Antiskyrmions and their electrical footprint in crystalline mesoscale structures of Mn$_{1.4}$PtSn

Skyrmionic materials hold the potential for future information technologies, such as racetrack memories. Key to that advancement are systems that exhibit high tunability and scalability, with stored information being easy to read and write by means of all-electrical techniques. Topological magnetic excitations such as skyrmions and antiskyrmions, give rise to a characteristic topological Hall effect. However, the electrical detection of antiskyrmions, in both thin films and bulk samples has been challenging to date. Here, we apply magneto-optical microscopy combined with electrical transport to explore the antiskyrmion phase as it emerges in crystalline mesoscale structures of the Heusler magnet Mn$_{1.4}$PtSn. We reveal the Hall signature of antiskyrmions in line with our theoretical model, comprising anomalous and topological components. We examine its dependence on the vertical device thickness, field orientation, and temperature. Our atomistic simulations and experimental anisotropy studies demonstrate the link between antiskyrmions and a complex magnetism that consists of competing ferromagnetic, antiferromagnetic, and chiral exchange interactions, not captured by micromagnetic simulations.


INTRODUCTION
The field of skyrmionics comprises new phases of magnetic systems, where the individual spins align as whirls [1][2][3][4], that hold the potential to advance the understanding of topology in condensed-matter physics. Fundamentally, magnetic textures, such as skyrmions [3,5], antiskyrmions [6,7] (ASKs) and related skyrmionic systems [8][9][10], are distinguished by the relative rotation of the individual magnetic moments with respect to one another in the presence of a magnetically ordered background. This topologically protected state can be used to robustly carry and store high-density information at fast speed with low power consumption, as proposed in 2 racetrack memory spintronic devices [11,12]. Solitary skyrmion states as well as crystal lattices of skyrmion systems have been reported for many magnetic crystals that break inversion symmetry such as bulk chiral magnetic compounds [5,13], multilayer heterostructures [14,15], thin films [16], oxides [17], and, more recently, tetragonal Heusler systems [6,18]. While experimental observations from, e.g., Lorentz transmission electron microscopy [6,19,20], neutron scattering [5], and magnetic force microscopy [21,22] have shown great advancement over the last decades, detection by electrical means is required for the realization of energy-efficient spintronic devices [11,12], probabilistic [23] and neuromorphic computing [24]. Furthermore, a pivotal point for applications is the ability to control and scale skyrmionics [25,26], or even to transform one species of magnetic texture into another in multiskyrmion systems [18,27,28].
The tetragonal Heusler compound Mn 1.4 PtSn [29] has recently gained a lot of interest as it hosts distinct topological states. Most excitingly, an ASK lattice can be established at room temperature. Besides the high tunability of the Heusler compounds [22,27,30], the competition of magnetic exchange interactions and anisotropy lead to an enhanced temperature range over which ASKs can be stabilized in comparison to skyrmion bulk systems. Bulk Mn 1.4 PtSn displays a complex correlation of exchange between two Mn sublattices that leads to a spin reorientation from a collinear to a noncollinear arrangement at temperatures T SR ≈ 170 K and below, which contributes to the anisotropy at low temperatures. At temperatures above the spin reorientation, sophisticated magnetic textures can be realized thanks to the combination of in-plane (perpendicular to the c axis) and anisotropic Dzyaloshinskii-Moriya interaction (DMI), crystal anisotropy along the tetragonal axis, and dipoledipole interaction due to large moments (> 4µ B ). Above T SR , in the absence of an external magnetic field, the compound prefers a spin-spiral state that is directly connected to the ratio of the exchange interaction and DMI. For a finite range of external magnetic fields, a long-range (> 100 nm) hexagonal ASK phase as well as a short-range noncoplanar state (< 1 nm) were found [6].
