Interplay between quantum fluctuations and reentrant superconductivity with a highly enhanced upper critical field in URhGe

The ferromagnetic superconductor URhGe has been known to exhibit an extremely large enhancement of the upper critical field when the field is confined and rotated in the $\left(ab\right)$ crystal plane. Our high-field ${}^{59}\mathrm{Co}$ NMR measurements on 10% Co-doped URhGe up to 30 T prove that this unconventional behavior of the superconductivity (SC) is associated with a strong anisotropy of field-dependent quantum fluctuations near a tricritical point. These fluctuations are rapidly suppressed by the field component along the $c$ axis, while being unaffected by any field component along the $a$ axis. The observed close interplay between the SC and quantum fluctuations strongly supports a pairing mechanism mediated by these fluctuations in URhGe.

Figure 1

(a) The ${\theta }_{ab}$ dependence of ${}^{59}\mathrm{Co}$ NMR spectra recorded with a NMR frequency of 125.45 MHz at 1.6 K. (b) Numerical simulations, based on diagonalization of the total Hamiltonian consisting of the Zeeman term and a quadrupolar term, which well reproduce the overall ${\theta }_{ab}$ dependence of the spectra, except for a small satellite peak observed in the interval ${\theta }_{ab}={30}^{\circ }$–${60}^{\circ }$ (inverted triangles). Here, the calculated bare NMR spectrum is convoluted with a Gaussian broadening function having a width of $\mathrm{\Delta }\sim 0.05$ T for $H\parallel a$, and $\sim 0.15$ T for $H\parallel b$, respectively. The major source of $\mathrm{\Delta }$ is magnetic, i.e., a distribution of the Knight shift values, since all seven NMR lines have a similar width.

Figure 2

Angular dependence of the Knight shift $K\left({\theta }_{ab}\right)$ in the $\left(ab\right)$ plane, extracted from the numerical simulations shown in Fig. 1. The solid line is the fit to Eq. (2) (see the text). The inset shows the TiNiSi-type crystal structure of URhGe.

Figure 3

The field and angle dependences of $1/T_2$ in the (a) $\left(bc\right)$ [14] and (b) $\left(ab\right)$ planes. The solid diamonds indicate the region where RSC has been observed from resistivity measurements at $T\sim 40$ mK in a single crystal of URhGe [17]. The solid triangles indicate the first-order-like metamagnetic transitions at $T\sim 500$ mK [17]. Since URhGe has a lower $H_{\mathrm{R}}$ (12 T) than ${\mathrm{URh}}_{0.9}{\mathrm{Co}}_{0.1}\mathrm{Ge}$ (13.4 T), the data for URhGe are plotted here with field values scaled proportionally to the $H_{\mathrm{R}}$ values in ${\mathrm{URh}}_{0.9}{\mathrm{Co}}_{0.1}\mathrm{Ge}$.

Figure 4

The $1/T_2$ data in the $\left(ab\right)$ plane are plotted against $H_b=Hcos{\theta }_{ab}$. The inset shows the $H_b$ dependence of the geometric average of the coherence length, evaluated from the resistivity data in several different ${\theta }_{ab}$, assuming constant anisotropy, in URhGe [10]. The quantity plotted is proportional to the inverse-square coherence length ${\varphi }_0/2\pi {\xi }^2$, given in units of magnetic field.