Ferromagnetic quantum criticality in Sm$_{1-x}$La$_{x}$NiC$_{2}$ (x$=$0.85, 0.92 and 0.96)

We report \textmu SR experiments on the ternary compounds Sm$_{1-x}$La$_{x}$NiC$_{2}$ (x$=$0.85, 0.92, and 0.96), possessing a non-centrosymmetric orthorhombic CeNiC$_{2}$ structure (Amm2). The end members of these compounds have the ferromagnetic (FM) and charge-density-wave states at x$=$0 and the superconducting (SC) state at x$=$1. A FM quantum criticality (QC) is anticipated to occur around x$=$0.92. The x$=$0.96 SC sample exhibits a linear T dependence of the muon relaxation rate $\lambda_{muon}$, giving no indication of time-reversal symmetry breaking unlike the x$=$1 sample. ZF-\textmu SR measurements of the x$=$0.85 FM sample show a steep increase of $\lambda_{muon}$ below 5 K without obvious muon-spin precession, suggesting the formation of an inhomogeneous, weak magnetic ordered state. Longitudinal field-\textmu SR experiments unveil an ordered volume fraction of about 56 {\%}. For a case of the putative x$=$0.92 QC compound, the static fraction is decreased to 15 {\%}, while $\lambda_{muon}$ extracted from the ZF-\textmu SR spectra display ``persisting spin dynamics''. This suggests that the x$=$0.92 sample is close to QCP.