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

P. V. Prakash Madduri  School of Physical Sciences, National Institute of
Science Education and Research, HBNI, Jatni-752050, India  Subir Sen  School
of Physical Sciences, National Institute of Science Education and Research,
HBNI, Jatni-752050, India  Bimalesh Giri  School of Physical Sciences,
National Institute of Science Education and Research, HBNI, Jatni-752050,
India  Dola Chakrabartty  School of Physical Sciences, National Institute of
Science Education and Research, HBNI, Jatni-752050, India  Subhendu K. Manna
School of Physical Sciences, National Institute of Science Education and
Research, HBNI, Jatni-752050, India  Stuart S. P. Parkin  Max Planck Institute
of Microstructure Physics, Weinberg 2, 06120 Halle, Germany  Ajaya K. Nayak
<EMAIL_ADDRESS>School of Physical Sciences, National Institute of Science
Education and Research, HBNI, Jatni-752050, India

###### Abstract

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 non-monotonic variation
in the characteristic relaxation time with distinct maxima at the phase
boundaries via helical $\longrightarrow$ antiskyrmion $\longrightarrow$ 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.

††preprint: APS/123-QED

##  I INTRODUCTION

Skyrmions represent a particular class of magnetic nano-domains characterized
by vortex-like localized spin configurations U.K.Robler , in contrast to the
conventional- helical, conical, and spontaneously magnetized spin structures.
The twist in the magnetization profile within each domain described by a
topological winding number gives robust protection to the spin configurations
against continuous deformation N.Nagaosa . Owing to their solitonic nature,
skyrmions can interact efficiently with the electrons and magnons in the host
material and result in several exotic electromagnetic phenomena such as
topological Hall effect A.Neubauer , thermally induced ratchet motion
M.Mochizuki , and skyrmion magnetic resonance Y.Onose . The competition
between the Heisenberg exchange and the Dzyaloshinskii-Moriya (DM) interaction
is one of the main reasons behind the observation of skyrmions in bulk and
thin-film magnetic systems A.Neubauer ; S.Muhlbauer ; C.Pappas ; W.Munzer ;
X.Z.Yu ; S.X.Huang ; H.Wilhelm ; S.Seki ; S.Heinze . Recently, a new type of
topological object named antiskyrmion was observed in an acentric tetragonal
Heusler-Mn-Pt(Pd)-Sn system with $D_{2d}$ crystal symmetry A.K.Nayak . Note
that the high $T_{C}$ of this material ($\approx$ 400 K) ensures that the
antiskyrmions get stabilized over a wide temperature range. In the archetypal
skyrmion hosting B20 materials, an externally applied magnetic field favours
conical helices propagating along the field direction. In contrast, a
particular pattern of DM vectors dictated by the $D_{2d}$ crystallographic
symmetry in the Mn-Pt(Pd)-Sn system precludes the emergence of the
longitudinal conical structure at finite fields.

In most cases, the presence of the skyrmion phase was identified using small-
angle neutron scattering S.Muhlbauer , Lorentz transmission electron
microscopy (LTEM) X.Z.Yu ; A.K.Nayak  and, indirectly by topological Hall
transport studies A.Neubauer ; N.Kanazava ; S.X.Huang ; S.X.Huang2 ; S.Sen .
It is well-known that ac-susceptibility is a potential technique to identify
various magnetic phases and to study their relaxation dynamics A.H.Morrish ;
M.Balanda ; S.J.Blundell . Linear and non-linear susceptibilities with various
ac-magnetic field amplitudes ($H_{ac}$) and frequencies ($f$) in the presence
and absence of superimposed dc-magnetic fields ($H_{dc}$) are widely used to
unambiguously identify Ferro-, Ferri-, Antiferro-magnetic ordering,
canonical-, cluster-, reentrant spin glass nature, spin-reorientation
transitions and superparamagnetism M.Balanda ; S.J.Blundell . Recently, it has
been shown that ac-susceptibility as a function of dc-magnetic field,
$\chi\left(H\right)$, at various fixed frequencies (f) of the oscillating
magnetic field can be used to study the relaxation dynamics of the modulated
magnetic phases in different skyrmion host materials that include metallic-
FeGe H.Wilhelm , MnSi A.Bauer , Fe1-xCoxSi L.J.Bannenberg , Mn1-xFexSi
L.J.Bannenberg2 , insulating Cu2OSeO3I.Levatic ; F.Qian  and semiconducting
GaV4S8A.Butykai . Peak/hump anomalies in the field evolution of ac-
susceptibility that characterize the antiskyrmion phase in the tetragonal
Heusler Mn-Pt(Pd)-Sn were recently reported S.K.Jamal . However, detailed
studies are required to understand the magnetic relaxation and magnetic phase
transitions in such a system. To this end, we have undertaken an in-depth
study of high-precision ac-susceptibility data on the antiskyrmion hosts-
Mn1.4PtSn and Mn1.4Pt0.9Pd0.1Sn.

Figure 1: (Color online) Magnetization as a function of temperature, $M(T)$,
(open squares, left Y-axis) measured at $H$ = 1 kOe in field-cooled mode and
the temperature dependent real part of zero-field ac-susceptibility curves
$\chi^{\prime}(T)$ shown with open circles (right Y-axis) for (a) Mn1.4PtSn
and (b) Mn1.4Pt0.9Pd0.1Sn. Isothermal field dependent ac-susceptibility curves
$\chi(H)$ taken at different temperatures in the range 10 K $\leq T\leq$ 350 K
for (c) Mn1.4PtSn and (d) Mn1.4Pt0.9Pd0.1Sn. The evolution of antiskyrmion
phase is inferred from the broad minimum surrounded by maxima in $\chi(H)$
curves. The low field maxima is well seen by the occurrence of a peak,
whereas, the high field maxima is visualized by a speared out hump kind of
behavior marked by arrows.

