Spin structure and magnetic frustration in multiferroic $R{\mathrm{Mn}}_2{\mathrm{O}}_5$ $(R=\mathrm{Tb},\mathrm{Ho},\mathrm{Dy})$

We have studied the crystal and magnetic structures of the magnetoelectric materials $R{\mathrm{Mn}}_2{\mathrm{O}}_5$ $(R=\mathrm{Tb},\mathrm{Ho},\mathrm{Dy})$ using neutron diffraction as a function of temperature. All three materials display incommensurate antiferromagnetic ordering below $40\mathrm{K}$, becoming commensurate on further cooling. For $R=\mathrm{Tb},\mathrm{Ho}$, a commensurate-incommensurate transition takes place at low temperatures. The commensurate magnetic structures have been solved and are discussed in terms of competing exchange interactions. The spin configuration within the $ab$ plane is essentially the same for each system, and the radius of $R$ determines the sign of the magnetic exchange between adjacent planes. The inherent magnetic frustration in these materials is lifted by a small lattice distortion, primarily involving shifts of the ${\mathrm{Mn}}^{3+}$ cations and giving rise to a canted antiferroelectric phase.