Domains within Domains and Walls within Walls: Evidence for Polar Domains in Cryogenic ${\mathrm{SrTiO}}_3$

Resonant piezoelectric spectroscopy shows polar resonances in paraelectric ${\mathrm{SrTiO}}_3$ at temperatures below 80 K. These resonances become strong at $T<40\mathrm{K}$. The resonances are induced by weak electric fields and lead to standing mechanical waves in the sample. This piezoelectric response does not exist in paraelastic ${\mathrm{SrTiO}}_3$ nor at temperatures just below the ferroelastic phase transition. The interpretation of the resonances is related to ferroelastic twin walls which become polar at low temperatures in close analogy with the known behavior of ${\mathrm{CaTiO}}_3$. ${\mathrm{SrTiO}}_3$ is different from ${\mathrm{CaTiO}}_3$, however, because the wall polarity is thermally induced; i.e., there exists a small temperature range well below the ferroelastic transition point at 105 K where polarity appears on cooling. As the walls are atomistically thin, this transition has the hallmarks of a two-dimensional phase transition restrained to the twin boundaries rather than a classic bulk phase transition.

Figure 1

Schematic diagram of the experimental arrangement for resonant piezoelectric spectroscopy (RPS). The same setup can be used for resonant ultrasonic spectroscopy (RUS) by applying the ac voltage across the top piezoelectric transducer rather than the electrodes coating two parallel surfaces of the sample. The applied ac voltage for RPS and RUS measurements on ${\mathrm{SrTiO}}_3$ was 25 V.

Figure 2

Low temperature RPS and RUS spectra of ${\mathrm{SrTiO}}_3$. Panel (a): Segments of RPS spectra in the frequency interval from 25 kHz to 100 kHz. The spectrum shown in blue was collected at $T^{*}\approx 80\mathrm{K}$. Disappearance of two mechanical resonances located at ${\nu }_1\approx 40\mathrm{kHz}$ and ${\nu }_2\approx 85\mathrm{kHz}$ with increasing temperature is shown in panels (b) and (c). Panel (d) shows segments of RUS spectra in the same frequency interval as in panel (a) for comparison. The spectrum shown in green was collected at the cubic-tetragonal ferroelastic transition temperature $T_c=105\mathrm{K}$. In all panels the left axis represents the amplitude and the right axis gives the temperature at which each spectrum was collected.

Figure 3

The temperature evolution of the frequency of the mechanical resonance ${\nu }_1$ (black, filled circles and empty, red circles) and ${\nu }_2$ (black, filled squares and empty, red squares) determined using RPS and RUS. The agreement of results from RPS and RUS confirms that Fano-like features observed in the RPS spectra correspond to mechanical resonances of ${\mathrm{SrTiO}}_3$, giving direct evidence of polarity in ${\mathrm{SrTiO}}_3$.

Figure 4

The temperature evolution of the amplitudes of the mechanical resonances ${\nu }_1$ and ${\nu }_2$ observed using RPS and RUS measurements on ${\mathrm{SrTiO}}_3$. (a) RPS (filled blue circles) and RUS (red empty circles) amplitudes of ${\nu }_1$. The inset in part (a) shows the temperature dependence of the central $1/2\leftrightarrow -1/2$ ${}^{87}\mathrm{Sr}$ NMR transition at 9.2 T for $\mathbf{B}\parallel [111]$ in ${\mathrm{SrTiO}}_3$ which is split into three components at $T<T^{*}\approx 75–80\mathrm{K}$. The black squares represent the central NMR component that exhibits tetragonal symmetry whereas the triangles are the frequencies of satellite peaks, which exhibit nontetragonal symmetry. (b) RPS (filled blue circles) and RUS (red empty circles) amplitudes of ${\nu }_2$. The uncertainty in RPS and RUS amplitudes is estimated to be 10%, which mainly stems from the mechanical coupling between the sample and transducers which may change by thermal contraction.