##  II Methods

Polycrystalline ingots of Mn1.4PtSn and Mn1.4Pt0.9Pd0.1Sn were synthesized
using arc melting technique. Details of the sample preparation and structural
characterization are provided in the supplementary information Supp . Nearly
disk-shaped samples of Mn1.4PtSn and Mn1.4Pt0.9Pd0.1Sn with a mass of 80.30 mg
and 100.2 mg, respectively, were taken for the ac-susceptibility measurements
performed on a Quantum Design Physical Property Measurement System. At fixed
temperatures (T), ac-susceptibility ($\chi$) as a function of the magnetic
field (H) was measured at an r.m.s field ($H_{ac}$) of 10 Oe and frequencies
($f$) ranging from 11 Hz to 9999 Hz. For each measurement, the sample was
‘zero-field-cooled’ from 400 K ($\approx T_{C}$) to the desired temperature at
a rate of 10 K/min. The in-phase and out-of-phase components of the
oscillating magnetization were measured and normalized by the ac drive
amplitude ($H_{ac}$) to obtain the real ($\chi^{\prime}$) and imaginary
($\chi^{\prime\prime}$) parts of the ac-susceptibility, respectively. Note
that during the $\chi(H)$ scan, the magnetic field was stabilized at each step
in the linear approach mode to achieve high precision data.

##  III RESULTS AND DISCUSSION

Mn1.4PtSn exhibits a $T_{C}$ of about 400 K and a low-temperature spin-
reorientation transition [$T_{SR}$] of $\approx$ 160 K, as depicted in the
temperature dependent magnetization $M(T)$ data plotted in Fig. 1(a). The
onset of $T_{SR}$ brings about a sudden change in the magnetization in the
$M(T)$ data and a peak/dip kind of behavior in the temperature-dependent real
part of ac-susceptibility $\chi^{\prime}(T)$ curve [Fig. 1(a)]. In case of
Mn1.4Pt0.9Pd0.1Sn, the $T_{C}$ is found at a temperature of about 400 K and
the $T_{SR}$ at 130 K [Fig. 1(b)]. It has been reported that a change in the
magnetic structure below the $T_{SR}$ hinders the nucleation process of the
antiskyrmion phase A.K.Nayak . Similar inference has also been made from the
field-dependent ac-susceptibility and magnetic entropy change measurements in
the same materials S.K.Jamal . Since the antiskyrmion phase in the present
system is well-established between the $T_{SR}$ and the $T_{C}$, the present
work mostly focuses on the ac-susceptibility analysis in this particular
temperature range. Figure 1(c) and (d) show the field dependent ac-
susceptibility [$\chi(H)$] data measured at different temperatures starting
from 10 K to 350 K. The presence of noticeable peak/dip kind of anomalies in
the $\chi(H)$ curves suggest that the applied dc-magnetic fields gradually
transform the system into different magnetically ordered states. As it can be
seen, the $\chi(H)$ curve initially starts decreasing with increasing field
before exhibiting a maxima in the field range of 1 kOe - 1.5 kOe for Mn1.4PtSn
and 0.6 kOe - 1.3 kOe for Mn1.4Pt0.9Pd0.1Sn (marked by arrows in the low-field
regime). Further increasing the field gives rise to a broad hump like behavior
(marked by arrows in the intermidiate-field regime) in the $\chi(H)$ data
before decreasing monotonically at higher fields. This broad hump in the
$\chi(H)$ data is well pronounced at temperatures close to the $T_{C}$. The
corresponding peak/hump positions signify lower ($H_{AskX}^{L}$) and upper
($H_{AskX}^{H}$) critical fields that bound the antiskyrmion (AskX) phase. For
$H<H_{AskX}^{L}$, the magnetic state is characterized by a helical spin
modulation (H), while for $H>H_{AskX}^{H}$, it is a collinear/field-polarized
(FP) state. It can be noted here that no peak/hump kind of behavior is found
for the $\chi(H)$ curves measured at 100 K and 2 K, signifying the absence of
antiskyrmion phase in this temperature regime.

Figure 2: (Color online) In-phase [$\chi^{\prime}(H)$] and out-of-phase
[$\chi^{\prime\prime}(H)$] components of $\chi(H)$ at $T$ = 300 K with
$H_{ac}$ = 10 Oe measured at different frequencies for Mn1.4PtSn [panels (a)
and (b)] and Mn1.4Pt0.9Pd0.1Sn [panels (c) and (d)]. Note that for the sake of
clarity, the successive $\chi^{\prime}(H)$ data starting from 8888 Hz, have
been given a constant upward shift of 2.5 m emu/g-Oe. Vertical dashed lines
represent phase crossover from H-AskX and AskX-FP states inferred from the
d$\chi^{\prime}$/d$H$ versus H as shown as example in the inset of (a). Shaded
regions highlights the co-existence of multiple phases.

The field dependence of $\chi^{\prime}(H)$ and $\chi^{\prime\prime}(H)$ data
measured at 300 K in presence of different frequencies are plotted in Fig. 2.
As highlighted by the shaded regions, both $\chi^{\prime}$ and
$\chi^{\prime\prime}$ exhibit hump kind of behaviour at the phase transition.
Note that the magnitude of $\chi^{\prime}(H)$ is almost independent of the
frequency (f) of the $H_{ac}$, especially when $f\lesssim$ 500 Hz (see Fig.
S3) Supp . At low frequencies the $\chi^{\prime\prime}(H)$ response is so weak
that the data are statistically scattered and no information can be gained. It
is also found that the $\chi^{\prime\prime}(H)$ signal is of about two orders
of magnitude smaller than that of $\chi^{\prime}(H)$ and falls close to the
instrument resolution limit.

Figure 3: (Color online) (a) Frequency dependence of $\chi^{\prime}$ at
different representative magnetic fields ($H$) measured at $T$ = 300 K for
Mn1.4PtSn. The fits (solid lines) to the data (open symbols) are based on the
Eq. (1) described in the text. Magnetic field dependence of the fitting
parameters for Mn1.4PtSn (b) $\chi_{T}-\chi_{S}$, (c) $B=\chi_{S}$, (d) the
characteristic relaxation time ($\tau_{0}$), and (e) the relaxation times
distribution parameter, $\alpha$. Vertical dashed lines serve as guide to eye
to indicate the phase boundaries between H, AskX and FP phases. (f)
Temperature dependence of relaxation time $\tau_{0}$ derived from (d).
$\tau_{0}$ at H $\rightarrow$ AskX and AskX $\rightarrow$ FP regions are
inferred from the lower and upper maxima in the $\tau_{0}(H)$ plots, whereas
the $\tau_{0}(T)$ in the AskX phase is extracted from $\tau_{0}(H)$ at H = 2
kOe. Note that the continuous lines represent fits to the observed data (open
sqaure symbols) using Arrhenius law.

