Weak ferromagnetism in two-dimensional bilayered ${\mathrm{Sr}}_{3-x}{\mathrm{Ca}}_x{\mathrm{Ru}}_2{\mathrm{O}}_7$

We have synthesized a bilayered ruthenate solid solution Sr${}_{3-x}$Ca${}_x$Ru${}_2$O${}_7$ $(0\mathrm{}<~x<~2.0)$, which is homologous to the copper-free layered-perovskite superconductor Sr${}_2$RuO${}_4$ ${(T}_c$=1.5 K). The $x$ dependence of the lattice parameters, magnetic susceptibility $\chi \mathrm{}\left(T\right)\mathrm{}$, magnetization $M\left(H\right)\mathrm{}$, and electrical resistivity $\rho \mathrm{}\left(T\right)\mathrm{}$ is systematically examined. Samples for all $x$ show magnetic ordering below 150 K. It is shown, however, that the magnetic correlations change with increasing Ca content. The lowest ferromagnetic-ordering temperature in this series is observed at 3.2 K in the sample with $x$(Ca) =1.0. This transition is described well as a weakly ferromagnetic ordering by self-consistent renormalization (SCR) theory of spin fluctuations. To the best of our knowledge, this is the first example of applying the SCR theory to a weak ferromagnet with a quasi-two-dimensional layered crystal structure.