Topology-guided rotational dynamics of magnetic soliton assemblies under pulsed current

Domain: physics
Published: 2026-04-07
Editorial outcome: major_revision

Abstract
Emergent topological spin textures, such as nanometric skyrmions and antiskyrmions, not
only exhibit a wealth of novel physical phenomena but also represent promising candidates for
next-generation spintronic devices due to their topological stability and low-current-driven
dynamics. Investigating their responses to electric currents is essential for uncovering unique
electromagnetic properties and advancing their integration into electronic technologies. While
considerable progress has been made through extensive research over the past decade, key
challenges still persist. In this study, we demonstrate the novel, current-driven oppositely rotating
dynamics of skyrmion and antiskyrmion assemblies in terms of Lorentz transmission electron
microscopy. Their senses of rotation are strongly dictated by their inherent topological charges and
remain largely unaffected by current direction, showing consistent behavior observed across
various sample geometries. Further experimental analyses have revealed a reduction in angular
velocity as (anti)skyrmions move away from the sample edge. The theoretical analyses, combined
with micromagnetic simulations, highlight the critical roles of boundary-induced confining
potentials and gyrotropic forces in steering the rotational dynamics of (anti)skyrmions. These
findings offer new insights into current-driven (anti)skyrmion dynamics and open promising
avenues for advancing topological concepts in confined spintronic systems.

Introduction
Magnetic skyrmions, characterized by their vortex-like noncollinear spin configurations,
occupy an important position in contemporarNy condensed matter physics due to their unique
topological properties and promising funct ional applications1,2. Skyrmions are found in diverse
magnetic systems and exhibit sizes raLnging from nanometers to micrometers3-5. Their topological
charge Q is defined as Q = 1I ∬m(r)∙(∂ m(r)×∂ m(r))d2r, where m(r) is the unit vector of
4π x y
magnetization. The Q typically takes integer values of 1 for skyrmions, with the sign set by the
core polarity1,2. Investigating the dynamics of skyrmions is essential for gaining insight into their
intricate topological entities and extending their applicability6,7. The dynamics of skyrmions are
driven by various external stimuli, including magnetic fields8, electric/thermal currents9-14,
magnons15,16 and acoustic waves17. Among these, current-induced dynamics are particularly well-
regarded and have been intensively studied due to their operational simplicity and superior energy
efficiency. The dynamics of skyrmions under current flows can be described by the Thiele
equation18-20, in which the gyrotropic vector G = 4πQẑ originates from the topological properties
of skyrmions. As a consequence, current-driven skyrmions acquire a finite transverse velocity
component, known as the skyrmion Hall effect1,18-20. This intriguing phenomenon has been
observed in both multilayer systems12,13 and bulk samples14,21, emerging as a defining feature of
skyrmions.

In realistic systems, skyrmions inevitably experience boundary-induced confinement,
which can strongly modify their dynamics under electric current flows. The interplay between
gyrotropic forces and confining potentials leads to rich transport behaviors distinct from those in
unbounded media22-23, highlighting the importance of confinement effects for both fundamental
studies and technological implementations.

In contrast to the extensive progress made in theR study of skyrmions, the investigations of
antiskyrmions remain relatively limited. Antiskyrm ions are characterized by a topological charge
of opposite sign24 to that of skyrmions Iand have been demonstrated to form in non-
centrosymmetric magnetic systems with D or S symmetry25,26, where anisotropic
2d 4
Dzyaloshinskii-Moriya interactions (DMI) give rise to complex spin textures27-29. Antiskyrmions
have been observed in Heusler compounds27,30 and schreibersite-type materials, including
(Fe₀.₆₃Ni₀.₃Pd₀.₀₇)₃P (F A NPP)31-33, as well as in multilayer films34,35. While theoretical models
predict that the dynamics of antiskyrmions should mirror those of skyrmions with reversed
trajectories due to the sign reversal of the topological charge36, experimental validation is still
scarce. To date, only experimentally confirmed current-driven motion of antiskyrmions is their
constrained sliding along stripe domains33,37, offering limited insight into dynamics guided by
topological properties. More complex behaviors—such as rotational motion—have neither been
experimentally demonstrated nor systematically explored. This lack of experimental insight leaves
a critical gap in understanding the topological properties of antiskyrmion dynamics, underscoring
the need for further investigation.

In this study, through in-situ Lorentz transmission electron microscopy (L-TEM), we have
identified novel current-induced rotations in skyrmion and antiskyrmion assemblies confined
within finite geometries. By altering the direction of the pulse current, we have demonstrated that
the consistent rotational dynamics of the assemblies are guided by their intrinsic topological
charges, independent of the current-flow orientation. The first experimental observation of current-
driven motion in antiskyrmion assemblies also validates the dynamic manifestation of its opposite
topology with skyrmions. Additionally, experimental observations in samples with different
geometries have confirmed that the rotation direction of (anti)skyrmions is independent of sample
geometry. In the circular shaped sample, the analyses Rof angular velocity across varying radial
distances have revealed a decrease in skyrmion velo city as they approach the center of the sample,
highlighting the significant impact of the bIoundary potential on rotational dynamics. Further
validation through micromagnetic simulations aligns with theoretical analyses, underscoring the
critical role of both the boundary and the gyrotropic force in governing skyrmion motion.

Results
Unidirectional rotatAion of (anti)skyrmion assemblies driven by pulse currents
Here, to highlight the differences in in-plane magnetization configurations between
skyrmions and antiskyrmions, we describe skyrmion as vortex-like and antiskyrmion as
antivortex-like magnetization textures, extending beyond the definition solely on topological
charge. A comprehensive classification and detailed depiction of these unique spin configurations
are provided in Supplementary Fig. S1.

The in-situ L-TEM experiments were performed on FNPP samples. Figure 1a shows the
schematic of the rectangular sample we prepared via a dual-beam system composed of a Focused
Ion Beam (FIB) and Scanning Electron Microscopy (SEM) as detailed in the Methods section. The
view plane for L-TEM observation is the (001) and its edges aligned along the <110> to coincide
with the wavevectors of stripy structures33. The L-TEM image listed in Fig. 1b shows the diverse
topological spin textures under an applied magnetic field along −z direction in the fabricated
sample. Both rectangular antiskyrmions and elliptic skyrmions with opposite helicities31-33 can be
stabilized, attributed to the anisotropic DMI in FNPP. The transport-of-intensity equation (TIE)
analyses in Fig. 1c point out the configurations of the in-plane components of each spin textures,
consistent with the previous results28, 31-33.

Figure 1d and e represent schematic diagrams summarizing the in-situ measurements.
The skyrmion assemblies (Fig. 1d) or antiskyrmion assemRblies (Fig. 1e) are respectively stabilized
under the external field. As defined in Supplementa ry Note1, when the magnetic field B is applied
along the −z axis, the created skyrmion assemIblies possess a topological charge of Q = +1, while
the antiskyrmion assemblies possess Q = −1. Both skyrmion and antiskyrmion assemblies display
rotational dynamics under pulsed currents, yet rotating senses are opposite: counterclockwise
(CCW) for skyrmions (Fig. 1d), while clockwise (CW) for antiskyrmions (Fig. 1e) when the
external field is kept along the −z axis. Importantly, the current flows, whether along the +x or −x
axis, do not affect the direction of skyrmion/antiskyrmion rotations, highlighting that the senses of
rotation are highly related to the topological charge. Additionally, as shown in the schematics
(Fig.1), (anti)skyrmions located near the sample center show slower motions than those nearer the
sample boundary, illustrating the significant impact of boundary effects on their rotational
dynamics.

Sequential L-TEM observations of rotational motions of skyrmion assemblies
Figure 2 provides detailed visuals and descriptions according to the in-situ L-TEM
observations under current flows. Both the influence of topological charge and current polarity are
examined in this part. Figure 2a presents the L-TEM image of a stabilized antiskyrmion assembly
under a magnetic field of 518 mT along −z axis. Three representative antiskyrmions are highlighted
using arrows of different colors, and their initial positions are marked with dashed boxes to
facilitate motion tracking. The antiskyrmion assembly is robust and reproducible without
spontaneous annihilation or significant rearrangement during L-TEM imaging, indicating that the
ambient thermal fluctuations alone are insufficient to induce any observable dynamics such as
Brownian motion38,39 in the present experimental setup. Upon application of multiple pulsed
current with 20-ns width along +x direction, the antiskyrmion assembly exhibits a CW rotation
(Fig. 2a−c, also see in Supplementary video 1). Our expeRrimental results mark the first documented
instance of antiskyrmion rotation induced by pu lsed currents, establishing a novel mode for
antiskyrmion’s dynamic physics.

By adjusting the sample tilting conditions and magnetic field strength to 523 mT, a
skyrmion assembly with Q = +1 is also stabilized (Fig. 2d−f). To facilitate clearer tracking the
assembly, two skyrmions with opposite contrasts are deliberately marked in black and orange, as
shown in Fig. 2d. These skyrmions exhibit a CCW rotation under identical pulsed current
excitation and complete one full loop after several pulses (also see in Supplementary video 2),
clearly demonstrating dynamic behavior opposite to that of antiskyrmions. Remarkably, the
rotation of the skyrmion assembly remains CCW despite the current reversal (Fig. 2g−i, also see
in Supplementary video 3). In earlier studies, geometric confinement was exploited to cause the
skyrmion Hall motion into arc-shaped trajectories, and the sense of rotation was inferred from
multiple arc segments obtained by reversing the current polarity40. In contrast, the current
presentation not only represents the first experimental observation of complete current-driven
rotational motions of (anti)skyrmions but further demonstrates that the rotation of (anti)skyrmions
is independent of the current direction, which differs from skyrmion Hall motion influenced by
both current and temperature gradients41,42. Additional experiments with varying pulse widths
(Supplementary Fig. S2) show that excessively long pulses suppress the rotational motion due to
enhanced Joule heating. Accordingly, the 20-ns pulse width used in Fig. 2 and Fig. 3 ensures
efficient current-driven excitation while minimizing thermal perturbations.

