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Evidence for broken time reversal symmetry (TRS) has been found in thesuperconducting states of the $R_5$Rh$_6$Sn$_{18}$ (R = Sc, Y, Lu) compoundswith a centrosymmetric caged crystal structure, but the origin of thisphenomenon is unresolved. Here we report neutron diffraction measurements ofsingle crystals with $R$=Lu, as well as measurements of the temperaturedependence of the magnetic penetration depth using a self-induced tunneldiode-oscillator (TDO) based technique, together with band structurecalculations using density functional theory. Neutron diffraction measurementsreveal that the system crystallizes in a tetragonal caged structure, and thatone of nominal Lu sites in the Lu$_5$Rh$_6$Sn$_{18}$ structure is occupied bySn, yielding a composition Lu$_{5-x}$Rh$_6$Sn$_{18+x}$ ($x=1$). The lowtemperature penetration depth shift $\Delta\lambda(T)$ exhibits an exponentialtemperature dependence below around $0.3T_c$, giving clear evidence for fullygapped superconductivity. The derived superfluid density is reasonably wellaccounted for by a single gap $s$-wave model, whereas agreement cannot be foundfor models of TRS breaking states with two-component order parameters.Moreover, band structure calculations reveal multiple bands crossing the Fermilevel, and indicate that the aforementioned TRS breaking states would beexpected to have nodes on the Fermi surface, in constrast to the observations.

Nodeless superconductivity in Lu$_{5-x}$Rh$_6$Sn$_{18+x}$ with broken time reversal symmetry