The $\chi^{\prime}(H)$ data at different frequencies over 786 Hz
$\leqslant~{}f~{}\leqslant$ 9999 Hz is re-plotted as $\chi^{\prime}(f)$ to
systematically investigate the frequency dependency of $\chi^{\prime}$ at
various static magnetic fields covering the helical, AskX and field-polarized
magnetic phases. First, we analyze the data for Mn1.4PtSn as shown in Fig.
3(a). The important observations are as follows:

(i) An inflection point in the $\chi^{\prime}(f)$ data indicates that the
relaxation in the antiskyrmion host-Mn1.4PtSn happens in a frequency window
covered in the present experiment.

(ii) In sharp contrast to the skyrmion hosts Fe1-xCoxSi L.J.Bannenberg ,
Cu2OSeO3 F.Qian , and GaV4S8 A.Butykai , where the entire magnetic relaxation
is observed in the frequency range over 0.1 Hz to 1 kHz, a frequency window
ranging from 0.5 kHz to 10 kHz is needed to study the magnetization dynamics
in the Mn1.4PtSn. A possible reason for the observed behavior could be
comparably high $T_{C}$ ($\approx$ 400 K) that signifies a large exchange
energy J in the present system. According to Heisenberg’s principle, higher
exchange energies leads to shorter switching times as
$\tau~{}\propto~{}\hslash/J$. Hence higher frequencies are needed in order to
probe the relaxation dynamics of the present system.

(iii) Noticeable changes are observed in the $\chi^{\prime}\left(f\right)$ as
one moves gradually from lower to higher magnetic fields. At any given
temperature, the inflection point in the $\chi^{\prime}(f)$ shifts to higher
frequencies with increasing fields [Fig. 3(a)].

We have attempted to fit the observed variation of
$\chi^{\prime}\left(f\right)$ and $\chi^{\prime\prime}\left(f\right)$ data
using the well-known Cole-Cole relation P.Debye ; Casimir ; Cole.Cole  as
given below,

$\chi^{\prime}(f)=\chi_{S}+\frac{A~{}\left[1+\left(2\pi
f\tau_{0}\right)^{1-\alpha}\sin\left(\dfrac{\pi\alpha}{2}\right)\right]}{1+2\left(2\pi
f\tau_{0}\right)^{1-\alpha}\sin\left(\dfrac{\pi\alpha}{2}\right)+\left(2\pi
f\tau_{0}\right)^{2\left({1-\alpha}\right)}}$ (1)
$\chi^{\prime\prime}(f)=\frac{A~{}\left[\left(2\pi
f\tau_{0}\right)^{1-\alpha}\cos\left(\dfrac{\pi\alpha}{2}\right)\right]}{1+2\left(2\pi
f\tau_{0}\right)^{1-\alpha}\sin\left(\dfrac{\pi\alpha}{2}\right)+\left(2\pi
f\tau_{0}\right)^{2\left({1-\alpha}\right)}}$ (2)

where A = $\chi_{T}-\chi_{S}$ with
$\chi_{S}\equiv\chi_{f\longrightarrow\infty}$,
$\chi_{T}\equiv\chi_{f\longrightarrow 0}$ represent adiabatic and isothermal
susceptibilities, $\tau_{0}$ is the average relaxation time, and ‘$\alpha$’
characterizes the width of the distribution in the relaxation times.

It can be clearly seen from Fig. 3(a) that the fits based on Eq. (1) best
represent the observed $\chi^{\prime}\left(f\right)$ data. The fit parameters
$A$, $B\equiv\chi_{S}$, $\tau_{0}$, and $\alpha$ are obtained at each field
step. The magnetic field variation of the fit parameters at different
temperatures are shown in Fig. 3 (b)-(e). The following conclusions can be
inferred from the field variation of the fit parameters based on the Cole-Cole
formalism.

(i) A non-monotonic variation is observed in the field dependency of $A$,
$B(\equiv\chi_{S})$, $\tau_{0}$, and $\alpha$. A broad hump kind of behaviour
in the field variations of the fit parameters at magnetic field H = $H_{1}$
[left dashed curves in Fig. 3(b)-(e)] and $H_{2}$ [right dashed curves in Fig.
3(b)-(e)] is found. These $H_{1}$ and $H_{2}$ are temperature sensitive and
show a good agreement with the $H_{AskX}^{L}$ and $H_{AskX}^{H}$ inferred from
the $\chi^{\prime}(H)$, $\chi^{\prime\prime}(H)$.

(ii) From figure 3(b), the parameter $\chi_{T}-\chi_{S}$ shows broad maxima at
H = $H_{1}$ and $H_{2}$. Higher values of ‘$A$’ highlight prominent magnetic
relaxation in the region of the phase coexistence (helical-to-AskX and AskX-
to-FP). Butykai et al. A.Butykai  attributed this kind of behaviour to the
defects in the magnetization profile arising from the incommensurate spin
modulations on the verge of magnetic phase transformation.

(iii) The presence of maxima in the $\tau_{0}(H)$ at H = $H_{1}$ and $H_{2}$
[see Fig. 3(d)] indicates slower magnetization dynamics in the vicinity of the
phase transformation that hints at a mixed phase of different magnetic
structures. $\tau_{0}$ at $H_{1}$ and $H_{2}$ is nearly 1.5 to 3 times higher
in comparison to the corresponding values in the pure magnetic phases. In case
of Fe1-xCoxSi L.J.Bannenberg , Cu2OSeO3 I.Levatic ; F.Qian  and GaV4S8
A.Butykai  the change in $\tau_{0}$ is found to be as large as in the order of
101 to 102. Slower relaxation phenomena characterized by higher relaxation
time in the vicinity of phase boundaries signifies possible origin of
irregularities in the spin-coordination between different magnetic textures.