In addition, Supplementary Fig. S3 compares skyrmion assemblies stabilized under
opposite perpendicular magnetic fields (along the +z direction). As a result of the reversed field, the
skyrmions carry topological charges Q = −1, opposite to the case in Fig. 2d−f. Correspondingly,
the direction of rotation under pulsed current excitation iRs reversed to CW (see also Supplementary
video 4), confirming the critical role of topologic al charge. The striking contrast in rotational
direction between skyrmions and antiskyrmIions underscores a strong correlation between the
rotational dynamics and the intrinsic topological properties of these spin textures. These results
advance the understanding and exploitation of topologically guided rotational mechanisms,
providing a novel pathway for current-driven spin textures without relying on thermal gradients or
external magnetic fields. Meanwhile, this marks the first experimental observation of current-
driven motion of antiskyrmions associated with their topological properties. The motion, which is
opposite to that of skyrmions, reflects their contrasting topological intrinsics.

The effect of confinement potential on rotational dynamics
When pulsed currents were applied along the short axis of a rectangular sample (see
Supplementary Fig. S4), rotational dynamics of skyrmions were observed. To further investigate
the geometric effect on the observed dynamics, a circular sample was fabricated using FIB. The
SEM image (Fig. 3a) illustrates the structure, which consists of two tungsten electrode pads
connected to a central FNPP region encircled by a tungsten ring. A protective carbon layer was
also deposited on top to prevent the curtaining effect during FIB thinning processes43. Due to
geometric confinement, the skyrmions within the central FNPP region (outlined by a black dashed
line) arrange into a circular pattern when stabilized under a perpendicular magnetic field (Fig. 3b).
Upon subsequent application of a pulsed current through the tungsten pads, skyrmions exhibited
CCW rotation (Supplementary Fig. S5 and Supplementary Video 5), in accordance with
observations described in Fig. 2d−f, thus confirming that the rotation direction of topological spin
textures does not depend on sample geometry. COMSOL simulations, as detailed in the Methods
section, indicate that the current inhomogeneity and temperature gradient under pulsed currents
(Supplementary Fig. S6) are minimal and negligible forR the rotational dynamics.

The angular velocity, ω, of skyrmions was further quantitatively estimated at four
different radial distances (schematically depIicted in Fig. 3c). For each radial position, angular
velocities were extracted from three representative skyrmions. The results with varying current
density in Fig. 3d demonstrate a decreasing trend in ω when the skyrmions approach the sample
center, with ω finally dropping to 0. The error bars reflect the experimental uncertainties like
statistical uncertainty and local pinning effects. Owing to the small sample tilting angle (∼5
degrees) and minimal thickness variation (< 50 nm) in our experiment, their influence on the
extracted angular velocities is negligible. The representative skyrmion trajectories under the same
current conditions are shown in Supplementary Fig. S7. Such a spatially correlated trend is difficult
to reconcile with random pinning or homogeneous damping effects, which would not naturally
produce a well-defined radial dependence. Due to the spin tilting at the sample boundaries in the
DMI system, the skyrmions experience a repulsive confinement potential from the boundary1,22.
Since the boundary potential energy experienced by skyrmions also exhibits a similar decreasing
trend with reducing radial distance, this indicates that the rotational velocity is strongly influenced
by the boundary confinement potential. Additionally, the current-density dependence of ω is also
shown in Fig. 3e. The increase in ω with current density is consistent with previous current-driven
experiments on skyrmions21,44,45, further highlighting the importance of the STT effect in initiating
this rotational motion. The absence of measurable texture distortions and excitation timescales far
exceeding intrinsic relaxation times suggest that effective inertia plays a negligible role in the
present regime.

Theoretical analyses and simulation results
Previous theoretical studies have extensively eRxplored skyrmion dynamics in proximity
to sample boundaries22,23, highlighting the critica l influence of boundary interactions. In this
section, we have summarized numerical simIulation results designed to elucidate the rotational
behavior of skyrmions under currents in confined geometries. Initially, the current-driven
dynamics of a single-skyrmion system has been simulated. Following this, the simulations have
been carried out for two-skyrmion and further multi-skyrmion systems (Fig. 4; see also
Supplementary Fig. S8−10). Comprehensive theoretical discussions and detailed derivations
supporting these simulations are provided in Supplementary Note2.

The simulation starts from a single skyrmion with a topological charge of Q = +1, as
depicted in Fig. 4a, which is first subjected to a DC current. Figure 4b presents the trajectory of
the skyrmion driven by a DC current applied along the horizontal direction, with its initial position
located at the system center. Under this current flow, the skyrmion moved toward the boundary,
eventually stopped at an equilibrium position, tracing a partial circular path. Following this, the
current was turned off to observe the relaxation dynamics, depicted in Fig. 4c. Remarkably, both
the driven and relaxation stages exhibited consistent CCW rotation, as explained quantitatively by
the Thiele equation (Eq. (1), also detailed in Supplementary Note2 and Supplementary Video 6).
This directional rotational behavior arises from gyrotropic dynamics characterized by the vector
G, whose sign is determined solely by the topological charge Q and is independent of the current
direction.

During the relaxation dynamics of the single skyrmion at r = r  0 without the STT
excitation (v = 0 in Eq.(1)), the Thiele equation simplifies to yield the angular velocity: ω =
G 1∂U
(r: radial distance from system center). To captuSre the most essential physical
G2 + (Dα)2r ∂r
mechanism while maintaining analytical transparency, we Eintentionally adopted a simplified but
effective form of the confinement potential, with UP = ∑ 1 k rn (detailed in Supplementary
n>2 n
n
Note2). Within this framework, near the system center (i.e. as r  0), the angular velocity converges
to a finite value ω = Gk2 . Figure E4d illustrates the simulated angular velocity as a function
r→0 G2+(Dα)2
of the normalized radial distance r/R (where R represents the system radius). Consequently, the
ω - r/R plot provides insights into the nonlinear confinement potential profile experienced by the
skyrmion, confirming the analytical predictions derived from the Thiele equation. The discrepancy
between Fig. 4d and experimental results shown in Fig. 3d arises from the absence of skyrmion–
skyrmion interactions in the single-skyrmion model, as discussed below.

In the case of a two-skyrmion system, the skyrmions eventually settled near the sample
center and exhibited zero angular velocity after several relaxation periods (see simulation results
in Supplementary Fig. S8 and Supplementary Video 7). The presence of skyrmion–skyrmion
interactions in multi-skyrmion systems ultimately drives the system into a static state with zero
angular velocity, as discussed in Supplementary S2.

These results suggest that the continuous rotation of a skyrmion arises from repeated
pulsed excitations followed by relaxation dynamics. During each pulse, the system is displaced
from equilibrium state, and in the subsequent relaxation state, boundary confinement and the
gyrotropic term dominate, resulting in a finite angular displacement. Repetition of this excitation–
relaxation cycle accumulates into a net rotational motion of a skyrmion.

The dynamics of skyrmion assemblies with a topological charge of Q = +1 under current
excitation were further investigated through comprehensive simulations. Initially, the simulations
under DC excitation (Supplementary Fig. S9 and Supplementary Video 8) revealed that the
skyrmion assembly firstly shifted to a new equilibrium position but subsequently ceased to move,
indicating the absence of sustained rotational dynamics. In contrast, under the pulsed current with
periodic excitation intervals (t
1
) followed by relaxa tion intervals (t
2
= 2t
1
), the simulation results
(Fig. 4e–g and Supplementary Video 9) consIistently revealed a CCW rotation of skyrmions after
each excitation-relaxation cycle, confirming the behavior discussed above. Conversely, when the
system configuration was altered to antiskyrmions possessing a topological charge of Q = −1, as
depicted in Fig. 4h−k and Supplementary Video 10, the direction of rotation changed to CW.
This
inversion aligns closely with our experimental observations in Fig. 2a−c, clearly demonstrating
the critical dependence of the rotational dynamics on the topological charge. Similar results under
current reversal are also shown in Supplementary Fig. S10 and Supplementary Videos 11−13,
which proves the dynamics are independent of the current direction. Furthermore, the theoretical
analysis agrees well with experimental results, revealing the validity and effectiveness of the
confinement-potential model adopted in this work.

To systematically examine the role of relaxation time, the pulse width was fixed at 500 ns
and the pulse interval was varied under continuous pulse trains. Representative frames extracted
from the recorded videos are shown in Supplementary Fig. S11. When the interval is 5 μs, the
rotational motions disappear, whereas the interval up to 10 μs restores pronounced rotational
dynamics. The results demonstrate that an adequate relaxation time is essential for enabling the
observed rotational motion, providing strong experimental support for our proposed excitation–
relaxation mechanism and theoretical framework.

To clarify the experimental and simulation results regarding skyrmion-assembly motion,
we provide a comprehensive theoretical derivation and validation based on the Thiele equation in
Supplementary Note2. Collectively, the theoretical analysis underscores three key conclusions: (i)
the direction of skyrmion rotational dynamics depends solely on the sign of the topological charge
Q, and is independent of current direction; (ii) the reRlaxation time between current pulses is
important for the continuous rotation and the nonlin ear boundary confinement potential constitutes
the primary driving force underlying thesIe rotational dynamics; and (iii) the equilibrium
established by skyrmion-skyrmion interactions leads to the gradual reduction in angular velocity
as skyrmion assemblies approach the sample center.

