The layered honeycomb magnet $\alpha -{\mathrm{RuCl}}_3$ has been proposed as a candidate to realize a Kitaev spin model with strongly frustrated, bond-dependent, anisotropic interactions between spin-orbit entangled $j_{\mathrm{eff}}=\frac{1}{2}{\mathrm{Ru}}^{3+}$ magnetic moments. Here, we report a detailed study of the three-dimensional crystal structure using x-ray diffraction on untwinned crystals combined with structural relaxation calculations. We consider several models for the stacking of honeycomb layers and find evidence for a parent crystal structure with a monoclinic unit cell corresponding to a stacking of layers with a unidirectional in-plane offset, with occasional in-plane sliding stacking faults, in contrast with the currently assumed trigonal three-layer stacking periodicity. We report electronic band-structure calculations for the monoclinic structure, which find support for the applicability of the $j_{\mathrm{eff}}=\frac{1}{2}$ picture once spin-orbit coupling and electron correlations are included. Of the three nearest-neighbor Ru-Ru bonds that comprise the honeycomb lattice, the monoclinic structure makes the bond parallel to the $b$ axis nonequivalent to the other two, and we propose that the resulting differences in the magnitude of the anisotropic exchange along these bonds could provide a natural mechanism to explain the previously reported spin gap in powder inelastic neutron scattering measurements, in contrast to spin models based on the three-fold symmetric trigonal structure, which predict a gapless spectrum within linear spin wave theory. Our susceptibility measurements on both powders and stacked crystals, as well as magnetic neutron powder diffraction, show a single magnetic transition upon cooling below $T_{\mathrm{N}}\approx 13$ K. The analysis of our neutron powder diffraction data provides evidence for zigzag magnetic order in the honeycomb layers with an antiferromagnetic stacking between layers. Magnetization measurements on stacked single crystals in pulsed field up to 60 T show a single transition around 8 T for in-plane fields followed by a gradual, asymptotic approach to magnetization saturation, as characteristic of strongly anisotropic exchange interactions.

Through a combination of neutron diffraction and Landau theory we describe the spin ordering in the ground state of the quadruple perovskite manganite ${\mathrm{CaMn}}_7{\mathrm{O}}_{12}$—a magnetic multiferroic supporting an incommensurate orbital density wave that onsets above the magnetic ordering temperature, $T_{\mathrm{N}1}=90$ K. The multi-$\mathbf{k}$ magnetic structure in the ground state was found to be a nearly-constant-moment helix with modulated spin helicity, which oscillates in phase with the orbital occupancies on the ${\mathrm{Mn}}^{3+}$ sites via trilinear magneto-orbital coupling. Our phenomenological model also shows that, above $T_{\mathrm{N}2}=48$ K, the primary magnetic order parameter is locked into the orbital wave by an admixture of helical and collinear spin density wave structures. Furthermore, our model naturally explains the lack of a sharp dielectric anomaly at $T_{\mathrm{N}1}$ and the unusual temperature dependence of the electrical polarization.

We report a neutron powder diffraction study of $R{\mathrm{Mn}}_7{\mathrm{O}}_{12}$ quadruple perovskite manganites with $R=\mathrm{La}$, 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 Néel temperature. The respective magnetic structures are collinear, with one uncompensated ${\mathrm{Mn}}^{3+}$ moment per formula unit as observed in bulk magnetization measurements. We show that both ${\mathrm{LaMn}}_7{\mathrm{O}}_{12}$ and ${\mathrm{NdMn}}_7{\mathrm{O}}_{12}$ undergo a second magnetic phase transition at low temperature, which introduces a canting of the $B$-site sublattice moments that is commensurate in ${\mathrm{LaMn}}_7{\mathrm{O}}_{12}$ and incommensurate in ${\mathrm{NdMn}}_7{\mathrm{O}}_{12}$. This spin canting is consistent with a magnetic instability originating in the $B$-site orbital order. Furthermore, ${\mathrm{NdMn}}_7{\mathrm{O}}_{12}$ 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.

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.

We present the magnetic structure of ${\mathrm{TmMn}}_3{\mathrm{O}}_6$, solved via neutron powder diffraction. We demonstrate that long-range magnetic order develops below 74 K, and at 28 K a spin-flop transition occurs driven by $f-d$ exchange and rare-earth single-ion anisotropy. In both magnetic phases, the magnetic structure may be described as a collinear ferrimagnet, contrary to conventional theories of magnetic order in the manganite perovskites. Instead, we show that these magnetic structures can be understood to arise due to ferro-orbital order, the $A, A^{\prime }$, and $A^{\prime \prime }$ site point symmetry, $mm2$, and the dominance of $A-B$ exchange over both $A-A$ and $B-B$ exchange, which together are unique to the $R{\mathrm{Mn}}_3{\mathrm{O}}_6$ perovskites.