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The comprehensive research of the electronic structure, thermodynamic andelectrical transport properties reveals the existence of inhomogeneoussuperconductivity due to structural disorder in Ca$_3$Rh$_4$Sn$_{13}$ dopedwith 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 relationbetween degree of an atomic disorder and separation of the {\ithigh-temperature} $T_c^{\star}$ and $T_c$-bulk phases. In these dirtysuperconductors 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-bandsuperconductivity or from the presence of strong inhomogeneity in thesesystems. The multiband model very well describes the $H-T$ dependencies, butthe present results as well as our previous studies give arguments for thescenario based on the presence of nanoscopic inhomogeneity of thesuperconducting state. We also revisited the nature of structural phasetransition at $T^{\star}\sim 130-170$ K and documented that there might beanother precursor transition at higher temperatures. The impact of the magneticCe-Ce correlations on the increase of $T_c$ in respect to the criticaltemperatures of Ca$_{3-x}$La$_x$Rh$_4$Sn$_{13}$ is also discussed.

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