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Possible singlet and triplet superconductivity on honeycomb lattice

We study the possible superconducting pairing symmetry mediated by spin and charge fluctuations on the honeycomb lattice using the extended Hubbard model and the random-phase-approximation method. From $2\%$ to $20\%$ doping levels, a spin-singlet $d_{x^{2}-y^{2}}+id_{xy}$-wave is shown to be the leading superconducting pairing symmetry when only the on-site Coulomb interaction $U$ is considered, with the gap function being a mixture of the nearest-neighbor and next-nearest-neighbor pairings. When the offset of the energy level between the two sublattices exceeds a critical value, the most favorable pairing is a spin-triplet $f$-wave which is mainly composed of the next-nearest-neighbor pairing. We show that the next-nearest-neighbor Coulomb interaction $V$ is also in favor of the spin-triplet $f$-wave pairing.