Superconductivity

The electrical resistivity of many metals and alloys drops suddenly to zero when the specimen
is cooled to a sufficiently low temperature. The specimen is said to have passed into
superconducting state. The temperature at which a phase transition takes place from normal
state to superconducting state is called critical or transition temperature Tc. In 1911, Heike
Kamerlingh Onnes firstly observed the disappearance of electrical resistance in mercury in a
bath of liquid Helium. In the superconducting state, the dc electrical resistivity is zero or close
to zero that persistence electrical currents are found to flow without attenuation for years and
years together.

Meissner Effect

A piece of material in the in the normal state is placed in an external magnetic field, which will
penetrate the material in the ordinary way. If the material is brought into the superconducting
state, by cooling it below critical temperature, the magnetic flux originally present is ejected
from the specimen. This is called the Meissner effect. The superconductor is characterized by
two specific properties:

(i) Resistivity is zero

(ii) Magnetic induction is zero

When a superconductor is placed in an external magnetic field Ba, the magnetic flux density
(B) inside is zero

𝐵 = 𝐵𝑎 + 𝜇0 𝑀

The strength of the magnetic field Ha is given by 𝐵𝑎 = 𝜇0 𝐻𝑎 , 𝜇0 is the permeability of the free
space.

So,

𝐵 = 𝜇0 𝐻𝑎 + 𝜇0 𝑀 = 𝜇0 (𝐻𝑎 + 𝑀)

But, B = 0. Hence, 𝜇0 (𝐻𝑎 + 𝑀) = 0

𝑀
So, 𝐻𝑎 = −𝑀 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝜒 = 𝐻 = −1
𝑎

This is maximum value of susceptibility of a diamagnet. Thus, superconductor is a perfect

diamagnetic material.
The magnetic field does not penetrate into the body of the superconductor.

Thermodynamics of the Superconducting Transition

The transition between the normal and superconducting states is thermodynamically reversible,
just like the transition between liquid and vapor phases of a substance is reversible.

We apply thermodynamics to the transition and obtain an expression for the entropy difference
between normal and superconducting states in terms of critical field curve Hc versus T. Type I
superconductor with a complete Meissner effect so that B = 0 inside the superconductor. We
will see the critical field Hc is a quantitative measure of the energy difference between the
superconducting and normal states at absolute zero. This energy at 0 K is called stabilization
energy. For type II superconductor, Hc is understood to be the thermodynamic critical field
related to the stabilization energy. The stabilization energy of the superconductors state with
respect to normal state can be determined calorimetrically or magnetically. The work done on
a superconductor when it is brought from a position at infinity where the applied field is zero
to a position ‘r’ in the field of permanent magnet.

𝐵𝑎
𝑤 = ∫ 𝑀 𝑑𝐵𝑎 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑝𝑒𝑐𝑖𝑚𝑒𝑛
0

As we have 𝑑𝑄 = 𝑑𝑈 + 𝑑𝑤 ⇒ 𝑇𝑑𝑠 = 𝑑𝑈 + 𝑑𝑤

Now, 𝑑𝑈 = 𝑇𝑑𝑠 − 𝑑𝑤 = 𝑇𝑑𝑠 − 𝑀. 𝑑𝐵𝑎 ;

𝑀 1
M related to Ba; = − 4𝜋 (𝐶𝐺𝑆)
𝐵𝑎

𝑀 1
=− (𝑆𝐼)
𝐵𝑎 𝜇𝑜

1
𝑆𝑜, 𝑑𝑈 = 𝑇𝑑𝑠 + 𝐵 𝑑𝐵 (𝐶𝐺𝑆)
4𝜋 𝑎 𝑎

1
𝑑𝑈 = 𝑇𝑑𝑠 + 𝐵 𝑑𝐵 (𝑆𝐼)
𝜇𝑜 𝑎 𝑎

At absolute zero, Tds = 0 the increase in energy density.

𝐵𝑎 2
𝑈𝑠 (𝐵𝑎 ) − 𝑈𝑠 (0) = (𝐶𝐺𝑆)
8𝜋

𝐵𝑎 2
𝑈𝑠 (𝐵𝑎 ) − 𝑈𝑠 (0) = (𝑆𝐼)
2𝜇𝑜

For a normal state of a metal, χ is vanishingly small, i.e., M ≈ 0.

