconductors below
critical temperature.

Critical temperature (Tc): It is a particular temperature at which the materials loses all its
resistance and acts as superconductors.

Examples for critical temperature:

Sl No Material Critical Temperature (Tc in K)

1 Mercury (Hg) 4.2 (-268.8 0C)
2 Lead (Pb) 7.2 (-265 0C)
3 Niobium (Nb) 9.3 (-263.70C)
4 Y-Ba-Cu-Oxide 92 (-1810C)
Note: Highest critical temperature ever recorded is 203 K (-700C) using Hydrogen
compound for research purpose.

Temperature Dependence of Resistivity of Superconductors:

The variation of resistivity versus temperature is as shown in

the graph. As temperature of a material is decreased, its
resistivity also decreased. At a particular temperature known as
critical temperature (T = Tc), the resistivity of the material
suddenly drops to zero and possess zero resistance to the
electric current. Hence the material acts a superconductor below
critical temperature. As the temperature of the material is
increased above the critical temperature, it acts as normal
conductor (R 0). Critical temperature is also called as
transition temperature

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MeissnerEffect:
When a normal conductor is
cooled in the presence of external
magnetic at a temperature above the
critical temperature (T >Tc), the magnetic
field lines will pass through the specimen
or material as shown in first figure.
But when the temperature of the
material is decreased below critical
temperature (T<Tc) then the normal
conductor loses all its resistance and
becomes superconductor and hence the
magnetic field lines (flux) will expel (moving away) near the vicinity of superconductor and
exhibits perfect diamagnetism as shown second figure. This effect is known as Meissner‟s
effect. The magnetic flux density inside the normal material is given by,
……….(1)
Where,  Magnetization
 External applied magnetic field (field strength)
 Absolute permeability of free space

When T<Tc then magnetic flux density inside the material is zero. Hence
Eqn (1)

i.e., Susceptibility indicates perfect diamagnetism.

Critical Field (Hc):

The minimum value of the applied magnetic field required to destroy superconductivity is called
the critical field (Hc). or
The minimum magnetic field required to switch a material from its superconducting state to its
normal state is called critical field (Hc).
Temperature dependence, of Critical field
Critical field is a function of temperature and it is given by
[ ( ) ] ------------ (1)

Where,  Temperature in K
 Critical Temperature in
 Critical Filed at 0 K.
Case (i) at T = Tc ,Hc = 0
Case (ii) at T= 0 K, Hc = H0
The graphical variation of Hc versus Temperature T is as shown in figure.

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Types of Superconductors:
Based on the critical field, superconductors are mainly classified in to two types. They are
1. Type–I superconductors
2. Type–II superconductors

Type–I superconductors:
The graph of Magnetization
(M) v/s applied magnetic field (H) for
Type-1 superconductor is as shown in the
figure.
As the value applied filed H, increases from
zero, magnetization also increases and
becomes maximum at critical field. That is
from 0 to Hc the magnetization varies
directly with the applied magnetic field and
the specimen acts like perfect diamagnetic
and obeys Meissner‟s effect.
As the magnetic is increased beyond Hc , then the field lines passes through the specimen and
the specimen acts as normal conductor.
Type -1 superconductor are also called as soft superconductors because they cannot withstand
for high magnetic fields and they cannot be used for making superconducting magnets.
Example: Mercury (Hg), Lead (Pb) and Zinc (Zn), other pure elements.

Type–II Superconductors:
The graph of magnetization (M) v/s applied
magnetic field (H) for Type -I1
superconductor is as shown in the figure.
Type- II superconductor possesses two
critical fields HC1 and HC2.
Between 0 to HC1, the specimen acts as
perfect diamagnetic and obeys complete
Meissner‟s effect. Thus the magnetization is
proportional to the applied magnetic field.