(iv) The relaxation time distribution parameter, $\alpha$, takes very small
values in the range 0.05 - 0.15. Low values of $\alpha$ suggest that the
relaxation process of the configurations happens nearly in unison. In other
words, topologically protected magnetic antiskyrmions, whether in the long-
range ordered lattice (Askx phase) phase or in the disordered state (H+Askx
phase / FP+Askx phase), undergo almost simultaneous relaxation. Nevertheless,
$\alpha$ tends to rise at the phase crossover regions.

(v) Smaller values of $\tau_{0}\simeq$ 10-5 s, compared to that of 10-3 s
reported for the skyrmion host Cu2OSeO3, reflects faster relaxation in
Mn1.4PtSn, presumably due to the large spin-orbit coupling introduced by the
presence of heavy element ‘Pt’ in the material. In the absence of a
quantitative proof, the premise that the strong spin-orbit coupling due to
‘Pt’ is the cause for the faster relaxation times in the present system
remains a conjecture only.

(vi) Temperature variation of the characteristic relaxation times
corresponding to Helical to AskX and AskX to FP phase are plotted in the upper
and lower panels of Fig. 3(f). In general, a field of 2 kOe covers the
antiskyrmion phase in the whole range of temperatures in the present study.
Hence, the temperature dependence of the relaxation time at $H$ = 2 kOe is
shown in the central panel of the Fig. 3(f). In all the cases, the relaxation
behavior can be described using the well-known Arrhenius law $\tau_{0}\propto
exp\left(E/k_{B}T\right)$, where $E$ characterizes the energy barrier over
which the thermal activation takes place. The fits based on the above relation
are shown as continuous lines to the data presented in symbols. The fitted
values yield activation energies (in the units of temperature, $E/k_{B}$) of
626(30) K, 882(7) K, 226.5(7.0) K across the H-AskX, AskX, and AskX-FP phases,
respectively. A higher value of the energy barrier in the core-antiskyrmion
lattice region might have resulted from the strong topological protection.
Lower values of E at H-AskX, AskX-FP phase boundaries signals at metastable
antiskyrmions, being protected by somewhat lesser energy barriers.

Figure 4: (Color online) Field dependent fitting parameters for
Mn1.4Pt0.9Pd0.1Sn (a) $\chi_{T}-\chi_{S}$, (b) $B=\chi_{S}$, and (e) $\alpha$
and (d) $\tau_{0}$. Dashed lines serve as guide to eye.

While the observation of antiskyrmion phase in Mn1.4PtSn system was briefly
mentioned earlier, a complete microscopic study was reported in case of
Mn1.4Pt0.9Pd0.1Sn A.K.Nayak . To further support the present analysis using
the Cole-Cole formalism, we have carried out a similar measurement protocol in
case of Mn1.4Pt0.9Pd0.1Sn. The frequency dependence of
$\chi^{\prime}\left(f\right)$ and $\chi^{\prime\prime}\left(f\right)$ data are
fitted using the Cole-Cole relation as discussed earlier Supp . Various fit
parameters obtained from the $\chi^{\prime}\left(f\right)$ fitting are plotted
in Fig. 4. As it can be seen, all the parameters $A$, $B$, $\alpha$, and
$\tau_{0}$ display a well-defined peak at the low-field regime indicating the
nucleation of antiskyrmion phase with application of magnetic fields.
Similarly, a hump kind of behaviour is observed at higher fields for $A$, $B$
and $\tau_{0}$, and a clear peak nature is found in case of the relaxation
time distribution parameter, $\alpha$. In addition, all the fit parameters
fall in the same range as that is realized in case of Mn1.4PtSn. All these
findings further validate the present method of characterizing antiskyrmion
phase in the tetragonal Heusler materials.

Figure 5: (Color online) Frequency dependence of $\chi^{\prime\prime}$ at
various representative magnetic fields measured at $T$ = 300 K for (a)
Mn1.4PtSn and (b) Mn1.4Pt0.9Pd0.1Sn. Solid balls represent experimental date
points, whereas, the lines indicate fitting to the data using Eq. 2.

It is well-known that the out-of-phase component of the ac-susceptibility in
the Debye model like relaxation shows a peak at a characteristic frequency
($f_{0}=1/\tau_{0}$). However, in the present case the
$\chi^{\prime\prime}(f)$ data show a shoulder nearly at $f$ where
$\chi^{\prime}(f)$ goes through an inflection point as shown in Fig. 5. A
complete range of $\chi^{\prime\prime}(f)$ data at different temperatures and
magnetic fields can be found in the supplementary information Supp . We have
attempted to fit the $\chi^{\prime\prime}\left(f\right)$ data using Eq. 2 as
depicted by solid lines in Fig. 5. We use the fit parameters obtained from the
$\chi^{\prime}(f)$ analysis as inputs for the fitting of
$\chi^{\prime\prime}(f)$ data. Then the parameters were relaxed to obtain the
best fitting. However, we could not fit the data in the whole frequency range
owing to the lack of complete peak kind of behaviour, in the case of both
Mn1.4PtSn and Mn1.4Pt0.9Pd0.1Sn. A good match between the experimental data
and the fitted curve is obtained only up to the frequency where
$\chi^{\prime\prime}(f)$ shows a shoulder like behavior. This is true for all
the temperatures and fields covered in the present study Supp . The fit
parameters obtained from the $\chi^{\prime\prime}(f)$ analysis fall in the
same order of magnitude as that found from the $\chi^{\prime}(f)$ fitting. A
comparison of fit parameters acquired from the $\chi^{\prime}(f)$ and
$\chi^{\prime\prime}(f)$ fittings at 300 K for both Mn1.4PtSn and
Mn1.4Pt0.9Pd0.1Sn is presented in Table I. Interestingly, both the samples
exhibit similar magnitude of the respective fit parameters, suggesting
presence of similar relaxation dynamics in the present systems. This can be
understand from the fact that both the samples display identical magnetic
ordering with almost equal $T_{C}$.

Table 1:  Comparison between fitted parameters of $\chi^{\prime}$ and
$\chi^{\prime\prime}$ at 300 K.

