Theory of Raman and resonant inelastic x-ray scattering from collective orbital excitations in ${\text{YTiO}}_3$

We present two different theories for Raman scattering and resonant inelastic x-ray scattering (RIXS) in the low-temperature ferromagnetic phase of ${\text{YTiO}}_3$ and compare this to the available experimental data. For description of the orbital ground state and orbital excitations, we consider two models corresponding to two theoretical limits: one where the $t_{2g}$ orbitals are degenerate and the other where strong lattice distortions split them. In the former model the orbitals interact through superexchange. The resulting superexchange Hamiltonian yields an orbitally ordered ground state with collective orbital excitations on top of it—the orbitons. In the orbital-lattice model, on the other hand, distortions lead to local $dd$ transitions between crystal-field levels. Correspondingly, the orbital response functions that determine Raman and RIXS line shapes and intensities are of cooperative or single-ion character. We find that the superexchange model yields theoretical Raman and RIXS spectra that fit very well to the experimental data.

Figure 1

Raman spectrum of ${\text{YTiO}}_3$ at $T=13\text{K}$ in $\left(z,z\right)$ geometry, taken from Ref. 21. A background is subtracted from the data. The sharp peak around 170 meV in the data is the two-phonon Raman signal and is not considered in our theory. The thin-solid line is the superexchange theory curve. The anisotropy of the local model is reflected in its Raman spectra: $\left(z,z\right)$ polarization (dashed line) gives a very different spectrum from $\left(x,x\right)$ polarization (dotted line). In the superexchange model, the $xx,yy$, and $zz$ polarizations are equivalent. It should be noted that the experimental Raman spectra are also of cubic symmetry (Ref. 21).

Figure 2

(Color online) RIXS spectra for a model of superexchange-driven orbital order with RIXS coupling to orbitons only via the single-site mechanism. Spectra (a) and (b) are for $\mathbf{q}$ directed along the [001] direction, and (c) and (d) for $\mathbf{q}$ along the [110] direction. Spectra (a) and (c) are for horizontal incoming polarization (electric field in the scattering plane) and (b) and (d) are for vertical incoming polarization (electric field perpendicular to the scattering plane). Note that the $\mathbf{q}=\mathbf{0}$ points are different in each spectrum because of the different experimental geometries, leading to different $P_{\Gamma }$. We only plotted the experimentally accessible part of the Brillouin zone.

Figure 3

RIXS spectra for a superexchange-driven orbital order with RIXS coupling to orbitons only via the single-site mechanism (solid line), compared to the experimental data (Ref. 23). The vector $\mathbf{q}$ is directed along the [001] direction, with $q_z$ as indicated in the figures. We took $J_{\text{orb}}=80\text{meV}$ and introduced a phenomenological HWHM broadening of $\gamma =0.4J_{\text{orb}}\approx 30\text{meV}$ for the orbitons as well as the HWHM experimental resolution of 27.5 meV. The elastic peak is fitted with a Gaussian (dashed-dotted line).

Figure 4

Spectra, obtained with the single-site mechanism, for the largest experimentally accessible momentum transfer directed along the [110] direction. The solid line indicates the case where the incoming polarization is horizontal and the dashed line is for vertical incoming polarization. The elastic peak has been removed. In the horizontally polarized case, the single orbiton peak is quite strong and should be visible in experiments if the system is superexchange driven and the RIXS signal is dominated by the single-site mechanism.

Figure 5

RIXS spectra for the local model (solid and dashed lines) compared to experimental data (Ref. 23). The vector $\mathbf{q}$ is directed along the [001] direction, where $q_z$ is indicated in the figures. The dashed curve shows the artificially optimized model with degenerate crystal-field levels. We introduced a phenomenological intrinsic HWHM broadening ($\gamma \approx 30\text{meV}$ [solid line] and $\gamma =100\text{meV}$ [dashed line]) for the final states and added experimental broadening.

Figure 6

(Color online) In the superexchange model, two-site RIXS processes locally modify the superexchange interaction, coupling the RIXS core hole to the $t_{2g}$ orbitals. Shown is an orbital superexchange process between two neighboring Ti ions in the presence of a core hole. On the left, one of the ions is excited by an incoming x-ray photon. After that, the $t_{2g}$ electrons undergo a superexchange process. On the right, the virtual state of the superexchange process is depicted. The presence of the core hole frustrates the superexchange process. Instead of the usual Hubbard $U$, the energy of the virtual state is lowered by the presence of the positively charged core hole. This modifies the superexchange constant $J_{SE}=4t^2/U$ at the core hole site.

Figure 7

(Color online) RIXS spectrum for two-site processes within the superexchange model. The color scale denotes the intensity. The figure shows $\mathbf{q}$ running from (0,0,0) to $\left(0,0,\pi \right)$. One-orbiton creation is not allowed for the superexchange modulation mechanism. The intrinsic energy broadening $\gamma$ of the orbiton states is $\gamma =0.4J_{\text{orb}}\approx 30\text{meV}$ and the added experimental resolution of 27.5 meV (Ref. 23) is approximately $0.34J_{\text{orb}}$.

Figure 8

Theoretical RIXS spectra for two-site processes, calculated within the superexchange model (solid line), compared to experiment (Ref. 23). $\mathbf{q}$ is directed along the [001] direction, with $q_z$ as indicated. We obtain a best fit for $J_{\text{orb}}=75\text{meV}$. The solid lines are cuts from the plot of Fig. 7, where we added a Gaussian fit to the elastic peak (dashed-dotted line). A phenomenological intrinsic orbiton broadening of $0.4r_1{J}_{SE}\approx 30\text{meV}$ is added as well as the experimental resolution of 27.5 meV (both HWHM) (Ref. 23).