Nodeless superconductivity in the cage-type superconductor Sc₅Ru₆Sn₁₈ with preserved time-reversal symmetry

Figure 1 shows the room temperature powder XRD pattern for Sc₅Ru₆Sn₁₈ with the scattering angle 2θ varying between 20°–80°. All the peaks in the pattern can be well indexed using a tetragonal structure with lattice constants a  =  1.387(3) nm and c  =  2.641(5) nm, which give a unit cell volume  =  5.080(5) ${\rm n}{{{\rm m}}^{3}}$ and a density ρ  =  7.8(3) g cm⁻³ (space group I4₁/acd.) The obtained values are in good agreement with those previously reported on the R₅M₆Sn₁₈ (R  =  rare earth, M  =  transition metal) family of compounds [19, 20].

Figure 2 shows the hysteresis curves of Sc₅Ru₆Sn₁₈ recorded at various temperatures below T_c, characterized by symmetric M–H loops. The M(H) curves exhibit a butterfly shape, typical of type-II superconductors. The symmetry of the hysteresis loop (or the lack of it) allows one to distinguish between pinning- and surface- (or geometrical barrier) induced hysteresis, with flux pinning known to produce symmetric hysteresis loops [21], as in our case. The lower-H_c1 and upper-H_c2 field values were determined from the magnetization data. The lower critical field, H_c1, is defined as the point where the field-dependent magnetization starts to deviates from linearity (see inset in figure 3) [22]. Since the actual magnetic field around the sample is larger than the applied magnetic field due to the demagnetizing effects by a factor of 1/(1  −  N), where N is the demagnetizing factor, we have to correct the H_c1 values for such an effect [23]. We have estimated the value of N from the rectangular prism approximation which is based on the dimensions of the crystal [24] and we get N  =  0.39. The main panel of figure 3 shows the temperature variation of corrected values of H_c1, which can be fitted by the relation [25]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{H}_{{\rm c1}}} (T)=~{{H}_{{\rm c}1}}(0)~\left[1-{{\left(\frac{T}{{{T}_{{\rm c}}}} \right)}^{2}} \right]\nonumber \end{align} \tag{ 1 }$$

where H_c1(0) is the lower critical field at zero-temperature. The fit gives H_c1(0)  =  157(9) Oe.

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The upper critical field is defined as the point where the magnetization curve touches the zero magnetic moment line (see inset in figure 4). The main panel in figure 4 shows the temperature variation of H_c2, which can be fitted using the Ginzburg–Landau equation given by [25]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{H}_{{\rm c}2}}\left(T \right)=~{{H}_{{\rm c}2}}\left(0 \right)~\frac{\left(1-{{q}^{2}} \right)}{\left(1+{{q}^{2}} \right)}\nonumber \end{align} \tag{ 2 }$$

where $q=~\frac{T}{{{T}_{{\rm c}}}}$ and H_c2(0) is the upper critical field at zero-temperature. The fit gives H_c2(0)  =  26(1) kOe. From this, we estimate a Ginzburg–Landau coherence length, $\xi (0)$  ≈  11.26(3) nm, by using the relation [25]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{H}_{{\rm c}2}} (0)=\frac{{{\varphi}_{0}}}{2\pi {{\xi}^{2}}(0)}\nonumber \end{align} \tag{ 3 }$$

where φ₀  =  2.07  ×  10⁻¹⁵ Tm², is the quantum of the magnetic flux. In addition, we can estimate the Ginzburg–Landau penetration depth using the relation [25]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{H}_{{\rm c}1}}(0)=\frac{{{\varphi}_{0}}}{2\pi {{\lambda}^{2}}(0)}\left[\ln \frac{\lambda (0)}{\xi (0)}+0.12 \right].\nonumber \end{align} \tag{ 4 }$$

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By using ${{H}_{{\rm c}1}}(0)$  =  157(9) Oe and $\xi (0)$  ≈  11.26(3) nm, we obtain $\lambda (0)=260(7)$ nm.