Between HC1 to HC2 the magnetization decreases with the increase in applied magnetic field.
Between HC1 to HC2 the specimen is electrically superconductor but not magnetically. This state
is called as mixed state or Vortex state because in this state the Meissner‟s effect is incomplete.
Above the value of critical field HC2, the specimen loses its magnetization and acts as normal
conductor.
Type- II superconductors are usually alloys or transition metals with high value of critical field.
Hence these superconductors are also known as Hard superconductors.
Example: Nb3Sn, Nb3Ge etc

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BCS Theory:
Three scientists Bardeen, Cooper and Schrieffer (BCS) explained the superconductivity on the
basis of quantum Theory.
Consider an electron approaching a
positive ion core and suffers attractive
coulomb interaction. Due to this attraction
ion core is set in motion and thus distorts the
lattice. Let a second electron come in the way
of distorted lattice and interaction between
the two occurs which lowers the energy of
the second electron.
The two electrons there- fore interact indirectly through the lattice distortion or the
phonon field which lowers the energy of the electrons. The above interaction is interpreted as
electron - Lattice - electron interaction through phonon field. It was shown by Cooper that, this
attractive force becomes maximum if two electrons have opposite spins and momentum. The
attractive force may exceed coulombs repulsive force between the two electrons below the
critical temperature. The first electron emits a phonon and second electron absorbs it. Hence
these two electron coupled together and forms Cooper pairs (bound pair of electrons).
Attraction force between two electrons in a Cooper pair is very weak and can be
separated by increase in temperature due to thermal or Lattice vibration.
Below the critical temperature the density of Cooper pair is large and move collectively
through the lattice, which minimize the collisions and resistance becomes zero. The movement
of Cooper pairs shows less velocity even for large current.

Flux quantization:
The flux quantization in superconductors is the phenomena where magnetic flux passing
through a super conducting loop is quantized in discrete unit of quantization of magnetic flux,
which is important for the development of flux quantization.
This quantization of magnetic flux is observed in superconductors. The quantization flux
is considered a key identification of electron pairing in super conductors. Flux quantization is
crucial in the operation of SQUIDs

Josephson junction:
Two superconductors separated by a thin layer of insulator is called as a weak link or Josephson
Junction.
The Cooper pairs tunnel through the insulating from one superconductor to another. The
following two are observed due to tunnelling of electrons pairs
1. DC Josephson Effect
2. AC Josephson Effect

DC Josephson Effect:

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It consists of two superconducting metals separated by a thin layer of insulator (oxides) of
thickness 10 Å to 20 Å. The movement of Cooper pairs in a superconductor is represented by a
wave function. This insulating layer introduces a phase difference between wave functions of
Cooper pairs on both sides of insulating layer. Because of this phase difference super current
flows across the junction, even when the applied voltage is zero. This is known as DC Josephson
effect.
The super current flowing through the junction is given by
IS = IC Sin0
Where, IC critical current at zero voltage condition. It depends on the thickness of
insulating layer and temperature.
0 the Phase difference between the wave function on either side of insulator.

Applications:

SQUID:

SQUIDs are the acronym for "Superconducting QUantum Interference Device".

It is used to measure tiny or weak magnetic fields of the order of 10-14 T and also used
as diagnostic tool in the detection of brain signals.
It mainly consists of superconducting ring with one or two Josephson junctions. The
magnetic flus interacting with superconducting loop is quantized in steps of 0= .
There are two types of SQUIDs
1. DC SQUIDs
2. AC SQUIDs (RF SQUIDs)

DC SQUIDS:
It works on the principle
of DC Josephson effect. It
consists of two josepson junctions
JA and JB. current entering the
superconducting ring (Iin) is
divided in to two parts IA and IB
as shown in the figure.
If an external magnetic
field is applied it modifies the
phase of two currents IA and IB which in turn modifies the output current (Iout) due to
interference. The output current varies periodically with applied magnetic field or
flux. Thus if the magnetic field (B) is applied to a SQUID, its output current changes.
This change in the current is used to measure the magnetic flux with help of feedback
current (voltage V).

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Numerical on Superconductivity
Solved Numerical:

1. The critical temperature of Nb is 9.15K. At 0 Kelvin, the critical field is 0.196T.

Calculate the critical field at 8K.
Solution:
Given Data: Critical magnetic field at zero Temperature Ho = 0.0196 T,
Critical temperature Tc = 9.15 K, and
Critical field at T = 8K ,Hc = ?
[ ( ) ]

0 ( ) 1
0.0046 T

2. A superconducting tin has a critical temperature of 3.7 K at zero magnetic field and a
critical field of 0.0306 Tesla at 0 K. Find the critical field at 2 K.
Solution:
Given Data: Critical field at zero Temperature Ho = 0.0306 T,
Critical temperature T c = 3.7k,
Critical field at T = 2K, HC=?
[ ( ) ]

0 ( ) 1
= 0.0216 T or 2.16 × 10-3T

3. Lead has a superconducting transition temperature of 7.26 K. If the initial critical field
at 0K is 50 × 103 Am-1 , Calculate the critical field at 6K.
Solution:
Given Data: Initial critical field at zero Temperature, Ho=50 × 103 Am-1 ,
Transition temperature Tc = 7.26K,
Critical field at T = 6K, HC= ?

