Defect dynamics and strain coupling to magnetization in the cubic helimagnet $\mathrm{C}{\mathrm{u}}_2\mathrm{OSe}{\mathrm{O}}_3$

Small but significant static and dynamic strain coupling effects have been detected in $\mathrm{C}{\mathrm{u}}_2\mathrm{OSe}{\mathrm{O}}_3$ through elastic and anelastic anomalies associated with magnetic phase transitions observed as a function of temperature (1.5–150 K) and magnetic field (0–300 mT). The magnetic transition near 60 K is accompanied by a small increase in single-crystal elastic constants which can be understood in terms of biquadratic coupling between shear strain and the magnetic order parameter, even though the shear strain itself is almost negligibly small. The conical-collinear transition is associated with distinct minima in the elastic properties, while weaker anomalies at lower fields may be related to changes in the configuration of magnetic domains. A distinctive acoustic loss peak at $\sim 42\text{K}$, independent of magnetic field, is attributed to freezing of a defect which is coupled with shear strain, has an associated activation energy of $\sim 5\text{kJ}\text{mo}{\mathrm{l}}^{-1}$, and may play a role in pinning the magnetic microstructures. Anomalies below $\sim 10\text{K}$ indicate the presence of some additional relaxation process which could signify a change in magnetic structure.

Figure 1

$T-H$ phase diagram for $\mathrm{C}{\mathrm{u}}_2\mathrm{OSe}{\mathrm{O}}_3$ with the magnetic field aligned parallel to [111], after Adams et al. [5] (filled symbols) and Ruff et al. [9] (open symbols). Superimposed on the phase diagram are dashed red lines to indicate regions of acoustic loss observed in the present paper.

Figure 2

Segments of RUS spectra of $\mathrm{C}{\mathrm{u}}_2\mathrm{OSe}{\mathrm{O}}_3$ collected during cooling showing a single resonance peak with frequency $\sim 1144\text{kHz}$. The $y$ axis was originally amplitude in volts from the detecting transducer, but the spectra have been offset in proportion to the temperature at which they were collected and the axis labeled as temperature.

Figure 3

Variations of $f^2$ (proportional predominantly to shear elastic constants) and $Q^{-1}$ (indicative of acoustic attenuation) for representative resonances as a function of temperature with different external magnetic field strengths. (a) Resonance peak with frequency close to 1104 kHz. (b) Resonance peak with frequency close to 657 kHz. (c) Magnification of the low-temperature region of (b) showing hysteresis of $f^2$ values obtained at temperatures below $\sim 42\text{K}$: red (heating), blue (cooling).

Figure 4

Values of $f^2$ (normalized) and $Q^{-1}$ as a function of magnetic field at selected temperatures. (a) 1144 kHz resonance. The $f^2$ values rescaled to give $\sim 1$ at the lowest temperature and then offset by some arbitrary amount up the $y$ axis for clarity. The minimum in $f^2$ near 100 mT at 1.5 K shifts to lower fields with increasing temperature. (b) $Q^{-1}$ for the same resonance as in (a) showing a small peak at $\sim 95\text{mT}/46.5\text{K},110\text{mT}/5\text{K},\sim 160\text{mT}/1.5\text{K}$. (c) The $f^2$ values for five different resonances, with rescaling as in (a). The general pattern of softening is the same as in (a) but differs in detail at different resonant frequencies. An additional, very weak anomaly appears to be present near 50 mT in some datasets (e.g., 5, 10, 16.5 K). (d) The $f^2$ and $Q^{-1}$ data for three resonances at 1.5 K, showing details of the different patterns of evolution with colors representing different frequencies rather than different temperatures.

Figure 5

Values of ${H_f}_{2\min }$ (colored markers and solid lines) superimposed on data for ${H_c}_1$ and ${H_c}_2$ from Fig. 1. Values shown for $T_{\mathrm{c}}$ are 3 K below the temperature at which the pronounced kink occurs in $f^2$ [Fig. 3]. Separate colored markers indicate the approximate position of slight softening with increasing field seen in $f^2$ data at low fields in Fig. 4. Limits of hysteresis seen in Fig. 6 are shown as filled squares joined by dotted lines.

Figure 6

Results for $f^2$ and $Q^{-1}$ from detailed analysis of resonances with frequencies near 1104 (left) and 1144 kHz (right). Original spectra collected by starting at 1.5 K, applying the specified field, and then following a sequence of heating (red) followed by cooling (blue) as set out in the Appendix.

Figure 7

Comparison of the excess elastic stiffening, $\mathrm{\Delta }|f^2|$, below $T_{\mathrm{c}}\approx 60\text{K}$ with the intensity of a superlattice reflection which varies with $M^2$ (green data points, from Fig. 4 of Adams et al. [5]) and $M^2$ directly, as measured in a field of 100 mT aligned parallel to [110] (purple squares, data from Fig. 3 of Živković et al. [27]). Magnetic data normalized to 1 at 10 K, and $\mathrm{\Delta }|f^2|$ data (619 kHz resonance for cooling, 592 kHz for heating) scaled to show that they are permissive of $\mathrm{\Delta }|f^2|\propto M^2$, at least between 60 and $\sim 42\text{K}$. Pink line in insert shows the fit of the Varshni equation for the baseline used to obtain values of $\mathrm{\Delta }|f^2|$.

Figure 8

Fit of Eq. (3) shown for the Debye loss peak. The dashed line is a fit to data between $\sim 45$ and $\sim 55\text{K}$ as a guide to the eye to show the approximate magnitude of anelastic stiffening associated with the Debye freezing process. This stiffening amounts to $\sim 230\text{kH}{\mathrm{z}}^2$, which is a change of $\sim 0.019\%$.