Finally, the Ginzburg–Landau parameter ($\kappa$) is obtained from the relation $\kappa =\frac{\lambda (0)}{\xi (0)}$. By using the values for $\lambda (0)$ and $\xi (0)$, we find κ  ≈  23(2)  >  $\frac{1}{\sqrt{2}}$, the latter indicating that Sc₅Ru₆Sn₁₈ is indeed a strong type-II superconductor. The zero-temperature thermodynamic critical field was estimated by using the relation [25],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{H}_{{\rm c}2}}(0)=\sqrt{2}\kappa {{H}_{{\rm c}}}(0).\nonumber \end{align} \tag{ 5 }$$

By using the values for H_c2(0)  =  26(1) kOe and κ  ≈  23(2), we get ${{H}_{{\rm c}}}(0)=$ 799(6) Oe.

The field dependence of the critical current density (J_c) was derived from the width ΔM of the magnetization curve, by using the Bean model [26, 27]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{J}_{{\rm c}}}=20\Delta M/[d(1-d/3b)]\nonumber \end{align} \tag{ 6 }$$

where d (mm) and b (mm) are the sample dimensions (d  <  b). In our case d  =  1.25 mm and b  =  1.40 mm. The critical current density (J_c) is expected to have a maximum at the lower critical field, whereas above this threshold it decreases rapidly with the increasing field. Figure 5(a) shows J_c as a function of the magnetic field at various temperatures. Above the threshold field, for each isothermal measurement, initially J_c shows an exponential decrease followed by a power-law variation, in good agreement with the previous reports [28, 29]. We obtain J_c(1.8 K) ~ 6(3)  ×  10⁸ A m⁻². The de-pairing current is given by the relation:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{J}_{{\rm d}}}=\frac{{{\varphi}_{0}}}{3\sqrt{3}{{\mu}_{0}}\pi {{\lambda}^{2}}\xi}.\nonumber \end{align} \tag{ 7 }$$

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By using the previously obtained values of ξ(0)  =  11.26(3) nm and λ(0)  =  260(7) nm, we find J_d  =  3.07(5)  ×  10¹⁰ A m⁻². The long coherence length implies a high pinning energy and flux-lines that do not move easily [30]. The material might have a large concentration of weak pinning centers, which ultimately leads to collective pinning. In the scenario of collective pinning, the critical current density depends strongly on the magnetic field and is typically small. This behavior has been observed, e.g., in case of layered inhomogeneous superconductors [31]. The irreversibility field (H_irr), at which the magnetic hysteresis disappears, is determined by the criterion J_c  =  2  ×  10⁷ A m⁻² [28]. The variation of the H_irr and H_c2 with temperature is shown in figure 5(b). We note that the irreversibility field is comparable to the upper critical field (H_c2), a rather plausible result considering that the material has a low T_c and a high coherence length [30].

To further confirm the superconducting transition temperature we measured the ac magnetic susceptibility in the temperature range 1.8–10 K, with H_ac  =  5 Oe and H_dc  =  0 Oe. Figure 6(a) shows the imaginary part of the ac susceptibility, while figure 6(b) shows the real part, corresponding to the out-of-phase and in-phase components, respectively (with respect to the ac field). The real part of the ac susceptibility versus temperature, represents the transition from the Meissner state (perfect shielding) to the complete penetration of the ac magnetic field inside the sample. On the other hand, the imaginary part of the ac susceptibility represents the ac losses occurring when the ac field penetrates the sample. As shown in figure 6(a), χ'' has a sharp transition near T_c  =  3.6 K. The transition in χ'' is independent of the frequency of the driving ac field. The appearance of a peak in χ'' is commonly associated with the superconducting transition. When the temperature is far below the critical temperature and the magnetic field is smaller than the lower critical field H_c1, the screening current generated by the ac field is confined to regions near the sample surface. Hence, no magnetic flux enters the sample and the ac losses are minimal; consequently χ'' is almost zero. As the temperature is raised, the magnetic field starts penetrating the sample and ac losses start increasing. The maximum in χ'' occurs when the ac field fully penetrates the sample.

