Spin-Lattice Coupling, Frustration, and Magnetic Order in Multiferroic $R{\mathrm{MnO}}_3$

We have performed high-resolution neutron diffraction and inelastic neutron scattering experiments in the frustrated multiferroic hexagonal compounds $R{\mathrm{MnO}}_3$ ($R=\mathrm{Ho},\mathrm{Yb},\mathrm{Sc},\mathrm{Y}$), which provide evidence of a strong magnetoelastic coupling in the whole family. We can correlate the atomic positions, the type of magnetic structure, and the nature of the spin waves whatever the $R$ ion and temperature. The key parameter is the position of the Mn ions in the unit cell with respect to a critical threshold of $1/3$, which determines the sign of the coupling between Mn triangular planes.

Figure 1

Sketch of the hexagonal MnO planes with out-of-plane exchange paths $J_{z1}$ (between ${\overrightarrow{S}}_4$ and ${\overrightarrow{S}}_3$) and $J_{z2}$ (between ${\overrightarrow{S}}_4$ and ${\overrightarrow{S}}_1$ as well as ${\overrightarrow{S}}_4$ and ${\overrightarrow{S}}_2$). Black (red) arrows depict the positions of the Mn spins within $z=0$ ($z=1/2$) Mn planes. ${\overrightarrow{S}}_1$, ${\overrightarrow{S}}_2$, and ${\overrightarrow{S}}_3$ are located at $(x,0,0)$, $(0,x,0)$, and $(-x,-x,0)$, respectively, while ${\overrightarrow{S}}_4$, ${\overrightarrow{S}}_5$, and ${\overrightarrow{S}}_6$ are at $(x,x,1/2)$, $(1-x,0,1/2)$, and $(0,1-x,1/2)$, respectively. Double arrows are used when several spin orientations can be stabilized. Inset: Sketch of the hexagonal MnO plane with in-plane exchange paths $J_1$ and $J_2$.

Figure 2

Refined Mn positions in the unit cell obtained from high-resolution neutron powder diffraction. Blue and red arrows indicate, respectively, the Néel and the spin reorientation temperatures. Lines are guides to the eyes. Right insets: Mn magnetic configuration for $T_{\mathrm{SR}}<T<T_N$. Left insets: Mn magnetic configuration $T<T_{\mathrm{SR}}$.

Figure 3

Maps of the dynamical structure factor in Ho, Yb, and ${\mathrm{YMnO}}_3$. The neutron intensity is plotted versus the energy transfer $\omega$ and wave vector $(1,0,Q_{\ell })$. Left panel: Experimental results from inelastic neutron scattering. Black lines highlight the low energy spin wave modes. Additional $Q$-independent lines in Yb and ${\mathrm{HoMnO}}_3$ maps are due to crystal field excitations. Right panel: Spin wave dispersion curves calculated using a Heisenberg Hamiltonian. Black arrows indicate the two pseudo Goldstone modes (see text).