Phonon softening and dispersion in EuTiO${}_3$

We measured phonon dispersion in single-crystal EuTiO${}_3$ using inelastic x-ray scattering. A structural transition to an antiferrodistortive phase was found at a critical temperature $T_0=287\pm 1$ K using powder and single-crystal x-ray diffraction. Clear softening of the zone boundary $R$-point $\mathbf{q}=\left(0.50.50.5\right)$ acoustic phonon shows this to be a displacive transition. The mode energy plotted against reduced temperature could be seen to nearly overlap that of ${\mathrm{SrTiO}}_3$, suggesting a universal scaling relation. Phonon dispersion was measured along $\Gamma$-$X$ $\left(000\right)\to \left(0.500\right)$. Mode eigenvectors were obtained from a shell model consistent with the q dependence of intensity and energy, which also showed that the dispersion is nominally the same as in ${\mathrm{SrTiO}}_3$ at room temperature, but corrected for mass. The lowest-energy optical mode, determined to be of Slater character, softens approximately linearly with temperature until the 70–100 K range where the softening stops, and at low temperature, the mode disperses linearly near the zone center.

Figure 1

(a)–(f) Selected phonon spectra at the $R$ point [$\mathbf{Q}=\left(4.50.51.5\right)$, with resolution $\delta \mathbf{Q}=\left(0.030.020.03\right)$] as a function of temperature as indicated. The solid lines are fits to a phonon mode (with no central component) as described in the text.

Figure 2

Energy of the soft mode at $\mathbf{Q}=\left(4.50.51.5\right)$, fitting the spectra using two methods. The open squares are from fitting a single mode (a Stokes/anti-Stokes pair), and the solid circles correspond to including a central resolution-limited peak. As shown in the inset, $T_0$ was determined from the Bragg peak intensity as a function of temperature measured at the $\mathbf{Q}=\left(3.50.50.5\right)$ $R$ point. The continuous curve corresponds to ${\mathrm{SrTiO}}_3$ (Ref. 19), for which $T_0=108$ K.

Figure 3

(a) Bottom panel: Phonon energy vs momentum $q$ along the $\Gamma -X$ direction. The dispersion away from the $\Gamma$ point ($q\ge 0.1$) was for $\left(HKL\right)=(4,0,1-q)$. For the near-$\Gamma$ dispersion, zones $(41-1)$ (open triangles and diamonds) and $(5-1-2)$ (solid triangles) were used as indicated in the legend. The smooth curves are model calculations, with the thickness of the curve proportional to the square root of the intensity in the $\left(401\right)$ zone. The inset shows the spectra at the anticrossing near 20 meV and $q=0.25$ at $T=10$ K. For clarity, the two distinct mode energies at $q=0.26$ have been slightly offset along the $q$ axis. (b) Top panel: Relative intensities of the transverse acoustic and high-energy modes along $(4,0,1-q)$, comparing the experimental values obtained from fits to the data for $T=250$ K, and calculated values from the shell model, which are the dashed lines.

Figure 4

Selected phonon spectra near the gamma point at $q=0.05$. Energy offsets were accounted for through fits of elastic intensity and acoustic modes on either side of $\hslash \omega =0$ where applicable. A smooth background was subtracted from the curves, whose intensities are corrected for the Bose factor. The intensities were further normalized by the Bose-factor corrected transverse acoustic mode intensity. The Q position used was $(5-1-2.05)$ except for $T=12$ K, which was taken at $\mathbf{Q}=(41-1.05)$. Fitted energy vs temperature is plotted in the inset for the $(41-1.05)$ and $(5-1-2.05)$ zones (circles and squares, respectively).