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The specific heat in the superconducting state is one of the key parameters to reflect closely the superconducting gap and its symmetry. Therefore, we measured and analyzed in detail the zero-field specific heat of Sc₅Ru₆Sn₁₈. The electronic specific heat (C_e/T) of the sample was obtained after subtracting the phonon contribution from the raw experimental data. As shown in the inset of figure 7, the normal-state specific heat was fitted using the relation C/T  =  γ  +  βT²  +  δT⁴ for 14 K²  ⩽  T²  ⩽  42 K² to obtain γ  =  36.93(6) mJ mol⁻¹ K⁻², β  =  2.5(5) mJ mol⁻¹ K⁻⁴ and δ  =  1.2(9) mJ mol⁻¹ K⁻⁶. The Debye temperature θ_D  =  205(1) K was calculated using the relation:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \beta =p(\frac{12}{5}){{\pi}^{4}}R\theta _{{\rm D}}^{-3}\nonumber \end{align} \tag{ 8 }$$

where p  =  29 is the number of atoms in one formula unit and R  =  8.314 J K⁻¹ mol⁻¹. The density of states at the Fermi level $N({{E}_{{\rm F}}})$ was estimated from [32]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \gamma =\frac{{{\left(\pi {{k}_{{\rm B}}} \right)}^{2}}}{3}N({{E}_{{\rm F}}})\nonumber \end{align} \tag{ 9 }$$

where k_B is the Boltzmann constant. By using the previously obtained γ value, we obtain $N({{E}_{{\rm F}}})$  =  15.24(6) states/eV per formula unit. The strength of electron-phonon coupling can be estimated by the McMillan equation [33]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\lambda}_{{\rm el{\rm {{\text {--}}}}ph}}}=\frac{1.04+{{\mu}^{*}}\ln (\frac{{{\theta}_{{\rm D}}}}{1.45{{T}_{{\rm c}}}})}{\left(1-0.62{{\mu}^{*}} \right)\ln \left(\frac{{{\theta}_{{\rm D}}}}{1.45{{T}_{{\rm c}}}} \right)-1.04}.\nonumber \end{align} \tag{ 10 }$$

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Here ${{\mu}^{*}}=0.13$ is the Coulomb repulsion parameter. By inserting the values for θ_D  =  205(1) K and T_c  =  3.5(1) K, we get ${{\lambda}_{{\rm el{\rm {{\text {--}}}}ph}}}$  =  0.64(4), which indicates that Sc₅Ru₆Sn₁₈ is a strongly coupled superconductor.

The calculated density of states N(E_F) and the effective mass of quasiparticles m^*, depend on the many-body electron-phonon interactions. These quantities are related to the bare band-structure of density of states N_band(E_F) and $m_{{\rm band}}^{*}$ by the following relations [34]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle N({{E}_{{\rm F}}})={{N}_{{\rm band}}}({{E}_{{\rm F}}})(1+{{\lambda}_{{\rm el}-{\rm ph}}})\nonumber \end{align} \tag{ 11 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{m}^{*}}=m_{{\rm band}}^{*}(1+{{\lambda}_{{\rm el{\rm {{\text {--}}}}ph}}}).\nonumber \end{align} \tag{ 12 }$$

By using the values of $N({{E}_{{\rm F}}})$ and ${{\lambda}_{{\rm el{\rm {{\text {--}}}}ph}}}$ in equation (11), we get ${{N}_{{\rm band}}}({{E}_{{\rm F}}})$  =  9.270(6) states eV⁻¹ f.u. By using $m_{{\rm band}}^{*}$  =  ${{m}_{{\rm e}}}$ (the free electron mass), in equation (12), we obtain 1.644 ${{m}_{{\rm e}}}$ for the mass of the quasiparticles.

The density of states can be used to estimate the Fermi velocity, v_F, which is related to $N({{E}_{{\rm F}}})$ by [35],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{v}_{{\rm F}}}=({{\pi}^{2}}{{\hbar}^{3}}/{{m}^{*2}}{{V}_{{\rm f}{\rm .u}.}})N ({{E}_{{\rm F}}}).\nonumber \end{align} \tag{ 13 }$$

