Momentum Dependence of Orbital Excitations in Mott-Insulating Titanates

High-resolution resonant inelastic x-ray scattering has been used to determine the momentum dependence of orbital excitations in Mott-insulating ${\mathrm{LaTiO}}_3$ and ${\mathrm{YTiO}}_3$ over a wide range of the Brillouin zone. The data are compared to calculations in the framework of lattice-driven and superexchange-driven orbital ordering models. A superexchange model in which the experimentally observed modes are attributed to two-orbiton excitations yields the best description of the data.

Figure 1

RIXS spectra of (a) ${\mathrm{LaTiO}}_3$ and (b) ${\mathrm{YTiO}}_3$ at room temperature and at $T=13\mathrm{K}$. The spectra were taken at an incident energy $E_i=452.0\mathrm{eV}$ (i.e., at the $t_{2g}$ level of the $L_3$ excitation of the Ti ions) in horizontal polarization of the incident light at a scattering angle $2\theta =90°$. The lines are the results of fits to the Voigt line shape. (c),(d) Evolution of the RIXS spectra as a function of $2\theta$. Note that the excitation momentum $q=2k_i\sin \theta$, where $k_i$ is the incident photon wave vector. The data along [100] ([110]) thus span a range of 11.5 to 25% (8.5 to 17.5%) of the diameter of the Brillouin zone. The weak features in the spectra arise from statistical fluctuations of the count rate, which were smoothed out by the detector readout algorithm.

Figure 2

Momentum ($q$) dependence of (a) the energy and (b) the amplitude of the RIXS signal of ${\mathrm{LaTiO}}_3$ and ${\mathrm{YTiO}}_3$ at $T=13\mathrm{K}$. $q$ is given as a fraction of the size of the Brillouin zone (BZ) along the high-symmetry directions quoted in the text. Note that the BZ boundary is at 50%. The $q=0$ data obtained from Raman light scattering [17] are also shown in (a). The lines are guides to the eye.

Figure 3

Comparison of the model calculations with the experimental RIXS data of ${\mathrm{YTiO}}_3$ for different scattering vectors $q$ along the [100] direction: (a) superexchange bond modulation, (b) local shakeup of collective orbital excitations, and (c) local crystal-field excitation model.