H(Oe) | A($\times$ 10-5 emu/g-Oe) |  $\tau_{0}$( $\times$ 10-5 s) | $\alpha$
---|---|---|---
$\chi^{\prime}$ | $\chi^{\prime\prime}$ | $\chi^{\prime}$ | $\chi^{\prime\prime}$ | $\chi^{\prime}$ | $\chi^{\prime\prime}$
0 a | 12.220  | 149.000 | 5.070  | 3.500 | 0.042  | 0.092
b | 54.193 | 122.358 | 1.536 | 6.623 | 0.081 | 0.080
500 a | 7.379 | 27.071 | 6.943 | 3.060 | 0.057 | 0.045
b | 15.122 | 40.136 | 10.979 | 3.501 | 0.103 | 0.083
1000 a | 8.051 | 36.280 | 8.287 | 1.930 | 0.048 | 0.080
b | 87.357 | 26.957 | 57.727 | 2.359 | 0.130 | 0.082
1500 a | 7.340 | 25.208 | 8.671 | 1.682 | 0.010 | 0.128
b | 10.476 | 12.345 | 1.480 | 1.214 | 0.086 | 0.081
2000 a | 3.784 | 17.365 | 5.198 | 1.185 | 0.036 | 0.034
b | 5.337 | 10.251 | 1.944 | 0.946 | 0.107 | 0.080
2500 a | 1.873 | 13.680 | 3.880 | 1.006 | 0.020 | 0.051
b | 3.998 | 8.724 | 2.516 | 0.872 | 0.104 | 0.083
3000 a | 2.761 | 10.721 | 5.437 | 1.048 | 0.040 | 0.051
b | 7.762 | 8.969 | 6.971 | 0.989 | 0.121 | 0.082
3500 a | 2.863 | 9.767 | 4.567 | 1.301 | 0.090 | 0.040
b | 3.803 | 8.113 | 4.383 | 0.824 | 0.092 | 0.088
4000 a | 2.835 | 7.383 | 3.389 | 1.222 | 0.020 | 0.070
b | 2.224 | 6.138 | 1.314 | 0.805 | 0.084 | 0.085
5000 a | 2.584 | 6.873 | 0.628 | 1.067 | 0.010 | 0.020
b | 1.253 | 6.297 | 1.470 | 0.854 | 0.082 | 0.084

aMn1.4PtSn, bMn1.4Pt0.9Pd0.1Sn
---

The deviation of the present $\chi^{\prime\prime}(f)$ data from the Debye
model like relaxation could stem from the eddy current losses that adversely
affect the susceptibility signals, particularly at higher frequencies and
higher dc fields. In general, the voltage induced in the pick-up coil by the
eddy currents generated in the sample is much smaller when compared to the
main voltage induced from the sample in response to the applied ac magnetic
field. As a result, the eddy current effect can be clearly seen when the
actual signal is very weak Kraftmakher . In the present case, the eddy current
effect is predominant in case of $\chi^{\prime\prime}(f)$ as the signal of
magnitude $\sim 10^{-5}-10^{-6}$ emu/Oe falls in the sensitivity limit of the
measuring instrumen. Similar magnitude of the eddy current effect of the order
$\sim 10^{-7}$$m^{3}$/mol or $\sim 10^{-5}$ emu/Oe is also found in the
$\chi^{\prime\prime}(f)$ data at 1 kHz frequency in case of the semiconducting
skyrmion host Fe1-xCoxSi L.J.Bannenberg . The eddy current effect in the
previously studied skyrmion hosting materials could not be seen prominantly
due to the fact that the magnetic relaxation in these systems happens in low
frequency regime spanning up to 1 kHz, whereas, the present antiskyrmion
materials require frequencies up to 10 kHz to study the ralaxation process.

Table 2: Comparison of relaxation times $\tau_{0}$ in different magnetic systems. | Magnetic system
---
$\tau_{0}$ (s)
Canonical/Cluster spin glassesJ.A.Mydosh  | 10-12 \- 10-8
SuperparamagnetsS.J.Blundell2  | 10-11 \- 10-9
Single molecular magentsL.Thomas  | 10-6
Skyrmion-Cu2OSeO3F.Qian (50 $\pm$ 10 nm) | 10-3 \- 10-2
Antiskyrmion-Mn1.4Pt(Pd)Sn (150 $\pm$ 20 nm) | 10-5 \- 10-4

It is customary here to compare various magnetic systems that display non-
trivial and finite-sized magnetic configurations such as spin glasses,
superparamagnetic particles, single molecular magnets, and skyrmion lattices
that exhibit slow magnetic relaxations and characterized by Debye relation.
Table 2 compares the typical relaxation times in such systems. In case of spin
glasses, the reorientation of individual magnetic moments or clusters during
the relaxation process causes additional frustrations. This result in further
rearrangements of the local spins in the system leading to slower
magnetization dynamics. In superparamagnets and single molecular magnets, slow
magnetic relaxation arises from the thermally-activated hopping of the
spins(macro) over energy barriers dictated by the uniaxial anisotropy which
separates parallel and anti-parallel configurations. A phase crossover from
helical spin modulations to skyrmion/antiskyrmion lattice associates with
topological point defects X.Z.Yu . It is reported that the reorientation of
large magnetic entities, such as modulations in the long-wavelength spin
helices, nucleation of skyrmion cores, and glassy nature of the skyrmion
phases Rajeswari  leads to slower magnetization dynamics in different skyrmion
hosts A.Bauer ; L.J.Bannenberg ; I.Levatic ; F.Qian ; A.Butykai . Similar
phenomena are also expected in the present antiskyrmion host Mn-Pt(Pd)-Sn
materials. Higher relaxation time ($\tau_{0}\approx 10^{-3}-10^{-5}$ s), i.e.,
slower relaxation process in the skyrmion/antiskyrmion systems when compared
to others signifies slower magnetic damping process, which suppresses the rate
at which an equilibrium configuration can be restored. In this regard,
resonance studies of the magnetization dynamics can help to understand the
stability and reliability of future skyrmion/antiskyrmion-based spintronics.

