Possible unconventional pairing in ${(\mathrm{Ca},\mathrm{Sr})}_3{(\mathrm{Ir},\mathrm{Rh})}_4{\mathrm{Sn}}_{13}$ superconductors revealed by controlling disorder

We study the evolution of temperature-dependent resistivity with added pointlike disorder induced by 2.5 MeV electron irradiation in stoichiometric compositions of the “3-4-13” stannides, ${\left(\text{Ca,Sr}\right)}_3{\left(\text{Ir,Rh}\right)}_4{\text{Sn}}_{13}$. Three of these cubic compounds exhibit a proposed microscopic coexistence of charge density wave (CDW) order and superconductivity (SC), while ${\text{Ca}}_3{\text{Rh}}_4{\text{Sn}}_{13}$ does not develop CDW order. As expected, the CDW transition temperature $T_{\text{CDW}}$ is universally suppressed by irradiation in all three compositions. The superconducting transition temperature, $T_c$, behaves in a more complex manner. In ${\text{Sr}}_3{\text{Rh}}_4{\text{Sn}}_{13}$, it increases initially in a way consistent with a direct competition of CDW and SC, but quickly saturates at higher irradiation doses. In the other three compounds, $T_c$ is monotonically suppressed by irradiation. The strongest suppression is found in ${\text{Ca}}_3{\text{Rh}}_4{\text{Sn}}_{13}$, which does not have CDW order. We further examine this composition by measuring the London penetration depth $\lambda \left(T\right)$, from which we derive the superfluid density. The result unambiguously points to a weak-coupling, full single gap, isotropic superconducting state. Therefore we must explain two seemingly incompatible experimental observations: a single isotropic superconducting gap and a significant suppression of $T_c$ by nonmagnetic disorder. We conduct a quantitative theoretical analysis based on a generalized Anderson theorem which points to an unconventional multiband $s^{+-}$-pairing state where the sign of the order parameter is different on one (or a small subset) of the smaller Fermi surface sheets but remains isotropic and overall fully gapped.

Figure 1

(a) Combined phase diagram of 3-4-13 compounds as determined from the measurements of ${\left({\mathrm{Ca}}_x{\mathrm{Sr}}_{1-x}\right)}_3{\mathrm{Rh}}_4{\mathrm{Sn}}_{13}$ (bottom axis, cross symbols) [29] and ${\left({\mathrm{Ca}}_x{\mathrm{Sr}}_{1-x}\right)}_3{\mathrm{Ir}}_4{\mathrm{Sn}}_{1}3$ (top axis, square symbols) systems [30]. The phase diagram for ${\left({\mathrm{Ca}}_x{\mathrm{Sr}}_{1-x}\right)}_3{(\mathrm{Rh},\mathrm{Ir})}_4{\mathrm{Sn}}_{13}$ was mapped using a combination of doping and pressure. The positions of the samples used in this study are shown by yellow-red stars. (b) Temperature-dependent resistivity of ${(\mathrm{Sr},\mathrm{Ca})}_3{(\mathrm{Rh},\mathrm{Ir})}_4{\mathrm{Sn}}_{13}$ samples selected for electron irradiation in this study. The inset zooms at the superconducting transition.

Figure 2

Total knock-out cross-sections for studied compounds as function of electron energy. Our operating energy of 2.5 MeV is marked by a dotted line. The difference between Ca/Sr pairs is negligible and is not large between Ir/Rh, being about 30 barn larger for Ir compounds.

Figure 3

The evolution of temperature-dependent resistivity of ${\text{Sr}}_3{\text{Ir}}_4{\text{Sn}}_{13}$ in pristine (blue line) and after electron irradiation of 1.14 (yellow), and 4.4 $\mathrm{C}/{\mathrm{cm}}^2$ (red). The green dashed line shows the resistivity difference between the pristine and 4.4 $\mathrm{C}/{\mathrm{cm}}^2$ curves, finding the Matthiessen's rule to be valid above $T_{\text{CDW}}$, but, expectedly, grossly violated below. The small cartoon in the top left corner indicates the sample's position on the generic phase diagram. The left inset shows the resistivity derivative $d\rho /dT$ in the vicinity of the CDW transition, where the arrows show the positions of the sharp features used to determine $T_{\text{CDW}}$. The right inset zooms into the region around the superconducting transition.

