Ultrasonic investigations of the spin ices ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ in and out of equilibrium

We report ultrasound studies of spin-lattice and single-ion effects in the spin-ice materials ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ (DTO) and ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ (HTO) across a broad field range up to 60 T, covering phase transformations, interactions with low-energy magnetic excitations, and single-ion effects. In particular, a sharp dip observed in the sound attenuation in DTO at the gas-liquid transition of the magnetic monopoles is explained based on an approach involving negative relaxation processes. Furthermore, quasiperiodic peaks in the acoustic properties of DTO due to nonequilibrium processes are found to be strongly affected by macroscopic thermal-coupling conditions: the thermal runaway observed in previous studies in DTO can be suppressed altogether by immersing the sample in liquid helium. Crystal-electric-field effects having a higher energy scale lead to a renormalization of the sound velocity and sound attenuation at very high magnetic fields. We analyze our observations using an approach based on an analysis of exchange-striction couplings and single-ion effects.

Figure 1

Temperature dependence of the sound-velocity changes $\Delta v/v$ in HTO and DTO for longitudinal, $c_{11}$ (top two curves) and $c_L$ (bottom two curves), and transverse, $c_T$ (curves in the middle), acoustic modes. The curves are offset along the $y$ axis for clarity. For DTO, the ultrasound frequencies of 97.7, 96.5, and 171 MHz were used for the $c_{11}$, $c_L$, and $c_T$ modes, respectively. For HTO, 50, 49, and 109.5 MHz were used for $c_{11}$, $c_L$, and $c_T$, respectively.

Figure 2

Temperature dependence of (a) the sound-velocity change $\Delta v/v$ and (b) the sound attenuation measured at 52 MHz for HTO and 86 MHz for DTO between 0.29 and 6.5 K. The data at the lowest temperatures were set to zero on the ordinates. Arrows in (a) indicate temperature-sweep directions.

Figure 3

Temperature dependence of (a) the sound velocity $\Delta v/v$ and (b) the sound attenuation of the transverse ultrasonic wave propagating along the [111] direction (acoustic $c_T$ mode) in HTO. The ultrasound frequency was 181 MHz. The insets show low-temperature data at 28 MHz obtained using the dilution refrigerator.

Figure 4

Field dependence of (a) the sound velocity $\Delta v/v$ and (b) the sound attenuation of the $c_T$ mode. The top two curves in (a) were obtained in the dilution refrigerator, whereas the bottom two curves were measured in the ${}^3\mathrm{He}$ cryostat (see text for details). Up and down sweeps are indicated by arrows. The ultrasound frequencies were 108, 108, 183, and 170 MHz from the top curve to the bottom curve, respectively. The field sweep rate was 0.015 T/min, except for the sample with $D$ = 0.65 at 0.29 K, which was 0.03 T/min. The curves are arbitrarily shifted along the $y$ axis for clarity.

Figure 5

Field dependence of the sound velocity $\Delta v/v$ for different field and sound propagation directions at 0.29 K under weak thermal-coupling conditions. The top curve shows $\Delta v/v$ for a quasilongitudinal ultrasonic wave propagating along [112] ($c_L^{\prime }$ mode). The magnetic field first was swept to 2 T, then to $-2$ T and back to 0 T. The bottom curve shows $\Delta v/v$ for the $c_{11}$ mode propagating along [001]. The sound frequencies for the $c_{11}$ and $c_L^{\prime }$ modes were 64.5 and 86 MHz, respectively. The inset shows various crystallographic directions mapped on a single tetrahedron of the pyrochlore structure.

Figure 6

Field dependence of (a) the sound velocity $\Delta v/v$ and (b) sound attenuation $\Delta \alpha$ of the transverse $c_T$ mode propagating along the [111] direction for 0.29 and 1.5 K. The ultrasound frequency was 181 MHz. Arrows indicate field-sweep directions.

Figure 7

Field dependence of (a) the sound velocity $\Delta v/v$ and sound attenuation $\Delta \alpha$ for the $c_L$ mode and (b) the magnetization in HTO measured at 1.4 K. The ultrasound frequency was 70.9 MHz. Thin arrows indicate field-sweep directions.

Figure 8

Field dependence of (a) the sound velocity $\Delta v/v$ and (b) sound attenuation $\Delta \alpha$ of the $c_L^{\prime }$ mode in DTO at 1.6 and 4.2 K. The ultrasound frequency was 53.4 MHz. Arrows indicate field-sweep directions. The data for different temperatures are shifted along the $y$ axis for clarity.

Figure 9

Calculated magnetic-field dependence of the relative change of the sound velocity $\Delta v/v$ at different temperatures for the field applied along the $\left[111\right]$ direction.

Figure 10

Calculated magnetic-field dependence of the sound attenuation $\Delta \alpha$ at different temperatures for the field applied along the $\left[111\right]$ direction.

Figure 11

Calculated magnetic-field dependence of $\Delta v/v$ in HTO from the field parameters of Ref. [55] and $H\parallel \left[111\right]$. The dip in the sound velocity is a typical feature, only weakly dependent on the choice of magnetoelastic couplings.