So I'm not going to talk about my research. I'm going to talk about a few of the broad ideas in cutting edge research into high temperature superconductors. The point is not really to teach you about high temperature superconductors (which is hardly useful information if you're not studying superconductors), but so you get the idea of what it looks like.
So. Superconductors. Below a certain temperature "Tc", superconductors conduct electricity perfectly, with no resistance. This property has obvious practical value, but unfortunately all known superconductors have a very low Tc. Even the so-called "high Tc superconductors" discovered in 1986 still have a Tc of about -140 Celsius.
We understand how low Tc superconductors work. The problem was solved in 1957, and the solution is called BCS theory. BCS theory provides a way for electrons to attract each other (despite being opposite charge), and the electron pairs form a Bose-Einstein condensate. The condensate of electron pairs is what makes a superconductor. High Tc superconductors also have condensed electron pairs, but BCS theory doesn't work, and no one knows why the electrons attract each other.
There are two major classes of high Tc superconductors, the cuprates and the iron-based superconductors.* Iron-based superconductors are currently a hot topic because they were just discovered in 2008. But I study cuprates. In particular, I spend most of my time on a material called Bi-2212, which is one of the most highly studied superconductors. I am not sure why it gets so much study, but I would guess that it is because it is cheap, easy to study, and (somewhat self-referentially) has been studied enough that it allows for an ever higher tower of knowledge.
*There are other high Tc superconductors, but I will not speak of them.
I study Bi-2212 using a technique called ARPES. In concept, ARPES is simple: shine light on the material, and look at the ejected electrons. In particular, we look at the direction that the ejected electrons go, and the energy. If we graph the energy of the electrons (vertical axis) vs the angle (horizontal axes), we get something like this:
From Ronning et al, Science 1998. Figure cropped for clarity. I marked the Fermi Surface in blue. Don't worry about what a quasiparticle is.
Experimental physics being what it is, we don't really see that whole picture there. We see tiny slices of it at a time. If I showed you some real data, it would be unrecognizable.
The way the electrons work, there are a bunch of quantum states for them to fill. The quantum states act like slots, and only one electron fits in each slot. The electrons only fill slots up to a certain energy, but there are more empty slots above that energy. The most interesting physics happens where the filled slots meet the empty slots, what's called the Fermi Surface. In fact, this is where the electron pairs live.
Long story short, when we look at the Fermi Surface of a superconductor, there is a small energy gap between the filled and empty slots. This gap in energy represents the energy required to pull the attractive electrons pairs apart. The fascinating thing about cuprate superconductors, is that the gap is not the same size everywhere on the Fermi Surface.
Also from Ronning et al. The size of the gap has been greatly exaggerated.
Yes, in fact, there are even points on the Fermi Surface where there is no gap at all! These points are called nodes. It would seem that at these nodes, there is no superconductivity happening. This is very different from conventional superconductors, which have gaps everywhere on the Fermi Surface.
And what about iron-based superconductors? Iron-based superconductors also have gaps everywhere, just like conventional superconductors. But it's not quite the same! We have reason to think that there are nodes in iron-based superconductors, but they cannot be seen directly because they are between Fermi Surfaces, rather than on the Fermi Surfaces. Of course, this is not a settled matter...
When you raise the temperature of a superconductor, the superconductivity disappears, and so does the gap. But the cuprates do something funny. The gap remains even at high temperatures, when the material is not a superconductor. Or at least, part of the gap does.
From Lee et al, Nature 2007. Figure cropped for clarity. The horizontal axis is the position in the Fermi Surface; the vertical axis is the size of the gap. The blue and green lines are at superconducting temperatures; the red line is above superconducting temperature.
The gap that remains when the material is no longer superconducting is called the "pseudogap". It's a silly name, since the gap is real. But is the gap there because of incipient superconductivity? Or is it an unrelated property of the material?
Check out the date of that paper. 2007. Scientists are still arguing over the pseudogap. I've seen several talks about it, talks which disagree with each other.
So, that's what superconductivity research looks like. Or at least, that's one very small part of it.