Flat bands as a route to high-temperature superconductivity in graphite

Superconductivity is traditionally viewed as a low-temperature phenomenon. Within the BCS theory this is understood to result from the fact that the pairing of electrons takes place only close to the usually two-dimensional Fermi surface residing at a finite chemical potential. Because of this, the critical temperature is exponentially suppressed compared to the microscopic energy scales. On the other hand, pairing electrons around a dispersionless (flat) energy band leads to very strong superconductivity, with a mean-field critical temperature linearly proportional to the microscopic coupling constant. The prize to be paid is that flat bands can probably be generated only on surfaces and interfaces, where high-temperature superconductivity would show up. The flat-band character and the low dimensionality also mean that despite the high critical temperature such a superconducting state would be more vulnerable to strong fluctuations than ordinary superconductors. Here we discuss the topological and non-topological flat bands discussed in different systems, and show that graphite is a good candidate for showing high-temperature flat-band interface superconductivity. The purpose of this chapter is to propose a route to increasing the critical temperature of superconductivity by searching for special electronic dispersion that would promote the superconducting strength. We first show that a huge increase in the (mean-field) critical temperature is possible, if a dispersionless energy spectrum, a flat band can be created in the system in the absence of the interaction leading to superconducting correlations. We then discuss a few known schemes to generate such (approximate or exact) flat bands.