Orbital ordering in spinel AV$_{2}$O$_{4}$ (A=Mn,Fe)

In spinel vanadates, AV$_{2}$O$_{4}$ (A:divalent ion), there is an orbital degree of freedom in the V$^{3+}$ ion, which contains two $d$ electrons in the triply degenerate $t_{2g}$ orbital. We measured the x-ray diffraction of the single crystal of MnV$_{2}$O$_{4}$, which exhibits simultaneous ferrimagnetic and a structural phase transition at 57 K, and found that a diamond-glide symmetry is broken in the ferrimagnetic tetragonal phase in this compound, indicating the antiferro-type ordering of the V $t_{2g}$ orbitals. This orbital ordering can be explained by the enhancement of the antiferro-orbital interaction caused by the ferromagnetic alignment of the V spins in the ferrimagnetic phase. We also measured the x-ray diffraction of the FeV$_{2}$O$_{4}$ single crystal, and found that a different type of orbital ordering from that of MnV$_{2}$O$_{4}$ occurs in this compound.