ac susceptibility study of magnetic relaxation phenomena in the antiskyrmion-hosting tetragonal Mn-Pt(Pd)-Sn system

Here, we report an exhaustive study of the frequency-dependent ac magnetic susceptibility of the $D_{2d}$-symmetric Heusler system Mn-Pt(Pd)-Sn that hosts antiskyrmions over a wide temperature range. Magnetic relaxation studies using Cole-Cole formalism reveal a Debye-type relaxation with a nearly negligible distribution in relaxation times. In contrast to the archetypical skyrmion hosts, the high Curie temperature $T_C$ of the present system ensures shorter switching times, and correspondingly, higher frequencies are required to probe the relaxation dynamics. We find a nonmonotonic variation in the characteristic relaxation time with distinct maxima at the phase boundaries via helical $⟶$ antiskyrmion $⟶$ field-polarized states, indicating slower magnetization dynamics over the region of phase coexistence. The temperature-dependent relaxation time across different phases is of the order of ${10}^{-5}–{10}^{-4}$ s and follows the well-known Arrhenius law with reasonable values of the energy barriers. The present study concerning the magnetization dynamics in the antiskyrmion host tetragonal Heusler system is an important contribution towards the basic understanding of the dynamical aspects of antiskyrmions for their potential applications.

Figure 1

Magnetization as a function of temperature $M\left(T\right)$ (open squares, left $Y$ axis) measured at $H$ = 1 kOe in the field-cooled mode and the temperature-dependent real part of zero-field ac susceptibility curves ${\chi }^{\prime }\left(T\right)$ shown with open circles (right $Y$ axis) for (a) ${\mathrm{Mn}}_{1.4}\mathrm{Pt}\mathrm{Sn}$ and (b) ${\mathrm{Mn}}_{1.4}{\mathrm{Pt}}_{0.9}{\mathrm{Pd}}_{0.1}\mathrm{Sn}$. Isothermal field-dependent ac susceptibility curves $\chi \left(H\right)$ taken at different temperatures in the range 10 K $\le T\le$ 350 K for (c) ${\mathrm{Mn}}_{1.4}\mathrm{Pt}\mathrm{Sn}$ and (d) ${\mathrm{Mn}}_{1.4}{\mathrm{Pt}}_{0.9}{\mathrm{Pd}}_{0.1}\mathrm{Sn}$. The evolution of the antiskyrmion phase is inferred from the broad minimum surrounded by maxima in $\chi \left(H\right)$ curves. The low-field maximum is well seen by the occurrence of a peak, whereas the high-field maximum is visualized by a spread out hump kind of behavior marked by arrows.

Figure 2

In-phase $\left[{\chi }^{\prime }\left(H\right)\right]$ and out-of-phase [${\chi }^{\prime \prime }\left(H\right)$] components of $\chi \left(H\right)$ at $T$ = 300 K with $H_{ac}$ = 10 Oe measured at different frequencies for (a) and (b) ${\mathrm{Mn}}_{1.4}\mathrm{Pt}\mathrm{Sn}$ and (c) and (d) ${\mathrm{Mn}}_{1.4}{\mathrm{Pt}}_{0.9}{\mathrm{Pd}}_{0.1}\mathrm{Sn}$. Note that for the sake of clarity, the successive ${\chi }^{\prime }\left(H\right)$ data starting from 8888 Hz have been given a constant upward shift of 2.5 m emu/g Oe. Vertical dashed lines represent the phase crossover from H-AskX and AskX-FP states inferred from $d{\chi }^{\prime }/dH$ versus $H$, as shown in the inset of (a). Shaded regions highlight the coexistence of multiple phases.

Figure 3

(a) Frequency dependence of ${\chi }^{\prime }$ at different representative magnetic fields $H$ measured at $T=300$ K for ${\mathrm{Mn}}_{1.4}\mathrm{Pt}\mathrm{Sn}$. The fits (solid lines) to the data (open symbols) are based on Eq. (1). Magnetic field dependence of the fitting parameters for ${\mathrm{Mn}}_{1.4}\mathrm{Pt}\mathrm{Sn}$ (b) $A={\chi }_T-{\chi }_S$, (c) $B={\chi }_S$, (d) the characteristic relaxation time ${\tau }_0$, and (e) the relaxation time distribution parameter $\alpha$. Vertical dashed lines serve as a guide to the eye to indicate the phase boundaries between the H, AskX, and FP phases. (f) Temperature dependence of relaxation time ${\tau }_0$ derived from (d). ${\tau }_0$ in the H $\to$ AskX and AskX $\to$ FP regions are inferred from the lower and upper maxima in the ${\tau }_0\left(H\right)$ plots, whereas ${\tau }_0\left(T\right)$ in the AskX phase is extracted from ${\tau }_0\left(H\right)$ at $H$ = 2 kOe. Note that the solid lines represent fits to the observed data (open squares) using Arrhenius's law.

Figure 4

Field-dependent fitting parameters for ${\mathrm{Mn}}_{1.4}{\mathrm{Pt}}_{0.9}{\mathrm{Pd}}_{0.1}\mathrm{Sn}$ (a) ${\chi }_T-{\chi }_S$, (b) $B={\chi }_S$, (c) $\alpha$, and (d) ${\tau }_0$. Dashed lines serve as guide to eye.

Figure 5

Frequency dependence of ${\chi }^{\prime \prime }$ at various representative magnetic fields measured at $T$ = 300 K for (a) ${\mathrm{Mn}}_{1.4}\mathrm{Pt}\mathrm{Sn}$ and (b) ${\mathrm{Mn}}_{1.4}{\mathrm{Pt}}_{0.9}{\mathrm{Pd}}_{0.1}\mathrm{Sn}$. Solid balls represent experimental data points, whereas the lines indicate fits to the data using Eq. (2).

Figure 6

$H-T$ phase diagram inferred from maxima in the ac susceptibility (circles) and the fit parameters of the Cole-Cole formalism (squares) as described in the text for ${\mathrm{Mn}}_{1.4}{\mathrm{Pt}}_{0.9}{\mathrm{Pd}}_{0.1}\mathrm{Sn}$. The inset shows $H-T$ phase diagram for ${\mathrm{Mn}}_{1.4}\mathrm{Pt}\mathrm{Sn}$. Open and closed symbols represent the lower and upper boundaries of antiskyrmion phase, respectively. $H_L^{\mathrm{ACS}}$ and $H_H^{\mathrm{ACS}}$ show the lower and upper phase boundaries of the antiskyrmion phase, respectively, obtained from the peak/hump anomalies in the field evolution of the ac susceptibility measurements. Similarly, $H_L^{\mathrm{Cole}-\mathrm{Cole}}$ and $H_H^{\mathrm{Cole}-\mathrm{Cole}}$ are lower and upper phase boundaries of the antiskyrmion phase obtained from the field-dependent fit parameters of the Cole-Cole expression using ${\chi }^{\prime }\left(H\right)$ data.