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Defects in the (d+id)-wave superconducting state in heavily doped graphene

A chiral time-reversal symmetry breaking (d+id)-wave superconducting state is likely to emerge in graphene doped close to the Van Hove singularity. As heavy doping procedures are expected to introduce defects, we here investigate the effects of microscopic defects on the (d+id)-wave superconducting state at the Van Hove singularity. We find that, while the superconducting order is reduced near a defect, the (d+id)-wave state remains intact and recovers in an exponential manner away from the defect. The recovery length is found to be on the order of one lattice constant for weak couplings. This is comparable to the recovery length of a conventional s-wave state, demonstrating that the unconventional (d+id)-wave state is quite resilient to defects. Moreover, we find no significant changes between a single site defect and more extended defects, such as a bivacancy. While the (d+id)-wave state is fully gapped, we also show that defects introduce localized midgap states with non-zero energies, which should be accessible via scanning probe experiments.