Magnetic order and ice rules in the multiferroic spinel FeV${}_2$O${}_4$

We present a neutron-diffraction study of FeV${}_2$O${}_4$, which is rare in exhibiting spin and orbital degrees of freedom on both cation sublattices of the spinel structure. Our data confirm the existence of three structural phase transitions previously identified with x-ray powder diffraction and reveal that the lower two transitions are associated with sequential collinear and canted ferrimagnetic transitions involving both cation sites. Through consideration of local crystal and spin symmetry, we further conclude that Fe${}^{2+}$ cations are ferro-orbitally ordered below 135 K and V${}^{3+}$ orbitals order at 60 K, in accordance with predictions for vanadium spinels with large trigonal distortions and strong spin-orbit coupling. Intriguingly, the direction of ordered vanadium spins at low temperature obey “ice rules” more commonly associated with the frustrated rare-earth pyrochlore systems.

Figure 1

Plots of raw NPD data (circles) measured at $T=200$, 125, 75, and 4 K for 2$\theta \le$ 85${}^{\circ }$. Solid lines are results of Rietveld refinements described in the main text. Differences between observed and calculated intensities are shown directly below the respective patterns. See the main text and Fig. 2 for more detail.

Figure 2

(a) An expanded view of the NPD data around the cubic (004) Bragg position for each of the patterns in Fig. 1. (b) Plot of the cubic lattice parameters extracted from NPD data. (c) An expanded view around the cubic (111) Bragg position for the same temperatures as (a). (d) A comparison of the temperature evolution of NPD (111) and (220) Bragg peak intensities above their values at 180 K. Overlaid is the scaled intensity of the (200) Bragg peak, as measured in a single-crystal sample.

Figure 3

Plots vs temperature of the interior O-M-O angles [$M\in ({\text{Fe}}^{2+},{\text{V}}^{3+})$] of the (a) FeO${}_4$ tetrahedra and (b) VO${}_6$ octahedra, shown schematically at the top. In pictures, bonds of equal length are shown with equivalent coloring. Arrows denote the largest distortion of each polyhedron: a compression of the FeO${}_4$ tetrahedron at 135 K along [001] and a trigonal distortion of the VO${}_6$ octahedron along the $\langle 111\rangle$ directions that exists at all temperatures.

Figure 4

(a) Plot of the ordered moment size on the Fe${}^{2+}$ and V${}^{3+}$ cation sites and (b) the angle between the V${}^{3+}$ moments and the axis defined by Fe${}^{2+}$ moments. (c) A sketch of the collinear ferrimagnetic state in the FCO phase, as seen along the cubic (001) direction and (d) a similar sketch of the canted state seen in the LTT phase. Details about the ordering patterns are given in the main text. (e) A view of V${}^{3+}$ moments in a low-temperature ordered state clearly demonstrating the two-in–two-out spin structure on the pyrochlore sublattice.