Lattice instabilities in bulk EuTiO${}_3$

The phase purity and the lattice dynamics in bulk EuTiO${}_3$ were investigated both microscopically, using x-ray and neutron diffraction, ${}^{151}$Eu-Mössbauer spectroscopy, and ${}^{151}$Eu nuclear inelastic scattering, and macroscopically using calorimetry, resonant ultrasound spectroscopy, and magnetometry. Furthermore, our investigations were corroborated by $ab\mathit{initio}$ theoretical studies. The perovskite symmetry, $Pm\overline{3}m$, is unstable at the $M$- and $R$-points of the Brillouin zone. The lattice instabilities are lifted when the structure relaxes in one of the symmetries: $I4/mcm$, $Imma$, $R\overline{3}c$ with relative relaxation energy around $-25\text{meV}$. Intimate phase analysis confirmed phase purity of our ceramics. A prominent peak in the Eu specific density of phonon states at $11.5\text{meV}$ can be modeled in all candidate symmetries. A stiffening on heating around room temperature is indicative of a phase transition similar to the one observed in SrTiO${}_3$, however, although previous studies reported the structural phase transition to the tetragonal $I4/mcm$ phase our detailed sample purity analysis and thorough structural studies using complementary techniques did not confirm a direct phase transition. Instead, in the same temperature range, Eu delocalization is observed which might explain the lattice dynamical instabilities.

Figure 1

Rietveld refinement (black line) of a typical EuTiO${}_3$ diffractogram (red points) and the corresponding refinement residuals (blue line) using (a) synchrotron radiation at $300\mathrm{K}$ and (c) neutrons at $180\mathrm{K}$. (b) The FWHM of certain reflections ($R_0$, $R_1$, and $R_2$) measured with two different instrumental resolutions (see text). Inset to (a) shows the temperature-dependent lattice parameter (point size defines error bars) in cubic symmetry.

Figure 2

Temperature-dependent${}^{151}$Eu-Mossbauer spectra measured on $\mathrm{EuTi}{\mathrm{O}}_3$ powder (black points) and the corresponding two component model (black line). The expected isomer shift relative to EuF${}_3$ is indicated with tics. The inset shows the temperature-dependent Lamb-Mössbauer factor $f_{\mathrm{LM}}$ of the majority phase, Eu(II), extracted from the spectra and the associated Debye model (black dashed line).

Figure 3

(a) Pair distribution function analysis (PDF) of neutron-scattering data, $Q=50\AA$ (black points), at $300\mathrm{K}$, and the corresponding refinement (red line). The refinement residual (blue line) is given in the same scale and represents a fair refinement. (b) A close up in the temperature-dependent PDF analysis (black lines) and a typical PDF analysis refinement (red line) around the first nearest-neighbor distances regime, indicated with dashed lines. Note that at a lattice parameter distance, $3.905\AA$ at room temperature (RT), Ti-Ti and O-O pair correlations coexist with Eu-Eu.

Figure 4

(a) The Eu specific density of phonon states in EuTiO${}_3$ at $210\mathrm{K}$ (black tics) and $360\mathrm{K}$ (red line-tics) measured using NIS, typical errorbar is given, and our theoretical spectra for different structures.[26] (b) Heat capacity data between 10 and $340\mathrm{K}$ measured on EuTiO${}_3$ using calorimetry (pointsize defines errorbar), line between points is guide to the eye. (c) The first $5\mathrm{meV}$ of the reduced ${}^{151}$Eu projected density of phonon states, $g\left(E\right)/E^2$, and the related Debye levels (dashed lines). (d) The real part, $\mathrm{Re}{Y}_{\left[100\right]}^{\mathrm{r}}$, and the imaginary part, (e), $\mathrm{Im}{Y}_{\left[100\right]}^{\mathrm{r}}$, of the complex Young's modulus in SrTiO${}_3$ obtained from Ref.45. (f) Speed of sound extracted from RUS (black points), pointsize defines errorbar, and the corresponding extracted from NIS (red circles) at 210, 300 and $360\mathrm{K}$. (g) The inverse quality factor of the resonance at $590\mathrm{kHz}$ between 200 and $370\mathrm{K}$.

Figure 5

Temperature dependence of the ac magnetic susceptibility measured on cooling in sample in $\mathrm{EuTi}{\mathrm{O}}_3$ using an oscillating magnetic field with frequency $f=20.4\mathrm{Hz}$, amplitude $H_{\mathrm{ac}}=10\mathrm{G}$, at $H_{\mathrm{dc}}=0\mathrm{G}$. Inset shows a $M-H$ curve measured on the same sample at $4.9\mathrm{K}$ with maximum applied magnetic field of $H_{\mathrm{dc}}=50\mathrm{kG}$.

Figure 6

(a) The atomic displacement parameters (ADPs) extracted in the harmonic approximation from neutron diffraction; typical error bar is given. (b) Probability density function (p.d.f.), distribution of the Eu atom in the $ab$ plane at 180 K. (c) The effective one-particle potential of the Eu atom along the $a$ direction ($y$ $=$ 0) at 320, 295, 250, and $180\mathrm{K}$ obtained from crystallographic structure analysis and the typical error bar. (d) The potential barrier $V_0$ and the displacement from equilibrium position $2d$ (dashed line is guide to the eye) extracted from the potential given in (c).