$A$-Type Antiferro-Orbital Ordering with $I{4}_1/a$ Symmetry and Geometrical Frustration in the Spinel Vanadate ${\mathrm{MgV}}_2{\mathbf{O}}_4$

We conduct a detailed structural analysis of the $S=1$ pyrochlore antiferromagnet ${\mathrm{MgV}}_2{\mathrm{O}}_4$, which exhibits an antiferromagnetic ordering marginally at $T_N=40\mathrm{K}$, triggered by a structural transition from cubic to tetragonal symmetry at $T_S=62\mathrm{K}$, using high resolution synchrotron x-ray diffraction and convergent beam electron diffraction. We reveal that the tetragonal phase below $T_S$ has the symmetry of $I{4}_1/a$ and that the distortion pattern of ${\mathrm{VO}}_6$ octahedra is consistent with $A$-type antiferro-orbital ordering with alternating stacking of layers with $yz/xy$ orbital chains and $zx/xy$ orbital chains along the tetragonal $c$ axis. This implies that an anisotropic coupling of V moments produced by the orbital ordering below $T_S$ primarily brings about the antiferromagnetic ordering.

Figure 1

(a) The temperature dependence of the magnetic susceptibility of a ${\mathrm{MgV}}_2{\mathrm{O}}_4$ single crystal with applied magnetic field of 1 T parallel to the [111] direction. The data were collected in the heating and sequential cooling processes after zero-field cooling. (b) CBED pattern along the [001] zone axis for room-temperature cubic ${\mathrm{MgV}}_2{\mathrm{O}}_4$ taken at an accelerating voltage of 98 kV. The GM lines for the 200 reflections can be clearly seen.

Figure 2

X-ray oscillation photographs of typical fundamental diffraction spots collected at (a) 65 K and (b) 60 K with the same oscillation. $hk{l}_c$ and $hk{l}_t$ represent indices based on the cubic and tetragonal spinel structures, respectively.

Figure 3

X-ray oscillation photographs collected at (a) 65 K and (b) 60 K with the same oscillation, indicating the appearance of extra diffractions. The insets of (b) show magnified views of the images around the extra reflections indexed as ${604}_c$ and ${806}_c$.

Figure 4

Observed (cross) and calculated (solid line) diffraction patterns from the Rietveld analysis of the powder SXRD data for the LTT phase of ${\mathrm{MgV}}_2{\mathrm{O}}_4$ at 45 K. Tick marks represent the positions of allowed Bragg reflections for the space group of $I{4}_1/a$. The difference line, observed minus calculated, is located at the bottom of the figure. The inset shows the diffraction patterns with vertical logarithmic scale around ${222}_t$ and ${114}_t$.

Figure 5

Schematic view of the 3D edge-sharing network of ${\mathrm{VO}}_6$ octahedra extracted from the experimentally determined LTT structure of ${\mathrm{MgV}}_2{\mathrm{O}}_4$. The two types of the color-coded ${\mathrm{VO}}_6$ octahedra in terms of orientation of the short and long in-plane bonds have different orbital occupancies of ($d_{xy}$, $d_{yz}$), and ($d_{xy}$, $d_{zx}$). The occupied orbitals of the vanadium ions are also shown in the ${\mathrm{VO}}_6$ octahedra.