Pressure dependence of ferroelectric quantum critical fluctuations

Phase transitions to a long-range-ordered state driven by a softened phonon mode are ubiquitous across condensed matter physics, but the evolution of such a mode as the system is tuned to or from the transition has never been explicitly measured until now. We report the effect of pressure on the soft mode associated with ferroelectricity in the archetypal quantum critical paraelectric ${\mathrm{SrTiO}}_3$. This is an ideal, clean, model system for exploring these effects, with pressure directly addressing the phonon modes only. We measure and report the effect of quantum critical fluctuations on the pressure and temperature dependence of the ferroelectric soft phonon mode as the system is tuned away from criticality. We show that the mean-field approximation is confirmed experimentally. Furthermore, using a self-consistent model of the quantum critical excitations including coupling to the volume strain and without adjustable parameters, we determine logarithmic corrections that would be observable only very close to the quantum critical point. Thus, the mean-field character of the pressure dependence is much more robust to the fluctuations than is the temperature dependence. We predict stronger corrections for lower dimensionalities. The same calculation confirms that the Lydanne-Sachs-Teller relation is valid over the whole pressure and temperature range considered. Therefore, the measured dielectric constant can be used to extract the frequency of the soft mode down to 1.5 K and up to 20 kbar of applied pressure. The soft mode is observed to stiffen further, raising the low-temperature energy gap and returning towards the expected shallow temperature dependence of an optical mode. This behavior is consistent with the existence of a ferroelectric quantum critical point on the pressure-temperature phase diagram of ${\mathrm{SrTiO}}_3$, which applied pressure tunes the system away from. This work represents an experimental measurement of the stiffening of a zone center soft phonon mode as a system is tuned away from criticality, a potentially universal phenomenon across a variety of phase transitions and systems in condensed matter physics.

Figure 1

Predicted phase diagram for ${\mathrm{SrTiO}}_3$ and related compounds [19]. The “tuning parameter” can be realized through either chemical doping or substitution, or by applying pressure to tune the frequencies of the phonon oscillators. ${\mathrm{SrTiO}}_3$ sits naturally slightly to the right of a quantum critical point on this diagram, and pressure serves to push it further away from the quantum critical region and towards the behavior of ${\mathrm{KTaO}}_3$ [8].

Figure 2

Temperature dependence of the dielectric constant of ${\mathrm{SrTiO}}_3$ at hydrostatic pressure values from ambient (blue) to 20 kbar (red). The inset shows the same data normalized to their 2 K values, revealing a clear evolution of the shape of the curves as pressure is increased.

Figure 3

(a) Zone-center soft mode phonon energy of ${\mathrm{SrTiO}}_3$ extracted from the dielectric constant via the Lydanne-Sachs-Teller relation, plotted against temperature for applied pressures from ambient (blue, lowest) to 20 kbar (red, topmost). The variation of the frequencies of other phonon branches and the refractive index with temperature and pressure were modeled as described in the text. Neutron scattering data from Yamada et al. [11] are shown as open circles and the corresponding data for ${\mathrm{KTaO}}_3$ from Shirane et al. [28] as crosses. (b) Calculated temperature dependence of the TO phonon energy for several pressures.

Figure 4

(a) Zone-center soft mode phonon energy of ${\mathrm{SrTiO}}_3$, plotted against pressure at fixed temperatures. The solid lines show fits to ${\omega }_0{(p-p_c)}^{0.5}$ with ${\omega }_o$ and $p_c$ free fitting parameters to guide the eye. The inset gives the square of the zero-temperature extrapolated phonon gap $\mathrm{\Delta }$ against pressure, with a linear fit shown as a solid line. (b) Calculated pressure dependence of the TO phonon energy for several temperatures. Inset – calculated pressure dependence of the TO phonon gap.