In conclusion, we have experimentally demonstrated novel current-induced rotational
dynamics of skyrmion and antiskyrmion assemblies confined within samples with different
geometries. The in-situ L-TEM results have revealed that the sense of rotation depends solely on
the intrinsic topological charge. Notably, this first experimental observation of antiskyrmion
assembly dynamics has directly confirmed their contrasting topological nature relative to
skyrmions. Further analyses have identified clear evidence for a decrease in the angular velocity
of spin textures toward the sample center, underscoring the crucial effect of confinement potentials
caused by the sample boundary. Complementary micromagnetic simulations and theoretical
modeling have emphasized the decisive roles of the nonlinear boundary potential and the
gyrotropic force in driving these unique rotational dynamics of topological spin textures as a
transient phenomenon. These results offer significant advances in exploring skyrmion dynamics.

Methods
Sample preparation
The single-crystalline bulk sample of (Fe Ni Pd ) P (FNPP) was synthesized using the self-
0.63 0.3 0.07 3
flux method, employing iron (Fe), nickel (Ni), palladium (Pd), and red phosphorus (P) sealed in
an evacuated quartz tube. For the in-situ L-TEM observations, sample plates were fabricated from
FNPP single crystals utilizing a FIB technique. The (001) plane was selected for the sample, with
the edges aligned along the <110> direction to align with the spiral wave vector q, optimizing the
observation of skyrmion dynamics. The dimensioRns of the rectangular samples were
approximately 6 µm in length, 4 µm in width, an d 160 nm in thickness. Additionally, a round-
shaped sample plate was fabricated, featurinIg two tungsten electrode pads and a central FNPP
circle encased by a tungsten ring. For preparation of the round-shaped sample, part of the ring
region was first removed by FIB milling, followed by tungsten deposition into the milled area. The
remaining FNPP in the ring region was then removed, leaving only the tungsten in that area. The
diameter of the inner FNPP circle was set to 6 µm, with a thickness of 160 nm.

L-TEM observations
The in-situ observations of skyrmions under pulsed current were conducted using JEM−2100 and
JEM−2800 microscopes (JEOL, Japan), equipped with a double-tilt current-capable holder (Fusion
Select, Protochips). In L-TEM, since the objective lens is essentially a strong electromagnetic lens
composed of current-carrying coils, the application of vertical magnetic fields was controlled by
adjusting the objective-lens current. For capturing L-TEM images with the magnetic field oriented
along the +z direction, the JEM−2800 microscope was utilized. Conversely, images with the
magnetic field applied along the −z direction were captured using the JEM−2100 microscope.
Pulsed voltages were generated using an arbitrary function generator (AFG31252, Tektronix,
USA), with each pulse manually triggered at intervals exceeding 0.1 seconds. To quantitatively
analyze the in-plane magnetic induction fields, the under- and over-focused L-TEM images were
processed using the QPt software package (HREM Co., Japan), based on the transport-of-intensity
equation.

Micromagnetic simulation
For the micromagnetic simulations, the GPU-accelerated software MuMax346-48 was used to model
the dynamic processes of skyrmions under pulsed curreRnt in a system featuring asymmetric DMI.
The simulations were governed by solving the La ndau-Lifshitz-Gilbert (LLG) equation, which
was augmented to include both adiabatic and Inon-adiabatic spin-transfer torques. In the FNPP, Ni,
Pd, and P do not contribute appreciably to the magnetic moment at room temperature31,32. Instead,
the magnetism is dominated by the Fe atoms. Therefore, a single-sublattice LLG framework was
used in the micromagnetic simulations to describe the system. The LLG equation is expressed as:
dM A α dM u βu
= -γ M × H + (M × )+ (M × ∂ M × M) - (M × ∂ M) (1)
dt 0 eff M dt M2 x M x
s s s
where M is the magnetization, M is the saturation magnetization, t represents time, γ is the
s 0
absolute gyromagnetic ratio, and α is the damping parameter. u represents the conduction electron
γ ℏ jP
velocity, related to the current density by u = | 0 |( ), where µ is the vacuum permeability
0
μ
0
e 2Ms
constant, ℏ is the reduced Planck constant, e is the electron charge, j is the applied current density,
and P is the spin polarization rate. The effective field is determined by H = -μ-1∂ε⁄∂M and the
eff 0
energy density ε is given by:
ε = A(∇m)2 + D[m∙(x̂ × ∂ m) - m∙(ŷ × ∂ m)]
x y
M
- K (m∙ẑ)2 - μ M m∙H - s m∙B (2)
u 0 s 2 d
This system energy encompasses the ferromagnetic exchange interaction with exchange stiffness
A, anisotropic DMI energy with the constant D, uniaxial magnetic anisotropy energy with the
constant K , Zeeman energy with the perpendicular field H, and demagnetization energy terms
u
associated with the saturation magnetization M . Because of the negligible skyrmion dynamics
s
caused by thermal fluctuations at room temperature in the present experimental setup, the
simulation has been employed at T = 0 K. For the setpoints of the parameters used in our simulation,
the data from previous experimental and simulation studies31,49,50 were adapted, as A = 8.1 pJ/m,
K = 31 kJ/m3, M = 417 kA/m, D = 0.2 mJ/m2, α = 0.03R and β = 0.01. In the simulation, 𝑥 and 𝑦
u s
were defined to correspond to the crystallogra phic [110] and [1ˉ10] directions of the
experimental sample respectively, ensuring Ithe consistency between the simulations and the
experimental geometry. The simulated area was set to a circular shape with open boundary
conditions and discretized into 480 × 480 × 20 cuboids with the mesh size of 8 nm. The
perpendicular field was set to 0.35 mT along the +z or −z direction. The density of the pulse current
was set to 7 × 1010 A A/m2 and injected along the +x or −x direction, closely aligning with
experimental values. Each current cycle consisted of a 400-ns excitation followed by an 800-ns
interval. No additional damping potentials were introduced in our simulation.