These ASKs exhibit an extraordinary stability to sample-thickness variations to at least a few hundred nanometers confirmed by Lorentz transmission electron microscopy [20] and x-ray studies [31]. The latter confirmed bulk skyrmions in few microns thick samples. The diameter of the ASKs scales linearly with thickness and was found to grow to multiple hundreds of nanometers in micron-sized samples [22,27]. For very thin platelets, the phase diagram becomes more complex. Multiple skyrmion-like textures that are different above and below T SR can be stabilized by varying the field orientation [18,30]. Surprisingly, neutron-scattering experiments on bulk Mn 1.4 PtSn did not resolve any ASKs [32]. Even more intriguing, a huge additional component in the Hall effect, associated with a topological origin, emerges in the low-temperature region of the phase diagram, just below T SR , exceeding expectations for skyrmions [33]. This is likely related to noncollinear magnetism. Therefore, it is still an open question if and how ASKs contribute to the Hall effect and, in particular, how strong that effect is.
The topological nature of skyrmions and ASKs should be observable in terms of a unique response to an external electric field, i.e., a topological Hall effect (THE) in electrical transport. The THE belongs to the family of Berry-curvature Hall effects [34,35], where emergent fields due to a real-space variation of the local magnetism, describable by the Berry curvature [36], are the origin of a transverse voltage. Closely related to the THE is the anomalous Hall effect (AHE), due to the interaction of localized and conduction electrons [37]. In a finite magnetic field, the Hall response is a voltage transverse to the applied current. This is sketched in Fig. 1a. Phenomenologically, the total Hall effect can be described by a superposition of three contributions [37]: The ordinary Hall effect (OHE) that scales with the external magnetic field, H; The AHE, which scales with the magnetization M, describable in terms of an intrinsic momentum-space Berry curvature and extrinsic scattering mechanisms, such as skew and side-jump scattering [37,38]; The THE, which can be described in terms of the chiral product: where S i is the local spin orientation. The latter may be utilized to identify the sign of the topological charge, being opposite for skyrmions and ASKs, respectively.
In this work, we successfully combined in-situ transport with magneto-optical microscopy applied to mesoscale structures fabricated from single-crystals by focused- In Fig. 2a, we sketch the spin configuration and the respective interactions present in Mn 1.4 PtSn. The combination of DMI, induced by the D2d symmetry, FM and AFM interaction, is an ideal foundation for diverse magnetic textures [33]. We find that the physics of topological textures in Mn 1.4 PtSn can be accurately modeled purely from the exchange interactions and magnetocrystalline anisotropy in three atomic layers of the two magnetic sublattices [10,42,43]. More significantly, the ratios of the exchange constants determine the underlying physics; therefore, we tuned The calculated out-of-plane (in-plane) component of the spin-spiral and ASK state is shown in the upper (lower) panel in Fig. 2b and 2c, respectively. The yielded magnetization hysteresis (Fig. 2d) is in line with our experimental observations of a finite hysteresis away from zero field for a transition from the spin-spiral ground state into a stable ASK phase (for further details see Supplementary Note 6). The calculated magnetization exhibits a weak region, that is a shallow slope change right before the saturation field, consistent with our THE observations (see Fig. 2d). The chiral product, χ ij leads to a topological winding number, w = 1/4π M · (δ xM × δ yM )dxdy, where δ i = δ/δ i , andM(r, t) = M(r, t)/|M| is the direction of the magnetization at each spatial position. The presence of w causes an emergent magnetic field, H e = 2 zxyM · (δ xM × δ yM ), originating from a real-space Berry curvature [34]. The so-called topological charge, q T , is negative for skyrmions and positive for ASKs.
Thereby, the THE is a direct link to the topology of the magnetic texture: where the Hall coefficient, R THE xy , is tied to the complex multiorbital electronic structure [44]. The composite nature of R THE xy and H e complicates the differentiation of skyrmionic configurations with distinct winding numbers across multiple materials.
However, in Mn 1.4 PtSn distinct magnetic textures are induced by the external field.
