Why does superconductivity occur?

[|mathematix|]

I can't get a definitive explanation of why superconductivity happens and I am getting mixed explanations from my textbooks.
I will tell you what I know and hopefully you can correct any misunderstandings I have:

• A metal consists of an ionic lattice because the electrons in the valence band have 'jumped' to the conduction band

• When a potential difference is applied across a conductor, electrons would move towards the positive potential but the lattice will become distorted as the ions are attracted to the electrons

• This creates a region that has dense positive charge. Electrons would be attracted towards this region. However, electrons repel each other and don't want to be close to each other, additionally, Wolfgang Pauli's exclusion principle states that no two fermions can occupy the same quantum state so every two electrons have to combine and form a Bose Einstein condensate as bosons can occupy the same quantum state.

• Cooper pairs arise because of the exchange of phonons. Phonons are a collection of excitations of atoms or molecules. The atoms or molecules have to be vibrating in some collective mode.

• The bond between two electrons in a boson is very weak, 10^-3 eV. Therefore, the temperature of the metal must be 10 Kelvin for the cooper pairs to exist (using E=kT where k is 10^-4 eV).

• Bosons do not interact with matter, hence, they aren't impeded by the lattice.

There are three things I do not understand about this explanation:
1) Pauli's principle
2) Why bosons can be in the same energy state and fermions can't
3) Phonons

Thank you!

 

[CompuChip]

1) and 2) are actually the same thing. Or, to be more precise, 2) follows from 1).
The explanation is quantum mechanical, I am not sure if you have any background in that so I will try to phrase it somewhat generally.

Pauli's exclusion principle says that fermions can't be in the same quantum state. The quantum state is usually represented by a number of (relevant) quantum numbers like spin and total energy (angular momentum is another one, but it is not relevant here). This means that the only way two fermions can have the same energy, is if they have different spins. Because in quantum mechanics energy levels are discrete, and the number of spin states is limited (2, for spin-1/2 particles) that means that once a lower energy level is filled up with the maximum number of 2n particles (for spin (2n-1)/2 particles) other fermions will have to go into the higher states.

The reason for Pauli's exclusion principle is pretty fundamental, and actually has to do with the reason that we have to consider both fermions and bosons. Basically, it comes down to a requirement on wave functions having to be symmetrical or anti-symmetrical. It then turns out that there is some quantum number (property) that we call spin, which takes on half-integer values for all anti-symmetric wave functions, or integer values for all symmetric wavefunctions. This property strongly influences the behaviour in terms of allowed states and interactions, and leading to things like the exclusion principle. This connection between spin and the statistics of the particle is quite deep. Useful to know is that most of the things that we see around us, such as the electrons, protons and neutrons in all materials, are fermions. Bosons are usually the particles that carry forces, such as the W and Z-bosons in the Standard Model of Physics which mediate the strong and weak nuclear force and electromagnetic force (my favourite image here is two skaters who move away from each other by throwing a ball without actually touching each other - here the skaters could be seen as the particles and the ball as the boson that transfers a - repulsive, in this case - force).

In solid state and condensed matter physics, it is common to consider certain interactions (i.e. exchanges of energy) as particles. Phonons are not "real" particles as opposed to e.g. the bosons that transfer nuclear and electromagnetic forces, but both mathematically and conceptually it is convenient to view them that way. In the case of Cooper pairs, it is important to realize that the two electrons in such a pair are not physically close to each other - they can be at completely different locations in the material but they are "bound" into a pair by this exchange.

 

[|mathematix|]

Thank you so much!