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

We report the first magnetocaloric and calorimetric observations of a magnetic-field-induced phase transition within a superconducting state to the long-sought exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state, first predicted over 50 years ago. Through the combination of bulk thermodynamic calorimetric and magnetocaloric measurements in the organic superconductor $\kappa \text{−}(\mathrm{BEDT}\text{−}\mathrm{TTF}{)}_2\mathrm{Cu}(\mathrm{NCS}{)}_2$ 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 $H_p$ is a transition 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 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.

Figure 1

Cartoon of the FFLO state showing the nodes in the order parameter as horizontal planes where we estimate the spin polarization to be $\approx 10\%$ at 25 T in the low temperature limit. The red arrows represent the net spin polarization. Although the diagram is schematic, all of the lengths are to scale; small boxes represent the unit cells of $\kappa \text{−}(\mathrm{BEDT}\text{−}\mathrm{TTF}{)}_2\mathrm{Cu}(\mathrm{NCS}{)}_2$, yellow slabs represent the least-conducting layers of the crystal, and red rectangles represent Josephson coreless vortices at about the right distance apart in a 25 T field. Full height of the crystal is $\approx 20\mathrm{nm}$.

Figure 2

Magnetic-field-induced change (solid—up, dotted—down) in the specific heat of $\kappa \text{−}(\mathrm{BEDT}\text{−}\mathrm{TTF}{)}_2\mathrm{Cu}{(\mathrm{NCS})}_2$ (scaled by temperature) for magnetic fields parallel to the superconducting planes ($\theta =0$). For ease of comparison, $\mathrm{\Delta }{C}_p/T$ is set equal to zero in the normal nonsuperconducting state ($H>H_{\mathrm{c}2}$). We observe a first-order phase transition between two different superconducting states at $H\approx H_p=20.7\mathrm{T}$, followed by a transition to the normal state at a temperature-dependent field $H_{c2}(T)$. Arrows represent $H_{c2}$ for 0.18 and 2.03 K.

Figure 3

Evolution of the heat capacity sweeps as a function of the angle at 0.3 K, with the plane parallel $\theta =0$ orientation shown in black at the bottom. The bump at 21 T, marking the transition at $H_p$ at 0°, is gone by 0.7° and is replaced by the beginning of the $H_{c2}$ transition by 1.6°, consistent with expectation for 2D FFLO superconductivity.

Figure 4

Phase diagram of $\kappa \text{−}(\mathrm{BEDT}\text{−}\mathrm{TTF}{)}_2\mathrm{Cu}(\mathrm{NCS}{)}_2$ for the parallel magnetic field ($\theta =0$). Solid black circles represent our calorimetric observations of the phase transitions between the lower- and higher-field superconducting phases at $H_p$ and squares, the normal and superconducting state at $H_{c2}(T)$. Points from an earlier calorimetric determination of $H_{c2}(T)$ [33] are shown as open blue squares. Also included are supporting determinations of both the $H_{c2}$ and $H_p$ phase boundaries by means of rf penetration measurements (green) [24] and NMR measurements [35, 36] (open purple and red symbols, respectively).

Figure 5

Magnetocaloric effect measurements at 200 mK. The temperature difference ${\mathrm{\Delta }T}_{\mathrm{ud}}$ between the up and down sweeps indicates that the system has become paramagnetic (yellow shading). On down sweeps (blue line), there is a brief increase in the temperature of the sample due to the release of latent heat from the sample at $H_p$. On up sweeps (red line), there is a corresponding decrease in the temperature of the sample due to the absorption of latent heat by the sample at $H_p$. The results indicate that the high-field state is higher entropy than the low-field state, as expected for high-field FFLO superconductivity.