4. Superconductivity

Question 1

What is superconductivity?

Answer 1

A phenomenon whereby certain pure metals, alloys and compounds the resistance to current flow sharply decreases to zero at a particular temperature.

Question 2

What 2 properties define superconductors

Answer 2

The combination of perfect conduction and perfect magnetic field expulsion is what defines a superconductor.

Question 3

What is the critical temperature Tc?

Answer 3

The temperature at which the transition to superconductivity occurs is known as the critical temperature, Tc.

Question 4

How is the magnetic field expelled from the centre of a superconducting sample?

Answer 4

Consider a solid sphere of superconducting material. If a magnetic field is then applied, currents are induced in the surface of the sphere, which exactly oppose the applied field and cause no magnetic field to penetrate the sample.

Question 5

What is the Meissner effect?

Answer 5

the expulsion of magnetic flux when a material becomes superconducting in a magnetic field.

Question 6

Describe the mechanism behind cooper pair formation

Answer 6

• As an electron moves through the positively charged ion cores of the lattice it attracts them and causes a distortion
• The distortion of the lattice causes the local area to gain a small net positive charge
• The area of positive charge can attract another electron towards the first
• This attraction can overcome the repulsive forces between the lectron and create a bindng between the two electrons
• The electrons can then travel through the lattice as a single entity, known as a Cooper Pair

Question 7

What is the coherence length?

Answer 7

The mean separation at which pair correlation becomes effective, or the mean separation between conduction electrons in a Cooper Pair in a metal. (Cooper pairs overlap greatly. In between one pair, there may be up to 107 other electrons which are themselves bound as pairs)

Question 8

What stops Cooper pairs from exchanging energy with the lattice?

Answer 8

The fact that all cooper pairs are in the same quantum state (are coherent) prevents a single pair from interacting with the lattice and exchanging momentum. For there to be any exchange with the lattice the cooper pair must be broken, this requires a certain amount of energy: the binding energy of the cooper pair. (this is why temp has to be low, otherwise cooper pairs would be broken)

Question 9

What principle do particles with half integer spins follow

Answer 9

Pauli exclusion principle, each energy level can only contain 2 particles of opposite spin

Question 10

What principle do particles with integer spins follow?

Answer 10

they dont obey the pauli exclusion principle, and thus any number of particles can occupy the same energy level

Question 11

What condition must be met for conduction to stop in a superconductor?

Answer 11

In order to stop the current, however, all of the pairs must be broken which would require a considerable combined effort. At the transition temperature, there are no more cooper pairs left

Question 12

What are two limitations on superconductivity except for a maximum temperature?

Answer 12

Superconductivity is destroyed by the application of a sufficiently large magnetic field (The critical magnetic field depends on how far below the critical temperature the material is) or if the flowing electrical current density exceeds a critical value.

Question 13

What is the difference between type I and type II superconductors

Answer 13

They differ in regard to their magnetic properties: type I have a sharp transition from the superconducting state where all magnetic flux is expelled to the normal state, where as type II exhibit similar behaviour by completely excluding a magnetic field below a lower critical field value and becoming normal again at an upper critical field. However, when the magnetic field is between these lower and upper critical fields, the superconductor enters a “mixed state” where there is partial penetration of flux.

Question 14

Describe the magnetic flux behaviour of type II superconductors in their ‘mixed state’

Answer 14

In order to lower the overall magnetic energy, the material allows bundles of flux to penetrate the sample. Within these filaments, the magnetic field is high and the superconductor reverts to normal conducting behaviour. Around each of the filaments is a circulating vortex of screening current which opposes the field inside the core. This arrangement ensures that the material outside these bundles remains in the superconducting state.

Question 15

How do the flux vortices arrange themselves in type 2 semiconductors’ mixed state?

Answer 15

The flux vortices often arrange themselves into regular periodic structures. They can be visualised by covering the surface with a coagulation of very fine ferromagnetic particles.

Question 16

Why does the flux penetrate the material in type II semiconductors?

Answer 16

In order to lower the magnetic energy

Question 17

Why are type 2 semiconductors still semiconducting in their mixed state, and how can this be ruined?

Answer 17

The sample as a whole continues to have zero resistance as current flows by the easiest path and as there are superconducting regions, current can still flow without energy loss. It must be noted however, that if the vortices move they will dissipate energy. For the superconductor to remain lossless, the vortices must be pinned in place by defects within the crystal structure of the material.

Question 18

Describe the relationship that dictates whether a material will be a type I or type II semiconductor

Answer 18

Penetration depth > Coherence length

The reason that some superconductors form a mixed state relies on the relationship between the coherence length, ξ, and the London penetration depth, λ. If the penetration depth, λ, is greater than the coherence length, ξ it is thermodynamically favourable for the magnetic field to penetrate the specimen and it will be type II. This is shown schematically in the diagram below.

Question 19

What causes electrons in a Cooper pair to be bound together in a conventional superconductor such as Nb or Sn?

Answer 19

An interaction between electrons and the vibrations of the lattice

Question 20

Why does a significantly high magnetic field collapse superconductivity?

Answer 20

Due to the fact that the Cooper pairs consist of electrons of opposite spin, and that an applied magnetic field has a tendency to align both electrons in the same direction, breaking down the cooper pairs.

Question 21

What is the susceptibility of a superconducting material?

Answer 21

-1, as the magnetisation is equal to the applied magnetic field (the self-field generated by the induced currents opposes the applied field)

Question 22

What phenomenon allows type 2 superconductors to superconduct at much greater temperatures in their mixed or vortex state?

Answer 22

The presence of inhomogeneities in the sample microstructure is able to pin the flux lines in the flux vortices in place, resisting their motion and hence resisting dissipation of energy.

Question 23

What force balance equation gives you the current carrying capacity of a type 2 superconductor

Answer 23

Fp = Jc x B

Fp= available pinning force density 
Jc= critical current density 
B= magnetic flux

Question 24

What is critical current density Jc a measure of?

Answer 24

How well flux is pinned, current density increases when flux vortices are pinned until the lorentz force (the combination of electric and magnetic force on a point charge due to electromagnetic fields) exceeds the pinning force

Question 25

What a big problem with the high temperture superconductors discovered?

Answer 25

Theyre ceramics, meaning theyre difficult to make and shape into wires.

Question 26

What is the equation for magetic moment m

Answer 26

m = iA

i= current
A=area

Question 27

What does the bean model do?

Answer 27

It gives a macroscopic explanation of the irreversible magnetization behavior (hysteresis) of hard Type-II superconductors (It also gives a way to predict the current in a superconductor given the external field

Question 28

What defines a superconductor in the critical state?

Answer 28

It is fully penetrated by manetic flux and hence carries a critical current everywhere

Question 29

what fraction is the field at the end of a long solenoid compared to that at its centre?

Answer 29

1/2

Question 30

What does integrating the M-H curve give you?

Answer 30

The magnetic contribution to the free energy per unit volume

Question 31

What are the two assumptions in the Bean mode model of current flow in bulk superconductors?

Answer 31

The Bean model assumes (i) that there are no local magnetic dipoles (i.e. unpaired spins)
present in the material so the currents can be treated as in the free space case (i.e. B = µ0 H)
and (ii) that the current flowing is either +Jc or –Jc where the field penetrates the material and
zero elsewhere.