Elastic and anelastic behavior associated with magnetic ordering in the skyrmion host ${\text{Cu}}_2{\text{OSeO}}_3$

Magnetic ordering in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ occurs without any detectable changes in the lattice symmetry but involves significant coupling with strain. The strain coupling effects in ${\text{Cu}}_2{\text{OSeO}}_3$ have been investigated with a focus on the skyrmion lattice by examining elastic and anelastic properties. Resonant ultrasound spectroscopy has been used to measure these properties of a ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ single crystal as a function of temperature and magnetic field. On heating, the skyrmion phase has been characterized by slightly softer elasticity compared to the helical phase. However, there were no obvious anomalies in elastic and anelastic properties associated with the boundary of the stability field of the skyrmion lattice. Evolution of elastic properties with magnetic field, passing through the stability field of the skyrmion lattice, showed a characteristic pattern of a glassy state, where an equilibrium state is never reached. These imply that coupling of the skyrmions with strain is extremely weak in ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$, leading to glassy or liquidlike behavior of skyrmions. Three Debye-like loss peaks were observed near $\sim 40$, $\sim 50$, and $\sim 60\mathrm{K}$. The relaxation mechanism for the 40 K loss peak has been found to have a single relaxation time. Overlapping acoustic loss peaks in the temperature interval $\sim 50–62\mathrm{K}$ suggest that the magnetic transitions with variable temperature in this temperature range involve freezing of some dynamic aspect(s) of the magnetic structure with an activation energy of $\sim 0.1–0.15$ eV.

Figure 1

(a) Schematic of the measurement setup for RUS experiments in low-temperature and magnetic-field environments. (b) An example of the asymmetric Lorentzian curve fit to a resonance peak, providing the peak frequency, $f$, and width at half maximum height, $\mathrm{\Delta }f$, of the peak. (c) Segments of RUS spectra collected from the single crystal of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ during cooling (blue lines), followed by heating (red lines) in a 25 mT field. The $y$ axis is really the amplitude from the amplifier but the spectra are stacked in proportion to the temperature at which they were collected; the axis was then relabeled as temperature to allow easy visualization of the evolution of individual resonance peaks with decreasing and increasing temperature.

Figure 2

Magnetic phase diagram of ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ in the vicinity of the transition temperature (after Chauhan et al. [13]). Blue dotted lines show the trajectories along which RUS spectra were collected with variable temperature at constant field or variable field at constant temperature. Filled circles mark the upper temperature limit of hysteresis in $f^2$ values between heating and cooling. Filled triangles represent the temperature at which small anomalies in $f^2$ and $Q^{-1}$ occur in zero field and in a 2 mT field. Filled squares show the upper magnetic field limit of hysteresis in $f^2$ values between increasing and decreasing fields.

Figure 3

Variations of $f^2$ and $Q^{-1}$ for resonances near 773, 861, and 1017 kHz in the RUS spectra collected at 25 mT. Filled symbols = heating; open symbols = cooling. Insets show details of $f^2$ variations in the vicinity of $T_{\mathrm{c}}$. Vertical dotted lines at 32 K mark the helical–conical transition temperature reported by Adams et al. [11]. Shaded areas indicate the expected stability field of the skyrmion lattice [13]. Black curves are fits of Eq. (1) to Debye-like peaks in $Q^{-1}$.

Figure 4

Variations of $f^2$ and $Q^{-1}$ for a resonance near 1017 kHz in RUS spectra collected during a sequence of heating followed by cooling in zero field (a) and during a sequence of cooling followed by heating in 2 mT (b) and 21 mT (c) fields. Filled symbols = heating; open symbols = cooling. Insets are expanded sections of the $f^2$ variation in the vicinity of $T_{\mathrm{c}}$, demonstrating the presence of a small but distinct dip. Shaded area indicates the expected stability field of the skyrmion lattice [13].

Figure 5

(a) Values of $T_{\mathrm{m}}$ determined from fitting of a loss peak at $\sim 40\mathrm{K}$ in the $Q^{-1}$ data from selected resonances collected during cooling in a 25 mT field. The inset shows details of $Q^{-1}$ variations of representative resonances. Arrhenius plot is shown in (b), where the slope of the straight line fit gives activation energy of $49\pm 7$ meV.

Figure 6

Enhanced views of variations of $f^2$ and $Q^{-1}$ for selected resonances in RUS spectra collected during cooling in zero field and in a 2 mT field. Steep softening accompanied by a steep increase in acoustic loss is clearly seen in a narrow temperature interval close to 59 K.

Figure 7

Evolution of $f^2$ values for resonances near 927 and 1240 kHz as a function of increasing (filled symbols) and decreasing (open symbols) magnetic field at 54, 56, 56.5, 57, 57.5, and 62 K. $f^2$ values have been multiplied by arbitrary scaling factors to allow for easy comparison of the trends of $f^2$ with magnetic field at different temperatures. Arrows indicate the direction of time evolution of the experiment.

Figure 8

Comparison of $f^2$ data for a resonance near 1017 kHz in a 2 mT field and in a 21 mT field shown in Fig. 4. Inset shows values of $E_{\mathrm{a}}/r_2\left(\beta \right)$ in four different magnetic fields determined from fitting of a peak in $Q^{-1}$ at $\sim 60\text{K}$ for this resonance.