Relaxation dynamics of modulated magnetic phases in the skyrmion host ${\mathrm{GaV}}_4{\mathrm{S}}_8$: An ac magnetic susceptibility study

We report on the slow magnetization dynamics observed upon the magnetic phase transitions of ${\mathrm{GaV}}_4{\mathrm{S}}_8$, a multiferroic compound featuring a long-ranged cycloidal magnetic order and a Néel-type skyrmion lattice in a relatively broad temperature range below its Curie temperature. The fundamental difference between ${\mathrm{GaV}}_4{\mathrm{S}}_8$ and the chiral helimagnets, the prototypical skyrmion host compounds, lies within the polar symmetry of ${\mathrm{GaV}}_4{\mathrm{S}}_8$, promoting a cycloidal instead of a helical magnetic order and rendering the magnetic phase diagram essentially different from that in the cubic helimagnets. Our ac magnetic susceptibility study reveals slow relaxation dynamics at the field-driven phase transitions between the cycloidal, skyrmion lattice and field-polarized states. At each phase boundary, the characteristic relaxation times were found to exhibit a strong temperature dependence, starting from the minute range at low temperatures, decreasing to the micro- to millisecond range at higher temperatures.

Figure 1

Static and ac susceptibility measured in ${\mathrm{GaV}}_4{\mathrm{S}}_8$, plotted against the external magnetic field. Panel (a) presents the static susceptibility, $\partial m/\partial H$, plotted in gray, along with the real component of the ac susceptibility, ${\chi }^{\prime }\left(H\right)$. Panel (b) displays the imaginary component of the ac susceptibility, ${\chi }^{\prime \prime }\left(H\right)$. Different colors represent data measured at various ac frequencies in the $f=0.1$-Hz–1-kHz range. Measured data are shifted proportionally with the sample temperature, which is indicated on the left side of each curve. The continuous lines connecting the dots are guides to the eye. The magnetic phases separated by the susceptibility peaks in the $B\text{−}T$ plane are indicated in the graphs. The paramagnetic state above $T_C=13\mathrm{K}$ is denoted as PM.

Figure 2

Panel (a) shows the magnetization curve $m\left(H\right)$ (right axis, gray curve), along with the static susceptibility $\partial m/\partial H$ and the ac susceptibility data measured with the highest modulation frequency, $f=1$ kHz at $T=11$ K (left axis, black and red curves, respectively). Panel (b) displays the imaginary component of the ac susceptibility at the same temperature and modulation frequency. The inset compares the adiabatic susceptibility, ${\chi }_{\infty }$ in the Cyc and SkL phases at various temperatures, obtained from the dc susceptibility curves as indicated by the black arrows in panel (a). The symbols signaling the main features of the susceptibility curves are the same as those used in Fig. 3.

Figure 3

Magnetic phase diagram of ${\mathrm{GaV}}_4{\mathrm{S}}_8$ established by static magnetization and ac susceptibility measured at $f=1$ kHz. Upward and downward pointing triangles represent the beginning and the end of the peaks in the static susceptibility, separating the pure Cyc, SkL, and FP phases from the regions of phase coexistence, indicated by pale shadowing. Full circles represent the location of the peaks in the static susceptibility, associated to the phase boundaries in earlier reports [20, 25, 29]. Red and blue stars display the location of the maxima in the real and imaginary parts of the ac susceptibility using a modulation frequency of 1 kHz.

Figure 4

Frequency dependence of the real (left column) and imaginary components (right column) of the susceptibility in various magnetic fields above the temperature of the triple point, at $T=11$ K (top), $T=10.75$ K (middle), and $T=10.5$ K (bottom row). Measured values are shifted with respect to the magnetic field. The color coding of the measured values represents the different ac frequencies in accordance with Fig. 1. Solid lines are fitted curves according to Eqs. (3) and (4) with the same set of parameters for the real and imaginary parts of the susceptibility. The blue and red bars next to the right axes represent the range of magnetic fields corresponding to the Cyc-SkL and SkL-FP phase transitions, respectively.

Figure 5

Panel (a) shows the fitted magnitude of the low-frequency susceptibility, $({\chi }_0-{\chi }_{\infty })$, as a function of the magnetic field at various temperatures. The fitted values are scaled to a common range using the factors indicated at three temperatures. Panels (b)–(d) present the distribution of relaxation times, $g[ln\left(\tau \right)]$, plotted as a function of the temperature and magnetic field. The distributions were calculated employing Eq. (2) with the ${\tau }_c$ and $\alpha$ parameters obtained from the fits to the frequency dependence of the complex susceptibility. Panels (b)–(d) present the relaxation times in the ranges of magnetic fields corresponding to the Cyc-SkL, SkL-FP, and Cyc-FP transitions, respectively. The distribution curves are shifted proportionally with the temperature along the $z$ axis, which is also indicated in the right side of the graphs. The curves are colored according to a color map representing decreasing temperatures ranging from $T=12$ K (red) to $T=6.5$ K (blue). The average relaxation time shows a dramatic decrease with decreasing temperatures along all the three phase boundaries, with values well above 10 s at their low-temperature limits.

Figure 6

Temperature dependence of the logarithmic average of the relaxation times obtained from the Cole-Cole fits, where the averaging was performed over the fitted values in the magnetic field region close to the phase transitions. The green, red, and blue circles correspond to the average relaxation times at the Cyc-FP, Cyc-SkL, and SkL-FP transitions. The error bars represent the standard deviation of the relaxation times on the logarithmic scale. The lines connecting the data points are guides to the eye. The dashed horizontal lines represent the measurement window defined by the inverse of the highest (1 kHz) and lowest (0.1 Hz) ac frequencies. The inset presents $ln{\tau }_{av}$ as the function of $1/T$ along the Cyc-SkL and SkL-FP phase boundaries. Relaxation time values close to the experimental window are plotted, as indicated by the black dotted frame. Linear fits to the data (solid black lines) yield the approximate average activation energies of 1100 and 1300 K near the Cyc-SkL and SkL-FP phase boundaries, respectively.

Figure 7

Frequency dependence of the real (left column) and imaginary components (right column) of the susceptibility in various magnetic fields below the temperature of the triple point at T = 9 K (top), T = 8 K (bottom row). Measured values are shifted with respect to the magnetic field. The color coding of the measured values represents the different ac frequencies in accordance with Fig. 1. Solid lines are fitted curves according to Eqs. (3) and (4) with the same set of parameters for the real and imaginary parts of the susceptibility. Gray dashed lines are separate fits to the real [(a) and (c)] and the imaginary components [(b) and (d)] of the susceptibility. The green bars next to the right axes emphasize that the range of magnetic fields at the given temperatures correspond to the Cyc-FP phase transition.