Figure 4

Temperature-dependent resistivity of ${\text{Sr}}_3{\text{Rh}}_4{\text{Sn}}_{13}$ before irradiation (blue curve), after $1.14\mathrm{C}/{\mathrm{cm}}^2$ irradiation (yellow), and after 4.44 $\mathrm{C}/{\mathrm{cm}}^2$ (red) irradiations. The green dashed line shows the resistivity difference between the pristine and 4.44 $\mathrm{C}/{\mathrm{cm}}^2$ curves, finding the Matthiessen rule valid above $T_{\text{CDW}}$, but violated below. The small cartoon in the top left corner indicates the sample position on generic phase diagram. The left inset shows the derivative of the resistivity in a region around the CDW transition with arrows pointing to $T_{\text{CDW}}$. This emphasizes the transition shift between the pristine state and after 4.44 $\mathrm{C}/{\mathrm{cm}}^2$ dose of irradiation. The right inset zooms into a region around the superconducting transition showing a nonmonotonic behavior of $T_c$: a small initial $T_c$ increase after $1.14\mathrm{C}/{\mathrm{cm}}^2$ irradiation, but only minimal changes in the behavior between 1.14 and 4.44 $\mathrm{C}/{\mathrm{cm}}^2$.

Figure 5

Temperature-dependent resistivity of ${\text{Ca}}_3{\text{Ir}}_4{\text{Sn}}_{13}$ before irradiation (blue line), after receiving 2.17 $\mathrm{C}/{\mathrm{cm}}^2$ of irradiation (yellow line), and then an additional 3.3 $\mathrm{C}/{\mathrm{cm}}^2$ for a total dose of 5.47 $\mathrm{C}/{\mathrm{cm}}^2$ of electron irradiation. The green dashed line shows the difference between the pristine and 5.47 $\mathrm{C}/{\mathrm{cm}}^2$ curves, showing deviation from Matthiessen's rule below the transition temperature $T_{\text{CDW}}$. The small cartoon in the top left corner indicates sample position on generic phase diagram. The left inset shows the derivative of the resistivity showing suppression and blurring of the CDW phase transition with irradiation. The right inset shows the shift in the superconducting transition temperature.

Figure 6

Temperature-dependent resistivity of ${\text{Ca}}_3{\text{Rh}}_4{\text{Sn}}_{13}$ in the pristine state before irradiation (blue curve) and after 2.1 $\mathrm{C}/{\mathrm{cm}}^2$ irradiation (red curve). The green dashed line shows the difference between the curves, finding strong Matthiessen's rule violation at all temperatures above the superconducting transition. The small cartoon in the top left corner indicates the position of the compound on the generic phase diagram, (the most right) with no CDW transition. The inset shows resistivity in the vicinity of the superconducting transition, revealing substantial $T_c$ suppression by electron irradiation.

Figure 7

(a) The evolution of the CDW transition temperature normalized by the value before irradiation, $T_{\text{CDW}}/T_{\text{CDW},0}$, and (b) similar plot of the superconducting transition temperature normalized by the value in pristine samples, $T_c/T_{c,0}$ with the induced disorder in units of defects per formula unit (dpf), see text for details. The $X$ and $Y$ scales on both graphs are the same for easy comparison. Note a significant increase of $T_c$ suppression in samples where CDW does not coexist with superconductivity.

