Competing superconducting and correlated phases in Bernal stacked bilayer graphene

Owing to its unique electronic properties and a high degree of tunability, Bernal-stacked bilayer graphene (BLG) has been the subject of research from many different perspectives. Recent experimental observations of superconductivity under a weak magnetic field or in proximity to a spin-orbit coupled substrate brought another new perspective for studying BLG. Moreover, correlated insulating phases at low but finite hole dopings have also been observed shortly after. The correlated insulating states, which are attributed to the formation of Wigner crystals, signify strong electronic repulsion that is very likely not in favor of superconductivity. Here we study the competition between the superconducting and correlated phases and try to shed light on the mechanism underlying both phenomena. We treat the Coulomb interaction and effective attraction between electrons on the same footing and consider the effect of screening and spin-orbit coupling due to the proximity effect. Our approaches can pave the road for studying the completion between superconductivity and correlated states in twisted bilayer graphene with more complicated electronic properties.