Energy of normal state is independent of field; 𝑈𝑁 (𝐵𝑎𝑐 ) = 𝑈𝑁 (0)

At this critical value Bac of the applied magnetic field, the energies are equal in normal and
superconducting states. 𝑈𝑁 (𝐵𝑎𝑐 ) = 𝑈𝑠 (𝐵𝑎𝑐 )

𝐵𝑎𝑐 2
𝑈𝑁 (𝐵𝑎𝑐 ) = 𝑈𝑠 (𝐵𝑎𝑐 ) = 𝑈𝑠 (0) + (𝐶𝐺𝑆)
8𝜋
 𝐵𝑎𝑐 2
𝑈𝑁 (𝐵𝑎𝑐 ) = 𝑈𝑠 (𝐵𝑎𝑐 ) = 𝑈𝑠 (0) + (𝑆𝐼)
2𝜇𝑜

𝐵𝑎𝑐 2
Again, 𝑈𝑁 (0) = 𝑈𝑠 (0) + (𝐶𝐺𝑆)
8𝜋

𝐵𝑎𝑐 2
So, ∆𝑈 = 𝑈𝑁 (0) − 𝑈𝑠 (0) = (𝐶𝐺𝑆)
8𝜋

Where, ∆U is the stabilization energy density of the superconducting state at absolute zero.

(105)2⁄ −3
For Al, Bac at absolute zero is 105 Gauss so that ∆U = 8𝜋 = 439 𝑒𝑟𝑔 𝑐𝑚 which
is in agreement thermal measurements 430 erg cm-3.

The free energy density FN of a non-magnetic metal is independent of B. At T < Tc the metal
is superconductor in zero magnetic field so that Fs (T, 0) < FN (T, 0). An applied magnetic field
𝐵𝑎 2 𝐵𝑎 2
increases Fs by (CGS) so that 𝐹𝑠 (𝑇, 𝐵𝑎 ) = 𝐹𝑠 (𝑇, 0) + . If Ba is larger than Bac, the free
8𝜋 8𝜋

energy density is lower in the normal state than in superconducting state and the normal state
is stable. The figure applies to UN and Us at T = 0.

Effect of Magnetic Field

Superconducting state of a metal exists in a particular range of temperature and magnetic field
strength. Superconductivity will disappear if the temperature of the specimen is raised above
its Tc or a sufficiently strong magnetic field is applied. The curves are nearly parabolic and can
be represented by
 𝑇 2
𝐻𝑐 = 𝐻0 [1 − ( ) ]
𝑇𝑐

Where Hc is the maximum critical field strength at temperature T, Ho is the maximum critical
field at absolute zero and Tc is the critical temperature.

Fig.: Variation of critical field as a function of temperature

 Type I and Type II superconductors

They are classified as soft (Type I) and hard (type II) superconductors. A superconductor which
exhibits complete Meissner effect is called Type I superconductor.

Fig.: (a) Magnetization curve for type I superconductors; (b) Magnetization curve for type II
superconductors

Type II superconductors exist in mixed state, with superconducting and normal regions. They
partially allow the magnetic flux and have zero electrical resistance.

Fig.: Magnetic phase diagram of type II superconductors

They are characterized by a lower critical field Hc1 at which magnetic flux enters the
superconductor and an upper critical field Hc2 at which superconductivity disappears.

In superconductors once the current starts flowing it can produce magnetic field without any
loss of energy. If the magnetic field exceeds Hc1, they become hard. Many materials were
synthesized for stable high field superconducting magnets. Now copper oxide is Type II
superconductor with Hc2 equals to 150T.

Lead is Type I superconductor with 𝐻𝑐 = 4.8 × 104 𝐴/𝑚 𝑎𝑡 4𝐾

A material can change from Type I to Type II superconductor.

𝐴
With 20% Indium added lead, 𝐻𝑐1 = 5.6 × 103 𝑚 𝑎𝑛𝑑 𝐻𝑐2 = 2.9 × 105 𝐴/𝑚, which is Type

II superconductor.

Vortex state: In a mixed state of type-II superconductor, the external magnetic field will
penetrate the thin normal regions uniformly, and the field will also penetrate somewhat into
the surrounding superconducting material. The term votex state describe the circulation of
superconducting currents in vortices throughout the bulk specimen. There is no chemical and
crystallographic difference between the normal and the superconducting regions in the voetex
state. A type-II superconductor is characterized by a vortex state stable over a certain range of
magnetic field strength; namely between Hc1 and Hc2.

