Origin of orbital ordering in YTiO3 and LaTiO3

The origin of orbital order in correlated transition-metal compounds is strongly debated. For the paradigmatic ${e}_{g}$ systems ${\mathrm{KCuF}}_{3}$ and ${\mathrm{LaMnO}}_{3}$, it has been shown that the electronic Kugel'-Khomskii mechanism alone is not sufficient to drive the orbital-ordering transition up to the high temperatures at which it is experimentally observed. In the case of ${t}_{2g}$ compounds, however, the role played by the superexchange interaction remains unclear. Here we investigate this question for two representative systems, the $3d\phantom{\rule{4pt}{0ex}}{t}_{2g}^{1}$ Mott insulators ${\mathrm{LaTiO}}_{3}$ and ${\mathrm{YTiO}}_{3}$. We show that the Kugel'-Khomskii superexchange transition temperature ${T}_{\mathrm{KK}}$ is unexpectedly large, comparable to the value for the ${e}_{g}^{3}$ fluoride ${\mathrm{KCuF}}_{3}$. By deriving the general form of the orbital superexchange Hamiltonian for the ${t}_{2g}^{1}$ configuration, we show that the ${\mathrm{GdFeO}}_{3}$-type distortion plays a key part in enhancing ${T}_{\mathrm{KK}}$ to about 300 K. Still, orbital ordering above 300 K can be ascribed only to the presence of a static crystal-field splitting.