In the lower panel of Fig. 2d, we plot q T against the external field. In the field-up sweep, q T in the ASK phase exhibits an opposite sign as compared to the net charge in the low-field spin-spiral phase and extends to higher fields as compared to the reoccurring q T in the down sweep (orange curve). The down sweep (black curve) remains flat and only exhibits a positive contribution at lower fields, originating from the reestablishing spin-spiral phase. Hence, the subtraction of field-up and -down sweep mostly cancels out the component from the ordered phase and should, therefore, yield the Hall effect due to ASKs directly. In Mn 1.4 PtSn, ASK sizes can reach a few hundred nanometers in diameter. For these huge objects the expected THE is rather small [39]. The hysteresis shows up for both quantities and a linear fit of the slope in the differences ∆ρ xy and ∆M just below the saturation field reveals a shoulder-like feature being discernible in both quantities. The insets of Fig. 2g present the contribution we associate with the presence of ASKs. Apparently, the shoulder can be addressed to an anomalous Hall component due to the emergence of the ASK textures. The magnetic moment scales with the volume of the material and the observed feature in the magnetization is hardly distinguishable from the noise background. We also know that ASKs only show up for micron-thick devices. Therefore, it is hard to trace the ASK feature for smaller thicknesses in the magnetization. Here, the Hall effect appears to be the ideal tool. In Supplementary Note 7 we provide subtraction results from our experimental transport data at various temperatures for device B with d = 1 µm. A clear feature can easily be traced upon varying the device thickness, temperature, and field orientation (see the following sections).
Thickness-and temperature-dependent Hall effect in the ASK phase While MOKE microscopy in combination with electrical transport allows for simultaneous measurements of the magnetic texture and the Hall effect in Mn 1.4 PtSn, it is limited in spatial resolution by the wavelength of the optical light used [41]. We, therefore, use magnetic force microscopy (MFM) to resolve the ASK lattice induced in samples of different thickness d. Figure 3a shows MFM images at zero and 0.55 T for a 5 µm and 1.5 µm thick sample. The spin-spiral domain bands and the ASK lattice are discernible with varying periodicity, depending on d (the lower d the smaller the magnetic periodicity). The fast Fourier transforms (FFTs), shown in the lower panels in Fig. 3a, highlight the change in topology as the system transforms from the spin-spiral into the hexagonal ASK phase. The size and periodicity of the ASKs are affected by the thickness as has been shown recently-the latter varies almost linearly with thickness [22,32].
This suggests that a THE induced by ASKs is expected to grow quadratically with decreasing d, as it is directly related to the ASK density, which is in turn proportional to the area occupied by each ASK. In Fig. 3b, we show ρ ASK xy at room temperature determined for various devices with different thicknesses between 2.4 and 0.8 µm. The overall magnitude varies between 40 and 50 nΩcm. We also studied the thickness dependence in more depth on transport devices E and G, presented in Fig. 3c. In these cases, we lowered the thickness by low-energy (5 kV) Ar-ion etching in steps and measured the room-temperature Hall effect after each thinning step. The overall trend of an increased response for smaller thickness is apparent. However, this may also originate from an increasing number of ASKs that can emerge in between the electrical Hall contacts due to their shrinking diameters upon device-thickness reduction. As was shown already for Néel skyrmions in Co nanolayers, the AHE of single skyrmions may be much stronger than the THE component [39]. Therefore, at this point, we can only provide an upper estimate of the THE: Namely, it must be smaller than a fraction of the magnitude of the detected ρ ASK xy . In Supplementary Note 7 we provide a theoretical estimate following the previous approaches [5,44] for a known ASK density. For our microscale devices the ASK size is approximately 100 nm, which would lead to a theoretically expected THE component of |ρ THE | ∼ 50 nΩcm.
Moreover, we find that ρ ASK xy remains as pronounced as at room temperature all the way down to temperatures close to T SR (see Fig. 3d and Supplementary Note 7).
Below T SR , it is strongly suppressed until it vanishes and additional step-like jumps in the Hall resistivity occur. The saturation field marks the upper boundary of the ASK field region (yellow). With decreasing temperature, it shifts towards higher values until saturating near T SR , where noncoplanar magnetism starts to take over. Its overall amplitude starts to subside around T SR as can be seen from Fig. 3e. Below 150 K, we cannot unambiguously link the observed deviations to skyrmions. To visualize the crossover around T SR we show the maximum ∆ρ xy plotted against the temperature in Fig. 3f. This demonstrates the gradual change of the overall behavior of the observed hysteresis near the spin-reorientation transition region (shaded region) associated with the onset of noncoplanar order and a reorientation of the magnetic easy axis. Further microscopic and spectroscopic studies are highly desirable in order to understand the details of the temperature dependence.

