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We investigate the superconducting properties of Sc$_{5}$Co$_{4}$Si$_{10}$using low-temperature resistivity, magnetization, heat capacity, and muon-spinrotation and relaxation ($\mu$SR) measurements. We find thatSc$_{5}$Co$_{4}$Si$_{10}$ {exhibits type-II} superconductivity with asuperconducting transition temperature $T_\mathrm{C}= 3.5 (1)$\,K. Thetemperature dependence of the superfluid density obtained from transverse-field$\mu$SR spectra is best modeled using an isotropic Bardeen-Cooper-Schrieffertype $s$-wave gap symmetry with $2\Delta/k_\mathrm{B}T_\mathrm{C} = 2.84(2)$.However, the zero-field muon-spin relaxation asymmetry reveals the appearanceof a spontaneous magnetic field below $T_\mathrm{C}$, indicating thattime-reversal symmetry (TRS) is broken in the superconducting state. Althoughthis behavior is commonly associated with non-unitary or mixed singlet-tripletpairing, our group-theoretical analysis of the Ginzburg-Landau free energyalongside density functional theory calculations indicates that unconventionalmechanisms are pretty unlikely. Therefore, we have hypothesized that TRSbreaking may occur via a conventional electron-phonon process.

Nodeless time-reversal symmetry breaking in the centrosymmetric superconductor Sc$_5$Co$_4$Si$_{10}$ probed by muon-spin spectroscopy