a, Superconducting T_cas a function of Ti content y in Sr₂(Ru_1−yTi_y)O₄ (top axis) and residual in-plane resistivity ρ_ab0 (bottom axis)¹⁵; the inset depicts the layered perovskite structure analogous to the cuprate superconductor (La,Ba)₂CuO₄. b, Previously published phase diagram of Ca_2−xSr_xRuO₄ based on magnetic susceptibility measurements, depicting the superconducting (SC) state for x = 2, paramagnetic insulator (P-I) and commensurate antiferromagnetic insulator (CAF-I) behaviour at small x, a glassy magnetic metallic (M-M) state at slightly higher x inferred from a peak in d.c. magnetization, and the ‘cluster-glass’ (C-G) region associated with the history dependence of d.c.-magnetization around x = 0.5. For 0.5<x<2.0, the ground state was presumed to be a paramagnetic metal (P-M). The in-plane conductivity and specific heat in this region exhibit Fermi liquid behaviour below T_FL (refs 8, 9, 10). T₀ denotes temperature for structural phase transition. c, Temperature dependence of the c-axis resistivity ρ_c(T) normalized to the room-temperature value, for several (Ca, Sr) systems (blue), (Ru, Ti) systems (red), and for pure Sr₂RuO₄ (green) plotted on the basis of published results^{8, 16}. Red (y = 0.09,0.05) and blue arrows (x = 1.5) indicate the magnetic ordering temperature determined by μSR and magnetic susceptibility. The upturn of ρ_c at low temperatures can be associated with static magnetic order, whereas superconducting Sr₂RuO₄exhibits a sharp reduction of ρ_c at low temperatures.

a–f, ZF- μSR time spectra observed in Ca_2−xSr_xRuO₄ systems with x = 0.5 (a), x = 0.9 (b), x = 1.5 (c) and x = 0.3 (e), and in Sr₂Ru_1−yTi_yO₄ systems with y = 0.09 (d) and y = 0.03 (f). g,h, LF- μSR time spectra in the (Ca, Sr) system with x = 1.5(g) and the (Ru, Ti) system with y = 0.09 (h). The solid lines in a, c and d represent fits to the zero-field muon spin relaxation function obtained for dilute-alloy spin glasses, equation (1) (ref. 20), and the solid lines in b, e and f represent fits to a root-exponential decay²⁰.

a,b, Generalized total muon spin relaxation rate observed in (Ca,Sr)₂RuO₄ systems (a) and in Sr₂(Ru,Ti)O₄ systems (b), which exhibit an increase with decreasing temperature due to slowing down of dynamic spin fluctuations above the ordering temperature T_N and a subsequent build-up of static local field below T_N. c, Inverse of the magnetic susceptibility χ observed in a low field (100 G) in a select subset of the (Ca, Sr) and (Ru, Ti) systems. d, Muon spin relaxation rate 1/T₁ observed in LF- μSR with a moderate strength of longitudinal field in a few select systems, which exhibit clear peaks indicating onset of static magnetic order. e, Overlay of the Bragg peak intensity I_B observed in elastic neutron scattering at an incommensurate wavevector (0.3, 0.3, 0) and the μSR relaxation rate Λ(T) observed in Sr₂Ru_0.91Ti_0.09O₄ (ref. 17). A linear comparison, expected for spatially inhomogeneous spin systems, shows better scaling than a quadratic comparison. The apparent difference in onset temperature of the increase is probably due to the difference in the time windows of neutron and muon measurements. f, Imaginary part of a.c.-magnetic susceptibility Imχ_a.c.(T) in Sr_1.5Ca_0.5RuO₄ and Sr₂Ru_0.91Ti_0.09O₄ measured with the a.c. field perpendicular to the ab-plane.