Reentrant Superconductivity Driven by Quantum Tricritical Fluctuations in URhGe: Evidence from ${}^{59}\mathrm{Co}$ NMR in ${\mathrm{URh}}_{0.9}{\mathrm{Co}}_{0.1}\mathrm{Ge}$

Our measurements of the ${}^{59}\mathrm{Co}$ NMR spin-spin relaxation in ${\mathrm{URh}}_{0.9}{\mathrm{Co}}_{0.1}\mathrm{Ge}$ reveal a divergence of electronic spin fluctuations in the vicinity of the field-induced quantum critical point at $H_R\approx 13\mathrm{T}$, around which reentrant superconductivity (RSC) occurs in the ferromagnetic heavy fermion compound URhGe. We map out the strength of spin fluctuations in the ($H_b,H_c$) plane of magnetic field components and show that critical fluctuations develop in the same limited region near the field $H_R$ as that where RSC is observed. This strongly suggests these quantum fluctuations as the pairing glue responsible for the RSC. The fluctuations observed are characteristic of a tricritical point, followed by a phase bifurcation toward quantum critical end points.

Figure 1

Temperature-field phase diagram of ${\mathrm{URh}}_{1-x}{\mathrm{Co}}_x\mathrm{Ge}$. The inset shows the magnetization curves at 2 K.

Figure 2

${}^{59}\mathrm{Co}\text{−}\mathrm{NMR}$ spectra in applied magnetic fields along the (a) $a$ and (b) $b$ axes at 1.6 K. The NMR spectra were recorded at fixed frequency by sweeping the magnetic field in equally spaced steps and by summing the Fourier transforms of the spin-echo signals measured with the $\pi /2\text{−}\tau \text{−}\pi /2$ pulse sequence at each field value. The blue lines show the results of NMR line simulation (see the text).

Figure 3

(a) Field dependence of the relaxation rate $1/T_2$. (b) The $H_c$ field component dependence of $1/T_2$ obtained by rotating the sample in applied fields of 4.8, 11.2, and 16 T. (c) Comparison of $1/T_1$ and $1/T_2$ for $H\parallel b$. The $1/T_2$ values were determined by fitting the $\tau$ dependence of the spin-echo intensity, measured at the center of the NMR spectrum, to an exponential function, $M(2\tau )\propto \exp (-2\tau /T_2)$. $1/T_1$ has been measured at the same position by the standard saturation-recovery method. All the data were taken at 1.6 K.

Figure 4

Map of the magnetic fluctuation amplitude in the ($H_b,H_c$) plane around $H_R$. This contour plot of $1/T_2$ at 1.6 K is constructed from 104 data points measured with different combinations of $H$ and $\theta$ values, between $H=4.7$ and 17 T and between $\theta =0°$ and 11°, respectively. White dashed lines show constant-$\theta$ scans. The open squares indicate the region where RSC has been observed from resistivity measurements at $T\sim 40\mathrm{mK}$ in a single crystal of URhGe [10]. The filled triangles indicate the first-order-like metamagnetic transitions at $T\sim 500\mathrm{mK}$ [10]. Since URhGe has a lower $H_R$ (12 T) than ${\mathrm{URh}}_{0.9}{\mathrm{Co}}_{0.1}\mathrm{Ge}$ (13.4 T), these data for URhGe are plotted here with the field values scaled proportionally to the $H_R$ values in ${\mathrm{URh}}_{0.9}{\mathrm{Co}}_{0.1}\mathrm{Ge}$.

Figure 5

(a) The $H_b\text{−}{H}_c$ phase diagram involves a first-order transition planner surface at $H_c=0$, which terminates at the QCP. The direction of the spontaneous FM moments (blue arrows) changes sign discontinuously when this surface is crossed. Dotted lines denote the crossover line passing through the QCP, which divides the paramagnetic state polarized along the $b$ axis (red arrows) from the spontaneous FM ordered state along the $c$ axis. (b) The first-order transition surface is delimited by a second-order FM transition (blue) line, continuously connected to the field-induced paramagnetic state at a single QCP (blue circle). (c) The first-order transition surface at $H_c=0$ may bifurcate into two wings at the TCP, which end up at zero temperature at two QCEPs (blue circles). (d) Considering only the $H_c=0$ plane, the TCP terminates the second-order phase transition line by a first-order transition (red dotted line).