Using a criterion given by Banerjee [Phys. Lett. 12, 16 (1964)] to distinguish first-order magnetic transitions from second-order ones, we demonstrate that not all the title perovskites exhibit a first-order transition at their Curie temperatures $T_C.$ We observe that, although ${\mathrm{La}}_{2/3}{\mathrm{Ca}}_{1/3}{\mathrm{MnO}}_3$ exhibits a first-order magnetic transition at $T_C,$ ${\mathrm{La}}_{2/3}{\mathrm{Sr}}_{1/3}{\mathrm{MnO}}_3$ exhibits a second-order one.

The crystallographic structure and the magnetic order of the distorted perovskite ${\mathrm{La}}_{1-x}{\mathrm{Sr}}_x{\mathrm{CoO}}_3$ $(0.10\mathrm{}<~x<~0.30)$ has been studied by neutron diffraction, high-resolution electron microscopy, and magnetic-susceptibility measurements. The results give direct evidence for an inhomogeneous distribution of the ${\mathrm{Sr}}^{2+}$ ions and the segregation of the material into hole-rich ferromagnetic regions and a hole-poor semiconducting matrix at lower values of x. The holes introduced by Sr doping are attracted to the ${\mathrm{Sr}}^{2+}$ ions where they stabilize to lowest temperatures an intermediate-spin state at neighboring trivalent cobalt. The antibonding e electrons so stabilized increase the mean unit-cell volume and are delocalized over the cobalt atoms of the cluster where they couple the localized $t^5$ configurations ferromagnetically. Long-range ferromagnetic order between clusters is realized even for Sr doping as low as $x=\mathrm{0.10.}\mathrm{}$ The transition to a spin glass state is observed only for Sr concentrations smaller than 0.10. The volume of a hole-rich cluster grows in a magnetic field, and the origin of the large negative magnetoresistance observed near $T_C$ for $0.15<~x<~0.25$ appears to be due to a growth of the clusters to a percolation threshold. For $x=0.30,$ the ${\sigma }^{*}$ band of the intermediate-spin state below $T_C$ is at the threshold of a transition from itinerant to polaronic conduction and, above $T_C,$ the system transforms smoothly to a cluster state.

We present Lu${}_2$MnCoO${}_6$, a multiferroic member of the double perovskites that we have investigated using x-ray and neutron diffraction, specific heat, magnetization, electric polarization, and dielectric constant measurements. This material possesses a net electric polarization strongly coupled to a net magnetization below 35 K, despite the antiferromagnetic ordering of the $S=3/2$ Mn${}^{4+}$ and Co${}^{2+}$ spins in an $\uparrow \uparrow \downarrow \downarrow$ configuration along the $c$ direction. We discuss the magnetic order in terms of a condensation of domain boundaries between $\uparrow \uparrow$ and $\downarrow \downarrow$ ferromagnetic domains, with each domain boundary producing an electric polarization due to spatial inversion symmetry breaking. In an applied magnetic field the domain boundaries slide, controlling the size of the magnetization, electric polarization, and magnetoelectric coupling.