The effect of disorder on quantum phase transition in the double layered ruthenates (Sr1-xCax)3Ru2O7

(Sr1-xCax)3Ru2O7 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 [Z. Qu et al., Phys. Rev. B 78, 180407(R) (2008)]. In this article, we present the scaling analyses of magnetization and the specific heat for (Sr1-xCax)3Ru2O7 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) \propto H^αf(H/T^δ) 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.

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