Band Structure in solids versus Superconductivity

[DaTario]

TL;DR Summary

Hi All, Is it possible to give a good explanation of superconductivity using the level (band) diagram of many regularly spaced nuclei as a backdrop? Would it be correct to say that the Bose-Einstein condensate travels through the lowest energy band, moving in the solid by tunneling through the barriers between the nuclei?

Hi All,

Is it possible to give a good explanation of superconductivity using the level (band) diagram of many regularly spaced nuclei as a backdrop? Would it be correct to say that the Bose-Einstein condensate travels through the lowest energy band, moving in the solid by tunneling through the barriers between the nuclei?

I am referring to the following diagram:

Best wishes,

DaTario

 

[doublereen]

Not quite... the diagram that you reference is a nice simplified illustration of average electron density over a lattice. It doesn't really do a good job of illustrating what the electrons are *doing*. What do I mean by this? Why is that important? Electrons in a superconductor are not acting like normal electrons. They associated with each other in just the right way to exempt themselves from the normal rules that govern electrons (fermion rules, or "Pauli exclusion", i.e. not being able to exist in the same place while being in the same quantum state). These superconducting electrons instead follow boson rules (which permits them to exist in the same quantum state as their neighbors), and as such, don't need energy to switch quantum states as they move.

The diagram that you refer to wasn't made to illustrate or capture that kind of behavioral difference between superconducting and regular-conducting electrons, but if I had to use your diagram to explain it, I would say something like what you said, i.e. the SC electrons move horizontally with ease, the conducting electrons always have to travel to the top to move horizontally, and then return to someplace lower in energy (releasing heat from resistance in the process).

 

[Vanadium 50]

Is it possible to give a good explanation of superconductivity using the level (band) diagram

Let me ask you a question that will help you answer it yourself. Are all materials with a band structure superconductors?

 

[DaTario]

Thank you, doublereen and Vanadium 50.

Let me ask you a question that will help you answer it yourself. Are all materials with a band structure superconductors?

I would answer 'no' to this question.
I am a bit familiar with part of the explanation of doublereen, specially when he says that electrons associate with others (possibly forming Cooper's pairs) and this association presents a bosonic nature which permit these pairs (many of them) to cool down to the same state. Being in these collective cooled state resistance practically disapears. These associations are established through interactions mediated by the solid network.

I think it is reasonable to assume that certain atomic lattices in solids do not allow strong correlations between electrons to appear and therefore do not allow Cooper pairs to form.

 

[Vanadium 50]

I would answer 'no' to this question.

Then doesn't that answer your larger question?

 

[DaTario]

Then doesn't that answer your larger question?

It seems to me that the figure I have shown in the OP works like a rather general scenario. My question in other words is: When a solid structure happens to be a superconductor, may the electrons' movement be described with the use of this diagram?

 

[Vanadium 50]

If any material, superconducting or not, has a band diagram, how can the diagram explain superconductivity.

Your message #6 is too vague to answer. Is there any utility of this diagram? Probably something somewhere.

 

[DaTario]

If any material, superconducting or not, has a band diagram, how can the diagram explain superconductivity.

Your message #6 is too vague to answer. Is there any utility of this diagram? Probably something somewhere.

Sorry, I don't think I made myself clear enough. I am referring to the possibility of using the diagram to say things like: "the pairs of electrons that are the charge carriers in the superconducting regime walk through the solid material, moving horizontally in this diagram, always occupying the first band. This implies that the electrons move necessarily through successive tunneling processes."
Something like.

 

[Vanadium 50]

This implies that the electrons move necessarily through successive tunneling processes."

Where are you getting this from?

 

[DaTario]

Where are you getting this from?

I am just testing one possibility to exemplify. My question is: does the diagram have any use in explaining the superconductive phenomena? The answer seems to be no.

Edit: I recognize the diagram shows a one dimensional configuration and perhaps it makes no sense to speak of superconductivity in one dimension. But it seems to be possible to consider an extended 2D or 3D diagram analogous to the one shown in OP.

 

[TeethWhitener]

1) Band structure diagrams are usually presented in reciprocal space rather than real space.

2) Band structure diagrams are generally one-electron mean field diagrams without explicit inclusion of electron correlation. Superconductivity is a two electron process.

3) Even if electron-electron interaction were included, you still wouldn’t get superconductivity, because two electrons cannot interact to form a lower energy state than two free electrons, because the interaction is repulsive everywhere. This means there needs to be a mediating potential, usually taken to be electron interaction with a phonon (though it can be any potential with an attractive component—some folks have started looking at electron plasmon interactions for example).

That said, the band structure does give us at least one piece of the puzzle. In BCS theory, the critical superconducting transition temperature is:
$$T_c = \Theta_D\exp\left(\frac{-1}{V_{ep}D(E_F)}\right)$$
Where $\Theta_D$ is the Debye temperature, $V_{ep}$ is the electron-phonon coupling, and $D(E_F)$ is the density of states at the Fermi energy. The band structure gives you $D(E_F)$, at least visually. You may hear people talking about flat bands when they talk about superconductivity; what they mean is that at the Fermi level, the curvature of the valence band is very low, which leads to a high density of states. This is a necessary, but not sufficient, condition for superconductivity. The other condition is a large electron-phonon interaction, which the band structure doesn’t tell you.

This indirectly answers your question about which band the superconducting electrons come from: they come from the top of the valence band.

 

[DaTario]

Thank you, very much, @TeethWhitener . I was already somewhat familiar with the idea that superconductivity involved interactions between electron pairs mediated by lattice phonons.

Thank you once more.