Point defects in the (d+id)-wave superconducting state of heavily doped graphene

Abstract: Previous studies have suggested that the material graphene might transition into an electron-electron interaction driven, unconventional, time-reversal-symmetry-breaking, (d+id)wave superconducting state upon either significant electron or hole doping, and, in particular, upon doping to the Van Hove singularity. As defects are likely to be introduced in the doping process, we are, in this text, concerned with the effects of defects on this superconducting state near the Van Hove singularity doping. To investigate the effects we use a mean-field treatment of a phenomenological resonant-valence-bond model. We find that the resonant-valence-bond amplitudes, which in the defect free graphene sheet are proportional to the superconducting pairing-potential, are suppressed near the defects, and that the recovery is well described by an exponential, yet anisotropic, recovery. In general, we find that the (d+id)-wave, superconducting state is quite resilient, and that even for strong defects, such as a vacancy, the recovery length is of the order of one lattice constant when extrapolated to weak pairing-potentials; this is compared to a conventional superconducting state of an attractive Hubbard model for which the same decay length is found to be of the order of a half lattice constant. For the defect free graphene sheet the (d+id)-wave state is a completely gapped state. The introduction of vacancies is, however, found to be accompanied by the appearance of midgap states. These states are shown to be localized around the vacancies. In accordance with the nature of this text, we will, for the benefit of students and non-experts, include an introductory section on the fundamental methods and concepts used. It gives a short and hopefully pedagogical introduction to the rudimentary concepts of solid state theory and the microscopic BCS theory of superconductivity.