NMR Study of the Possible FFLO State in CeCoIn$_5$

We report nuclear magnetic resonance (NMR) measurements of the heavy-fermion superconductor CeCoIn$_5$ in the vicinity of the superconducting critical field $H_{C2}$ for a magnetic field applied perpendicular to the $c$ axis. A possible inhomogeneous superconducting state, Fulde-Ferrel-Larkin- Ovchinnikov (FFLO), is stabilized in this part of the phase diagram. In 11 T applied magnetic field, we observe clear signatures of the two phase transitions. We show that a higher temperature transition is clearly associated with the transition to the homogeneous superconducting state. The low temperature phase cannot be characterized by a simple FFLO state, viewed as a stack of static, and spatially well separated, superconducting and normal regions. Alternative explanations of the nature of the low temperature phase are discussed.