Where $\hbar$  =  Planck's constant/2π, ${{V}_{{\rm f}{\rm .u}.}}$  =  ${{V}_{{\rm cell}}}$/2 is the volume per formula unit. Using the values of m^*, ${{V}_{{\rm cell}}}$ and $N({{E}_{{\rm F}}})$, we get ${{v}_{{\rm F}}}$  =  1.94(7)  ×  10⁷ cm s⁻¹ for Sc₅Ru₆Sn₁₈. We can further estimate the mean free path $(l)$ of the superconducting carriers by using the relation $l={{v}_{{\rm F}}}\tau$, where $\tau$ is the mean free scattering time, given by $\tau ={{m}^{*}}/{{n}_{{\rm s}}}{{e}^{2}}{{\rho}_{0}}$, where ${{\rho}_{0}}$ is the residual resistivity and ${{n}_{{\rm s}}}$ is the superconducting carrier density. The latter is given by ${{n}_{{\rm s}}}={{m}^{*3}}v_{{\rm F}}^{3}/3{{\pi}^{2}}{{h}^{3}}$, assuming a spherical Fermi surface [35]. Using the value of ${{v}_{{\rm F}}}$, we get ${{n}_{{\rm s}}}$  =  7.05(2)  ×  10²⁶ carriers m⁻³. By combining these expressions we get [35],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle l=3{{\pi}^{2}}\left(\frac{\hbar}{{{e}^{2}}{{\rho}_{0}}} \right){{\left(\frac{\hbar}{{{m}^{*}}{{v}_{{\rm F}}}} \right)}^{2}}.\nonumber \end{align} \tag{ 14 }$$

Putting ${{\rho}_{0}}$  =  200 µ$\Omega$ cm (resistivity data not shown here) and the above estimated values of ${{m}^{*}}$ and ${{v}_{{\rm F}}}$, we get l  =  8.14(5) nm.

The normalized electronic specific heat C_e/γT is plotted versus the normalized temperature T/T_c as shown in figure 7. The normalized specific heat jump at T_c is $\frac{\Delta {{C}_{{\rm e}}}}{\gamma {{T}_{{\rm c}}}}=1.6$ for γ  =  36.93(6) mJ mol⁻¹ K⁻². Since such a value is higher than the BCS value of 1.43 (for a weakly coupled superconductor), it again indicates a strong electron-phonon coupling in Sc₅Ru₆Sn_18.

The temperature dependence of specific heat in the superconducting state is best modelled by an s-wave single-gap BCS expression of the normalized entropy S [34]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{S}{\gamma {{T}_{{\rm c}}}}=-\frac{6}{{{\pi}^{2}}}\left(\frac{\Delta (0)}{{{k}_{{\rm B}}}{{T}_{{\rm C}}}} \right)\int_{0}^{\infty}{\left[\,f~\ln \left(\,f \right)+\left(1-f \right)~\ln (1-f) \right]}{\rm d}x\nonumber \end{align} \tag{ 15 }$$

where $\newcommand{\e}{{\rm e}} f\left(\xi \right)={{\left[\exp \left(\frac{E\left(\xi \right)}{{{k}_{{\rm B}}}T} \right)+1 \right]}^{-1}}$ is the Fermi function, $E\left(\xi \right)=\sqrt{{{\xi}^{2}}+{{\Delta}^{2}}(q)}$, where $\xi$ is the energy of electrons in the normal state, measured with respect to the Fermi energy, $x=\xi /\Delta (0)$, $q=T/{{T}_{{\rm c}}}$ and $\Delta (q)=\Delta (0)~\tanh \left[1.82{{\left(1.018\left(\left(\frac{1}{q} \right)-1 \right) \right)}^{0.51}} \right]$ is the temperature dependence of the superconducting gap. The normalized electronic specific heat is given by the expression [34]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{{C}_{{\rm e}}}}{\gamma {{T}_{{\rm c}}}}=q\frac{{\rm d}}{{\rm d}q}\left(\frac{S}{\gamma {{T}_{{\rm c}}}} \right).\nonumber \end{align} \tag{ 16 }$$

The electronic specific heat (${{C}_{{\rm e}}}$) in the superconducting state is described by equation (16), while it is equal to $\gamma {{T}_{{\rm c}}}$ in the normal state. The specific heat data were fitted to equation (16) as shown in the figure 7. The fit gives a superconducting gap, $\Delta (0)$  =  0.64(1) meV.

