Calorimetric Measurements of Magnetic-Field-Induced Inhomogeneous Superconductivity Above The Paramagnetic Limit

We report the first magneto-caloric and calorimetric observations of a magnetic-field-induced phase transition within a superconducting state to the long-sought exotic "FFLO" superconducting state first predicted over 50 years ago. Through the combination of bulk thermodynamic calorimetric and magnetocaloric measurements in the organic superconductor $\kappa$ - (BEDT-TTF)$_2$Cu(NCS)$_2$, as a function of temperature, magnetic field strength, and magnetic field orientation, we establish for the first time that this field-induced first-order phase transition at the paramagnetic limit $H_p$ for traditional superconductivity is to a higher entropy superconducting phase uniquely characteristic of the FFLO state. We also establish that this high-field superconducting state displays the bulk paramagnetic ordering of spin domains required of the FFLO state. These results rule out the alternate possibility of spin-density wave (SDW) ordering in the high field superconducting phase. The phase diagram determined from our measurements --- including the observation of a phase transition into the FFLO phase at $H_p$ --- is in good agreement with recent NMR results and our own earlier tunnel-diode magnetic penetration depth experiments, but is in disagreement with the only previous calorimetric report.