COMSOL simulation
The current distribution and temperature variation in the samples under pulsed current were
simulated using COMSOL Multiphysics. For these simulations, the "Joule Heating" module was
employed, incorporating the "Electric Currents" and "Heat Transfer in Solids" interfaces, along
with a Multiphysics node for "Electromagnetic Heating". A time-dependent Solver was selected to
meticulously track the temperature evolution throughout the pulsed current application. The
geometrical dimensions of the simulated structures were meticulously matched to those of the
actual experimental setup. Specifically, the two tungsten electrode pads were modeled with
dimensions of 2 µm × 6 µm × 0.16 µm. The diameter of the inner FNPP circle (colored in Brown)
was set to 4 µm, while the surrounding tungsten ring (colored in orange) was assigned a width of
1 µm. The sample thickness was consistently modeled at 160 nm. Given that FNPP is primarily
iron-based, the material parameters employed in the COMSOL simulations were chiefly derived
from those of iron.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this paper, the authors highlighted the critical influence of boundary induced confining potential and gyro tropic forces in steering the rotational dynamics of (anti)skyrmions. The experiments are carefully designed. The results are interesting. The rotational dynamics are intriguing. However, I strongly recommend the authors clarify the following.\n1. In previous experimental work, the nonreciprocal transportation and circulation of skyrmions have been discussed in detail (Nano Letters 22 (23), 9638-9644, (2022)). The authors should clarify the improvements of the current work with those in the literature.\n2. In this work, the (anti)skyrmions are driven by current pulse. It has been reported that Skyrmion Brownian dynamics also have similar behaviors (Applied physics letters, 117, 082402 (2020), Applied Physics Letters 119, 072402 (2021)). It is also well known that the Joule heating is evitable for current pulses. The authors should clarify the thermal effect, spin-transfer-torque, and boundary effects.", "reviewer_2": "Please see the attachment", "reviewer_3": "The manuscript by Z. Li et al reports current-driven rotational dynamics of skyrmion and antiskyrmion assemblies in confined geometries, observed via in-situ Lorentz transmission electron microscopy, and supported by micromagnetic simulations as well as Thiele-equation-based analysis. The authors demonstrate that the sense of rotation is determined solely by the sign of the topological charge, independent of current polarity and sample geometry, and attribute the phenomenon to the interplay between gyrotropic forces and boundary-induced confinement potentials. The work addresses an important and timely problem in skyrmionics, particularly given the scarcity of experimental studies on antiskyrmion dynamics. The experimental data are generally convincing, and the antiskyrmion results are potentially impactful.\nHowever, for publication in Nature Communications, the manuscript requires substantial revision. Several claims are overstated, the physical interpretation lacks sufficient quantitative support, and key assumptions are not adequately justified. I therefore recommend a major revision. Below, I outline my concerns as major and minor points, all of which should be addressed before the manuscript can be considered for publication.\nMajor comments:\n1. Conceptual framing: topology-governed dynamics\nThe central claim that the observed rotational dynamics are governed by topology is conceptually overstated. While the direction of rotation is indeed determined by the sign of the topological charge via the gyrotropic term, the presence and magnitude of the rotation clearly (to me) depend on the boundary-induced confinement potential. In the absence of confinement, the Thiele equation predicts a transverse deflection rather than sustained rotation. Throughout the manuscript, the authors frequently suggest that topology alone governs the dynamics, whereas both the experimental results and theoretical framework indicate that topology sets the chirality, not the dynamics itself. The authors should therefore explicitly distinguish between:\n- topology determining the sense of rotation, and\n- confinement providing the restoring force that enables rotational motion\nFollowing that, the title, abstract, and conclusions should be softened accordingly.\n2. Boundary confinement potential: origin and quantification\nThe boundary-induced confinement potential plays a central role in the interpretation of the manuscript, yet its physical origin and functional form are not sufficiently discussed. Several key questions remain unresolved, including whether the confinement is dominated by magnetostatic energy, DMI-induced edge canting, or effective pinning arising from thickness variations. The manuscript assumes a nonlinear confinement potential but does not experimentally extract or independently validate its radial dependence. Moreover, the comparison between the experimental angular velocity profiles (Fig. 3d) and the simulated results (Fig. 4d) remains largely qualitative, with noticeable discrepancies that are acknowledged but not adequately addressed. I therefore urge the authors to:\n- Provide a clear physical discussion of the confinement mechanism in FNPP,\n- Attempt a semi-quantitative extraction of the effective potential from experimental data, or explicitly state the limitations of the model and thereby point out the reasons for such quantitative mismatch\n3. A key result of the study is that sustained rotation is observed only under pulsed current, whereas a DC current leads to a static equilibrium. Although this behavior is reproduced in the simulations, the underlying physical mechanism is not clarified. In particular, the possible roles of relaxation dynamics, effective inertia, or transient overshoot are not discussed. Moreover, the choice of pulse parameters (t₁, t₂, and their ratio) appears ad hoc, with no experimental investigation of alternative parameter regimes. The authors should therefore either provide a more detailed physical explanation of why pulsed excitation is essential, or clearly state this as an empirical observation limited to the explored parameter range.\n4. The experimental observation of current-driven antiskyrmion rotation represents a major strength of the manuscript. However, although recent experimental reports on current-driven antiskyrmion motion (such as sliding or confined motion) are cited, they are not discussed in sufficient depth. The manuscript would benefit from a clearer articulation of what is fundamentally new relative to these prior works, whether it is the rotational motion itself, the topology-controlled chirality, a comparative assessment of mobility, or the use of pulsed-current operation.\n5. The micromagnetic simulations employ a single-sublattice ferromagnetic LLG framework to model FNPP, a structurally and magnetically complex, anisotropic, multi-element system. However, the validity of this approximation, as well as its limitations, is not discussed. More importantly, thermal fluctuations are neglected, despite the experiments being conducted at room temperature. In addition, the asymmetric DMI term is introduced without explicitly relating it to the underlying symmetry. A more careful justification of the modelling assumptions used to explain the experimental observations is therefore needed.\n6. There are also a few quantitative rigors of the experimental analysis, as listed below:\n- Angular velocities are reported without error bars or uncertainty estimates\n- The number of tracked skyrmions at each radial position is limited\n- Possible systematic effects (projection in L-TEM, drift, defocus variation) are not discussed\nAt a minimum, the authors should comment on experimental uncertainties and the reproducibility of the results.\nMinor comments:\n1. The manuscript is a bit long, with repetitive background material that could be shortened.\n2. Several grammatical issues and typographical errors remain (e.g. figure captions, plural agreement).\n3. Some figure captions lack sufficient standalone clarity.\n4. Data availability is stated, but no guidance is given on accessing raw L-TEM videos or tracking data.\n5. The discussion of potential device implications remains speculative and could be better focused or shortened.\nIn summary, the manuscript presents high-quality experimental data and reports an important observation, particularly concerning antiskyrmion dynamics. Nevertheless, to meet the standards of Nature Communications, the authors need to address the points raised above. Once these issues are satisfactorily resolved, the work would represent a strong and impactful contribution to the field of topological spin textures and merit consideration for publication.", "reviewer_attached_report": "Paper NCOMMS 25-92720-T\n- What are the noteworthy results?\nThis work is perhaps one of the first that report on rotational dynamics of antiskyrmions in confined geometries. And the results are quite new and unexpected. This adds more options for new applications in future devices.\n- Will the work be of significance to the field and related fields? How does it compare to the established literature? If the work is not original, please provide relevant references.\nYes, the work is original and presents new interest results and will be valuable to the literature. The supplementary material is quite substantial.\n- Does the work support the conclusions and claims, or is additional evidence needed?\n- Are there any flaws in the data analysis, interpretation and conclusions? - Do these prohibit publication or require revision?\nI think the paper is reasonably complete. It can improve, of course, but at this time it does represent a worthy contribution to the field since it is the first of its kind.\n- Is the methodology sound? Does the work meet the expected standards in your field?\nThe methods are consistent with present-day standards, namely the use of micromagnetic simulations, Lorentz transmission electron microscopy and theoretical analysis via the Thiele equation.\n- Is there enough detail provided in the methods for the work to be reproduced?\nThere is enough detail in the paper to help others reproduce their results.\nOverall I recommend acceptance of the paper.\nSome specific queries for the authors:\nThe skyrmion confined potential U does not depend on the azimuthal angle  and is monotonically increasing for increasing radial distance. What evidence, experimental or theoretical, can be given to support this important statement?\nThe assumption of the form of U according to Eq (26) of the supplementary section might also be explained at least qualitatively. Is the assumption of radial dependence is unique?\nI suggest giving the original source of the Thiele equation. This is to enable others to assess the limitations of the Thiele equation.\nIn lines 173 and 210 of the main text the description ‘stabilized’ is mentioned. Can this be described in more detail? Does stability refer to lifetime of the skyrmion?\nCOMSOL is used in the paper. Is it as ‘reliable’ as micromagnetic simulations? What specific elements of COMSOL were contributed by the present authors?\nThe decrease in rotational velocity of the skyrmions in lines 218 ff is attributed to the boundary confining potential. Might some other damping potential exist? It might be due to other causes. Please comment on this possibility."}, "author_response": "Authors’ response to referees’ comments\nComments from Reviewer # 1:\nOverall comments: In this paper, the authors highlighted the critical influence of\nboundary induced confining potential and gyro tropic forces in steering the rotational\ndynamics of (anti)skyrmions. The experiments are carefully designed. The results are\ninteresting. The rotational dynamics are intriguing.\nResponse: We sincerely thank the referee for the positive and encouraging comments.\nComment 1. In previous experimental work, the nonreciprocal transportation and\ncirculation of skyrmions have been discussed in detail (Nano Letters 22 (23), 9638-\n9644, (2022)). The authors should clarify the improvements of the current work with\nthose in the literature.