Orbitally driven trimerization in LiVO$_2$ and LiVS$_2$: a ``partial Mott transition"

Layered triangular-lattice transition-metal compounds often display interesting magnetic and electronic properties. Here we studied the formation of the trimerized spin-singlet state of the V$^{3+}$ ($S$=1) in vanadates LiVO$_2$ and LiVS$_2$ and their electronic structure with a special orbital order, using constrained LSDA+$U$ calculations combined with lattice optimization. The obtained results show that the trimerization distortion in LiVO$_2$ increases as the effective $U$ decreases, and the calculated distortion of $\sim$0.3 \AA~ at the small $U$=1 eV agrees well with the experiments, indicating that LiVO$_2$ is close to a metal-insulator transition. The corresponding distortion in LiVS$_2$ is even stronger, being $\sim$0.4 \AA~ at the $U$=1 eV, which is due to enhanced electron delocalization via increased V-S covalency, in spite of a lattice expansion. This agrees with the experimental finding that LiVS$_2$ has a metal-insulator transition. The calculated energy gain associated with the trimerization well accounts for the observed structural phase transition temperature in LiVO$_2$ and LiVS$_2$. We conclude that the trimerization in LiVO$_2$ and LiVS$_2$ is due to a partial delocalization of the orbitally ordered electrons---a ``partial Mott transition,'' occurring not in the whole system but in small clusters (here in trimers). This situation is contrasted with that in NaVO$_2$, which is further away from the localized-itinerant crossover and thus remains insulating with different orbital ordering.