Evidence for Spin Singlet Pairing with Strong Uniaxial Anisotropy in ${\mathrm{URu}}_2{\mathrm{Si}}_2$ Using Nuclear Magnetic Resonance

In order to identify the spin contribution to superconducting pairing compatible with the so-called “hidden order”, ${}^{29}\mathrm{Si}$ nuclear magnetic resonance measurements have been performed using a high-quality single crystal of ${\mathrm{URu}}_2{\mathrm{Si}}_2$. A clear reduction of the ${}^{29}\mathrm{Si}$ Knight shift in the superconducting state has been observed under a magnetic field applied along the crystalline $c$ axis, corresponding to the magnetic easy axis. These results provide direct evidence for the formation of spin-singlet Cooper pairs. Consequently, results indicating a very tiny change of the in-plane Knight shift reported previously demonstrate extreme uniaxial anisotropy for the spin susceptibility in the hidden order state.

Figure 1

(a) Temperature dependence of the upper critical field $H_{\mathrm{c}2}$ plotted for two single crystals ${\mathrm{URu}}_2{\mathrm{Si}}_2$: ${}^{29}\mathrm{Si}$-enriched (diamond) and nonenriched (triangle) [25], with external fields along the $c$ axis. Temperature scans of ${}^{29}\mathrm{Si}$ NMR spectra were performed under magnetic fields (42 and 12.5 kOe) as indicated by the horizontal two-dotted chain arrows. (b) ${}^{29}\mathrm{Si}$ NMR spectrum under $H^c=42\mathrm{kOe}$ at $T=1.2\mathrm{K}$. $f_0$ is the center frequency of the ${}^{29}\mathrm{Si}$ spectrum. Full width at half maximum of the spectrum was about 3.7 kHz.

Figure 2

${}^{29}\mathrm{Si}$ NMR spectra in the superconducting state with external fields $H^c=7.5\mathrm{kOe}$ and 12.5 kOe. The shaded area represents the field distribution ($\delta H$) due to the triangular vortex lattice, where the spectra are simulated with the field distribution convoluted with a Lorentzian broadening function. The inset shows the vortex lattice at applied field $H_0=12.5\mathrm{kOe}$.

Figure 3

Temperature dependence of differential ${}^{29}\mathrm{Si}$ Knight shift: $\mathrm{\Delta }K\equiv K-K(T_{\mathrm{SC}})$ in both the normal and superconducting states. Temperatures on the horizontal scale are scaled to $T_{\mathrm{SC}}=1.2\mathrm{K}$ at $H^c=12.5\mathrm{kOe}$. The dotted (dot-dash) curve represents a simulation using the SC gap function of the chiral $d$-wave model (BCS $s$-wave model).

Figure 4

Temperature dependence of ${}^{29}\mathrm{Si}$ Knight shift with a field along each crystalline axis. The $c$-axis data below $T=1.2\mathrm{K}$ were taken with the present sample, while the others were taken with the sample used in Ref. [22]. The spin components of the Knight shift ($K_{\mathrm{spin}}^{a,c}$) estimated from $\mathrm{\Delta }K$, the reduction of $K$ below $T_{\mathrm{SC}}$, are much smaller than the nearly temperature-independent orbital component ($K_{\mathrm{orb}}^{a,c}$).

Figure 5

Temperature dependence of nuclear spin-lattice relaxation rate $1/T_1$ with a field along the $c$ axis. The data above $T=1.5\mathrm{K}$ were obtained with the previous sample [22]. $1/T_1$ was measured both in the normal state ($H^c=42\mathrm{kOe}$) and in the superconducting state ($H^c=12.5\mathrm{kOe}$). The solid line represents the Korringa law: $1/T_1\propto T$. The dotted (dot-dash) line was calculated via a chiral $d$-wave (conventional $s$-wave) model with a SC gap of $2\mathrm{\Delta }/k_{\mathrm{B}}{T}_{\mathrm{sc}}=3.52$.