Changes in elastic moduli as evidence for quadrupolar ordering in the rare-earth frustrated magnet ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

Numerous materials feature unexplained phases with invisible or hidden order of electronic origin. A particularly mysterious case is that of ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, which avoids magnetic order to the lowest temperatures, but nevertheless has an unexplained second-order phase transition near $T=0.5$ K. Our ultrasound measurements of ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ provide direct evidence of a huge softening followed by strong hardening of the structural lattice below $T=0.5$ K. In the absence of magnetic order at this temperature, our results provide conclusive evidence for the proposed quadrupolar order and emphasize the importance of higher-order multipolar interactions in rare-earth frustrated magnets.

Figure 1

Temperature dependence of the (a) ac susceptibility and (b) specific heat of the full volume of the EP2 and EP3 samples used for ultrasound experiments. The ac-susceptibility data were obtained in field-cooled condition after adiabatic demagnetization of a paramagnetic salt from 1.8 K at 5 T. The inset in (a) shows the low-temperature susceptibility in an enlarged and semilogarithmic scale.

Figure 2

Temperature dependence of the relative elastic moduli, (a) $\mathrm{\Delta }{c}_{11}/c_{11}$, (b) $\mathrm{\Delta }{c}_{44}/c_{44}$, and (c) $\mathrm{\Delta }{c}_{\tau }/c_{\tau }=\mathrm{\Delta }(c_{11}-c_{12})/(c_{11}-c_{12})$ measured at various magnetic fields in ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ for sample (a), (b) EP2 and (c) EP3. Up and down temperature sweeps are shown. Field dependence of the elastic modulus, $\mathrm{\Delta }{c}_{11}/c_{11}$, for (d) sample EP3, (e) sample EP2, and (f) the sound attenuation, $\mathrm{\Delta }\alpha$ in sample EP2. The magnetic field is applied along the [110] direction. Vertical arrows indicate the anomalies in the elastic moduli. The curves for different (a)–(c) magnetic fields and (d)–(f) temperatures are arbitrarily shifted along the $y$ axis for clarity. The ultrasound frequency used was (a), (d) 47 MHz, (b) 30 MHz, (c) 32 MHz, and (e), (f) 41.4 MHz.

Figure 3

Temperature dependence of the elastic moduli $\mathrm{\Delta }{c}_{11}/c_{11}$ and $\mathrm{\Delta }(c_{11}-c_{12})/(c_{11}-c_{12})$ (black symbols) obtained in zero magnetic field. The ultrasound frequency was 45 MHz for $\mathrm{\Delta }{c}_{11}/c_{11}$ and 32 MHz for $\mathrm{\Delta }(c_{11}-c_{12})/(c_{11}-c_{12})$. The red line shows the best fit of the data using Eq. (1) with $G=232$ K and $K=-0.055$ K (see text for details). Inset: Positions of the anomalies observed in the acoustic properties of ${\mathrm{Tb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ plotted in the $H\text{−}T$ plane for magnetic field applied along the [110] direction. The dashed lines are guides to the eye and separate a dynamical Jahn-Teller regime from the antiferroquadrupolar (AFQ) order, possible antiferromagnetic (AFM) state, and unknown low-temperature field-induced phases beyond 0.1 T. The hatched areas indicate a strong hysteresis in the acoustic properties.