\nResponse 1: We thank referee for the helpful comment. In the literature raised by\nreferee, skyrmion circulation was realized through specially designed geometries that\ncause the skyrmion Hall motion into arc-shaped trajectories. The apparent circular\nmotion was inferred from multiple arc segments obtained by reversing the current\npolarity. Such dynamics are essentially geometry-modified skyrmion Hall motion,\nwhere the gyrotropic force dominates current-driven translational motion of skyrmions,\nresulting in the current-direction dependent dynamics of skyrmion.\nIn contrast, our work reports a current-direction independent rotational motion of\nskyrmion assemblies. Crucially, this rotation is governed by combination of the\ngyrotropic force with the boundary confinement potential. As a result, the rotation\ndirection is dictated solely by the topological charge, representing a different dynamical\nregime in our present work beyond previously reported geometry-assisted circulation.\nIn addition, while the prior study focused exclusively on skyrmions, our present\nwork has demonstrated the current-driven rotational dynamics for both skyrmion and\nantiskyrmion assemblies. We have clarified that antiskyrmions rotate in the opposite\nsense to skyrmions’ rotation under the same current excitations, offering the first\ndynamic experimental verification of their opposite topological charges—a key issue\nthat has not been addressed in previous experiments.\nIn the revised manuscript, we have cited the paper raised by the referee and added\na few sentences to discuss the differences of our present work from the literature (see\nline 170-177 in the revised manuscript), as described below:\n“In earlier studies, geometric confinement was exploited to cause the skyrmion\nHall motion into arc-shaped trajectories, and the sense of rotation was inferred from\nmultiple arc segments obtained by reversing the current polarity. In contrast, the current\npresentation not only represents the first experimental observation of complete current-\ndriven rotational motions of (anti)skyrmions but further demonstrates that the rotation\nof (anti)skyrmions is independent of the current direction, which differs from skyrmion\nHall motion influenced by both current and temperature gradients”\nComment 2. In this work, the (anti)skyrmions are driven by current pulse. It has been\nreported that Skyrmion Brownian dynamics also have similar behaviors (Applied\n1\nphysics letters, 117, 082402 (2020), Applied Physics Letters 119, 072402 (2021)). It is\nalso well known that the Joule heating is evitable for current pulses. The authors should\nclarify the thermal effect, spin-transfer-torque, and boundary effects.\nResponse 2: We thank the referee for raising this important comment. Regarding\nthermal effects and Brownian dynamics, we note that the skyrmion Brownian motion\nreported in Applied Physics Letters 117, 082402 (2020) and 119, 072402 (2021) arises\nfrom stochastic thermal fluctuations and is intrinsically random, lacking the\ndeterministic rotational direction. In contrast, the rotational dynamics observed in our\npresent work are highly reproducible over many current pulses and exhibit a well-\ndefined rotational sense strictly correlated with the topological charge. Such robust\nrotation sense and repeatability are fundamentally different from thermally driven\nBrownian motion.\nRegarding the thermal effect on skyrmion dynamics caused by Joule heating in the\nthin sample, we roughly estimated the temperature change in our present sample during\nthe current flows, through detailed COMSOL simulations (see Supplementary Section\nS3). The COMSOL results show that the maximum temperature change during a current\npulse is less than ~2 K, and no appreciable temperature gradient across the sample.\nTherefore, the thermally induced effect can be excluded in our experiment setup. The\ncorresponding discussions have been added in the revised manuscript (see line 154- 158\nand line 209-211 in the revised manuscript), as described below:\n“The antiskyrmion assembly is robust and reproducible without spontaneous\nannihilation or significant rearrangement during L-TEM imaging, indicating that the\nambient thermal fluctuations alone are insufficient to induce any observable dynamics\nsuch as Brownian motion in the present experimental setup” (line 154- 158)\n“COMSOL simulations, as detailed in the Methods section, indicate that the current\ninhomogeneity and temperature gradient under pulsed currents (Extended Data Fig. 5)\nare minimal and negligible for the rotational dynamics.” (line 209- 211)\nThe role of spin-transfer torque (STT) effect on (anti)skyrmion dynamics should\nbe essential for providing the initial excitation that displaces (anti)skyrmions from their\nequilibrium state. In the subsequent relaxation state, boundary confinement and the\ngyrotropic term dominate, resulting in a finite angular displacement. Repetition of this\nexcitation–relaxation cycle accumulates into a net rotational motion of a skyrmion. This\nmechanism has been proved in the combination of our in-situ real-space observations,\ntheoretical analyses, and micromagnetic simulations, which is fundamentally distinct\nfrom both Brownian dynamics and current-driven Hall motions of (anti)skyrmions.\nThe corresponding discussions have been added in the revised manuscript (line 286-291 in\nthe revised manuscript), as described below:\n“These results suggest that the continuous rotation of a skyrmion arises from\nrepeated pulsed excitations followed by relaxation dynamics. During each pulse, the\nsystem is displaced from equilibrium state, and in the subsequent relaxation state,\nboundary confinement and the gyrotropic term dominate, resulting in a finite angular\ndisplacement. Repetition of this excitation–relaxation cycle accumulates into a net\nrotational motion of a skyrmion.”\nOverall comments: This work is perhaps one of the first that report on rotational\ndynamics of antiskyrmions in confined geometries. And the results are quite new and\nunexpected. This adds more options for new applications in future devices. The work\nis original and presents new interest results and will be valuable to the literature.\nThe supplementary material is quite substantial. I think the paper is reasonably complete. It\ncan improve, of course, but at this time it does represent a worthy contribution to the\nfield since it is the first of its kind. The methods are consistent with present-day\nstandards, namely the use of micromagnetic simulations, Lorentz transmission electron\nmicroscopy and theoretical analysis via the Thiele equation. There is enough detail in\nthe paper to help others reproduce their results.\nOverall I recommend acceptance of the paper.\nResponse:We sincerely thank the referee for the positive and encouraging assessment\nof our work.\nComment 1. The skyrmion confined potential U does not depend on the azimuthal\nangle θ and is monotonically increasing for increasing radial distance. What evidence,\nexperimental or theoretical, can be given to support this important statement?\nResponse 1: We thank the referee for this constructive and insightful comment. In our\ntheoretical analysis, we intentionally adopted a simplified but effective form of the\nconfinement potential to capture the most essential physical mechanism underlying the\nobserved rotational dynamics.\nWe fully acknowledge that, in real experimental systems, imperfections such as\ndefects, edge roughness, and local inhomogeneities may introduce weak angular\nvariations in the confinement potential, even when no deliberate angular anisotropy is\nintroduced during sample fabrication. To minimize the influence of such extrinsic\nangular inhomogeneities, we therefore fabricated circular samples with axial symmetry,\nin which the boundary conditions are, to first order, isotropic in the angular direction.\nImportantly, the rotational dynamics observed in circular samples are fully\nconsistent with those obtained in rectangular samples, demonstrating that the essential\nfeatures of the rotation are robust irrespective of sample geometry. In parallel, our\nmicromagnetic simulations were performed using a model that incorporates an\nangularly isotropic confinement potential only. Under this assumption, the simulations\nsuccessfully reproduce all key experimental observations, including the rotation sense,\nthe radial dependence of the angular velocity, and the independence of rotation direction\nfrom current polarity.\nThese combined experimental and numerical results indicate that the dominant and\ndecisive mechanism governing the rotational dynamics arises from angularly isotropic\nboundary confinement potential. Possible angular-dependent variations, while\ninevitably present in real systems, play a secondary role and do not alter the primary\nphysical conclusions of this work. We have revised the corresponding parts in the main\ntext to more clearly stress out this point (see line 269-272 and line 318-320 in the\nrevised manuscript), as described below:\n3\n“To capture the most essential physical mechanism while maintaining analytical\ntransparency, we intentionally adopted a simplified but effective form of the\nconfinement potential, with (detailed in Supplementary S2).” (line\n1 n\n269-272) U = ∑n>2 nknr\n“Furthermore, the theoretical analysis agrees well with experimental results,\nrevealing the validity and effectiveness of the confinement-potential model adopted in\nthis work” (line 318-320)\nComment 2. The assumption of the form of U according to Eq (26) of the\nsupplementary section might also be explained at least qualitatively. Is the assumption\nof radial dependence unique?\nResponse 2: We thank the referee for this important and constructive comment. The\nform of the confinement potential ( ) introduced in Eq. (26) is not meant to represent\na unique microscopic expression, but rather a minimal and physically reasonable\n𝑈 𝑟\nphenomenological description that captures the essential features required to analyze\nthe observed rotational dynamics.\nIn our study, the rotational motion of (anti)skyrmion assemblies is governed by\ntopological dynamics, which are valid in the continuum limit where the magnetization\nvaries smoothly in space. Within this framework, the confinement potential should be\nexpressed as a regular, singularity-free analytic function. The polynomial expansion in\nEq. (26) therefore provides a general and flexible representation of such a smooth radial\npotential near the sample center, without assuming a specific microscopic origin.\nHere we intentionally focus on the case where the confinement potential is\nazimuthally symmetric and monotonically increasing with distance from the center, and\nwhere no additional local minima exist apart from the global minimum at = 0. Local\nminima away from the center, while capable of trapping skyrmions, would suppress\n𝑟\ncollective rotational motion rather than promote it. Since our experiments show robust\nglobal rotation of skyrmion assemblies, such trapping-dominated scenarios are not\nrelevant to the present system.\nAs the referee correctly pointed out, the previous wording in Supplementary\nSection S2 (line 170 in the original version) could be misleading. We have therefore\nrevised the manuscript to explicitly clarify this assumption. Specifically, we now state\nat the beginning of Supplementary Section S2 (line 63 and line 172) that strong local\nminima in the confinement potential would inhibit rotational dynamics, and that we\ntherefore consider an azimuthally symmetric confinement potential ( ) without\nangular dependence.\n𝑈 𝑟\nWith these clarifications, we believe the physical motivation and scope of\napplicability of the assumed confinement potential are now clearly stated and accessible\nto a broad readership.\nComment 3. I suggest giving the original source of the Thiele equation. This is to\nenable others to assess the limitations of the Thiele equation.\nthe Thiele equation is listed in the Supplementary Note S2 and we have explicitly cited\nthe original source of the Thiele equation when it is introduced. The Thiele equation is\nderived from the Landau–Lifshitz–Gilbert (LLG) equation under the assumption of a\nrigid spin texture, an approximation that is well justified for the skyrmions studied here.\nIn our analysis, the skyrmion dynamics observed experimentally are systematically\nexamined within the Thiele framework by incorporating a nonlinear boundary\nconfinement potential as well as skyrmion–skyrmion interactions into the force term.