The Sommerfeld constant ($\gamma$) is calculated by fitting the $\frac{{{C}_{{\rm e}}}}{T}$ versus T data for various fields using the relation [36]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{{C}_{{\rm e}}}}{T}=\gamma +\frac{{{A}_{1}}}{T}\exp \left(\frac{-{{A}_{2}}{{T}_{{\rm c}}}}{T} \right)\nonumber \end{align} \tag{ 17 }$$

where ${{A}_{1}}$ and ${{A}_{2}}$ are constants.

The single gap s-wave superconductivity in Sc₅Ru₆Sn₁₈ is also confirmed by the magnetic-field dependence of the electronic specific-heat coefficient (Sommerfeld constant), γ(H). In a single-gap type-II superconductor, the Sommerfeld constant is proportional to the vortex density. When the applied magnetic field is increased, the density of vortices increases too, which, in turn, result in an increase of the quasiparticle density of states. Consequently, γ turns out to be proportional to the applied magnetic field in case of a nodeless and isotropic s-wave superconductor [34, 36, 37]. On the other hand, in case of nodes in the superconducting gap, Volovik predicted a nonlinear relation given by $\gamma \propto ~{{H}^{0.5}}$ [38]. The field dependence of $\gamma$ is shown in figure 8. The linear relation between $\gamma$ and $H$ confirms the single-gap s-wave superconductivity in Sc₅Ru₆Sn₁₈.

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The condensation energy $U\left(0 \right)$ can be estimated from the relation:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle U(0)=\frac{1}{2}{{\Delta}^{2}}\left(0 \right)~{{N}_{{\rm band}}}({{E}_{{\rm F}}})=\frac{3\gamma {{\Delta}^{2}}\left(0 \right)}{4{{\pi}^{2}}k_{{\rm B}}^{2}}.\nonumber \end{align} \tag{ 18 }$$

Using the previously obtained values of $\gamma$  =  36.93(6) mJ mol⁻¹ K⁻² and $\Delta (0)$  =  0.64(1) meV, we get $U(0)$  =  156.8(9) mJ mol⁻¹.

To further confirm the characteristics of the superconducting gap function, we analyzed ${{C}_{{\rm e}}}(T)$ by fitting the data by means of different functional forms, i.e., ${{e}^{-b/T}}$, ${{T}^{2}}$ and ${{T}^{3}}$, corresponding to the expected temperature dependence of a superconducting gap which is isotropic, has line nodes or linear point nodes, respectively. As can be seen in figure 9, well below $T/{{T}_{{\rm c}}}$  =  0.8, data are best modelled by the exponential function, $\frac{{{C}_{{\rm e}}}}{\gamma T}=a{{{\rm e}}^{-b{{T}_{{\rm c}}}/T}}$. The quadratic and cubic fits instead are rather poor. Similar exponential fits have been used to model the superconducting gap of other superconductors with comparable transition temperatures [9, 39, 40]. Thus, the above analysis confirms that Sc₅Ru₆Sn₁₈ has an isotropic gap and s-wave pairing.

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We can estimate the London penetration depth at T  =  0 K, ${{\lambda}_{{\rm L}}}(0)$, using the relation [25]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \lambda _{{\rm L}}^{2} (0)=\frac{{{m}^{*}}{{c}^{2}}}{4\pi n{{e}^{2}}}=~\frac{3\pi {{c}^{2}}{{\hbar}^{3}}}{4{{m}^{*2}}{{e}^{2}}v_{{\rm F}}^{3}}\nonumber \end{align} \tag{ 19 }$$

where c is the speed of light in vacuum. Putting the values of all parameters, we get ${{\lambda}_{{\rm L}}}(0)$  =  245(6) nm. The BCS coherence length ${{\xi}_{0}}$ can be estimated from ${{v}_{{\rm F}}}$ and the superconducting energy gap $\Delta (0)$ using the relation [41]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\xi}_{0}}=\frac{\hbar {{v}_{{\rm F}}}}{\pi \Delta (0)}.\nonumber \end{align} \tag{ 20 }$$

Putting the values of ${{v}_{{\rm F}}}$  =  1.94(7)  ×  10⁷ cm s⁻¹ and $\Delta (0)$  =  0.64(1) meV, we get ${{\xi}_{0}}$  =  63.72(9) nm. We find that BCS coherence length (${{\xi}_{0}}$) is much larger than the mean free path l  =  8.14(5) nm, ${}^{l}\diagup{}_{{{\xi}_{0}}}\;$  =  0.12 $\ll$ 1, which indicates that the superconductivity in Sc₅Ru₆Sn₁₈ is in the moderate dirty limit.

