A complex structure of the superconducting order parameter in $Ln_2$C$_3$ ($Ln$ = La, Y) is demonstrated by muon spin relaxation ($mu$SR) measurements in their mixed state. The muon depolarization rate [$sigma_{rm v}(T)$] exhibits a characteristic temperature dependence that can be perfectly described by a phenomenological double-gap model for nodeless superconductivity. While the magnitude of two gaps is similar between La$_2$C$_3$ and Y$_2$C$_3$, a significant difference in the interband coupling between those two cases is clearly observed in the behavior of $sigma_{rm v}(T)$.
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