Magnetic phase separation in double layer ruthenates Ca$_3$(Ru$_{1-x}$Ti$_x$)$_2$O$_7$

Ti doping of a small concentration in the double-layered ruthenate Ca$_3$(Ru$_{1-x}$Ti$_x$)$_2$O$_7$ was previously found to induce an unusual magnetic phase transition from a metallic antiferromagnetic state formed from anti-parallel stacking of ferromagnetic bilayers (AFM-b) to a nearest-neighbor antiferromagnetic state (G-AFM) with Mott insulating properties; the critical Ti concentration for the transition is near $x$ = 0.03. In this article, we conducted systematic studies on this magnetic transition near the critical composition through detailed magnetization measurements. We found that no intermediate magnetic phases exist between AFM-b and G-AFM states; this is contrasted with manganites where a similar magnetic phase transition takes place through the presence of several intermediate magnetic phases. The AFM-b-to-G-AFM transition in Ca$_3$(Ru$_{1-x}$Ti$_x$)$_2$O$_7$ happens through a phase separation process; the AFM-b and G-AFM phases coexist in the 2-5% Ti range. Another distinct characteristic of this magnetic transition is that it is triggered by low concentration impurity doping (< 5%), whereas similar magnetic transitions in manganites are tuned by 50-70% chemical substitutions. We have discussed the possible origin of such an unusual magnetic transition and com