We report a muon spin rotation/relaxation ($mu$SR) study of single-crystalline samples of the $alpha$-RuCl$_3$ honeycomb magnet, which is presumed to be a model compound for the Kitaev-Heisenberg interaction. It is inferred from magnetic susceptibility and specific-heat measurements that the present samples exhibit successive magnetic transitions at different critical temperatures $T_{rm N}$ with decreasing temperature, eventually falling into the $T_{rm N}=7$ K antiferromagnetic (7 K) phase that has been observed in only single-crystalline specimens with the least stacking fault. Via $mu$SR measurements conducted under a zero external field, we show that such behavior originates from a phase separation induced by the honeycomb plane stacking fault, yielding multiple domains with different $T_{rm N}$s. We also perform $mu$SR measurements under a transverse field in the paramagnetic phase to identify the muon site from the muon-Ru hyperfine parameters. Based on a comparison of the experimental and calculated internal fields at the muon site for the two possible spin structures inferred from neutron diffraction data, we suggest a modulated zig-zag spin structure for the 7 K phase, with the amplitude of the ordered magnetic moment being significantly reduced from that expected for the orbital quenched spin-1/2 state.
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