Elastic and anelastic relaxations associated with phase transitions in ${\mathrm{EuTiO}}_3$

Elastic and anelastic properties of single crystal samples of ${\mathrm{EuTiO}}_3$ have been measured between 10 and 300 K by resonant ultrasound spectroscopy at frequencies in the vicinity of 1 MHz. Softening of the shear elastic constants $C_{44}$ and $\frac{1}{2}\left(C_{11}-C_{12}\right)$ by $\sim 20–30\%$ occurs with falling temperature in a narrow interval through the transition point, $T_{\mathrm{c}}=284$ K, for the cubic-tetragonal transition. This is accounted for by classical coupling of macroscopic spontaneous strains with the tilt order parameter in the same manner as occurs in ${\mathrm{SrTiO}}_3$. A peak in the acoustic loss occurs a few degrees below $T_{\mathrm{c}}$ and is interpreted in terms of initially mobile ferroelastic twin walls, which rapidly become pinned with further lowering of temperature. This contrasts with the properties of twin walls in ${\mathrm{SrTiO}}_3$, which remain mobile down to at least 15 K. No further anomalies were observed that might be indicative of strain coupling to any additional phase transitions above 10 K. A slight anomaly in the shear elastic constants, independent of frequency and without any associated acoustic loss, was found at $\sim 140$ K. It marks a change from elastic stiffening to softening with falling temperature and perhaps provides evidence for coupling between strain and local fluctuations of dipoles related to the incipient ferroelectric transition. An increase in acoustic loss below $\sim 80$ K is attributed to the development of dynamical magnetic clustering ahead of the known antiferromagnetic ordering transition at $\sim 5.5$ K. Detection of these elastic anomalies serves to emphasize that coupling of strain with tilting, ferroelectric, and magnetic order parameters is likely to be a permeating influence in determining the structure, stability, properties, and behavior of ${\mathrm{EuTiO}}_3$.

Figure 1

Segments of a selection of RUS spectra collected during heating of crystal 2. The $y$ axis should really be amplitude in volts from the detector transducer, but each spectrum has been offset in proportion to the temperature at which it was collected and the axis labeled as temperature. There is an obvious steep minimum in resonance frequencies near 280 K, but additional anomalies are also evident as changes in trend at $\sim 140$ K and $\sim 80$ K. Two trends for the temperature dependence below $\sim 280$ K have been picked out, and the relevant peaks are labeled 1 and 2. Peak 1 has stiffening as $T$ → 280 K from below, and peak 2 shows softening. The trends of all other peaks in the spectra can be represented as showing some combination of the trends of these two.

Figure 2

Variations of $f^2$ and $Q^{-1}$ obtained from the fitting of selected resonance peaks in RUS spectra collected from both single crystals. Values of the single crystal elastic constants that determine each resonance scale with $f^2$. None of the resonances is a pure mode, in the sense of being determined by only one or two elastic constants, but the pattern shown by resonance frequencies in (a) is likely to be a good approximation for the variation of “$C_{44}$” for a tetragonal crystal containing multiple ferroelastic twins. The pattern of resonance frequencies in (b) is believed to be a reasonable approximation for “$\frac{1}{2}\left(C_{11}-C_{12}\right)$.” Note that $f^2$ values for different resonances have been scaled to produce close overlap below $T_{\mathrm{c}}$; values of their actual resonance frequencies at room temperature are given in the figure keys. The dashed vertical line is at 284 K.