In the comments for my last post, Hamilton asked the following question:

What does the band structure for a superconductor look like?

I’m not an expert on this topic,  but I thought I’d share what I know. Take it with a grain of salt. I also wanted to warn you all that my site will be going down for maintenance this Sunday. I apologize for the inconvenience. I don’t know if this will affect my regular Sunday post.

The current most popular theory of superconductors is BCS theory, which is incomplete. BCS theory says that at extremely low temperatures, electrons in a superconductor pair-bond into what are called “cooper pairs,” which are bosonic. As bosons, the cooper pairs aren’t bound by the Pauli exclusion principle, and they ignore band stracture. Instead, they obey Bose-Einstein statistics, meaning that almost all of them are in the lowest energy state.

In this extremely low energy state, each cooper pair has a very large wavefunction, and these wavefunctions overlap to form a large superposition state. We can sort of think of them as becoming an electron superluid, which can travel through the material with no resistance. Alternatively, we can sort of think of them as forming one single electron that permeates the entire material. In either case, the result is superconductivity.

This effect does not fully explain high temperature superconductors, which are largely still a mystery.

tl;dr, band theory can’t explain superconductivity. We need additional mathematical machinery.

EDIT: I’ve been informed by one of my friends who works on superconductors that I’m somewhat incorrect that band theory doesn’t play a role at all. Only electrons at the Fermi level can form cooper pairs. When they do, they “reserve” their place in the band structure so that no other electrons can inhabit those states. Over time, more and more states become forbidden as electrons form cooper pairs and go to the lowest energy state. This creates a large band gap. Eventually, the band gap is so large that if an electron changes its energy, there is absolutely nowhere to go.

Materials electrically resist flow because electrons hit the atomic nuclei inside the material. However, this changes an electron’s energy. If the electrons don’t have energy states available to them, they’re simply forbidden from hitting anything. This causes electrons to flow without resistance.

Hope this helped! I’ll try and look into BCS theory so that I can maybe write a longer post about it.