Figure 10(a) shows the TF-µSR spectra collected above (5 K) and below (1.5 K) the superconducting transition temperature, T_c. The fast decay of muon-spin polarization below T_c indicates an inhomogeneous field distribution due to the flux line lattice (FLL) in the vortex state. The time-domain spectra were fit by using the following model [9]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle G\left(t \right)={{A}_{1}}\cos \left({{\gamma}_{\mu}}{{B}_{1}}t+\Phi \right){{{\rm e}}^{-\frac{{{\left(\sigma t \right)}^{2}}}{2}}}+{{A}_{2}}\cos ({{\gamma}_{\mu}}{{B}_{2}}t+\Phi).\nonumber \end{align} \tag{ 21 }$$

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Here A₁ and A₂ are the initial muon-spin asymmetries, whereas B₁ and B₂ are the local fields sensed by muons implanted in the sample and the sample holder. ${{\gamma}_{\mu}}$  =  2π  ×  135.53 MHz T⁻¹ is the muon gyromagnetic ratio, Φ is the phase factor and σ is the Gaussian relaxation rate. Since the sample holder is non-magnetic, B₂ coincides with the applied magnetic field.

In the superconducting state, the Gaussian relaxation rate σ contains contributions from both the FLL (σ_sc) and the nuclear magnetic moments (${{\sigma}_{{\rm n}}}$), such that:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\sigma}^{2}}=~\sigma _{{\rm sc}}^{2}+\sigma _{{\rm n}}^{2}.\nonumber \end{align} \tag{ 22 }$$

By using equation (22), one can estimate the FLL-related relaxation rate ${{\sigma}_{{\rm sc}}}=\sqrt{{{\sigma}^{2}}-\sigma _{{\rm n}}^{2}},$ since ${{\sigma}_{{\rm n}}}$ is expected to be temperature independent in the measured temperature range and is determined from the measurements made above T_c. Considering that ${{\sigma}_{{\rm sc}}}$ is directly related to the superfluid density, the temperature dependence of ${{\sigma}_{{\rm sc}}}$ provides hints on the superconducting gap and its symmetry. Further, ${{\sigma}_{{\rm sc}}}$ can be modeled by [42–44]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{{\sigma}_{{\rm sc}}}(T)}{{{\sigma}_{{\rm sc}}}(0)}=1+2{{\left\langle \int_{{{\Delta}_{k}}}^{\infty}{\frac{EdE}{\sqrt{{{E}^{2}}-\Delta _{k}^{2}}}\frac{\partial f}{\partial E}} \right\rangle}_{{\rm FS}}}\nonumber \end{align} \tag{ 23 }$$

where f and E are the same as defined in equation (15). The curved brackets represent an average over the Fermi surface (FS). The superconducting gap is defined by ${{\Delta}_{k}}(q)$  =  $\Delta (q){{g}_{k}}$, ($q=T/{{T}_{{\rm c}}}$), which contains an angular dependent part ${{g}_{k}}$ and a temperature dependent part $\Delta \left(q \right)$, which can be approximated as $\Delta (q)=\Delta(0)~\tanh \left[1.82{{\left(1.018(\left(\frac{1}{q} \right)-1) \right)}^{0.51}} \right]$. As shown in figure 10(b), the temperature dependence of the normalized FLL-related relaxation rate ${{\sigma}_{{\rm sc}}}$(T)/${{\sigma}_{{\rm sc}}}$(0), can be fitted by a single-gap isotropic s-wave using equation (23) (for single gap s-wave model, ${{g}_{k}}=1$), with $\Delta (0)$  =  0.64(1) meV and ${{\sigma}_{{\rm sc}}}$(0)  =  0.178 $\mu {{{\rm s}}^{-1}}$. This implies a ratio $2\Delta (0)/{{k}_{{\rm B}}}{{T}_{{\rm c}}}=$ 4.25(4), i.e., higher than 3.53 expected for a weakly coupled BCS superconductor. This further confirms the strong electron–phonon coupling in Sc₅Ru₆Sn₁₈ and is consistent with the results obtained from specific-heat data.

