ABSTRACT
We show that rhombohedral graphite may support surface superconductivity with an unusual relation between the BCS coupling constant and the order parameter. This feature results from the properties of the states localized on the graphite surfaces. In a description including only the nearest neighbour coupling of the graphene layers, the surface states are topologically protected and have a flat band dispersion. We show that including higherorder couplings destroys this flat band character and leads to a particle-hole symmetry breaking quadratic dispersion with a large effective mass. Employing this dispersion, we then study its effect on superconductivity and find two regimes of parameters, depending on the relation between the strength of the coupling constant and †Deceased
the details of the quadratic dispersion. For low coupling strengths, superconductivity is localized on the surfaces, but the order parameter is exponentially suppressed as in a conventional BCS superconductor, whereas for large coupling strengths, we obtain surface superconductivity with a linear relation between the order parameter and the coupling constant. Our results may explain the recent findings of graphite superconductivity with a relatively high transition temperature. 9.1 IntroductionSuperconductivity is a ubiquitous phenomenon in metals: according to a commonly held view, all metals become either superconducting or magnetic at low enough temperatures. However, the corresponding transition temperatures may be (so far) unobservably low. In conventional superconductors, such as Al, Hg, or Nb, the transition temperature depends exponentially on the inverse of the BCS coupling constant, and whereas the coupling constant itself may be fairly large (the relevant energy scale connected with it may be many times larger than the thermal energy at room temperature), the resulting transition temperature typically does not much exceed 10 K. This property is intrinsically related to the quadratic dependence of electrons’ energy on momentum, which leads to a logarithmic divergence in the BCS self-consistency equation. With a higher-order dispersion around the Fermi energy, the relation between the magnitudes of the critical temperature and the coupling constant becomes stronger, boosting the superconductivity.The extreme case would be a completely dispersionless energy spectrum, the so called “flat band.” Fermionic systems with dispersionless branches of excitation spectrum have quite unusual properties; nowadays they attract lots of research interest. Flat bands were predicted in many condensed matter systems, see for example [1-4]. In some cases, the flat bands are protected by topology in momentum space; they emerge on the surfaces of gapless topological matter [5] such as surfaces of nodal superconductors [6-8], graphene edges [6], surfaces of multilayered graphene structures [9-12], and in the cores of quantized vortices in topological superfluids and superconductors [5,13,14]. The singular density of states (DOS) associated with the dispersionless spectrum was recently shown by
us to essentially enhance the transition temperature opening a new route to room-temperature superconductivity. The problem is to find the metal with such a higher-order dispersion around the Fermi sea. Along with our collaborators, we have shown [10,15] that within the nearest-neighbour approximation, rhombohedral graphite has topologically protected
surface states with a flat band at the Fermi energy, and these surface states support high-temperature superconductivity, where the superconducting order parameter is concentrated around the surfaces. Such a superconductor may also carry a large surface supercurrent with a critical value proportional to the large critical temperature. The corresponding critical temperature depends linearly on the pairing interaction strength, and can be thus considerably higher than the usual exponentially small critical temperature in the bulk. A flat band forms out of a low dispersive band that appears on the surface of a multilayered graphene structure with rhombohedral stacking with a large number of layers. Surface superconductivity is favorable already for a system having N ≥ 3 layers, where the normal-state spectrum has a power-law dispersion ξp ∝ |p|N as a function of the in-plane momentum p. The DOS ν ξ ξ( )p p( - /∝ 2 )N N has a singularity at zero energy which results in a drastic enhancement of the critical temperature. This singularity in DOS has also been reported to produce unconventional behavior of quantum Hall effect and singularity in the cyclotron mass of three-layer graphene [16].However, next-nearest neighbour hoppings which are present in real rhombohedral graphite can break the exact topological protection and, therefore, the flat-band mechanism of superconductivity at sufficiently low values of the coupling constant can be destroyed. Here we study the detailed effect of these higher-order interactions and show that, though they indeed break the flat-band scenario for weak superconducting coupling, they provide another mechanism of surface superconductivity which is of the BCS type but has a much larger coupling constant than the usual superconductivity in bulk graphite. This large coupling constant comes from a large DOS associated with a heavy effective mass of surface quasiparticles that is clearly distinguishable on the background of the flat band which would exist without the higher-order interactions. Both these mechanisms favor the high-temperature superconductivity. Our results provide a criterion for
the parameters needed to obtain the highest critical temperature. They may be relevant in explaining the recent experimental findings [17-21] reporting the observation of even room-temperature superconductivity in doped graphite.