Figure 6: (Color online) $H-T$ 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 Mn1.4Pt0.9Pd0.1Sn. The inset shows
$H-T$ phase diagram for Mn1.4PtSn. Open and closed symbols represent the lower
and upper boundaries of antiskyrmion phase, respectively. $H_{L}^{ACS}$ and
$H_{H}^{ACS}$ show the lower and upper phase boundaries of the antiskyrmion
phase, respectively, obtained from the peak/hump anomalies in the field-
evaluation of the ac susceptibility measurements. Similarly, $H_{L}^{Cole-
Cole}$ and $H_{H}^{Cole-Cole}$ are lower and upper phase boundaries of the
antiskyrmion phase obtained from the field-dependent fit parameters of the
Cole-Cole expression using $\chi^{\prime}(H)$ data.

$H$-$T$ phase diagrams are established from the above-discussed features of
$\chi^{\prime}(H)$ and the field-dependent fit parameters of the Cole-Cole
relation on the polycrystalline samples Mn1.4Pt0.9Pd0.1Sn and Mn1.4PtSn as
shown in Fig. 6. It may be noted here that the $H_{L}^{ACS}$($H_{L}^{Cole-
Cole}$) and $H_{H}^{ACS}$($H_{H}^{Cole-Cole}$) are the lower and upper phase
boundaries of the antiskyrmion phase which describe the crossover from the
helical state to the antiskyrmion region and the antiskyrmion region to the
field-polarized state, respectively. The phase boundaries obtained from the
Cole-Cole analysis in the present system broadly matches with that estimated
from the magnetic entropy study S.K.Jamal  and the Lorentz transmission
electron microscopy imaging (LTEM) technique A.K.Nayak . The discrepancy seen
in the variation of the upper critical fields at low temperature regime mostly
originates from the non-identical orientations of the crystallites in the
polycrystalline samples used for different measurements. This is due to the
fact that the angle between the applied magnetic field and the
crystallographic orientation of the crystallites determines the critical field
required for the stabilization of antiskyrmions A.K.Nayak . Hence, depending
upon the orientation of the crystallites the critical field for the nucleation
of antiskyrmions may vary for different pieces cut from the same sample. The
discrepancy found for the lower critical fields in the present study to that
of LTEM investigation A.K.Nayak  can be understood as follows. In the phase
diagram inferred from the LTEM studies, typically magnetic field was applied
at room temperature to stabilize the antiskyrmion lattice and then ‘field-
cooled’ the sample to the required temperatures. Subsequently, the magnetic
field was slowly varied in different steps, i.e., either reduced to zero or
increased to obtain a field-polarized state and simultaneously monitored the
emergence of different magnetic phases. Whereas, in the present case, the
sample was ‘zero-field-cooled’ from 400 K to the required temperature and the
magnetic field strength is continuously raised starting from a zero value to
see the field evolution of helical, antiskyrmion and field-polarized phases.
In addition, the Mn-Pt(Pd)-Sn samples used in the LTEM study A.K.Nayak  are
[001] oriented 100 nm thin-crystalline films prepared by the Focused Ion Beam
(FIB) technique. Whereas, the present ac-susceptibility measurements are
carried out on the polycrystalline bulk samples. Therefore, it is expected
that the phase diagram for the bulk polycrystalline samples may differ
slightly in comparison to the single grain thin-crystalline-film sample.

##  IV CONCLUSION

In summary, we have presented a detailed study on the frequency-dependent ac-
susceptibility measurements on the $D_{2d}$ symmetry based tetragonal Heusler
system that hosts antiskyrmion lattice over a wide temperature range. The
magnetic relaxation follows a Debye relation with nearly negligible
distribution in the relaxation times. At the boundaries between the different
phases via helical, antiskyrmion, and field-polarized states, the
characteristic relaxation time, the isothermal and adiabatic susceptibilities
show a non-monotonic variation. Maxima in the relaxation times that develop in
the vicinity of the phase crossover indicate slower magnetization dynamics.
Temperature-dependent relaxation times across different magnetic phases follow
an Arrhenius kind of slowing down with reasonable values of the energy
barriers. The observation of higher relaxation times in comparison to other
particulate storage mediums such as super-paramagnets, and single molecular
magnets signifies slower damping process in the present antiskyrmion host Mn-
Pt(Pd)-Sn system.

###### Acknowledgements.

This work was financially supported by the Science and Engineering Research
Board (SERB) under research grant (No. ECR/2017/000845), Department of Science
and Technology (DST)-Ramanujan research grant (No. SB/S2/RJN-081/2016) and
Nanomission research grant [SR/NM/NS-1036/2017(G)] of the Government of India.
AKN acknowledges the Max Plank Society for support under the Max Plank-India
partner group project. P.V.P.M. and S.S. contributed equally to this work.

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##  V SUPPLEMENTARY INFORMATION

###  V.1 Experimental methods: Sample preparation and Structural
Characterization

Polycrystalline ingots of Mn1.4PtSn, Mn1.4Pt0.9Pd0.1Sn were prepared by arc
melting technique. The constituent elements were taken according to their
stoichiometric ratio and melted together into an ingot (under high pure argon
atmosphere) inside the Arc-melting furnace. For better homogeneity, the ingots
were melted several times by flipping them upside down. Because of volatile
nature of manganese, some extra amount of Mn was added to compensate the
weight loss during melting. As the ingots are practically quenched after the
melting, annealing of arc melted ingots is mandatory for better homogeneity
and phase formation. Hence the arc-melted ingots were sealed inside quartz
ampoule under 10-5 mbar pressure. The quartz tube containing the sample was
kept in PID-controlled resistive furnace for heat treatment at 1073 K for 7
days. Subsequently, the samples were quenched in ice-water mixture. Structural
characterization and phase purity conformation of the annealed sample were
characterized by the x-ray powder diffraction (XRD) using a Cu-$K_{\alpha}$
source.

