Order-parameter evolution in the Fulde-Ferrell-Larkin-Ovchinnikov phase

We report on the temperature dependence of the spatially modulated spin-polarization amplitude $\mathrm{\Delta }{K}_{\text{spin}}$, which is a hallmark of the superconducting Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. For that, we use ${}^{13}\mathrm{C}$ nuclear magnetic resonance (NMR) spectroscopy performed on the organic conductor ${\beta }^{\prime \prime }\text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{SF}}_5{\mathrm{CH}}_2{\mathrm{CF}}_2{\mathrm{SO}}_3$. From a comparison of our experimental results to a comprehensive modeling of the ${}^{13}\mathrm{C}$ NMR spectra, we determine the evolution of $\mathrm{\Delta }{K}_{\text{spin}}$ upon condensation of the FFLO state. Further, the modeling of the spectra in the superconducting phase allows to quantify the decrease of the average spin susceptibility, stemming from the spin-singlet coupling of the superconducting electron pairs in the FFLO state of ${\beta }^{\prime \prime }\text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{SF}}_5{\mathrm{CH}}_2{\mathrm{CF}}_2{\mathrm{SO}}_3$.

Figure 1

(a) Crystal structure (left) of ${\beta }^{\prime \prime }$-ET, projected along the [100] direction. The ET molecules, stacked along [100] and viewed along [010] (right), are ${}^{13}\mathrm{C}$ labeled on their central positions. (b) and (c) Representative ${}^{13}\mathrm{C}$ NMR spectra in the normal-conducting and FFLO state, respectively, with the site-resolved contributions to the spectral modeling shown as dashed lines. The modeled spectrum of the line with $n=4$ and $i=l$ is highlighted by the red bold line, with the site-specific first spectral moment $K_{\text{orb},n=4}+\lambda K_{\text{spin},n=4}-D_{n=4,i=l}$ and the local-polarization modulation $2\lambda K_{\text{spin},n=4}\mathrm{\Delta }{K}_{\text{spin}}$ indicated by the blue diamond and the red horizontal bar, respectively [see Eq. (2)]. In (c), the intrinsic linewidth of the site-specific contributions to the modeled spectrum is minimized to highlight the underlying double-horn shape, resulting from the one-dimensionally modulated spin polarization.

Figure 2

(a) Superconducting phase diagram of ${\beta }^{\prime \prime }$-ET for magnetic fields aligned parallel to the conducting layers. The phase boundaries are inferred from specific-heat measurements (black circles) [5], the measurements of ${}^{13}\mathrm{C}$ NMR spectra at ${\mu }_0{H}_{\parallel }=10.5$ T, and various temperatures in this work are indicated by the red horizontal line. (b) Waterfall diagram of the experimental ${}^{13}\mathrm{C}$ NMR spectra at various temperatures between 0.3 and 20 K (same as the temperatures in Fig. 3). (c) Modeled spectra as described in the main text. Below 1.4 K, indicated by the brown dashed-line spectrum, a decrease of all line shifts is observed, and accompanied by the emergence of inhomogeneous line broadening. As in Fig. 1, the spin-polarization modulation amplitude and the first spectral moment of the line with $n=4$ and $i=l$ are indicated by the red horizontal bar and the blue diamond, respectively.

Figure 3

Temperature dependence of the modulation amplitude $\mathrm{\Delta }{K}_{\text{spin}}$ (red triangles) and relative reduction $\lambda$ of the average spin part of the Knight shift (black circles) of ${\beta }^{\prime \prime }$-ET at ${\mu }_0{H}_{\parallel }=10.5$ T.