The effective penetration depth (${{\lambda}_{{\rm ef\,f}}}$) for small applied fields, such that ${{H}_{{\rm applied}}}/{{H}_{{\rm c}2}}=0.0115\ll 1$, is given by the relation [9, 45]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\sigma _{{\rm sc}}^{2}(0)}{\gamma _{\mu}^{2}}=0.003\,71\frac{\varphi _{0}^{2}}{\lambda _{{\rm ef\,f}}^{4}}.\nonumber \end{align} \tag{ 24 }$$

Using ${{\sigma}_{{\rm sc}}}$(0)  =  0.178 $\mu {{{\rm s}}^{-1}}$, we get ${{\lambda}_{{\rm ef\,f}}}$  =  774 nm. The effective penetration depth $({{\lambda}_{{\rm ef\,f}}})$ and London penetration depth ${{\lambda}_{{\rm L}}}(0)$ are related by ${{\lambda}_{{\rm ef\,f}}}=~{{\lambda}_{{\rm L}}}(0)~\sqrt{1+\frac{{{\xi}_{0}}}{l}}~$ [25]. Using ${{\lambda}_{{\rm ef\,f}}}$  =  774 nm, ${{\xi}_{0}}$  =  63.72(9) nm and l  =  8.14(5) nm, we get, ${{\lambda}_{{\rm L}}}(0)$  =  258(7) nm, which is comparable to the value estimated using m^* and ${{v}_{{\rm F}}}$ and the magnetization measurements. This confirms the validity of our fitting model.

To probe the occurrence (or absence) of TRS breaking in Sc₅Ru₆Sn₁₈, we performed ZF-µSR experiments. The availability of 100% spin-polarized muon beams, along with the large muon gyromagnetic ratio, make ZF-µSR a very useful technique to detect the spontaneous internal fields, as has been shown in previous studies of Y₅Rh₆Sn₁₈ [18] and Sr₂RuO₄ [4]. In general, in absence of an external magnetic field, the onset of the superconducting phase does not induce any changes in the ZF-muon spin-relaxation rate. However, in the case of TRS breaking, the presence of tiny spontaneous currents leads to the associated weak magnetic fields, which are detected by ZF-µSR as an increase in the muon spin-relaxation rate. Since the expected effects are rather small, we measured carefully the muon spin-relaxation rates both above and below T_c.

The ZF-µSR time-domain data for three representative temperatures (1.5, 15 and 30 K, i.e., below and above T_c) are shown in figure 11(a). Muons stopped in the silver sample holder give rise to a time independent background signal. Since no precession signals were observed in the entire 1.5–30 K temperature range, we rule out the possibility of significant internal fields and consequently, of any long range magnetic order in Sc₅Ru₆Sn₁₈. In this scenario, the only possibility is that the muon-spin relaxation is due to the static, randomly distributed local fields, arising from the nuclear moments close to the muon-stopping site. In this case, the muon-spin depolarization can be described by the Kubo–Toyabe (KT) function:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{G}_{{\rm KT}}}(t)=\frac{1}{3}+\frac{2}{3}\left(1-{{\sigma}^{2}}{{t}^{2}} \right)\exp \left(\frac{-{{\sigma}^{2}}{{t}^{2}}}{2} \right).\nonumber \end{align} \tag{ 25 }$$

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After subtracting the background due to the silver sample holder, the ZF-µSR spectra could be analyzed by using the function:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle A (t)={{A}_{0}}{{G}_{{\rm KT}}} (t)~\exp (-{{\lambda}_{1}}t).\nonumber \end{align} \tag{ 26 }$$

Here, A₀ is the initial asymmetry at t  =  0. G_KT can be treated as a temperature-independent parameter (within the µSR time window [46]). λ₁ is the muon-spin relaxation rate, considered to reflect the fluctuations of surrounding electronic spins lying close to the muon. Solid lines in figure 11(a) are the best-fit results by using equation (26). Figure 11(b) shows the temperature dependence of λ₁, as obtained from the fits of the time spectra. Here λ₁ shows a slight enhancement with decreasing temperature, starting from T  >  T_c. This would mean that some unpaired electronic spins still remain, thus causing the observed muon-spin depolarization behavior.

