Domain Dynamics in Quantum-Paraelectric ${\mathrm{SrTiO}}_3$

Twin dynamics forced by acoustic waves shows several linear and nonlinear response modes below $T_c=106\mathrm{K}$. In the quantum paraelectric state a “quantum domain glass” at $25\mathrm{K}<T<40\mathrm{K}$ shows intense relaxation and temperature hysteresis. Domains float collectively in a complex, smooth landscape with long relaxation times. In the “quantum domain solid” state below 25 K new phenomena occur. A temperature-dependent memory effect of the elastic response after anneal at 36 K depends on the lowest temperature reached in the quantum domain solid state below 25 K. The glassiness of twin boundary dynamics vanishes for temperatures approaching absolute zero.

Figure 1

Temperature spectra of low-amplitude (${ϵ}_0={10}^{-7}$) ultrasonic internal friction $\delta$ (a) and Young’s modulus $E$ (b) for several cooling-heating scans below $T_c\approx 106\mathrm{K}$. Cooling and Heating 2 scans are performed with the cryocompressor switched on, Heating 1 and Heating 3—with the compressor switched off. During Heating 3, strain amplitude dependences of the internal friction were measured at selected temperatures (data shown in Fig. 2). The inset in (b) shows, on an expanded scale, the internal friction $\delta$ and Young’s modulus $E$ below 40 K during the Heating 1 scan. For the Heating 2 scan (green curve) only each fourth experimental point is shown above 90 K in (a) and (b) to make the points of Heating 1 scan (black curve) visible.

Figure 2

Strain amplitude dependence of the internal friction $\delta$ measured at temperatures between 72 and 108 K (a), 46 and 72 K (b), and 18 and 46 K (c) during heating from 18 K. Solid lines connect experimental points in panels (a),(b), and (c). All strain amplitude dependences include direct and reverse runs (increasing and decreasing strain amplitude). The jerks during increasing and decreasing strain amplitudes are seen in (b) and (c). Arrows in Fig. 2 (for jerky curves at 42 and 46 K) indicate increase and decrease of the strain amplitude, mark the difference between the data for direct and reverse runs, and indicate the strain amplitude hysteresis between ca. 40 and 70 K. Panel (d) shows the strain amplitude dependences between 18 and 36 K on an expanded scale. Solid lines are their fittings with power law. The inset shows the glassy strain exponent $\mu$ vs temperature; the dotted line in the inset is a guide to the eye.

Figure 3

(a) Schematic time-temperature diagram of experimental protocol used to study the “memory” effect in the temperature spectra of Young’s modulus, induced by annealing the sample at a temperature $T_{\text{anneal}}$, which includes (i) cooling from above $T_c$ to a variable temperature $T_{\min 1}$, (ii) heating to a temperature $T_{\text{anneal}}$ and isothermal holding at $T_{\text{anneal}}$ during time $t_{\text{anneal}}$, (iii) cooling to $T_{\min }=17\mathrm{K}$ and (iv) heating above $T_c$. (b) Young’s modulus $E$ vs temperature for the continuous cooling-heating cycle; (c) annealing experiment following the protocol in Fig. 3 with $T_{\min 1}=T_{\text{anneal}}=36\mathrm{K}$; (d),(e),(f), and (g) annealing experiments following the same protocol with $T_{\min 1}=17$, 22, 26, and 31 K, respectively, $T_{\text{anneal}}=36\mathrm{K}$ and $t_{\text{anneal}}=2\mathrm{h}$. Panel (h) shows the fraction of the Young’s modulus increase during isothermal annealing recovered as “memory effect” during consecutive heating as a function of the temperature of precooling $T_{\min 1}$; letters (c),(d),(e),(f), and (g) close to the data points denote the panels from which the data were taken.

Figure 4

Examples of strain amplitude dependences of the internal friction $\delta$ for selected temperatures of 94, 56, and 30 K, representative for viscous twin boundary motion, depinning, and glassy dynamics, respectively.