Microscopic Study of the Fulde-Ferrell-Larkin-Ovchinnikov State in an All-Organic Superconductor

Quasi-two-dimensional superconductors with a sufficiently weak interlayer coupling allow magnetic flux to penetrate in the form of Josephson vortices for in-plane applied magnetic fields. A consequence is the dominance of the Zeeman interaction over orbital effects. In the clean limit, the normal state is favored over superconductivity for fields greater than the paramagnetic limiting field, unless an intermediate, inhomogeneous state is stabilized. Presented here are nuclear magnetic resonance (NMR) studies of the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state for ${\beta }^{\prime \prime }\text{−}(\mathrm{ET}{)}_2{\mathrm{SF}}_5{\mathrm{CH}}_2{\mathrm{CF}}_2{\mathrm{SO}}_3$. The uniform superconductivity-FFLO transition is identified at an applied field value of 9.3(0.1) T at low temperature ($T=130\mathrm{mK}$), and evidence for a possible second transition between inhomogeneous states at $\sim 11\mathrm{T}$ is presented. The spin polarization distribution inferred from the NMR absorption spectrum compares favorably to a single-$Q$ modulation of the superconducting order parameter.

Figure 1

(top) Crystal structure, viewed along [100] (left) and [010] (right) directions. Inequivalent molecules are segregated in stacks aligned with $a$. The ${}^{13}\mathrm{C}$ sites are indicated by the colors orange or red, with the red molecules sandwiched by the negatively charged ${\mathrm{SO}}_3$ groups (grey spheres). The associated band structure and Fermi surface are described in the Supplemental Material [21]. (bottom) ${}^{13}\mathrm{C}$ NMR spectrum, recorded at $B=11.9\mathrm{T}$ and $T=1.7\mathrm{K}$, corresponding to the normal state, with the color-coding indicating the contributions from the red and orange sites. The applied field is directed precisely in the plane formed by the ET layers, approximately perpendicular to $\mathbf{b}$.

Figure 2

(a) Angle dependence of the reflection signal, with the anomaly at $\theta =0$ associated with the in-plane condition. Hysteresis between clockwise (CW) and counterclockwise (CCW) rotation is reduced upon increasing the field strength. (b) Field/temperature superconductor/normal state phase boundary, as inferred from specific heat measurements (black circles) [11], and the NMR results reported here (blue hexagons). The vertical red bars correspond to the range of field strength for our measurements. (c) Spectral evolution with field, recorded at $T=130\mathrm{mK}$ and (d) $T=1.7\mathrm{K}$. At the lower temperature, significant line broadening is exhibited for fields greater than 9.5 T, which we take to be the onset of the FFLO phase. For $T=1.7\mathrm{K}$, only the spectrum at 9.5 T takes on a different, transitional, form from what appears at lower and higher fields.

Figure 3

(a) ${}^{13}\mathrm{C}$ NMR shift vs $B$. The solid red line corresponds to the Zeeman effect on a $d$-wave model for circular Fermi surface and weak coupling (see text). Its extrapolation to $B=0$ leads to a 150 ppm estimate for the orbital shift. (b) Spin-lattice relaxation rate, $T_1^{-1}(B)$. The data recorded at $T=1.7\mathrm{K}$ exhibit an increase at 9.3 T greater than the normal state value, consistent with an incursion into the FFLO phase [12, 21]. The curves connecting the data points are guides to the eye.

Figure 4

Temperature dependence of the ${}^{13}\mathrm{C}$ spectrum, carried out at an applied field $B=11.9\mathrm{T}$. The variation is due to an increasing hyperfine field upon warming.

Figure 5

Example of spectrum simulation in the FFLO state, compared to recorded spectrum (blue). The simulation is a sum (green) of four Gaussian-broadened contributions (red, orange) arising from a single-$Q$ sinusoidal modulation of the SC order parameter [21].