Possible Fulde-Ferrell-Larkin-Ovchinnikov superconducting state in CeCoIn₅

We report specific heat measurements of the heavy fermion superconductor CeCoIn₅ in the vicinity of
the superconducting critical field H _c2, with magnetic fields in the [110], [100], and [001] directions, and at temperatures down to 50 mK. The superconducting phase transition changes from second to first
order for fields above 10 T for H ║ [110] and H ║ [100]. In the same range of magnetic fields, we observe
a second specific heat anomaly within the superconducting state. We interpret this anomaly as a
signature of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) inhomogeneous superconducting state. We obtain similar results for H ║ [100], with the FFLO state occupying a smaller part of the phase
diagram.