\nFurthermore, to properly describe the pulsed-current excitation used in the experiments,\nwe explicitly consider both current-on ( ≠ 0) and current-off ( = 0) regimes within\nthe same formalism. This extended treatment allows us to capture the excitation–\n𝑠 𝑠\n𝑣 𝑣\nrelaxation process induced by the current pulses and provides a consistent theoretical\nexplanation for the experimentally observed rotational dynamics.\nComment 4. In lines 173 and 210 of the main text the description ‘stabilized’ is\nmentioned. Can this be described in more detail? Does stability refer to lifetime of the\nskyrmion?\nResponse 4: In our manuscript, the term “stabilized” does not refer to the intrinsic\nlifetime of individual (anti)skyrmions. Instead, it means a metastability for skyrmions\nat a nonequilibrium state under specific condition of external magnetic field.\nSpecifically, “stabilized” state means that the (anti)skyrmions and their assemblies\nare robust and can be reproduced under specific field conditions in the absence of\ncurrent excitation, during L-TEM imaging. Such metastability of (anti)skyrmions\nallows reliable tracking of their current-driven rotational dynamics. We have added a\nfew sentences in the revised manuscript to clarify this point and avoid ambiguity (see\nline 154-158 in the revised manuscript), as described below:\n“The antiskyrmion assembly is robust and reproducible without spontaneous\nannihilation or significant rearrangement during L-TEM imaging, indicating that the\nambient thermal fluctuations alone are insufficient to induce any observable dynamics\nsuch as Brownian motion in the present experimental setup”\nComment 5. COMSOL is used in the paper. Is it as ‘reliable’ as micromagnetic\nsimulations? What specific elements of COMSOL were contributed by the present\nauthors?\nResponse 5:COMSOL is as reliable as micromagnetic simulation. In the present work,\nCOMSOL was used to solve coupled electric-current and heat-transport equations\nbased on experimentally realistic sample geometries, material parameters, and pulsed-\ncurrent conditions. These simulations allow us to assess whether current inhomogeneity\nor thermal gradients could contribute to the observed rotational dynamics. Our results\nshow that both the current density and temperature distribution remain spatially\nuniform within the skyrmion-hosting region, thereby excluding these extrinsic effects\nas dominant driving mechanisms.\nWe used home-build models to closely match the experimental configuration\n5\nthrough COMSOL calculation. These calculations provide an essential supporting role,\nstrengthening the conclusion that the observed rotational dynamics originate from\ntopological gyrotropic effects and boundary confinement, rather than Jule heat.\nCorresponding clarifications have been added to the revised manuscript. (see line 209-\n211 in the revised manuscript), as described below:\n“COMSOL simulations, as detailed in the Methods section, indicate that the current\ninhomogeneity and temperature gradient under pulsed currents (Extended Data Fig. 5)\nare minimal and negligible for the rotational dynamics.”\nComment 6. The decrease in rotational velocity of the skyrmions in lines 218 ff is\nattributed to the boundary confining potential. Might some other damping potential\nexist? It might be due to other causes. Please comment on this possibility.\nResponse 6: We appreciate the referee’s thoughtful comment. In principle, additional\ndamping mechanisms—such as enhanced Gilbert damping or defect-induced pinning—\ncould influence skyrmion dynamics. However, experimentally, the angular velocity\ndecreases systematically and monotonically with decreasing radial distance,\nindependent of current polarity and sample geometry. Such a spatially correlated trend\nis difficult to reconcile with random pinning or homogeneous damping effects, which\nwould not naturally produce a well-defined radial dependence.\nIn our micromagnetic simulations, no additional damping potentials were\nintroduced. Nevertheless, owing solely to boundary confinement and skyrmion–\nskyrmion interactions, the angular velocity of skyrmions exhibits a clear decreasing\ntrend toward the center of the sample, in quantitative agreement with the experimental\nresults. This behavior is further supported by our Thiele-equation-based analysis in\nSupplementary Note 2.\nTaken together, these experimental and micromagnetic simulation results indicate\nthat these effects are not the dominant origin of the observed decrease in rotational\nvelocity. The corresponding discussions have been added in our main text. (see line\n221-223 and line 410 in the revised manuscript), as described below:\n“Such a spatially correlated trend is difficult to reconcile with random pinning or\nhomogeneous damping effects, which would not naturally produce a well-defined radial\ndependence.” (line 221-223)\n“No additional damping potentials were introduced in our simulation” (line 410)\nComments from Reviewer # 3:\nOverall comments: The manuscript by Z. Li et al reports current-driven rotational\ndynamics of skyrmion and antiskyrmion assemblies in confined geometries, observed\nvia in-situ Lorentz transmission electron microscopy, and supported by micromagnetic\nsimulations as well as Thiele-equation-based analysis. The authors demonstrate that the\nsense of rotation is determined solely by the sign of the topological charge, independent\nof current polarity and sample geometry, and attribute the phenomenon to the interplay\nbetween gyrotropic forces and boundary-induced confinement potentials. The work\naddresses an important and timely problem in skyrmionics, particularly given the\n6\nscarcity of experimental studies on antiskyrmion dynamics. The experimental data are\ngenerally convincing, and the antiskyrmion results are potentially impactful.\nHowever, for publication in Nature Communications, the manuscript requires\nsubstantial revision. Several claims are overstated, the physical interpretation lacks\nsufficient quantitative support, and key assumptions are not adequately justified.\nI therefore recommend a major revision. Below, I outline my concerns as major and\nminor points, all of which should be addressed before the manuscript can be considered\nfor publication.\nResponse: We thank the referee for the careful reading of our manuscript and for the\npositive and thoughtful assessment. In response, the comments have been addressed\npoint by point, and the manuscript has been revised accordingly.\nMajor comments:\nComment 1. Conceptual framing: topology-governed dynamics\nThe central claim that the observed rotational dynamics are governed by topology is\nconceptually overstated. While the direction of rotation is indeed determined by the\nsign of the topological charge via the gyrotropic term, the presence and magnitude of\nthe rotation clearly (to me) depend on the boundary-induced confinement potential. In\nthe absence of confinement, the Thiele equation predicts a transverse deflection rather\nthan sustained rotation. Throughout the manuscript, the authors frequently suggest that\ntopology alone governs the dynamics, whereas both the experimental results and\ntheoretical framework indicate that topology sets the chirality, not the dynamics itself.\nThe authors should therefore explicitly distinguish between:\n- topology determining the sense of rotation, and\n- confinement providing the restoring force that enables rotational motion\nFollowing that, the title, abstract, and conclusions should be softened accordingly.\nResponse 1: We thank the referee for this insightful comment and fully agree with the\ncentral point raised by referee.\nAccording to the referee’s comments, we have modified the title, abstract, and\nconclusions in the revised manuscript. We have replaced the wording “topology-\ngoverned” with “topology-guided”.\nComment 2. Boundary confinement potential: origin and quantification\nThe boundary-induced confinement potential plays a central role in the interpretation\nof the manuscript, yet its physical origin and functional form are not sufficiently\ndiscussed. Several key questions remain unresolved, including whether the\nconfinement is dominated by magnetostatic energy, DMI-induced edge canting, or\neffective pinning arising from thickness variations. The manuscript assumes a nonlinear\nconfinement potential but does not experimentally extract or independently validate its\nradial dependence. Moreover, the comparison between the experimental angular\nvelocity profiles (Fig. 3d) and the simulated results (Fig. 4d) remains largely qualitative,\nwith noticeable discrepancies that are acknowledged but not adequately addressed. I\ntherefore urge the authors to:\n7\n- Provide a clear physical discussion of the confinement mechanism in FNPP,\n- Attempt a semi-quantitative extraction of the effective potential from experimental\ndata, or explicitly state the limitations of the model and thereby point out the reasons\nfor such quantitative mismatch\nResponse 2: We thank the referee for the comment. In DMI-stabilized magnetic\nsystems, as discussed in theearlier works (Nature Nanotech 8, 742-747 (2013), Nature\nNanotech 8, 899-911 (2013)), the DMI induces spin tilting at sample boundaries. The\nexchange interaction between the skyrmion and the boundary-induced tilting generates\na repulsive force, whereas magnetostatic energy tends to produce an attractive\ninteraction. The competition between these two effects gives rise to an effective\nboundary confinement potential. In our experiments, we consistently observe a\nrepulsive interaction between skyrmions and the sample boundary, indicating that the\nexchange-mediated interaction associated with spin canting at the boundary dominates\nin the present system. The corresponding discussion has been added to revised\nmanuscript (see line 223-225 in the revised manuscript), as described below:\n“Due to the spin tilting at the sample boundaries in the DMI system, the skyrmions\nexperience a repulsive confinement potential from the boundary.”\nOn the other hand, the discrepancy between Fig. 3d and Fig. 4d arises because Fig.\n3d shows experimental results obtained from skyrmion assemblies, whereas Fig. 4d\ncorresponds to simulations considering a single skyrmion. For a single skyrmion during\nthe relaxation stage, the boundary confinement drives rotation toward the sample center\nwith a gradually decreasing angular velocity, consistent with experiments. However, in\nthe absence of skyrmion–skyrmion interactions, the skyrmion does not come to stop\nnear the center but continues rotating with a finite angular velocity.\nWhen skyrmion assemblies are considered in the theoretical analysis and\nmicromagnetic simulations, the additional skyrmion–skyrmion interactions play a\ncrucial role. Therefore, near the sample center, the combined effect of weakened\nboundary confinement and skrymion-skyrmion interactions suppresses the angular\nvelocity, leading to the vanishing of angular velocity. This behavior is consistently\nreproduced by our theoretical modeling (Section Ⅱ and Ⅲ in Supplementary S2),\nmicromagnetic simulations (Extended Data Fig. 7-9), and experimental observations.\nThe corresponding discussions have been added in the revised manuscript (see line 277-\n280 and line 283-285 in the revised manuscript), as described below:\n“The discrepancy between Fig. 4d and experimental results shown in Fig. 3d arises\nfrom the absence of skyrmion–skyrmion interactions in the single-skyrmion model, as\ndiscussed below.” (line 277-280)\n“The presence of skyrmion–skyrmion interactions in multi-skyrmion systems\nultimately drives the system into a static state with zero angular velocity, as discussed\nin Supplementary S2.” (line 283-285)\nComment 3. A key result of the study is that sustained rotation is observed only under\npulsed current, whereas a DC current leads to a static equilibrium. Although this\nbehavior is reproduced in the simulations, the underlying physical mechanism is not\n8\nclarified. In particular, the possible roles of relaxation dynamics, effective inertia, or\ntransient overshoot are not discussed. Moreover, the choice of pulse parameters (t₁,\nt₂, and their ratio) appears ad hoc, with no experimental investigation of alternative\nparameter regimes. The authors should therefore either provide a more detailed physical\nexplanation of why pulsed excitation is essential, or clearly state this as an empirical\nobservation limited to the explored parameter range.