Avoided quantum criticality and cluster-glass formation in itinerant ferromagnet Sr$_{1-x}$(La$_{0.5}$K$_{0.5}$)$_x$RuO$_3$

We demonstrate that the cluster-glass state emerges as ferromagnetic quantum criticality is avoided in itinerant ferromagnet Sr1-x(La0.5K0.5)xRuO3. In this compound, the ferromagnetic order is suppressed by increasing x, and then disappears at the critical concentration: x=0.5. In this x range, the present study reveals that no prominent feature is ascribed to the quantum critical fluctuations in specific heat. Instead, ac magnetic susceptibility exhibits a broad peak due to spontaneous spin freezing, and the peak temperature depends significantly on the frequency of the applied ac magnetic field. Furthermore, specific heat is enhanced within a wide temperature range, whereas specific heat shows no salient anomaly associated with spin freezing. These features are characteristics of the formation of cluster-glass; in particular, the observed frequency variations in ac magnetic susceptibility are well described by the Vogel-Fulcher law. We compare the features concerning the suppression of the ferromagnetic order in this doped compound with those in isostructural Ca- and La-doped SrRuO3, and suggest that a local correlated disorder effect and the very small coherence of itinerant Ru 4d electrons are responsible for the cluster-glass formation instead of the quantum phase transition in Sr1-x(La0.5K0.5)xRuO3.