Interplay of superconductivity and spin-density-wave order in doped graphene

We study the interplay between superconductivity and spin-density-wave order in graphene doped to 3/8 or 5/8 filling (a van Hove doping). At this doping level, the system is known to exhibit weak-coupling instabilities to both chiral d+id superconductivity and to a uniaxial spin density wave. Right at van Hove doping, the superconducting instability is strongest and emerges at the highest T _c, but slightly away from van Hove doping, a spin density wave likely emerges first. We investigate whether at some lower temperature superconductivity and spin density waves coexist. We derive the Landau-Ginzburg functional describing interplay of the two order parameters. Our calculations show that superconductivity and spin-density-wave order do not coexist and are separated by first-order transitions, either as a function of doping or as a function of T.