Probing the superconducting ground state of the rare-earth ternary boride superconductors $R{\mathrm{RuB}}_2$ ($R=$ Lu,Y) using muon-spin rotation and relaxation

The superconductivity in the rare-earth transition-metal ternary borides $R{\mathrm{RuB}}_2$ (where $R=\text{Lu}$ and Y) has been investigated using muon-spin rotation and relaxation. Measurements made in zero field suggest that time-reversal symmetry is preserved upon entering the superconducting state in both materials; a small difference in depolarization is observed above and below the superconducting transition in both compounds, however, this has been attributed to quasistatic magnetic fluctuations. Transverse-field measurements of the flux-line lattice indicate that the superconductivity in both materials is fully gapped, with a conventional $s$-wave pairing symmetry and BCS-like magnitudes for the zero-temperature gap energies. The electronic properties of the charge carriers in the superconducting state have been calculated, with effective masses $m^{*}/m_{\mathrm{e}}=9.8\pm 0.1$ and $15.0\pm 0.1$ in the Lu and Y compounds, respectively, with superconducting carrier densities $n_{\mathrm{s}}=\left(2.73\pm 0.04\right)\times {10}^{28}{\mathrm{m}}^{-3}$ and $\left(2.17\pm 0.02\right)\times {10}^{28}{\mathrm{m}}^{-3}$. The materials have been classified according to the Uemura scheme for superconductivity, with values for $T_{\mathrm{c}}/T_{\mathrm{F}}$ of $1/(414\pm 6)$ and $1/(304\pm 3)$, implying that the superconductivity may not be entirely conventional in nature.

Figure 1

Crystal structure of the $R{\mathrm{RuB}}_2$ ternary borides. The $R$ atoms (large spheres) form zigzag chains that run parallel to the $b$ crystallographic axis. The B atoms (small spheres) form weakly interacting dimers, with the Ru atoms (medium spheres) isolated.

Figure 2

Temperature dependence of the magnetic susceptibility $\chi$ for (a) ${\mathrm{LuRuB}}_2$ and (b) ${\mathrm{YRuB}}_2$. The samples were cooled in zero field to 1.8 K, at which point a field of 1 mT was applied. Data were collected upon zero-field-cooled warming (ZFCW) and during a subsequent field-cooled cooling (FCC).

Figure 3

Time evolution of the spin polarization of muons implanted under zero-field conditions in (a) ${\mathrm{LuRuB}}_2$ and (b) ${\mathrm{YRuB}}_2$ at temperatures above and below $T_{\mathrm{c}}$. The time-independent background due to muons stopping in silver has been subtracted, and the data normalized to the initial asymmetry—the muons are 100% spin polarized at $t=0$ s. The solid lines are the results of fitting the data to Eq. (2).

Figure 4

Temperature dependence of the electronic relaxation rate in ${\mathrm{LuRuB}}_2$ (triangles) and ${\mathrm{YRuB}}_2$ (circles), collected in ZF and in an applied longitudinal field of 10 mT. The solid lines are guides to the eye, indicating the exponential decay of $\mathrm{\Lambda }$ in ZF as $T$ is increased.

Figure 5

Representative TF-$\mu \mathrm{SR}$ polarization signals collected (a) above and (b) below $T_{\mathrm{c}}$ in ${\mathrm{LuRuB}}_2$ under an applied field of 30 mT. A nondecaying background oscillation due to muons stopping in the silver has been subtracted, and the data normalized to the initial asymmetry. The solid lines are fits using Eq. (3).

Figure 6

TF-$\mu \mathrm{SR}$ effective depolarization rates in (a) ${\mathrm{LuRuB}}_2$ and (b) ${\mathrm{YRuB}}_2$, calculated from the ${\sigma }_i$ (insets) as described in the text. The solid line is a fit to Eq. (7), which is valid as there is a simple numerical coefficient relating ${\sigma }_{\mathrm{eff}}$ and ${\lambda }^{-2}$.

Figure 7

Temperature dependence of the superfluid density as a function of the reduced temperature $T/T_{\mathrm{c}}$ for ${\mathrm{LuRuB}}_2$ (triangles) and ${\mathrm{YRuB}}_2$ (circles). The data overlay each other, reflecting the high degree of similarity in the order parameters of the two materials. The solid lines are fits using Eq. (7).

Figure 8

The results of the $\mu \mathrm{SR}$ experiments summarized in the “Uemura plot,” which describes a universal scaling between $T_{\mathrm{c}}$ and $T_{\mathrm{F}}$ in different classes of superconductors. The Lu and Y ternary borides find themselves halfway between the conventional and unconventional regions, in the vicinity of the borocarbide and hexaboride superconductors.