Figure 8

Normalized suppression of the superconducting transition temperature, $T_c/T_{c0}$, as function of the dimensionless scattering rate ${\gamma }^{\lambda }$ evaluated from resistivity and London penetration depth, $\lambda \left(0\right)$ using Eq. (1). To set the scale, grey dashed line shows the predictions of Abrikosov-Gor'kov theory for a line-nodal $d$-wave superconductor with nonmagnetic impurities [81, 86, 87]. Purple stars are experimental data for the nodal iron-based superconductor $\mathrm{Ba}{\mathrm{Fe}}_2{\left({\mathrm{As}}_{1-x}{\mathrm{P}}_x\right)}_2$ [10], red-yellow squares are for the superconducting Dirac semimetal $\mathrm{Pd}{\mathrm{Te}}_2$ [78], maroon cross-pentagons are for CDW superconductor ${\text{2H-NbSe}}_2$ [20]. The data for 3-4-13 compounds are shown by red rhombi for ${\text{Ca}}_3{\text{Rh}}_4{\text{Sn}}_{13}$, blue squares for ${\text{Ca}}_3{\text{Ir}}_4{\text{Sn}}_{13}$, yellow-blue circles for ${\text{Sr}}_3{\text{Rh}}_4{\text{Sn}}_{13}$ and yellow triangles for ${\text{Sr}}_3{\text{Ir}}_4{\text{Sn}}_{13}$.

Figure 9

(Main) Superfluid density in ${\text{Ca}}_3{\text{Rh}}_4{\text{Sn}}_{13}$ calculated from the London penetration depth, $\lambda \left(T\right)$, measured using the tunnel-diode resonator technique. The thick orange line shows standard isotropic $s$-wave BCS behavior, while the dashed line shows the expectation for a nodal $d$-wave order parameter. The lower inset shows a BCS fit of $\mathrm{\Delta }\lambda \left(T\right)$ to the single-gap expression shown. The only fitting parameter was $\lambda \left(0\right)=330$ nm, while the $T_c$ and the weak-coupling gap ratio, ${\mathrm{\Delta }}_0/T_c=1.764$ was fixed. The obtained $\lambda \left(0\right)=330\text{nm}$ was used to construct the ${\rho }_s\left(T\right)={(1+\mathrm{\Delta }\lambda \left(T\right)/\lambda \left(0\right))}^{-2}$ shown in the main panel. The upper inset shows the sharp transition of our high-quality sample and $\lambda \left(T_c\right)$ consistent with the expected skin depth of the normal state.

Figure 10

Calculated normalized superfluid density to examine the influence of gap anisotropy. (a) $g$-wave correction to $s$-wave pairing, as defined in Eq. (4), with relative strengths $\delta =-3.66$ (blue) and $-1.08$ (black), needed to reproduce the observed suppression of $T_c$. The 3D structure of the gap in the reciprocal space is shown as insets. The standard $s$-wave, ${\mathrm{\Psi }}_{\mathit{k}}={\mathrm{\Psi }}_0$, and $d$-wave, ${\mathrm{\Psi }}_{\mathit{k}}={\mathrm{\Psi }}_0(k_x^2-k_y^2)$, cases are shown by dashed lines. Clearly, the computed superfluid density is far from the experimental data shown in Fig. 9. (b) Two-band $A_{1g}^{++}$ superconductor with two different gap magnitudes, ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$. To reproduce the observed $T_c$ suppression, the gap ratio should be ${\mathrm{\Delta }}_1/{\mathrm{\Delta }}_2=2.78$, see text. There are many sets of the interaction matrix elements to obtain that value at $T_c$ but with different temperature dependencies of ${\mathrm{\Delta }}_1/{\mathrm{\Delta }}_2$ below $T_c$ (see inset). We show the computed ${\rho }_s\left(T\right)$ for several choices, but none of them is consistent with the experimental data. See Appendix pp2 for more details of the computations.

Figure 11

Fermi surface contours at various $k_z$ in ${\text{Ca}}_3{\text{Rh}}_4{\text{Sn}}_{13}$ obtained within density functional theory (DFT). Different colors denote different bands crossing the Fermi energy (see Appendix pp3 for details).

Figure 12

(Top) Band structures and (bottom) partial density of states of the eight bands across the Fermi level in ${\text{Ca}}_3{\text{Rh}}_4{\text{Sn}}_{13}$.