Room-temperature x-ray diffraction pattern along with the Rietveld refinement
fit is shown in Fig.7. It is observed that Mn1.4PtSn crystallizes to a non-
centrosymmetric tetragonal lattice with space group I$\bar{4}$ 2m (space group
no: 121). The lattice parameters obtained from the Rietveld analysis are $a$ =
$b$ = 6.3478 $\pm$ 0.0005 Å and $c$ = 12.2006 $\pm$ 0.0007 Å with $c/a$ =
1.9220. The sample is polycrystalline in nature with crystallite sizes of the
order of several tens of microns. For this reason, when we tried to find out
the average crystallite size from the Rietveld refinement using the instrument
resolution file (IRF), we obtained negligible size broadening.

Figure 7: (Color online) Room temperature powder XRD pattern for Mn1.4PtSn.

Compositional homogeneity of the MnPtSn sample was confirmed by Energy-
dispersive x-ray spectroscopy (EDX) analysis (See figure 8). The EDX analysis
yields a composition of 41.14 $\pm$ 1.50 $at.\%$ for Mn, 28.36 $\pm$ 1.00
$at.\%$ for Pt and 30.50 $\pm$ 1.00 $at.\%$ for Sn, finally resulting a
composition of Mn1.4PtSn.

Figure 8: (Color online) EDX spectrums measured on Mn1.4PtSn sample.

###  V.2 AC susceptibility at frequencies in the range 11 Hz $\leq$ f $\leq$
666 Hz

Fig.9 testifies that the $\chi^{\prime}$(H, T = 300 K) and
$\chi^{\prime\prime}$(H, T = 300 K) at frequencies ($\leq$ 500 Hz) are almost
independent of frequency of the ac signal. This is also true at other
temperatures. Note that, the $\chi^{\prime\prime}$ signal is nearly an order
of magnitude smaller when compared to that of at higher frequencies and falls
in the instrument resolution limit, leading to the scattering in the data.

Figure 9: (Color online) Field-dependent (a) Real part, $\chi^{\prime}$ and
(b) imaginary part, $\chi^{\prime\prime}$ of the susceptibilities at various
ac-field frequencies in the range 11 Hz $-$ 666 Hz measured at 300 K.

###  V.3 Theoretical considerations

When probing a system with an oscillating magnetic field of certain frequency
which is comparable to the characteristic frequency (equivalently timescale)
of the magnetic relaxation of the system, then there will be some phase lag
and hence dissipation. In such a case the total response ($\chi$) can be
conveniently expressed by the sum of in-phase, $\chi^{\prime}$(real) and out-
of-phase, $\chi^{\prime\prime}$ (imaginary) components as

$\chi=\chi^{\prime}+i\chi^{\prime\prime}$ (3)

Where the $\chi^{\prime}$ reflects the sensitivity of a material to the
applied field, also known as reversible magnetization, $\chi^{\prime\prime}$
represents the dissipation of absorbed energy from the high frequency ac field
i.e., irreversible magnetization A.H.Morrish . This irreversibility results
from the relaxation processes of various origin such as irreversible movement
of domain walls, hysteresis loss in ferromagnets, magnetic phase transition,
spin reorientation, spin-spin relaxation, spin-lattice relaxation to name a
few. Since relaxation is very much sensitive to the magnetic phases, study of
relaxation provides an insight into the magnetic structures, spin-spin and
spin-lattice interactions in a material A.H.Morrish . By considering analogy
to the well-known Debye model of dielectric relaxation P.Debye , Casimir and
du Pré Casimir  derived a thermodynamic model of relaxation for a magnetic
system according to which the expression for complex susceptibility is given
by,

$\chi(\omega)=\chi_{S}+\frac{\chi_{T}-\chi_{S}}{1+(i\omega\tau_{0})}$ (4)

Where $\chi_{T}\equiv\chi(\omega\longrightarrow 0)$ represents isothermal
susceptibility in the limit of the lowest frequencies,
$\chi_{S}\equiv\chi(\omega\longrightarrow\infty)$ is the adiabatic
susceptibility in the limit of the highest frequency for which the spin system
remains isolated from surroundings and $\tau_{0}$ is the characteristic
relaxation time constant and $\omega=2\pi f$. The above Eq. (4) is called the
‘Debye relation’. The real ($\chi^{\prime}$) and imaginary
($\chi^{\prime\prime}$) components of the ac susceptibility from Eq. (4) can
be written as,

$\displaystyle\chi^{\prime}(\omega)$
$\displaystyle=\chi_{S}+\frac{\chi_{T}-\chi_{S}}{1+(\omega^{2}\tau_{0}^{2})}$
(5a) $\displaystyle\chi^{\prime\prime}(\omega)$
$\displaystyle=(\chi_{T}-\chi_{S})+\frac{\omega\tau}{1+(\omega^{2}\tau_{0}^{2})}$
(5b)

A frequency dependent $\chi^{\prime}$ and a non-zero value of
$\chi^{\prime\prime}$ are always consequence of one or more relaxation
processes with a characteristic relaxation time constant ($\tau_{0}$). Eq. (4)
represents the exponential time dependence of the relaxation with the
assumption that the magnetization is a slowly relaxing ‘entity’ with a single
time scale. But when the entities interact with each other it can produce
clustering effects which lead to a distribution in the relaxation times. When
a single relaxation no longer governs the system dynamics, interactions in the
spin systems and presence of cooperative effects leads to a spread in the
relaxation times. This is often accounted for by introducing a
phenomenological parameter ‘$\alpha$’ in the Eqn. (4) as,

$\chi(\omega)=\chi_{S}+\frac{\chi_{T}-\chi_{S}}{1+(i\omega\tau_{0})^{1-\alpha}}$
(6)

The resulting expression represents a generalized Debye model, also known as
Cole-Cole relation Cole.Cole . A zero value of ‘$\alpha$’ signifies a single
relaxation process, while close to unity represents an infinitely broad
distribution of relaxation times. The frequency dependence of the real and
imaginary components from Eq.(6) can be explicitly written as

