Orbital structures in spinel vanadates $A$V${}_2$O${}_4$ ($A$ $=$ Fe, Mn)

Spinel FeV${}_2$O${}_4$ exhibits successive structural phase transitions, reflecting the interplay between the Fe${}^{2+}$ (3$d^6$) and V${}^{3+}$ (3$d^2$) ions, both of which have orbital and spin degrees of freedom. The temperature-dependent orbital shapes of Fe${}^{2+}$ and V${}^{3+}$ were investigated by means of single-crystal structure analysis, and were compared with those in MnV${}_2$O${}_4$, where only the V${}^{3+}$ ions are Jahn-Teller active. The highest-temperature transition from the cubic to the high-temperature tetragonal phase was driven by a ferroic Fe${}^{2+}$ 3$z^2$-$r^2$ orbital order (OO). At 110 K, where the ferrimagnetic transition takes place, the magnetic order modified the orbital shape through intratomic spin-orbit coupling, causing an orthorhombic distortion. The V${}^{3+}$ orbital order (V-OO) contributed to the lowest temperature transition from the orthorhombic to the low-temperature tetragonal phase. The V-OO in FeV${}_2$O${}_4$ was qualitatively different from that in MnV${}_2$O${}_4$. We propose that ferro-OO contains a complex orbital in FeV${}_2$O${}_4$ in contrast to the V-OO of real orbitals observed in MnV${}_2$O${}_4$.

Figure 1

(a) Schematic illustration of the $A{B}_2$O${}_4$ spinel oxides. FeV${}_2$O${}_4$ has orbital and spin degrees of freedom both at the A site (Fe${}^{2+}$) and at the B site (V${}^{3+}$). MnV${}_2$O${}_4$ has orbital-active ions at only the B-site V${}^{3+}$. Schematic illustrations of the phase diagrams with crystallographic strain for (b) FeV${}_2$O${}_4$ and (c) MnV${}_2$O${}_4$. The red arrows indicate the magnetization ($M$) direction. The phases are labeled paramagnetic (PM), collinear-ferrimagnetic (C-FM), and noncollinear-ferrimagnetic phase (NC-FM).

Figure 2

The 3D plots of IP images around (a) the (0 0 8)${}_c$ and (b) the (4 0 0)${}_c$ reflections for a 20 μm crystal at 40 K (LT tetragonal phase) in FeV${}_2$O${}_4$. The reflections were indexed in the cubic setting. The peak height ratios were 82:12:6 for {8 0 0}${}_c$ and 98:1:1 for {4 0 0}${}_c$, showing that an almost twin-free state was achieved. In addition, the strongest peak was well separated from the other two minor peaks, which arose from the second and third domains. The crystal structure analysis was carried out by masking the most relevant peak. (c) Comparison of the calculated ($F_{\mathrm{cal}}$) and the experimental ($F_{\mathrm{obs}}$) structure factors for FeV${}_2$O${}_4$ at 40 K (LT tetragonal).

Figure 3

(a) Temperature dependence of the magnetization ($M$) and (b) lattice constants for FeV${}_2$O${}_4$.

Figure 4

IP images for the cubic and tetragonal phases in (a) FeV${}_2$O${}_4$ and (b) MnV${}_2$O${}_4$. (c) Temperature dependence of the integral intensity of the (4 0 −2) forbidden reflections. In MnV${}_2$O${}_4$ an abrupt increase in intensity was observed below $T_s$, which indicates the breaking of the $d$-glide symmetry, whereas there was no change in the intensity for the tetragonal phase in FeV${}_2$O${}_4$. This indicates that the space group of the tetragonal phase in MnV${}_2$O${}_4$ should be $I$4${}_1/a$, whereas FeV${}_2$O${}_4$ maintains the highest symmetry in the tetragonally distorted spinel structure $I$4${}_1/amd$.

Figure 5

(a) Schematic illustration of the $Q_2$ and $Q_3$ ($E_g$), and the $Q_4$, $Q_5$, and $Q_6$ ($T_{2g}$) normal modes of an octahedron. $Q_2$ and $Q_3$ correspond to the orthorhombic and tetragonal distortion, respectively. The average of $Q_4$, $Q_5$, and $Q_6$ is the trigonal distortion $\pm Q_t$, which corresponds to elongation or compression along the ⟨111⟩ direction. The temperature dependence of each of the modes ($Q_2$, $Q_3$, $Q_t$) is shown for (b), (c) FeV${}_2$O${}_4$, and (d), (e) MnV${}_2$O${}_4$.

Figure 6

$Q_2$-$Q_3$ mode analyses of (a) an FeO${}_4$ tetrahedron and (b) a VO${}_6$ octahedron. (c) The $u_2$-$u_3$ plot of the global distortion for FeV${}_2$O${}_4$. The $u_2$-$u_3$ values were calculated from the lattice constants.

Figure 7

(a) Schematic illustration of the V-site pyrochlore lattice. (b) Distortion of the VO${}_6$ octahedron for MnV${}_2$O${}_4$ in the $Q_2$-$Q_3$ plane. For the tetragonal phase where $c$ < $a$ (20 K) there were two sites with different local distortions, which alternately stack along the $c$ axis (shown in purple and yellow) in (a).