0 ( ) 1

0 ( ) 1

4. At the temperature of 6 K critical magnetic field is 5x103 Am-1, Calculate the transition
temperature when a critical magnetic field is 2x104 A/m.
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Solution:
Given Data: Initial critical field at T = 6 K is ,Hc =5 × 103 Am-1 ,
Field at 0 K, Ho = 2 × 104 Am-1 ,
Transition temperature Tc= ?,

0 ( ) 1

[ –( ) ]

0 ( ) 1
Tc = 6.93K

5. The superconducting transition temperature of Lead is 7.26K. The initial critical field at
0K is 64×103 Am-1. Calculate the critical magnetic field at 5K.
Solution:
Given Data: Critical magnetic field at 0 K, Ho =64 × 103 Am-1
Transition temperature Tc = 7.26 K,
Critical magnetic field at T = 5K, HC=?

0 ( ) 1

0 ( ) 1

6. A superconducting material has a critical temperature of 3.7 K in zero magnetic field

and a critical magnetic field of 0.02 Am-1 at 0 K. Find the critical magnetic field at 3 K.
Solution:
Given Data: critical Field at 0 K, Ho = 0.02 T
Transition temperature Tc = 3.7 K,
Critical field at T = 3K, HC= ?
: [ ( ) ]

[ ( ) ]

7. A superconducting sample has a critical temperature of 3.722 K in zero magnetic field

and a critical field of 0.0305 T at 0 K. Evaluate the critical field at 2 K.
Solution:
Given Data: Field at zero Temperature, Ho =0.0305T
Transition temperature Tc = 3.722 K,
Critical field at T = 2K, HC=?

0 ( ) 1

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0 ( ) 1

8. The material lead (Pb) behaves as a superconductor at a temperature of T c = 7.26 K.

Ifthe value of critical magnetic field of lead at 0 K is Hc = 8 × 10 5 A/m, find the
critical magnetic field of lead at 4 K.
Solution:
Given Data: Field at zero Temperature, Ho =8 × 105 A/m
Transition temperature Tc = 7.26 K,
Critical field at T = 4K, HC=?

0 ( ) 1

0 ( ) 1

9. Determine the transition temperature and critical field at 4.2 K for a given specimen
of a superconductor if the critical fields are 1.41 × 10 5 and 4.205 × 10 5 A/m at 14.1 K
and 12.9 K, respectively.
Solution:
Given Data: and =? At =4.2K
If =1.41 × A/m at
= 4.205 × A/m at

0 ( ) 1

[ –( ) ] ------------- (1)

0 ( ) 1

[ –( ) ] ------------- (2)
To find TC
Equation (1) / (2) => =14.66K
By using Eqn (1) or (2)
[ –( ) ]
5
= 18.81 x 10 A/m
Caluclation of using = 18.81 x 10 5 A/m and =4.2K

[ . / ]

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0 –( ) 1
\

10. The critical field for lead is 1.2 × 10 5 A/m at 8 K and 2.4 × 10 5 A/m at 0 K. Find the
critical temperature of the material.

Given Data: Field at 0 K Temperature, Ho =2.4 × 10 5 A/m

Critical temperature Tc= ?
Critical field at T = 8K, HC=1.2 × 10 5 A/m
Solution: [ ( ) ]

[ ( ) ]

( )

=> =11.31K

Exercise Problems:

1. Lead has superconducting transition temperature of 7.26K. If the initial critical field at 0 K
is 75 × 10 3 Am -1 Calculate the critical field at 7K.
2. A superconducting tin has a critical temperature of 3.4K at zero critical magnetic field and
a critical magnetic field of 0.0206 tesla at 0K. Find the critical magnetic field at 5K.
3. The superconducting transition temperature of Lead is 7.26K. Calculate the initial critical
field at 0K.Given the critical magnetic field at 8K as 33.644 x 10 3 Am -1
4. Calculate the ratio of critical fields for a superconductor at 7K and 5K given thecritical
temperature 8K.
5. The critical field for niobium is 1 × 105 Am-1 at 8K and 2 x 105 Am-¹ at 0 K.Calculate the
transition temperature of the element.

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