Microscopic studies of the normal and superconducting state of Ca${}_3$Ir${}_4$Sn${}_{13}$

We report on muon spin rotation ($\mu$SR) studies of the superconducting and magnetic properties of the ternary intermetallic stannide Ca${}_3$Ir${}_4$Sn${}_{13}$. This material has recently been the focus of intense research activity due to a proposed interplay of ferromagnetic spin fluctuations and superconductivity. In the temperature range $T=1.6$–200 K, we find that the zero-field muon relaxation rate is very low and does not provide evidence for spin fluctuations on the $\mu$SR time scale. The field-induced magnetization cannot be attributed to localized magnetic moments. In particular, our $\mu$SR data reveal that the anomaly observed in thermal and transport properties at $T^{*}\approx 38$ K is not of magnetic origin. Results for the transverse-field muon relaxation rate at $T=0.02$–12 K suggest that superconductivity emerges out of a normal state that is not of a Fermi-liquid type. This is unusual for an electronic system lacking partially filled $f$-electron shells. The superconducting state is dominated by a nodeless order parameter with a London penetration depth of ${\lambda }_{\mathrm{L}}=385\left(1\right)$ nm and the electron-phonon pairing interaction is in the strong-coupling limit.

Figure 1

Neutron powder diffraction of Ca${}_3$Ir${}_4$Sn${}_{13}$ measured at $T=300$ K (red circles). The data is fitted with a single $Pm\overline{3}n$ phase and scattering from the vanadium sample container (black line). Green and grey dashes indicate Bragg positions of Ca${}_3$Ir${}_4$Sn${}_{13}$ and vanadium, respectively. The largest differences between measured and refined intensities (blue line) appear from tiny shifts of the most intense Ca${}_3$Ir${}_4$Sn${}_{13}$ Bragg peaks.

Figure 2

(a) Magnetization $M\left(B\right)$ as a function of the effective magnetic field $B$ at $T=1.9$ K. A diamagnetic Meissner state is found at low fields, with a crossover to the mixed state at the lower critical field $B_{\mathrm{c}1}=530$ G. (b) $T$ dependence of the ZFC and FC magnetic susceptibility $\chi \left(T\right)$ with the superconducting transition at $T_{\mathrm{c}}=6.8$ K. The difference between ZFC and FC points to vortex pinning. (c) An anomaly of the resistivity $\rho \left(T\right)$ is clearly visible at $T^{*}\approx 38$ K for all applied current densities.

Figure 3

Asymmetry $A\left(t\right)$ of ZF (circles) and wLF measurements (diamonds; $B_{\mathrm{wLF}}=30.5$ G) at temperatures below (filled symbols) and above (open symbols) $T^{*}\approx 38$ K. The wLF quenches the nuclear moments, which reduces the muon relaxation rate. Lines depict fits to a static Gaussian Kubo-Toyabe relaxation function. The upper three data sets are shifted by 0.05 with respect to the lower series.

Figure 4

ZF (circles) and wLF (diamonds; $B_{\mathrm{wLF}}=30.5$ G) muon relaxation rate $\sigma \left(T\right)$ for $T=1.6$–200 K. We attribute the slight increase in ${\sigma }_{\mathrm{ZF}}$ at $T_{\mathrm{diff}}\approx 120$ K to muon diffusion for $T>T_{\mathrm{diff}}$. In the range $T=1.6$–100 K, we find a constant low ${\sigma }_{\mathrm{ZF}}=0.054\left(2\right)\mu$s${}^{-1}$ (red; shading, $\pm$1 SD). This is supported by ${\sigma }_{\mathrm{wLF}}=0.013\left(10\right)\mu$s${}^{-1}$ (blue), which is even lower and constant over the entire temperature range. An increase in the muon relaxation rate would be expected in a scenario with fluctuating or quasistatic magnetism at $T^{*}$.

Figure 5

TF measurements with an FC applied magnetic field of $B_{\mathrm{TF}}=622$ G. Only a marginal relaxation is found in the normal state [$T=10$ K $>T_{\mathrm{c}}$; (blue) diamonds]. However, $\sigma$ increases strongly deep inside the superconducting state [$T=0.02$ K $\ll T_{\mathrm{c}}$; (red) circles] as a result of the field distribution from the ordered vortex lattice in the mixed state.

Figure 6

(a) FC transverse-field muon relaxation rate $\sigma \left(T\right)={[{\sigma }_{\mathrm{sc}}^2\left(T\right)+{\sigma }_{\mathrm{bg}}^2]}^{1/2}$, consisting of a superconducting contribution ${\sigma }_{\mathrm{sc}}\left(T\right)$ for $T<T_{\mathrm{c}}$ and a constant background ${\sigma }_{\mathrm{bg}}$. Inset: Diamagnetic shift $\Delta B\left(T\right)=B_{\mathrm{int}}\left(T\right)-B_{\mathrm{eff}}$. (b) $1/{\lambda }^2\left(T\right)\propto n_{\mathrm{s}}\left(T\right)$ with the weak-coupling prediction from BCS theory [lower (blue) curve] and a nodeless strong-coupling superconducting order parameter [upper (red) curve].

Figure 7

(a) Electrostatic potential along the $(h,0,0)$ (red), $(h,h,0)$ (blue), and $(h,h,h)$ (green) high-symmetry directions. Four candidates of muon sites are found at (1/2, 0, 0) (red arrow), (1/2, 1/2, 0) (blue arrow), and $(h,h,h)$ with $h=0.19\left(2\right)$ and 0.43(1) (green arrows). Ca${}^{2+}$, Ir${}^{4+}$, and Sn${}^{4+}$ were assumed for the depicted calculation. (b) A $T$-dependent diamagnetic shift in the normal state is not expected for a Fermi-liquid ground state.