| Summary: | Motivated by recent developments in the experimental study of ultracold atoms in graphene-like honeycomb optical lattices, we investigate superconductivity of the attractive Kane-Mele-Habbard (KMH) model with the next-nearest-neighbor (NNN) hoping at half filling. The mean-field approximation is used to study the phase diagram which interpolates the trivial and the non-trivial topological states. It is shown that: (a) when the NNN hoping is taken into account, one has to introduce two mean-field gap equations for the two sublattices, instead of a single gap when the NNN hopping is neglected, and (b) in the non-trivial topological region the phase diagram with the NNN hopping is significantly different compared to the phase diagram calculated previously, but without the NNN term. We also discuss the superconducting instability of the attractive KMH model that is driven by condensation of Cooperons.
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