Magneto-orbital ordering in the divalent <i>A</i>-site quadruple perovskite manganites <i>A</i>Mn₇O₁₂ (<i>A</i> = Sr, Cd, and Pb)

Through analysis of variable temperature neutron powder diffraction data, we present solutions for the magnetic structures of SrMn₇O₁₂, CdMn₇O₁₂, and PbMn₇O₁₂ in all long-range ordered phases. The three compounds were found to have magnetic structures analogous to that reported for CaMn₇O₁₂. 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. CdMn₇O₁₂ 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 CaMn₇O₁₂. Furthermore, we show that the model is universal to the A²⁺ quadruple perovskite manganites synthesised to date, and that it is tunable by selection of the A-site ionic radius.