Orbital reformation with vanadium trimerization in $d^2$ triangular lattice LiVO${}_2$ revealed by ${}^{51}$V NMR

LiVO${}_2$ is a model system of a valence bond solid (VBS) in a $3d^2$ triangular lattice. The origin of VBS formation has remained controversial. We investigate the microscopic mechanism by elucidating the $d$ orbital character via on-site ${}^{51}$V NMR measurements in a single crystal up to 550 K across a structural transition temperature $T_c$. The Knight shift, $K$, and nuclear quadrupole frequency, $\delta \nu$, show that the $3d$ orbitals with local trigonal symmetry are reconstructed into a $d_{yz}{d}_{zx}$ orbital order below $T_c$. Together with the NMR spectra with threefold rotational symmetry, we confirm a vanadium trimerization with $d$-$d$ $\sigma$ bonds. The Knight shift extracts the large Van Vleck orbital susceptibility, ${\chi }^{\mathrm{VV}}=3.6\times {10}^{-4}$, in a paramagnetic state above $T_c$, which is comparable to the spin susceptibility. The results suggest that the orbitally induced Peierls transition in the proximity of the frustrated itinerant state is the dominant driving force of the trimerization transition.

Figure 1

(a) Layered triangular lattice of LiVO${}_2$ viewed from the $c$ axis with an orbital texture obtained in the present study; only two of four leaves in $d_{yz}$ and $d_{zx}$ orbitals are shown for clarity. Thick solid and dotted lines represent unit cells of an original $R\overline{3}m$ lattice above $T_c$ and a $\surd 3a\times \surd 3a$ superlattice below $T_c$, respectively.[12, 14] (b) VO${}_6$ unit trigonally elongated along the $c$ axis above $T_c$. (c) $d$ levels in a trigonal field: lowest $a_{1g}$ singlet, middle $e_g^{\prime }$ doublet, and higher $e_g$ doublet. (d) Minority $d_{xy}$ orbitals dominate the anisotropy of NMR spectra below $T_c$. The $x$, $y$, and $z$ directions are taken parallel to the VO bond directions and identical to the $Y-X$, $X+Y$, and $Z$ axes, where $X$, $Y$, and $Z$ are the principal axes of the local symmetry that determines the $K$ and $\delta \nu$ tensor. The $b^{*}$ axis is defined perpendicular to the $a$ and $c$ axes. The $b^{\prime }$ axis is located 30${}^{\circ }$ from the $a$ axis.

Figure 2

Frequency-swept ${}^{51}$V NMR spectra in LiVO${}_2$ (300 K, $B_0$ $=$ 6.105 T). The magnetic field is rotated around the (a) $a$, (b) $b^{*}$, and (c) $b$ axes. Dotted guidelines show the sinusoidal fitting[28] at the bottom of the peak position by taking the vertical axis as a rotation angle. Blue, red, and green lines are assigned to V1, V2, and V3 in Fig. 1c, respectively.

Figure 3

Angle dependence of (a)–(c) the nuclear quadrupole splitting frequency $\delta \nu$ and (d)–(f) the Knight shift $K$ obtained for the $b^{*}c$, $ca$, and $b^{\prime }c$ rotations. $X$ and $Z$ are defined by a maximum and minimum of $K$ and $\delta \nu$, respectively. Experimental data are well fitted into the Volkov's sinusoidal formula, $y=y_0^{\alpha }+y_1^{\alpha }\cos 2(\theta -{\varphi }^{\alpha })$ ($y=\delta \nu$, $K$), with fitting parameters $y_0^{\alpha }$, $y_1^{\alpha }$, and $\varphi$ for the rotation axes $\alpha =a,b^{*}$, and $b$.[28]

Figure 4

Temperature dependence of the ${}^{51}$V NMR spectra measured at (a) the $c$ axis and (b) 54${}^{\circ }$ from the $c$ axis in LiVO${}_2$. The spectrum pointed out by the arrow comes from the impurity LiV${}_2$O${}_4$.