Through analysis of variable-temperature neutron powder-diffraction data, we present solutions for the magnetic structures of ${\mathrm{SrMn}}_7{\mathrm{O}}_{12}$, ${\mathrm{CdMn}}_7{\mathrm{O}}_{12}$, and ${\mathrm{PbMn}}_7{\mathrm{O}}_{12}$ in all long-range ordered phases. The three compounds were found to have magnetic structures analogous to that reported for ${\mathrm{CaMn}}_7{\mathrm{O}}_{12}$. They all feature a higher-temperature lock-in phase with commensurate magneto-orbital coupling and a delocked multi-k magnetic ground state where incommensurate magneto-orbital coupling gives rise to a constant-moment magnetic helix with modulated spin helicity. ${\mathrm{CdMn}}_7{\mathrm{O}}_{12}$ represents a special case in which the orbital modulation is commensurate with the crystal lattice and involves stacking of fully and partially polarized orbital states. Our results provide a robust confirmation of the phenomenological model for magneto-orbital coupling previously presented for ${\mathrm{CaMn}}_7{\mathrm{O}}_{12}$. Furthermore, we show that the model is universal to the $A^{2+}$ quadruple perovskite manganites synthesized to date and that it is tunable by selection of the $A$-site ionic radius.

• Received 7 July 2017
• Revised 17 August 2017

DOI:https://doi.org/10.1103/PhysRevB.96.054448