Lattice dynamics of the hybrid improper ferroelectrics (Ca,Sr)3Ti2O7

The structure and lattice dynamics of the hybrid improper ferroelectric compound ${\mathrm{Ca}}_{3\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$ (for $x=0,0.6,$ and 0.9) have been studied with a combination of diffraction, inelastic scattering experiments, Raman spectroscopy, and calorimetry measurements, as well as first-principles simulations. Using inelastic neutron scattering, we have measured the phonon density of states (DOS) for $x=0.9$, which revealed a strong broadening but little change in phonon energies on heating from 10 K to 728 K across the ferroelectric phase transition temperature, ${T}_{FE}$. Using inelastic x-ray scattering, the momentum-resolved phonon dispersions were measured from 80 K to 950 K on a single crystal (for $x=0.6$), and also revealed a strong phonon broadening but a small energy shift for acoustic modes on heating across ${T}_{FE}$. Our Raman measurements (for $x=0.6$) showed robust rotational and oxygen breathing modes but soft tilt modes, consistent with previous measurements on similar compounds. Our density functional calculations achieve good agreement with both the phonon DOS and dispersions measured. We did not observe any unusual quadratic dispersion for $c$-polarized transverse acoustic modes, at odds with a recently predicted quasi-2D character, for either undoped ($x=0$) or doped ($x=0.6$) compounds.