Hyper-Raman tensors of the zone-center optical phonons in ${\mathrm{SrTiO}}_3$ and ${\mathrm{KTaO}}_3$

We measure the relative intensities of the seven hyper-Raman (HR) lines of ${\mathrm{SrTiO}}_3$ and ${\mathrm{KTaO}}_3$ at room temperature as a function of the scattering geometry. From our experimental data we deduce relative values of the HR tensor elements. There are two different HR tensor elements for each TO and LO component of the three $F_{1u}$ phonons and one for the silent mode of symmetry type $F_{2u}$. Taking the phonon eigenvectors and longitudinal effective charges from improved versions of Cowley’s shell model, we obtain the derivatives ∂${\chi }_{\mathrm{ijl}}^{\mathrm{}\left(2\right)\mathrm{}}$/∂$u_m$(κ) and ∂${\chi }_{\mathrm{ijl}}^{\mathrm{}\left(2\right)\mathrm{}}$/∂$E_m$ of the second-order susceptibility with respect to the displacements of the five sublattices and the macroscopic electric field of the LO phonons, respectively. We interpret the parameters ∂${\chi }_{\mathrm{ijl}}^{\mathrm{}\left(2\right)\mathrm{}}$/∂$u_m$(κ) in terms of a nonlinear bond-polarizability model and confirm the dominant role of the Ti-O and Ta-O interactions in the nonlinear-optical response.