Originally, superconductivity was discovered in metals (mercury in particular), which are characterized by having roughly one free electron per ion in the crystal. This leads to a typical density of 10^23 free electrons in a cm^3. As time advanced new groups of superconductors were discovered: for example the heavy fermion compounds, the high-Tc cuprates and the pnictides. Each group has its own unique phenomenology and intriguing open questions.

Recently a new group of superconductors is being discovered. Namely, superconductors that emerge from systems of extremely low density, such as semiconductors and semimetals. These kind of systems are ideally insulating and become conducting only upon doping. This leads to a completely different regime of density range - something like 10^17 to 10^20 free electrons per cm^3, equivalent to one free electron per thousand, or even million, ions in the crystal. In this limit it is very easy to show that standard BCS theory can not explain how these materials become superconducting (mainly because the density of states is just too small!). Thus, there is a truly fundamental question: how do these materials become superconducting?

Before I get to that question let me first elaborate a bit more on the materials. The first member of this group of superconductors, SrTiO3, was discovered in 1964 (not so recent...). For many years it was the only member and was considered an interesting anomaly. However, over the past decade at least four new superconductors were discovered, including doped Bi2Se3, the half Heusler compounds (e.g. YPtBi, LaPtBi ...) and pure bismuth. This makes it clear that this is not an anomaly uniquely related to SrTiO3, but truly a phenomena. The figure below (stolen from a perspective in Science magazine, written by Kamran Behnia from CNRS France, regarding the discovery of superconductivity in Bismuth) shows the transition temperature of various low density superconductors vs. density.

Now to the question: "how do these materials become superconducting?" ...