\nResponse 3: We thank the referee for this insightful comment, which touches on the\nmost essential part of this work. Under DC current, spin-transfer torque drives the\nskyrmions until gyrotropic force, damping, and boundary confinement balance, leading\nto a stationary state. In contrast, pulsed excitation repeatedly drives the system to a\nnonequilibrium state during the current “on”, followed by a relaxation stage in which\nboundary confinement and the gyrotropic term dominate. This relaxation generates a\ntransverse velocity component and a finite angular displacement, which accumulates\ninto net rotation over repeated cycles. This mechanism is supported by our\nmicromagnetic simulations (Extended Data Fig. 7–8) and analytical treatment based on\nthe Thiele equation (Supplementary Note 2). The essential role of relaxation time has\nbeen clarified in the revised main text (see line 286-291 in the revised manuscript), as\ndescribed below:\n“These results suggest that the continuous rotation of a skyrmion arises from\nrepeated pulsed excitations followed by relaxation dynamics. During each pulse, the\nsystem is displaced from equilibrium state, and in the subsequent relaxation state,\nboundary confinement and the gyrotropic term dominate, resulting in a finite angular\ndisplacement. Repetition of this excitation–relaxation cycle accumulates into a net\nrotational motion of a skyrmion.”\nImportantly, this interpretation is directly supported by additional experiments\nemploying different pulse widths and relaxation intervals. When the relaxation interval\nwas fixed at 100 ms and the pulse width gradually increased (Fig. R1), the excessively\nlong pulse durations led to pronounced Joule heating. This heating markedly affected\nthe magnetic domain configurations, as evidenced by skyrmion elongation and partial\nannihilation in Fig. R1d–f and Fig. R1g–i.\nTo minimize these thermal effects and sample damage, the current density had to\nbe substantially reduced. However, at pulse widths of approximately 2 μs, the reduced\ncurrent density was no longer sufficient to drive skyrmion motion. These results clearly\nindicate that overly long current pulses are unfavorable for sustaining the observed\nrotational dynamics. The corresponding discussions have been added in the revised\nmanuscript (see line 177-181 in the revised manuscript), as described below:\n“Additional experiments with varying pulse widths (Extended Data Fig. 1) show\nthat excessively long pulses suppress the rotational motion due to enhanced Joule\nheating. Accordingly, the 20 ns pulse width used in Fig. 2 and Fig. 3 ensures efficient\ncurrent-driven excitation while minimizing thermal perturbations.”\nexcitation with a fixed pulse interval and varying pulse widths. a–c,\nCounterclockwise rotational motion of the skyrmion assembly driven by pulsed\ncurrents with a pulse width of 500 ns. d–f, Counterclockwise rotational response under\na pulse width of 1 μs, where noticeable elongation of skyrmions appears due to an\nincrease of Joule heat associated with the longer pulse duration. g–i, No rotational\nmotions of skyrmions under pulsed currents with a pulse width of 2 μs.\nTo systematically explore the role of relaxation time in skyrmion rotation, we fixed\nthe pulse width at 500 ns and varied the pulse interval while applying continuous pulse\ntrains. The representative frames extracted from in-situ L-TEM videos are shown in the\nrevised manuscript. We find that when the pulse interval is as short as 5 μs, the skyrmion\nassemblies don’t move. In contrast, increasing the interval to 10 μs restores clear\nrotational dynamics.\nThese results demonstrate that an adequate relaxation time is essential for enabling\nthe observed rotational motion, providing strong experimental support for our proposed\nexcitation–relaxation mechanism and theoretical framework. Therefore, in our\nexperiments, the pulse parameters were deliberately chosen to ensure sufficient\nrelaxation time while maintaining efficient current-driven excitation and minimizing\nthermal effects on the magnetic textures and the sample.\nwith a fixed pulse width of 500 ns and varying pulse intervals. a–c,\nRepresentative snapshots of the skyrmion assembly extracted at selected time points for a pulse interval\nof 5 μs, showing negligible motion. d–f, Corresponding snapshots for a pulse interval\nof 10 μs, where clear rotational dynamics of the skyrmion assembly are restored.\nThe corresponding data and discussion have been added to the revised manuscript.\n(see line 321-327 in the revised manuscript), as discussed below:\n“To systematically examine the role of relaxation time, the pulse width was fixed\nat 500 ns and the pulse interval was varied under continuous pulse trains.\nRepresentative frames extracted from the recorded videos are shown in Extended Data\nFig. 10. When the interval is 5 μs, the rotational motions disappear, whereas the interval\nup to 10 μs restores pronounced rotational dynamics. The results demonstrate that an\nadequate relaxation time is essential for enabling the observed rotational motion,\nproviding strong experimental support for our proposed excitation–relaxation\nmechanism and theoretical framework.”\nIn metallic ferromagnets, skyrmion and antiskyrmion dynamics are well described\nby the Landau–Lifshitz–Gilbert equation and its collective-coordinate reduction, the\nThiele equation, which is first order in time and contains no intrinsic mass term.\nAlthough an effective skyrmion mass has been proposed in theories beyond the Thiele\napproach (e.g., Phys. Rev. B 90, 174434), it is generally attributed to strong texture\ndistortions or coupling to internal modes. Within the spatial and temporal resolution of\nour L-TEM measurements, no pronounced transient oscillations or overshoots are\nobserved, indicating that effective inertia is not a dominant factor in the present\nparameter regime.\nFurthermore, our pulse excitation and relaxation times (tens of nanoseconds to\nmicroseconds) are much longer than the intrinsic precessional and internal relaxation\ntimes of the magnetic texture. Under these conditions, the skyrmion velocity quickly\nrelaxes to a force-balanced state during each cycle, strongly suppressing inertial\nmemory or transient overshoot effects. The corresponding discussion has been added\nto revised manuscript (see line 231-233 in the revised manuscript), as described below:\n“The absence of measurable texture distortions and excitation timescales far\nexceeding intrinsic relaxation times suggest that effective inertia plays a negligible role\nin the present regime.”\nComment 4. The experimental observation of current-driven antiskyrmion rotation\nrepresents a major strength of the manuscript. However, although recent experimental\nreports on current-driven antiskyrmion motion (such as sliding or confined motion)\nare cited, they are not discussed in sufficient depth. The manuscript would benefit from a\nclearer articulation of what is fundamentally new relative to these prior works, whether\nit is the rotational motion itself, the topology-controlled chirality, a comparative\nassessment of mobility, or the use of pulsed-current operation.\nResponse 4: We thank the referee for raising this important point regarding the\n11\ndistinction from prior studies. In previous experimental studies on current-driven\nantiskyrmion dynamics, the reported motion has been largely limited to constrained\nsliding along helical domains. In such scenarios, the motion is highly restricted and\nexhibits limited characteristic topological signatures, such as Hall-like transverse\ndeflection. Consequently, no experimental evidence has been provided to directly\ndemonstrate the topological contrast between antiskyrmions and skyrmions through\ntheir dynamics, let alone more complex motion modes.\nIn contrast, the present work reports the first real-space experimental observation\nof rotational dynamics in antiskyrmion assemblies driven by electric current. This\nrotation represents a qualitatively new dynamical regime that goes beyond confined\ntranslational motion and is also highly related to its topological properties. More\nimportantly, we demonstrate that antiskyrmions rotate opposite to skyrmions under\nidentical excitation, providing the first dynamic experimental confirmation of their\nopposite topological charges. This topology-dependent contrast cannot be inferred from\nstripe-confined sliding motion alone and has not been experimentally established in\nprior studies.\nTherefore, the key novelty of our work lies in revealing a previously unobserved\ntopology-controlled rotational sense of antiskyrmion dynamics and establishing a direct\nlink between their motion and intrinsic topological properties. This extends current-\ndriven antiskyrmion dynamics from restricted, geometry-guided motion to topology-\nrelated collective behavior.\nWe have revised the manuscript accordingly to clarify these distinctions and to\nbetter position our results relative to existing experimental reports (see line 68-74 in the\nrevised manuscript), as described below:\n“To date, only experimentally confirmed current-driven motion of antiskyrmions\nis their constrained sliding along stripe domains33,37, offering limited insight into\ndynamics guided by topological properties. More complex behaviors—such as\nrotational motion—have neither been experimentally demonstrated nor systematically\nexplored. This lack of experimental insight leaves a critical gap in understanding the\ntopological properties of antiskyrmion dynamics, underscoring the need for further\ninvestigation.”\nComment 5. The micromagnetic simulations employ a single-sublattice ferromagnetic\nLLG framework to model FNPP, a structurally and magnetically complex, anisotropic,\nmulti-element system. However, the validity of this approximation, as well as its\nlimitations, is not discussed. More importantly, thermal fluctuations are neglected,\ndespite the experiments being conducted at room temperature. In addition, the\nasymmetric DMI term is introduced without explicitly relating it to the underlying\nsymmetry. A more careful justification of the modelling assumptions used to explain\nthe experimental observations is therefore needed.\nResponse 5: We thank the referee for these instructive comments. In FNPP, the\nmagnetic framework is dominated by the Fe₃P-based backbone, which exhibits\nferromagnetic order at room temperature. The substitution of Ni and Pd primarily serves\nto tune the saturation magnetization, magnetic anisotropy, and spin–orbit–related\n12\ncoupling strengths, rather than introducing additional independent magnetic sublattices.\nParticularly, Ni₃P is paramagnetic at room temperature, and Pd is non-magnetic.\nConsequently, the dominant magnetic moments in FNPP originate from the Fe atoms.\nUnder these conditions, it is physically reasonable to treat FNPP as an effective single-\nsublattice ferromagnet when modeling its long-wavelength magnetic dynamics.\nMoreover, the skyrmions and antiksyrmions investigated in this work possess\ncharacteristic length scales on the order of hundreds of nanometers, far exceeding the\natomic lattice spacing. In this regime, the magnetization dynamics are governed by\ncontinuum energetics rather than atomistic details. Therefore, a continuum\nmicromagnetic description based on the LLG equation provides an appropriate and\nwidely adopted framework. Within this approximation, the system can be discretized\ninto nanoscale finite cells, each representing an effective magnetic moment, and the\ninteractions between these elements capture exchange, anisotropy, DMI,\ndemagnetization, and current-induced torques. The corresponding discussion has been\nadded to revised manuscript (see line 381-384 in the revised manuscript), as described\nbelow:\n“In the FNPP, Ni, Pd, and P do not contribute appreciably to the magnetic moment\nat room temperature. Instead, the magnetism is dominated by the Fe atoms. Therefore,\na single-sublattice LLG framework was used in the micromagnetic simulations to\ndescribe the system.”