London Equation (Two-fluid model of superconductor)

According to London’s theory, there are two types of conduction electrons in a superconductor,
the super conducting electrons and the normal electrons. At 0K, a superconductor contains only
super conducting electrons, but as the temperature increases the ratio of normal electrons to the
super conducting electrons increases until at the transition temperature all the electrons are
normal.

Let nn, vn, and ns, vs are the densities and velocities of normal and superconducting electrons.

𝑛 = 𝑛𝑛 + 𝑛𝑠

𝐽 = 𝐽𝑛 + 𝐽𝑠 = 𝑒𝑛𝑛 𝑣𝑛 + 𝑒𝑛𝑠 𝑣𝑠

In superconducting metal, the superconducting electrons encounter no resistance to their

motion. Equation of motion of superconducting electron

𝑑𝑣𝑠
𝑚 ( ) = 𝑒𝐸 ⇒ 𝑚𝑣𝑠 = 𝑒𝐸
𝑑𝑡
 𝑑𝐽𝑠 𝑑𝑣𝑠
𝐽𝑠 = 𝑒𝑛𝑠 𝑣𝑠 ⇒ = 𝑒𝑛𝑠
𝑑𝑡 𝑑𝑡

𝑒𝐸 𝑛𝑠 𝑒 2 𝐸
So, 𝐽𝑠̇ = 𝑒𝑛𝑠 𝑣𝑠̇ = 𝑒𝑛𝑠 =
𝑚 𝑚

This is first London’s equation.

The corresponding equation for normal electrons is:

𝑛𝑛 𝑒 2 𝜏𝐸
𝐽𝑛 = 𝜎𝑛 𝐸 =
𝑚

Electric field is necessary for establishing a steady current. If E = 0, Jn = 0 unlike the

superconducting state.

From Maxwell’s equations,

𝐵̇ = − 𝑐𝑢𝑟𝑙 𝐸

𝑐𝑢𝑟𝑙 𝐻 = 𝐽𝑠 + 𝐷̇

⇒ 𝑐𝑢𝑟𝑙 𝐵 = 𝜇𝑜 (𝐽𝑠 + 𝐷̇); 𝐽𝑠 𝑖𝑠 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑑𝑒𝑛𝑠𝑖𝑡𝑦

Unless the fields are varying very rapidly with time, the displacement current Ḋ is negligible
in comparison with Js.

So,
𝐵̇ = −𝑐𝑢𝑟𝑙 𝐸 (1)

𝑛𝑠 𝑒 2 𝐸 𝑚
Also, 𝐽𝑠̇ = 𝑠𝑜, 𝐸 = 𝐽̇ 𝑎𝑛𝑑 𝑐𝑢𝑟𝑙 𝐵 = 𝜇𝑜 𝐽𝑠 𝑠𝑜, 𝑐𝑢𝑟𝑙 𝐵̇ = 𝜇𝑜 𝐽𝑠̇
𝑚 𝑛𝑠 𝑒 2 𝑠

Using these above relations, equation (1) becomes

𝑚
𝐵̇ = − ( ) 𝑐𝑢𝑟𝑙 𝐽𝑠̇
𝑛𝑠 𝑒2

𝑚
⇒ 𝐵̇ = − ( 2
) 𝑐𝑢𝑟𝑙 𝑐𝑢𝑟𝑙 𝐵̇
𝜇𝑜 𝑛𝑠 𝑒

𝑚
⇒ 𝐵̇ = −𝛼 𝑐𝑢𝑟𝑙 𝑐𝑢𝑟𝑙 𝐵̇ Where 𝛼 =
𝜇 𝑜 𝑛𝑠 𝑒 2

But, 𝑐𝑢𝑟𝑙 𝑐𝑢𝑟𝑙 𝐵̇ = 𝑔𝑟𝑎𝑑 𝑑𝑖𝑣 𝐵̇ − 𝛻 2 𝐵̇

From Maxwell’s equation, 𝑑𝑖𝑣 𝐵̇ = 0, 𝑆𝑜 𝑐𝑢𝑟𝑙 𝑐𝑢𝑟𝑙 𝐵̇ = −𝛻 2 𝐵̇

𝐵 ̇
Hence, 𝐵̇ = 𝛼 𝛻 2 𝐵̇ ⇒ 𝛻 2 𝐵̇ = 𝛼

Meissner effect shows that the flux density in a superconductor is not only constant but also
zero.

