Magnetic ground state of Rh-doped Kondo semiconductor CeRu$_2$Al$_{10}$ [Ce(Ru$_{1-x}$Rh$_x$)$_2$Al$_{10}$] is investigated by muon-spin relaxation method. Muon-spin precession with two frequencies is observed in the $x$ = 0 sample, while only one frequency is present in the $x$ = 0.05 and 0.1 samples, which is attributed to the broad static field distribution at the muon site. The internal field at the muon site is enhanced from about 180 G in $x$ = 0 sample to about 800 G in the Rh-doped samples, supporting the spin-flop transition as suggested by macroscopic measurements, and the boundary of different magnetic ground states is identified around $x$ = 0.03. The drastic change of magnetic ground state by a small amount of Rh-doping (3%) indicates that the magnetic structure in CeRu$_2$Al$_{10}$ is not robust and can be easily tuned by external perturbations such as electron doping. The anomalous temperature dependence of internal field in CeRu$_2$Al$_{10}$ is suggested to be attributed to the hyperfine interaction between muons and conduction electrons.
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