Role of orbital filling on nonlinear ionic Raman scattering in perovskite titanates

The linear and nonlinear phononic interactions between an optically excited infrared (IR) or hyper-Raman mode and a driven Raman mode are computed for the $d^0 \left({\mathrm{CaTiO}}_3\right)$ and $d^1 \left({\mathrm{LaTiO}}_3\right)$ titanates within a first-principles density functional framework. We calculate the potential energy surface expanded in terms of the $A_g$ or $B_{1g}$ mode amplitudes coupled to the $A_u$ or the $B_{3u}$ mode and determine the coupling coefficients for these multimode interactions. We find that the linear-quadratic coupling dominates the anharmonicities over the quadratic-quadratic interaction in the perovskite titanates. The IR and Raman modes both modify the electronic structure with the former being more significant but occurring on a different time scale; furthermore, the coupled-mode interactions lead to sizable perturbations to the valence bandwidth ($\sim 100\text{meV}$) and band gap ($\sim 50$  meV). By comparing the coupling coefficients of undoped ${\mathrm{CaTiO}}_3$ and ${\mathrm{LaTiO}}_3$ to those for electron-doped $\left({\mathrm{CaTiO}}_3\right)$ and hole-doped $\left({\mathrm{LaTiO}}_3\right)$ titanates, we isolate the role of orbital filling in the nonlinear coupling process. We find that with increasing occupancy of the $d$ manifold, the linear-quadratic interaction decreases by approximately 30% with minor changes induced by the cation chemistry (that mainly affect the phonon mode frequencies) or by electron correlation. We identify the importance of the Ti-O bond stiffness, which depends on the orbital filling, in governing the lattice anharmonicitiy. This microscopic understanding can be used to increase the nonlinear coupling coefficient to facilitate more facile access of nonequilibrium structures and properties through ionic Raman scattering processes.

Figure 1

(a) Structure model of orthorhombic ${\text{CaTiO}}_3$ and ${\text{LaTiO}}_3$. The green spheres denote the $A\text{-site}$ cations, Ca or La. The blue and red spheres denote the Ti and oxygen atoms, respectively. The octahedra tilting $\left(\varphi \right)$ and rotational $\left(\theta \right)$ angles are defined as $\varphi =({180}^{\circ }-\angle (\mathrm{Ti}-\mathrm{O}-\mathrm{Ti}))/2$, and $\theta =({90}^{\circ }-\angle \left(\text{OOO}\right))/2$, where the Ti-O-Ti and interoctahedral O-O-O (denoted OOO) angles are shown following the convention introduced in Ref. [28]. (b) Illustration of the (upper drawings) three Raman mode displacements (directions indicated by gray arrows) that are anharmonically coupled through a nonlinear interaction (black arrow) to the different symmetry (lower drawings) hyper-Raman and IR modes explored in this work.

Figure 2

Lattice dynamical properties of ${\mathrm{CaTiO}}_3$ and ${\mathrm{LaTiO}}_3$ computed at the $\text{PBEsol}+U$ level. (a) Frequencies for the Raman active modes. The black and red points are for CTO and LTO, respectively. The selected $A_g$ and $B_{1g}$ Raman modes used to evaluate the nonlinear coupling strength correspond to the open symbols. (b) Phonon density of states (pDOS) for CTO and LTO. The black curve is the total pDOS, whereas the solid blue curve shows the contribution from the Ti atoms. The Ti contribution to the lattice dynamical properties of CTO and LTO largely spans the frequency range from $150–550{\mathrm{cm}}^{-1}$, with small mixing from Ca/La at low frequency.

Figure 3

Energy profiles for the nonlinear coupling between two phonon modes. (a)–(d) Energy profiles for the two $A_g$ modes under the excitation of the IR $B_{3u}$ mode in CTO and LTO. (e)–(h) Energy profiles for the $B_{1g}$ mode and the two $A_g$ modes under the excitation of the hyper-Raman $A_u$ mode. Note that in the top panels all curves are overlapping, indicating that the $B_{1g}$ mode is decoupled from the $A_u$ mode by symmetry. Note that here we show the $A_g$(1) mode for CTO and the $A_g$(2) mode for LTO, because the corresponding couplings are stronger. The inset in the left panel shows the predicted $Q_R^{*}$ dependence on the IR mode amplitude from Eq. (2) (curves) and the DFT computed data (symbols).

Figure 4

Electronic densities of states (DOS) for the (a) and (e) equilibrium ${\text{CaTiO}}_3$ and ${\text{LaTiO}}_3$ structure, (b) and (f) singly excited $A_g\left(1\right)$ mode, (c) and (g) singly excited $B_{3u}$ mode, and (d),(h) coupled excited modes with corresponding amplitudes given in parentheses for each mode as $(Q\left(A_g\left(1\right)\right),Q\left(B_{3u}\right))$ in units of $\AA {\mathrm{amu}}^{1/2}$ for CTO [panels (b)–(d)] and LTO [panels (f)–(h)]. In panels (a) and (e), the total and the atom-projected DOS are provided. The equilibrium DOS is also shown in the gray shaded region. The inset in (g) schematically denotes the distortion induced by the $B_{3u}$ mode.

Figure 5

(a)–(d) Structural parameters and (e)–(g) lattice dynamical properties with respect to the number of electrons in the Ti $3d$ manifold. $d^0$ corresponds to undoped ${\text{CaTiO}}_3$ (black squares) and holed-doped ${\left[\text{LTO}\right]}^{1+}$ (red circles), whereas $d^1$ corresponds to undoped ${\text{LaTiO}}_3$ and electron-doped ${\left[\text{CTO}\right]}^{1-}$. A quantitative measure of changes in these parameters with the orbital filling is given as a percentage relative to the original bulk values of CTO. Panel (g) clearly shows that the additional one electron quantitatively reduces the nonlinear coupling strength $g$ by $\sim 32\%$.

Figure 6

${\mathrm{CaTiO}}_3$ and ${\mathrm{LaTiO}}_3$ equations of state. The third- through fifth-order terms of CTO are more than one order-of-magnitude larger than those of LTO.

Figure 7

Effect of electron correlation to the linear (a),(b) and nonlinear (c) phonon properties of ${\text{LaTiO}}_3$. There is no clear trend for the changes in nonlinear coupling strength with correlation strength; the variation over the $U$ range explored is within 5%.