d-wave superconductivity on the honeycomb bilayer

We introduce a microscopic model on the honeycomb bilayer, which in the small-momentum limit captures the usual (quadratic dispersion in kinetic term) description of bilayer graphene. In the limit of strong interlayer hopping it reduces to an effective honeycomb monolayer model with also third neighbor hopping. We study interaction effects in this effective model focusing on possible superconducting instabilities. We find d_{x^2-y^2} superconductivity in the strong coupling limit of an effective tJ-model-like description that gradually transforms into d + id time-reversal symmetry breaking superconductivity at weak couplings. In this limit the small momentum order parameter expansion is (k_x + i k_y)^2 [or (k_x - ik_y)^2] in both valleys of the effective low-energy description. The relevance of our model and investigation for the physics of bilayer graphene is also discussed.