Effect of disorder on quantum phase transition in the double layered ruthenates (Sr _1-xCa _x) ₃Ru ₂O ₇

(Sr _1-xCa _x) ₃Ru ₂O ₇ is characterized by complex magnetic states, spanning from a long-range antiferromagnetically ordered state over an unusual heavy-mass nearly ferromagnetic (NFM) state to an itinerant metamagnetic (IMM) state. The NFM state, which occurs in the 0.4 >x> 0.08 composition range, freezes into a cluster spin glass (CSG) phase at low temperatures. In this article, we present the scaling analyses of magnetization and the specific heat for (Sr _1-xCa _x) ₃Ru ₂O ₇ in the 0.4 >x> 0.08 composition range. We find that in a temperature region immediately above the spin freezing temperature T _f, the isothermal magnetization M(H) and the temperature dependence of electronic specific heat C _e(T) exhibit anomalous power-law singularities; both quantities are controlled by a single exponent. The temperature dependence of magnetization M(T) also displays a power-law behavior, but its exponent differs remarkably from that derived from M(H) and C _e(T). Our analyses further reveal that the magnetization data M(H,T) obey a phenomenological scaling law of M(H,T) Hαf(H/Tδ) in a temperature region between the spin freezing temperature T _f and the scaling temperature T _scaling. T _scaling systematically decreases with the decease of Ca content. This scaling law breaks down near the critical concentration x= 0.1 where a CSG-to-IMM phase transition occurs. We discussed these behaviors in term of the effect of disorder on the quantum phase transition.