Solid State - Energy Bands And The Dispersion Relation

If we can describe conductivity of materials with energy bands, what do superconductors' band structures look like?

The band structure of a superconductor looks sort of like a semiconductor in that there is a gap in the density of states.  However it is different from a semiconductor in that:The gap is very small.  A typical semiconductor may have a gap of 1 eV, whereas a superconductor has a gap on the scale of meV's (1000x smaller).The gap is centered around EF (the Fermi level) by definition.  A semiconductor can have EF anywhere relative to the valence and conduction bands, depending on the details of the band structure.The gap in a superconductor is a property of the superconducting phase, whereas the gap in semiconductors and insulators is there at all temperatures.  Above the superconducting transition temperature, superconductors (usually) have normal metallic band structure.Stating this pictorially, this is the dispersion relation (E vs k)  in the superconducting state (red dotted line is above Tc; black/blue and green/blue is below Tc):And this is what the density of states looks like experimentally:Inset: STM spectrum showing density of states in NbN at low temperature.  Main plot shows temperature dependence of superconducting gap (roughly the distance between two peaks in inset, at least at low temperature), showing that it goes to zero at Tc.  Image source: Page on iop.orgAs for understanding zero resistivity of superconductors from the band structure, I have trouble with this myself.  The most intuitive explanation (for me) is here: Inna Vishik's answer to How does the Cooper-pairing mechanism give rise to zero electrical resistance?

What is band theory of solid?

It starts from the Pauli exclusion principle, which says that no two identical fermions can be in the same state in the same place at the same time.That means only two electrons (one with spin up and the other with spin down) can occupy the lowest energy level in a solid — the one with half a wavelength in the width, length and height of the crystal. The next two electrons go into the next level (with a full wavelength in each direction), and so on until you get up to the Fermi energy, which is where you run out of electrons and the crystal is electrically neutral.Before you get there, however, you usually come to an energy level for which half a wavelength is equal to the spacing between atoms in the crystal. At that energy there are two different states with the same wavelength but different energies: one lined up with nodes at the positive ions and the other with antinodes at the ions. The latter has a lower electrostatic energy because the negative electrons are more likely to be on top of the positive ions.That means an energy “gap” opens up at that wavelength (and another when there’s a full wavelength between ions, and so on). These gaps are located at wavelengths (which are inversely proportional to the momenta of the electrons) called “zone boundaries”.All the possible states in each zone (between two such gaps) form one band of states. If all the states in that band are filled, there is nowhere for any of those electrons to go unless they “jump” up into a higher band, which usually takes more energy that they can get from mere thermal excitations. If the last electron just completes one band, you get an insulator (or, if the gap is small enough, a semiconductor). But if the last electron goes in at the middle of a band, it has lots of convenient nearby energy levels to visit, so it is basically free to move under the influence of electric fields, so you have a metal.Band theory is how you calculate the dependence of energy on momentum (or frequency on inverse wavelength, which is the same thing) — this is known as the dispersion relation, [math]\omega(\vec{k})[/math]. It requires a detailed understanding of the crystal structure and the electromagnetic interactions between all its components.

What is an intuitive explanation of Bloch's theorem?

Bloch's thoerem lets us write the solutions for a wavefunction in a periodic potential as a periodic function [math]u(\mathbf{r})=u(\mathbf{r}+\mathbf{a})[/math] (where [math]\mathbf{a}[/math] is any lattice vector of the periodic potential) multiplied by a plane wave [math]e^{i\mathbf{k.r}}[/math]:[math]\phi(\mathbf{r})=u(\mathbf{r})e^{i\mathbf{k.r}}[/math]The intuition is that we can calculate wavefunctions inside crystals (the periodic potential) and treat them like plane waves, but instead of the momentum, we have the "crystal momentum" [math]\mathbf{k}[/math].  The periodic part is needed so that the solution "fits" inside the crystal.Here is a simulation of a Bloch wave.  Its actually a photon travelling through a photonic crystal.  The "atoms" of the photonic crystal are shown by the white circles.  The peaks of the Bloch wave are red and the troughs blue.  Notice how much it resembles a plane-wave travelling towards the bottom-right corner.  Notice also the periodic patterns, letting the Bloch wave respect the crystal's symmetry.A Bloch wave in a crystal.  See how much it's like a plane wave?The Bloch waves will, however, have completely different energy and momenta to their free space counterparts.  The relationship between energy and momentum is the dispersion relation [math]E(\mathbf{p})[/math] or [math]\omega(\mathbf{k})[/math], and its very simple in free space (quadratic for electrons, linear for photons).  A crystal severely modifies this simple dispersion relation.  In fact, you will find that the different solutions or modes will tend to form into energy bands (examples include the famous valence and conduction bands in semiconductors), and this is how the bandstructure is calculated.(OK, so this is probably not so intuitive unless you already know a lot about plane-waves.)