Not only Ḃ but B itself must die away rapidly below the surface.

𝐵 𝜕 2 𝐵(𝑥) 𝐵(𝑥)
𝛻2𝐵 = ⇒ =
𝛼 𝜕𝑥 2 𝛼
Where B(x) is the flux density at a distance x inside the metal.

The solution of this equation is of the form of-

𝑥
𝐵(𝑥) = 𝐵𝑎 𝑒𝑥𝑝 (− )
√𝛼
𝑚
If we replace Ḃ by B in the equation, Ḃ = − 𝑛 2
𝑐𝑢𝑟𝑙 𝐽𝑠̇
𝑠𝑒

𝑚
We get 𝐵 = − 𝑛 2
𝑐𝑢𝑟𝑙 𝐽𝑠
𝑠𝑒

𝑛𝑠 𝑒 2
And also, we have 𝐽𝑠̇ = 𝐸
𝑚

𝑥 𝑚
Now 𝐵(𝑥) = 𝐵𝑎 𝑒𝑥𝑝 (− 𝛼) 𝑤ℎ𝑒𝑟𝑒, 𝛼 = 𝜇 𝑛 𝑒2
√ 𝑜 𝑠

Where Ba is the flux density of applied field at the surface. The flux decays exponentially in a
super conductor falling 1/e of its value at the surface at a distance 𝑥 = √∝. This distance is
called penetration depth λ.

𝑚
So, 𝜆 = √𝛼 = √𝜇 𝑛 𝑒2
𝑜 𝑠

Now, we can write London equation,

𝐵 𝐸
𝑐𝑢𝑟𝑙 𝐽𝑠 = − 𝑎𝑛𝑑 𝐽𝑠̇ =
𝜇𝑜 𝜆2 𝜇𝑜 𝜆2
 Coherence Length

A massive Type II super conductor when in a magnetic field is considered as collection of

filaments (parallel to the magnetic field) of a normal conductor and a super conductor in an
alternating pattern. The distance from the interface between a normal conductor and super
conductor over which the density of Cooper pairs rises from zero to its maximum value is
termed as coherence length.

The maximum distance up to which the states of paired electrons (cooper pair) are correlated
to produce superconductivity is called coherence length (𝜀𝑜 ).

𝜆
The ratio of London’s penetration depth (λ) to the coherence length (𝜀𝑜 ) is 𝐾 = (Landau
𝜀𝑜

factor) parameter.

1
For type I superconductor; 𝐾 <
√2

1
For type II superconductor; 𝐾 >
√2

The coherence length is related to the band gap

𝑣𝐹 ћ
𝜀𝑜 =
2𝛥

where 2Δ is the energy gap and vF is the electron velocity at the fermi surface.

Cooper Pair

In Bardeen, Cooper and Schrieffer (BCS) theory, the many electron system is described in
terms of one particle orbital. The pair of electrons are occupied in one particle orbital. If the
orbital with wave vector K and spin up is occupied and then the orbital with vector -K and spin
down is also occupied. If k1↑ is vacant, then -k1↓ is also vacant. These pairs are called Cooper
pairs. They have spin zero and have many attributes of bosons.

For temperature T < Tc, the lattice – electron interaction is stronger than the electron – electron
coulomb force. At this state, the Cooper pairs of electrons will have a property of sailing
through lattice without any exchange of energy i.e., the Cooper pairs are not scattered by the
lattice points.
The electron – electron attraction results from electron – lattice interaction. Electron 1 is
negatively charged. It attracts positive ions towards itself. The electron 1 is screened by the
positive ions. The screening may reduce the effective charge of the electron. The ions may over
respond and produce a net positive charge. If this happens, then electron 2 will be attracted
towards electron 1. This leads to a net attractive interaction and formation of Cooper pairs.

An electron deforms the lattice in its vicinity, exciting a phonon that travels through the crystal.
This phonon is attracted by a second electron, getting thus coupled to the first electron.

Fig.: Electron – electron interaction through lattice phonons.

If k1 and k1’ denote the initial and final wavevectors of the first electron, and k2 and k2’ denote
the initial and final wavevector of the second electron. Then the momentum conservation
requires 𝑘1 − 𝑘1′ = 𝑘2 − 𝑘2′ = 𝑞, where, q is the wave vector of the phonon involved in the
process.