\nBesides, our experiments show that skyrmions and antiskyrmions stabilized at\nroom temperature exhibit no spontaneous motion in the absence of pulsed current\nexcitation. This indicates that ambient thermal fluctuations alone do not induce\nappreciable dynamics in our system. Furthermore, COMSOL simulations under our\nexperimental conditions demonstrate that the temperature increase in the FNPP sample\nduring the current excitation is less than 2 K and remains spatially uniform, thereby\nexcluding effective thermal gradients. Consequently, thermal effects are not expected\nto play a dominant role in the observed rotational dynamics. Therefore, thermal\nfluctuations were neglected in micromagnetic simulations to focus on the intrinsic\nmechanisms of the current-driven rotation. The corresponding discussion has been\nadded to revised manuscript (see line 154-158, line 209-211 and line 397-399 in the\nrevised manuscript), as described below:\n“The antiskyrmion assembly is robust and reproducible without spontaneous\nannihilation or significant rearrangement during L-TEM imaging, indicating that the\nambient thermal fluctuations alone are insufficient to induce any observable dynamics\nsuch as Brownian motion in the present experimental setup.” (line 154-158)\n“COMSOL simulations, as detailed in the Methods section, indicate that the current\ninhomogeneity and temperature gradient under pulsed currents (Extended Data Fig. 5)\nare minimal and negligible for the rotational dynamics.” (line 209-211)\n“Because of the negligible skyrmion dynamics caused by thermal fluctuations at\nroom temperature in the present experimental setup, the simulation has been employed\nat T = 0 K.” (line 397-399)\ncrystal symmetry enters the micromagnetic description primarily through the form of\nthe energy terms rather than through atomistic details. Accordingly, the anisotropic\nDMI implemented in our simulation is expressed as:\nHere, axes and 𝜀 𝑑 𝑚ar𝑖e = de 𝐷 f [ in 𝒎 ed ∙ (t 𝒙̂ o × co 𝜕 r𝑥re 𝒎 sp)o − nd 𝒎 to ∙ ( t 𝒚 h ̂ e × c 𝜕 ry𝑦s 𝒎 ta ) ll ] ographic and\ndirections of the experimental sample, respectively. This construction explicitly\n𝑥 𝑦 [110]\nbreaks the rotational symmetry between the two in-plane directions, indirectly\n[1¯10]\nreflecting the anisotropic crystal symmetry of the material. As a direct consequence,\nthe simulated spin textures naturally develop elliptical skyrmions and antiskyrmions, in\nexcellent agreement with the experimentally observed domain morphologies. The\ncorresponding discussion has been added to revised manuscript (see line 402-405 in the\nrevised manuscript), as described below:\n“In the simulation, and were defined to correspond to the crystallographic\nand directions of the experimental sample respectively, ensuring the\nx y\nconsistency between the simulations and the experimental geometry.”\n[110] [1¯10]\nComment 6. There are also a few quantitative rigors of the experimental analysis, as\nlisted below:\n- Angular velocities are reported without error bars or uncertainty estimates\n- The number of tracked skyrmions at each radial position is limited\n- Possible systematic effects (projection in L-TEM, drift, defocus variation) are not\ndiscussed\nAt a minimum, the authors should comment on experimental uncertainties and the\nreproducibility of the results.\nResponse 6: We thank the referee very much for this constructive comment. We have\nrevised Fig. 3d and Fig. 3e to include uncertainties and reproducibility. For each radial\nposition, angular velocities were extracted from three representative skyrmions and are\nnow presented with error bars. The observed scatter in angular velocity originates from\nexperimental uncertainties like statistical uncertainty and local pinning effects. These\ncontributions are reflected in the error bars. Nevertheless, the overall radial dependence\nremains robust, supporting the generality of our conclusions.\nThe potential sample drift was carefully minimized in our experiment. All L-TEM\nmeasurements were performed at relatively low magnification (< 8k), and the sample\nwas allowed to stabilize prior to data acquisition. Under these conditions, no\nmeasurable drift exceeding the spatial resolution was observed, ensuring a negligible\ninfluence on the extracted angular velocities.\nAlthough real samples inevitably exhibit spatial thickness variations, which may\nintroduce local defocus differences, in our experiment the thickness variation\nintroduced during FIB preparation was below 50 nm across the sample. Compared with\nthe applied defocus value (~600 μm) used for magnetic contrast, this corresponds to a\nrelative variation below 10⁻⁴ and thus has a negligible influence on the quantitative\nanalysis.\nrotation angle. However, the relatively small tiling (~5°) angles in our experiment\nminimizes the effect. The tilt introduces a height difference of approximately 0.9 μm\nacross a 10 μm field of view, corresponding to a relative defocus variation of ~10⁻³,\nwhich remains negligible compared with the large defocus employed. Furthermore,\nprojection effects associated with sample tilt introduce only a minor correction to the\nin-plane rotation angle. For example, a true rotation angle of 10° would be modified by\napproximately 0.04°, which is well below the spatial and angular resolution of the L-\nTEM measurements and therefore does not affect the statistical analysis.\nThese discussions on experimental uncertainty have now been added to the revised\nmanuscript to further clarify the robustness of our quantitative analysis. (see line 213-\n214 and line 216-219 in the revised manuscript), as described below:\n“For each radial position, angular velocities were extracted from three\nrepresentative skyrmions.” (line 213-214)\n“The error bars reflect the experimental uncertainties like statistical uncertainty and\nlocal pinning effects. Owing to the small sample tilting angle ( 5 degrees) and minimal\nthickness variation (< 50 nm) in our experiment, their influence on the extracted angular\n∼\nvelocities is negligible.” (line 216-219)\nMinor comments:\nComment 7. The manuscript is a bit long, with repetitive background material that\ncould be shortened.\nResponse 7: Thank the referee very much for the helpful comment. We have revised\nthe manuscript to reduce repetition in the background material and to improve\nconciseness accordingly.\nComment 8. Several grammatical issues and typographical errors remain (e.g. figure\ncaptions, plural agreement).\nResponse 8: Thank the referee very much for the careful reading, and then the good\ncomment. In response, we have thoroughly revised the text to correct grammatical\nerrors, typographical mistakes, and issues related to plural agreement, including those\nin the figure captions. The manuscript has been carefully edited throughout to improve\nclarity and consistency.\nComment 9. Some figure captions lack sufficient standalone clarity.\nResponse 9: Thank the referee very much for the helpful comment. Accordingly, we\nhave carefully revised the figure captions to enhance their standalone clarity and ensure\nthat each figure can be understood without direct reference to the main text.\nComment 10. Data availability is stated, but no guidance is given on accessing raw L-\nTEM videos or tracking data.\nResponse 10:Thank the referee very much for the helpful comment. We have added a\nnew supplementary file to provide a clearer description of the uploaded raw video data\n15\nand to clarify their correspondence with the figures presented in the main text.\nComment 11. The discussion of potential device implications remains speculative and\ncould be better focused or shortened.\nResponse 11: Thank the referee very much for the helpful comment. In response, we\nhave streamlined the discussion of potential device implications, removed speculative\nstatements and refocused the text on the fundamental physical insights.\nComments: In summary, the manuscript presents high-quality experimental data and\nreports an important observation, particularly concerning antiskyrmion dynamics.\nNevertheless, to meet the standards of Nature Communications, the authors need to\naddress the points raised above. Once these issues are satisfactorily resolved, the work\nwould represent a strong and impactful contribution to the field of topological spin\ntextures and merit consideration for publication.\nResponse: We sincerely thank the referee for this constructive summary and positive\nassessment of our work. Following the valuable comments, we have carefully revised\nthe manuscript to address all major and minor concerns. We hope the revised version\nnow meets the standards of Nature Communications and appreciate the opportunity for\nfurther consideration.\n16"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have revised the manuscript carefully. All my previous concerns are cleared. I suggest to publish the manuscript.", "reviewer_2": "I am quite pleased with the manuscript in its revised form. There is ample new material contained in it. I recommend acceptance.\nFor the sake of the readers and particularly experimentalists who would like to follow this work, I suggest giving some more details in the Methods section on (a) the FIB method for preparing samples and (b) how the magnetic fields are created within the confines of the electron microscope.\nI do not need to review the manuscript anymore from this point.", "reviewer_3": "The authors have addressed more or less all of the concerns raised in my previous review and have made the necessary revisions accordingly. The responses provided are satisfactory, and the modifications incorporated into the manuscript have improved its overall clarity and quality. In light of these revisions, I believe that the manuscript now meets the standards required for publication. Therefore, in my opinion, the current version of the manuscript can be considered suitable for publication in its present form in Nature Communications."}, "author_response": "Authors’ response to referees’ comments\nComments from Referee # 1:\nComments: The authors have revised the manuscript carefully. All my previous\nconcerns are cleared. I suggest to publish the manuscript.\nResponse: We sincerely thank the referee for the positive assessment and greatly\nappreciate the referee’s support for the publication.\nComments from Referee # 2:\nComments: I am quite pleased with the manuscript in its revised form. There is ample\nnew material contained in it. I recommend acceptance.\nFor the sake of the readers and particularly experimentalists who would like to\nfollow this work, I suggest giving some more details in the Methods section on (a) the\nFIB method for preparing samples and (b) how the magnetic fields are created within\nthe confines of the electron microscope.\nI do not need to review the manuscript anymore from this point.\nOverall I recommend acceptance of the paper.\nResponse:We sincerely thank the referee for the positive assessment of our work and\nconstructive comments. We have added more details in the Methods section to more\nclearly explain these points, as described below:\n“Additionally, a round-shaped sample plate was fabricated, featuring two tungsten\nelectrode pads and a central FNPP circle encased by a tungsten ring. For preparation of\nthe round-shaped sample, part of the ring region was first removed by FIB milling,\nfollowed by tungsten deposition into the milled area. The remaining FNPP in the ring\nregion was then removed, leaving only the tungsten in that area.” (line 316-321 in main\ntext)\n“In L-TEM, since the objective lens is essentially a strong electromagnetic lens\ncomposed of current-carrying coils, the application of vertical magnetic fields was\ncontrolled by adjusting the objective-lens current.” (line 326-329 in main text)\nComments from Referee # 3:\nComments: The authors have addressed more or less all of the concerns raised in my\nprevious review and have made the necessary revisions accordingly. The responses\nprovided are satisfactory, and the modifications incorporated into the manuscript have\nimproved its overall clarity and quality. In light of these revisions, I believe that the\nmanuscript now meets the standards required for publication. Therefore, in my opinion,\nthe current version of the manuscript can be considered suitable for publication in its\npresent form in Nature Communications.\nResponse:We thank the referee for the positive assessment and greatly appreciate the\nreferee’s support for the publication.\n1"}]

Decision letter