We investigate the effect of enhancement of superconducting transition temperature $T_c$ by nonmagnetic atom disorder in the filled skutterudite-related tetragonal ${\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ compound doped with Ca. We documented experimentally that ${\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ and its Ca-doped alloys are electronically inhomogeneous at the nanoscale, when Ca content is smaller than $\sim 1.5$ in the ${\mathrm{Y}}_{5-x}{\mathrm{Ca}}_x{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ system, while for $x\ge 1.5$ much stronger chemical phase inhomogeneity is observed with an about 20% volume fraction of the second 3:4:13 cubic phase, which we interpret as a fluctuation of stoichiometry within the bulk sample. Then the enhancement of $T_c$ vs $x$ could be modeled by a mechanism proposed in recent theoretical reports for increasing the mean-field transition temperature $T_c$ in the presence of nonmagnetic disorder. The increase in disorder with doping increases the sample inhomogeneity and causes a systematic increase in $T_c$, while for two-phase $x\ge 1.5$ samples the critical temperature $T_c^{★}\approx 2\times T_c$ rapidly increases. Based on band structure calculations performed under pressure, we demonstrate how the change in density of states would affect $T_c$ of ${\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$. We obtained the Grüneisen parameter ${\gamma }_G$ larger for the inhomogeneous phase with respect to ${\gamma }_G$ of the bulk $T_c$ phase and attribute the enhancement of $T_c$ to larger stiffening of the high-temperature $T_c^{★}$ phase.

For a number of skutterudite-related cage compounds, such as cubic $R_3{M}_4{\mathrm{Sn}}_{13}$ ($R=\mathrm{Ca}$ or La, $M=\text{transition}$ $d$-electron metal) or tetragonal ${\mathrm{Y}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$, we have previously reported the impact of various atomic defects in the crystal lattices on the normal-state and superconducting properties of these clathrates. In these quasiskutterudites the nanoscale disorder and/or local inhomogeneity in composition lead to an abnormal increase in the superconducting transition temperature $T_c$. Here we investigate the impact of atomic defects in the superconductors ${\mathrm{Lu}}_{5-\delta }{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ (tetragonal symmetry $I{4}_1/acd$). We have documented that ${\mathrm{Lu}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ does not crystallize in the assumed stoichiometry but forms a 5-6-18 phase with a Lu deficiency ($\delta \approx 0.5$). Our comprehensive investigations of electronic structure and thermodynamic and electrical transport properties documented a $T_c$ increase in the more disordered sample (sample 1). The quality of two investigated samples (sample 1 and sample 2) with similar stoichiometry but different local inhomogeneity was obtained by microanalysis and electron transmission microscopy observations. The band structure calculations show a hybridization pseudogap in the electronic bands of stoichiometric ${\mathrm{Lu}}_5{\mathrm{Rh}}_6{\mathrm{Sn}}_{18}$ at about 0.3 eV in respect to the Fermi level ${\epsilon }_F$. The ab initio calculations predict the scenario that vacancies could shift this pseudogap towards ${\epsilon }_F$, which is manifested by the Mott variable-range hopping behavior in the resistivity $\rho \sim T^{-1/4}$ of more homogeneous sample 2.