Josephson Effect

The Josephson effect describes the tunneling of Cooper pairs from a superconductor through
the layer of an insulator into another superconductor.

The nature of Josephson’s effects can be understood in a way as described below.

A current is made to flow in a bar of superconductor. A voltmeter is connected across the ends
of the bar as shown in Fig. (a) below.
 Fig.: Josephson’s Effect
The voltmeter indicates a drop-in voltage as zero across the superconductor according to our
expectation. Suppose the bar is cut into two pieces and the two pieces are separated by say 1
cm as shown in Fig. (b). No current will flow and the voltmeter will indicate a voltage equal to
the open circuit voltage of the current source. If the distance between the pieces is reduced to
1 nm, the voltmeter suddenly shows zero voltage showing thereby that a current flow across
the gap in a superconducting way. This is known as the d.c. Josephson's effect. Another effect
which is observed if that the voltmeter indicates a voltage, but at the same time a very high
frequency electromagnetic radiation emanates from the gap, indicating the presence of a very
high frequency alternating current in the gap. This phenomenon is known as a.c. Josephson’s
effect.
The accomplishments of Josephson’s effects are:

• A d.c. current flows across the junction in the absence of any magnetic or electric field.
• When a d.c. voltage is applied across the junction, it causes r.f. current oscillations
across the junction. This effect has been utilised for the precise determination of the
value of h/e. Further, an r.f. voltage applied with d.c. voltage can then cause a flow of
d.c. current across the junction.
• When a d.c. magnetic field applied through a superconducting circuit containing two
junctions, it causes the maximum supercurrent to show interference effects as a function
of magnetic field intensity. This effect can be utilised in sensitive magnetometers.
 Superfluidity
The ability of the fluid to flow through fine capillaries without outward friction refers to
superfluidity. It is one of the interesting properties of superfluid, which occurs in certain
substances and under some special conditions. This property occurs in extra-terrestrial systems
like neutron stars. There exists some evidence, which shows that some other terrestrial systems
also hold this property like excitons. Excitons are bosonic quasi-particles formed by bound
electron-hole pairs and can therefore condense and exhibit superfluidity at sufficiently low
temperature. When stirred, a superfluid forms vortex that continue to rotate indefinitely.
Examples of superfluid include helium-3 (or 3He) and helium-4 (or 4He). For temperatures
below 2.17 K, helium-4 becomes a superfluid. Helium-3 becomes a superfluid only below
0.0025 K.

The unique properties of superfluid are caused by the absence of interparticle attraction. The
attraction between helium molecules can be eliminated primarily due to its unique atomic
structure: the combination of small atomic size and even distribution of electrons.

Discovery of Superfluid Liquids

The phenomenon of superfluid helium is a stimulating property, which one can observe directly
in ultracold atomic gases and helium isotopes. Helium-3 and Helium-4 are the two stable
isotopes of helium, which remain liquid at low pressures to absolute zero. Both isotopes exhibit
the property of superfluidity. Helium-4 was liquefied in 1908, but in 1936 and 1937 scientists
recognized that below the temperature of 2.17 degrees absolute (lambda point), it possessed
properties different from any other substance i.e., the thermal conductivity of the low-
temperature phase (He-II) is very large, but with anomalously low viscosity.

In 1938, Pyotr Kapitza in Moscow and John Allen and Don Misener at the University of
Cambridge simultaneously performed a direct measurement of the behaviour of the viscosity
of the helium contained in a thin tube as a function of temperature. Both groups found a drop
in viscosity in He-II, which appeared discontinuously at the lambda point. On the basis of the
analogy with superconductivity, Kapitza coined the term superfluidity for this behaviour.

Relation between Superfluidity and Superconductivity

According to modern science, superconductivity is like a superfluid phenomenon that arises in

electrically charged systems. The term superconductivity means that an electric current can
flow without any resistance like a super liquid, which can flow down a narrow channel without
any friction. The phenomenon of superconductivity exists in various metals like Nb, Al, and
Sn at low temperatures. It occurs due to the movement of conduction electrons without any
resistance in a metal. So, one can understand this phenomenon as superfluidity of the
conduction electrons.

Superfluidity is related to the congregation of atoms into the same quantum state – Bose
Einstein condensation (BEC).

Applications- Superfluidity find applications in Cryogenics, Quantum Computing , etc.