The effective increase in atomic scale disorder by doping and superconductivity in Ca$_3$Rh$_4$Sn$_{13}$

The comprehensive research of the electronic structure, thermodynamic and electrical transport properties reveals the existence of inhomogeneous superconductivity due to structural disorder in Ca$_3$Rh$_4$Sn$_{13}$ doped with La (Ca$_{3-x}$La$_x$Rh$_4$Sn$_{13}$) or Ce (Ca$_{3-x}$Ce$_x$Rh$_4$Sn$_{13}$) with superconducting critical temperatures $T_c^{\star}$ higher than those ($T_c$) observed in the parent compounds. The $T-x$ diagrams and the entropy $S(x)_T$ isotherms well document the relation between degree of an atomic disorder and separation of the {\it high-temperature} $T_c^{\star}$ and $T_c$-bulk phases. In these dirty superconductors with the mean free path much smaller than the coherence length, the Werthamer-Helfand-Hohenber theoretical model does not well fits the $H_{c2}(T)$ data. We suggest that this can result from two-band superconductivity or from the presence of strong inhomogeneity in these systems. The multiband model very well describes the $H-T$ dependencies, but the present results as well as our previous studies give arguments for the scenario based on the presence of nanoscopic inhomogeneity of the superconducting state. We also revisited the nature of structural phase transition at $T^{\star}\sim 130-170$ K and documented that there might be another precursor transition at higher temperatures. The impact of the magnetic Ce-Ce correlations on the increase of $T_c$ in respect to the critical temperatures of Ca$_{3-x}$La$_x$Rh$_4$Sn$_{13}$ is also discussed.