$\displaystyle\chi^{\prime}(f)$
$\displaystyle=\chi_{S}+\frac{A~{}\left[1+\left(2\pi
f\tau_{0}\right)^{1-\alpha}\sin\left(\dfrac{\pi\alpha}{2}\right)\right]}{1+2\left(2\pi
f\tau_{0}\right)^{1-\alpha}\sin\left(\dfrac{\pi\alpha}{2}\right)+\left(2\pi
f\tau_{0}\right)^{2\left({1-\alpha}\right)}}$ (7a)
$\displaystyle\chi^{\prime\prime}(f)$
$\displaystyle=\frac{A~{}\left[\left(2\pi
f\tau_{0}\right)^{1-\alpha}\cos\left(\dfrac{\pi\alpha}{2}\right)\right]}{1+2\left(2\pi
f\tau_{0}\right)^{1-\alpha}\sin\left(\dfrac{\pi\alpha}{2}\right)+\left(2\pi
f\tau_{0}\right)^{2\left({1-\alpha}\right)}}$ (7b)

Where A = ($\chi_{T}$ \- $\chi_{S}$ ) and $\tau_{0}$ is the average relaxation
time. Fits have been attempted to the $\chi^{\prime}(f)$ data at different but
fixed magnetic fields over the temperature range 200 K-350 K, based on the
eq.(7a) keeping the A, $\chi_{S}$, $\tau_{0}$, $\alpha$ as free fitting
parameters. The fits (continuous lines) best represent the measured data
(symbols) as shown in the figures 10-13 for Mn1.4PtSn and 14-17 for
Mn1.4Pt0.9Pd0.1Sn. Similarly, imaginary part of ac susceptibility data
[$\chi^{\prime\prime}(f)$] as a function frequency is fitted using the eq.(7b)
by taking the same values of parameter as initial values that are used to fit
$\chi^{\prime}(f)$ data (as discussed in the main texts) as shown in figures
18-21 for Mn1.4PtSn and 22-25 for Mn1.4Pt0.9Pd0.1Sn.

Figure 10: (Color online) Frequency dependence of $\chi^{\prime}$ at various
representative magnetic fields (H) in the case of T = 200 K for Mn1.4PtSn. The
fits (solid lines) to the data (open symbols) are based on the Eq.(7a)
described in the text. Figure 11: (Color online) Frequency dependence of
$\chi^{\prime}$ at various representative magnetic fields (H) in the case of T
= 250 K for Mn1.4PtSn. The fits (solid lines) to the data (open symbols) are
based on the Eq.(7a) described in the text. Figure 12: (Color online)
Frequency dependence of $\chi^{\prime}$ at various representative magnetic
fields (H) in the case of T = 300 K for Mn1.4PtSn. The fits (solid lines) to
the data (open symbols) are based on the Eq.(7a) described in the text. Figure
13: (Color online) Frequency dependence of $\chi^{\prime}$ at various
representative magnetic fields (H) in the case of T = 350 K for Mn1.4PtSn. The
fits (solid lines) to the data (open symbols) are based on the Eq.(7a)
described in the text. Figure 14: (Color online) Frequency dependence of
$\chi^{\prime}$ at various representative magnetic fields (H) in the case of T
= 200 K for Mn1.4Pt0.9Pd0.1Sn. The fits (solid lines) to the data (open
symbols) are based on the Eq.(7a) described in the text. Figure 15: (Color
online) Frequency dependence of $\chi^{\prime}$ at various representative
magnetic fields (H) in the case of T = 250 K for Mn1.4Pt0.9Pd0.1Sn. The fits
(solid lines) to the data (open symbols) are based on the Eq.(7a) described in
the text. Figure 16: (Color online) Frequency dependence of $\chi^{\prime}$ at
various representative magnetic fields (H) in the case of T = 300 K for
Mn1.4Pt0.9Pd0.1Sn. The fits (solid lines) to the data (open symbols) are based
on the Eq.(7a) described in the text. Figure 17: (Color online) Frequency
dependence of $\chi^{\prime}$ at various representative magnetic fields (H) in
the case of T = 350 K for Mn1.4Pt0.9Pd0.1Sn. The fits (solid lines) to the
data (open symbols) are based on the Eq.(7a) described in the text. Figure 18:
(Color online) Frequency dependence of $\chi^{\prime\prime}$ at various
representative magnetic fields (H) in the case of T = 200 K for Mn1.4PtSn. The
fits (solid lines) to the data (open symbols) are based on the Eq.(7b)
described in the text. Figure 19: (Color online) Frequency dependence of
$\chi^{\prime\prime}$ at various representative magnetic fields (H) in the
case of T = 250 K for Mn1.4PtSn. The fits (solid lines) to the data (open
symbols) are based on the Eq.(7b) described in the text. Figure 20: (Color
online) Frequency dependence of $\chi^{\prime\prime}$ at various
representative magnetic fields (H) in the case of T = 300 K for Mn1.4PtSn. The
fits (solid lines) to the data (open symbols) are based on the Eq.(7b)
described in the text. Figure 21: (Color online) Frequency dependence of
$\chi^{\prime\prime}$ at various representative magnetic fields (H) in the
case of T = 350 K for Mn1.4PtSn. The fits (solid lines) to the data (open
symbols) are based on the Eq.(7b) described in the text. Figure 22: (Color
online) Frequency dependence of $\chi^{\prime\prime}$ at various
representative magnetic fields (H) in the case of T = 200 K for
Mn1.4Pt0.9Pd0.1Sn. The fits (solid lines) to the data (open symbols) are based
on the Eq.(7b) described in the text. Figure 23: (Color online) Frequency
dependence of $\chi^{\prime\prime}$ at various representative magnetic fields
(H) in the case of T = 250 K for Mn1.4Pt0.9Pd0.1Sn. The fits (solid lines) to
the data (open symbols) are based on the Eq.(7b) described in the text. Figure
24: (Color online) Frequency dependence of $\chi^{\prime\prime}$ at various
representative magnetic fields (H) in the case of T = 300 K for
Mn1.4Pt0.9Pd0.1Sn. The fits (solid lines) to the data (open symbols) are based
on the Eq.(7b) described in the text. Figure 25: (Color online) Frequency
dependence of $\chi^{\prime\prime}$ at various representative magnetic fields
(H) in the case of T = 350 K for Mn1.4Pt0.9Pd0.1Sn. The fits (solid lines) to
the data (open symbols) are based on the Eq.(7b) described in the text.