A key result of the ZF-µSR data is the almost temperature independent λ₁ below ~10 K, especially when crossing T_c i.e. upon entering the superconducting phase. In conventional BCS type s-wave superconductors, no effects are expected in the ZF-µSR data in the superconducting state. On the other hand, when TRS is broken, ZF-µSR time spectra are modified due to the appearance of spontaneous magnetic fields below T_c. This is typically the case for the Cooper pairs with a p-wave symmetry, as, e.g. in Sr₂RuO₄ [4]. As shown in figure 11(b), since the time spectra do not show any relevant changes (within statistical errors), we must conclude that a TRS breaking is unlikely in the superconducting state of Sc₅Ru₆Sn₁₈.

Note that the TRS breaking depends on the pairing symmetry of the electrons in the superconducting state. The TRS can be broken if the superconducting state has degenerate representations, as is the case of triplet superconducting states [4]. However, in the case of singlet-superconducting states, where the superconducting state has non-degenerate representations, the TRS may not be broken. While TRS breaking is possible in systems with strong spin–orbit coupling such as Y₅Rh₆Sn₁₈ [18], it might still be conserved in systems with weak spin–orbit coupling, such as Sc₅Ru₆Sn₁₈.

In case of type-II superconductors, the breaking of Cooper pairs upon applying an external magnetic field may happen via two mechanisms: orbital or Pauli paramagnetic limiting field effects [34]. In case of an orbital pair-breaking mechanism, Cooper pairs break when the field-induced kinetic energy of the pair exceeds the superconducting condensation energy. On the other hand, in case of the Pauli paramagnetic effect, it is energetically favorable for electron spins to point in the direction of the applied magnetic field, thus decoupling from their partner, which results in the splitting of the Cooper pair [34]. For a BCS superconductor the orbital limit of the upper (i.e. orbital) critical field is given by the Werthamer–Helfand–Hohenberg (WHH) expression [47, 48]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H_{{\rm c}2}^{{\rm orbital}}\left(0 \right)=-0.693{{T}_{{\rm c}}}{{\left(\frac{{\rm d}{{H}_{{\rm c}2}}(T)}{{\rm d}T} \right)}_{T={{T}_{{\rm c}}}}}.\nonumber \end{align} \tag{ 27 }$$

The slope ${{\left(\frac{{\rm d}{{H}_{{\rm c}2}}(T)}{{\rm d}T} \right)}_{T={{T}_{{\rm c}}}}}$ is estimated from the ${{H}_{{\rm c}2}}-T$ phase diagram (see figure 4) and is equal to 8.1(1) kOe K⁻¹ in our case, which gives an orbital upper limiting field, $H_{{\rm c2}}^{{\rm orbital}}\left(0 \right)\approx$ 19.18 kOe. The Pauli limiting field, $H_{{\rm c}2}^{{\rm P}}\left(0 \right),$ within the BCS theory is given by [49, 50]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle H_{{\rm c}2}^{{\rm P}} (0)=1.86~{{T}_{{\rm c}}}.\nonumber \end{align} \tag{ 28 }$$

For ${{T}_{{\rm c}}}$  =  3.5 K, $H_{{\rm c}2}^{{\rm P}}(0)$  ≈  65.1 kOe. The Maki parameter [51], ${{\alpha}_{{\rm M}}}$, is used to measure the relative strength of the orbital and Pauli limiting field values and is given by the expression:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\alpha}_{{\rm M}}}=\sqrt{2}\frac{H_{{\rm c}2}^{{\rm orbital}}(0)}{H_{{\rm c}2}^{{\rm P}}(0)}.\nonumber \end{align} \tag{ 29 }$$

From this relation, we get ${{\alpha}_{{\rm M}}}$  ≈  0.29, which means that in our case, $H_{{\rm c}2}^{{\rm P}}(0)$ is much larger than $H_{{\rm c}2}^{{\rm orbital}}(0)$, hence implying that the upper critical field is limited by the orbital effects and the Pauli paramagnetic effect is negligible as indicated by a small value of Maki parameter. For Sc₅Ru₆Sn₁₈, the calculated ${{H}_{{\rm c}2}}(0)$ is close to the orbital limiting field and is much smaller than the Pauli limiting field.