## Superconductivity

## Introduction

- The electrical conductivity of metals drops suddenly to zero, when the specimen is cooled to a sufficiently low temperature. This zero resistivity or infinite conductivity is "super conductivity and the materials are called super conductors.
- It was first observed in 1911 by a Dutch physicist Kamerlingh Onnes in the course of his experiments on electrical conductivities of different metals at low temperatures.

- He observed that when pure Mercury is cooled, its resistivity vanishes at 4.2K.
- The temperature at which the normal metal transforms to super conducting state is called " Transition temperature " and it varies from metal to metal.

*General properties of superconductors:*

The properties of superconducting metals can be changed by varying temperature, magnetic field, electric field, isotopic mass, atomic structure, size and frequency.

- Superconductors have zero resistivity and infinite conductivity.
- All the superconducting materials are diamagnetic in nature.
- Copper, Gold, Silver are the best conductors of electricity but not good superconductors.
- Superconductivity has been observed only for those having valence electrons between 2 to 8.
- Elastic properties and thermal properties do not change in the transition,
- Seebeck effect, Peltier effect, Thomson effect are disappeared in superconducting state.

*Critical magnetic field:*

- Superconductivity can be destroyed or its normal resistance can be restored, if the material is exposed to magnetic field of sufficient strength.
- When the magnetic field is gradually increased, at a particular value, superconductivity is destroyed.
- The field at which super conductivity is destroyed is called " Critical Magnetic field ".
- Critical Magnetic field is a function of temperature. Its value decreases, when then the temperature is increased from T=0 K to T=Tc. for example,

The variation of critical field with temperature in given by

critical magnetic field is maximum at zero kelvin and zero at transition temperature.

*Critical current :*

- The minimum current that can be passed in a super conductor without destroying its super conductivity is called "Critical current".
- When the current increases beyond this value, the production of magnetic field also increases and super conductivity disappears.

This is silsbee's rule.

*Meissner effect :*

- Superconductors expels out magnetic flux completely, below the transition temperature Tc, this phenomenon is known as Meisnner effect.
- When a long cylinder superconductor is cooled in the magnetic field to below its transition temperature, the lines of induction will be pushed out the material due to infinite conductivity.

If B=0 according to Meissner effect

susceptibility is negative only for diamagnetic materials, so superconductors exhibit perfect diamagnetism. This is called Meisnner effect.

*Type I and Type II superconductors :*

__Type I superconductors :__- Type I superconductors are soft superconductors.
- They exhibit perfect Meisnner effect.
- They have only one critical field value Hc, below Hc the specimen is a superconductor and beyond Hc they are normal conductors.
- Examples are Gallium, Zinc, Aluminium, Mercury.

*Type II superconductors :*

- These superconductors are hard conductors.
- They have two critical field values of Hc1 and Hc2.
- Below Hc1 material is diamagnetic and it exhibits superconductivity. Hc1 is called Lower critical field. When the field is increased beyond Hc1 magnetic flux starts penetrating until Hc2. Hc2 is called Upper critical field.
- Above Hc2 the specimen is in normal state.
- In between Hc1 and Hc2 the substance is in mixed state.
- Examples are Zrconium (Zr), Niobium (Nb)

*London's equations*

__London's first equation :__

In 1935 London brothers, derived two field equations to explain the phenomenon of superconductivity. London's theory assumed that there are two types of conduction electrons in a superconductor namely normal electrons and super electrons. Superconductors are not subjected to any lattice scattering and they are accelerated in the direction opposite to the field.

current density of super electrons is

Force experienced by each electrons with the application of field

if E=0 then ................................................

This equation shows that superconducting specimen have finite current density in the absence of electric field.

London's first equation is modified as London's second equation by using Maxwell's equations. They are

__Note:__London's first equation is modified as London's second equation by using Maxwell's equations. They are

__London's second equation:__

The electric field E, when it is made to zero, it leads to another important result. When it is combined with Maxwell's equations.

put E=0 then dB=0 which implies B=constant.

It indicates that B is constant inside a superconductor. This result is not in agreement with the Meisnner effect. He modified the equation by taking London's first equation.

It indicates that B is constant inside a superconductor. This result is not in agreement with the Meisnner effect. He modified the equation by taking London's first equation.

by applying curl on both sides we get

according to Maxwell's equations

by integrating the equation with respect to time and taking constant equals to zero, we get

__London's penetration depth:__

According to London's second equation we have

applying curl on both sides we get

we also know that

by equating both the above equations we get

where .............. is called penetration depth.

The distance from the surface of superconducting specimen at which current density fall by 1/e times of its original value.

penetration depth is a function of temperature.

__Note:__penetration depth is a function of temperature.

*B,C,S theory:*

The most successful microscopic theory developed by Barden, Cooper, Schrieffer in 1957 is known as B,C,S theory.

- Let us consider an electron passing through a lattice of positive ions.
- Because it is negatively charged it is attracted by the positive ions and hence its effective charge is greatly reduced.
- Now suppose another electron, passes by the side of that assembly of positive charges, then it gets attracted to the deformed lattice and the electron in the that deformed lattice.
- In the language of field theory these two electrons are paired by the exchange of virtual phonon via lattice deformation. Let k1 be the wave vector of first electron and k2 be the wave vector of second electron. so,

Here the first electron looses its momentum, while the second electron gains it and so the overall momentum remains constant.

- The energy of cooper pairs in the bounded state is less than the energy of unbounded electrons.
- Cooper pairs has energy less than 2Ef and the pairing between the two electrons can be broken if we go beyond the transition temperature.
- Then these cooper pairs are called Bosons.
- In the distribution of states, they wont follow Pauli's exclusion principle.

*Flux Quantization:*

- When a hollow cylinder or ring of a superconducting material, is placed in an external magnetic field and then cooled below its transistion temperature, finite amount of magnetic flux will remain even in the absence of magnetic field with in the material.
- This is due to circulating current in superconducting cylinder which is calle " Persistent Current ".
- The magnitude of magnetic flux decreases when the persistent current decreases.
- The magnitude of the magnetic flux is given by

*Josephson effect:*

Josephson theoritically showed that and electron pair can also tunnel from one superconductor to the other, when these are coupled together with fine insulating layer. This effect is called Josephson effect.

The insulating barrier must be very weak so that there is a very low probability of finding the cooper pair in that region. The thickness should be less than 50 angstroms.

__DC Josephson effect:__- When two superconductors are coupling together a direct current flows through the junction without applying any field. This effect is called DC Josephson effect.
- The current across the junction is given by

Here the ............... angle is the phase difference across the junction. The value of .................... depends on the thickness of the sheet and temperature.

__AC Josephson effect:__- When voltage is applied across the junction of two superconductors, an alternative current is produced across the junction. This effect is called AC Josephson effect.
- This results in an additional phase difference introduced by cooper pair across the junction. In such case energy difference is of cooper pair is given by

we also know the energy difference is

by equating both the above equations we get