The magnetic structures of rare-earth quadruple perovskite manganites $R$Mn$_7$O$_{12}$

We report a neutron powder diffraction study of $R$Mn$_7$O$_{12}$ quadruple perovskite manganites with $R$ = La, Ce, Nd, Sm, and Eu. We show that in all measured compounds concomitant magnetic ordering of the $A$ and $B$ manganese sublattices occurs on cooling below the N$\mathrm{\acute{e}}$el temperature. The respective magnetic structures are collinear, with one uncompensated Mn$^{3+}$ moment per formula unit as observed in bulk magnetisation measurements. We show that both LaMn$_7$O$_{12}$ and NdMn$_7$O$_{12}$ undergo a second magnetic phase transition at low temperature, which introduces a canting of the $B$ site sublattice moments that is commensurate in LaMn$_7$O$_{12}$ and incommensurate in NdMn$_7$O$_{12}$. This spin canting is consistent with a magnetic instability originating in the $B$ site orbital order. Furthermore, NdMn$_7$O$_{12}$ displays a third magnetic phase transition at which long range ordering of the Nd sublattice modifies the periodicity of the incommensurate spin canting. Our results demonstrate a rich interplay between transition metal magnetism, orbital order, and the crystal lattice, which may be fine tuned by cation substitution and rare earth magnetism.