# Band Gap in Superconductors

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1. Jun 18, 2015

### Edward888

Hi all,

I am currently writing a report about superconductors, and am currently reading about how the band gap shows that single electrons are not the charge carriers responsible for superconductivity. However, I was confused when I read that electrons are fermions and as such there are no band gaps. I was wondering if someone could possible explain what this means? As I understand it, there is no band gap in a normal metal, but there is in an insulator since otherwise it would conduct electricity, and there are also energy gaps between the 1s, 2s, 3p orbitals etc.

Thanks in advance!

2. Jun 18, 2015

### DrDu

Wher did you read this?

3. Jun 19, 2015

### radium

Take a look at the BCS state and the Hamiltonian. In MF it has a pairing potential which does not conserve charge. That the BCS order parameter. If you do a bogoliubov transformation to diagonalize H in term of factors u and v (the amplitudes for electrons and cooper pairs in the BCS state) you will find that the bogoliubov quasiparticles are like free particles but with a gap, the superconducting gap. This gap is the BCS order parameter, the phase transition happens at the critical temperature when it goes to zero.
The fact that the gap is the pairing potential is significant. The cooper instability causes a bound state to form with a localized potential which is caused by the electron electron interaction mediated by phonons and phonons are vibrational modes of the lattice.

So the pairing potential corresponds to the gap which corresponds to the cooper pair bound state (the thing attracting the electrons).

4. Jun 23, 2015

### Edward888

5. Jun 23, 2015

### radium

Well back to your original question, a "normal" metal (i.e. Fermi liquid) has no band gap. However, the BCS state is most definitely not a normal metal. It has no resistance and its charge carriers aren't even fermions, they are cooper pairs. Cooper pairs are bosons. Bosons have a property where they can all condense into the ground state. This is the Bose Einstein condensate for free bosons. If you have interacting bosons, likewise you can get a superfluid. That's really pretty much what BCS is, the electrons form cooper pairs and undergo condensation to the ground state which is gapped.

An additional characteristic of BCS is if you lower the temperature down below Tc, you will expel all magnetic fields. Fields can only penetrate to a given London penetration depth which can be determined from the London equations. That is where there meissner effect comes from. At a critical value of the field, the superconductivity is destroyed.

6. Jun 23, 2015

### ZapperZ

Staff Emeritus
In addition, the gap is NOT a "band gap", ie it is not from the band structure of the material. It is a gap in the electronic single-particle state. So do not use the term "band gap" in your report or you will be making an unintentional error.

In the future, ALWAYS cite you sources and references before anyone asks for them.

Zz.

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