Emergence of a new state below T SR
Below T SR , the overall Hall response decreases more rapidly and the hysteresis in field narrows (see Fig. 4a). We present a detailed angular study on device B in Supplementary Note 8. As we tilt the magnetic field away from H c, the hysteresis subsides (it is absent in the high-temperature curves shown in Fig. 4b).
For temperatures near and below T SR and the field being oriented away from the c direction, we observe a prominent hump-like enhancement of ρ xy (H) close to the saturation field (see for example the maximum in ρ xy (H) at 180 K and H b in Fig. 4b). This feature was already explored for bulk samples [29,33]. Its origin is a THE linked to the establishment of a noncoplanar spin structure with a strong real-space Berry curvature. Excitingly, for fields along the b axis, ρ xy (H) starts acquiring a negative high-field slope and even changes sign at low temperatures ( Fig. 4b). At 2 K, we observe a surprisingly large transverse transport signal even without any out-of-plane field component. For angles within only a few degrees off the in-plane orientation (i.e., θ = 90 • ) it is accompanied by a new hysteresis that exhibits opposite sign as compared to the one for H c. As we fine-tune the angular step width, we observe a hysteresis that extends to fields even larger than 4 T, being extremely sensitive to minute changes in θ (see Supplementary Note 8). Moreover, step-like changes in the hysteretic part of ρ xy (H) occur (see for example the 150 K curve in Fig. 4b). Hence, our data indicates an intriguing magnetic behavior for low temperature and field aligned within the ab planes, likely originating from the noncoplanar magnetism with a much stronger magnetocrystalline anisotropy for low temperatures. As we show in Fig. 4d and Supplementary Note 8, both the AHE and the OHE deviate from the conventional cos θ dependence, observable at room temperature, as temperature is tuned to T SR and below (see red dashed fits). This further indicates an enhanced saturation field for the in-plane field orientation. These temperature-and angle-dependent changes are independent of the device thickness (confirmed for devices A, B, and C).
To further explore the temperature-dependent change in the magnetic exchange interactions we conducted fixed-frequency FMR experiments on a thin (∼ 800 nm) lamella sample. In Fig. 4e, we present FMR spectra recorded at 260 and 10 K for two field directions each. The spectra were recorded while sweeping the field from the negative field-polarized state to the positive side and back, i.e., following the Hall-effect hysteresis curves. For further details, see also Supplementary Note 9.
We also observe the hysteresis around 0.5 T for H c in the FMR data (see black curve in Fig. 4e and inset). Furthermore, at 260 K, we observe one narrow resonance mode attributed to the field-polarized state. As we tilt the field towards H b, the resonance preserves its narrow shape and the nearly isotropic angular dependence, varying between 0.62 and 0.75 T. This suggests that above T SR , the magnetocrystalline anisotropy is weak. The maximum resonance field (corresponding to a magnetically hard axis) occurs around 24 • . An approximate fit to the angle-dependent data yields a weak effective in-plane-anisotropy field, which is comparable in magnitude to the ASK compound Fe 1.9 Ni 0.9 Pd 0.2 P [28]. At temperatures well below T SR , the main resonance mode becomes broad for H c and asymmetric. In Fig. 4f, the error bars represent the resonance linewidth (half width at half maximum). Such a linewidth broadening is typical for the so-called field dragging, i.e., when H is not parallel to M due to a nonuniform magnetization for example. We observe that for high tilt angles close to the b axis the resonances become narrower and shift to much lower fields of 0.1 T. The overall angular dependence indicates that the anisotropy is much stronger below T SR . These significant changes in the magnetic anisotropies can explain the previously reported vanishing of ASKs and emergence of Bloch-type skyrmions at low temperatures [30]. In addition, the observed hysteresis in transport at high fields indicates new hard-magnetic behavior for the in-plane field orientation.
This, therefore, may provide the right environment of the establishment of other distinct skyrmionics, e.g., AFM bimerons [45].

CONCLUSION
In summary, we demonstrated the direct detection of the Hall resistivity due to the emergence of ASKs in the Heusler compound Mn 1.4 PtSn by combining electrical transport with magneto-optical microscopy, in line with our theoretical model. Our studies of transport, magnetization, and ferromagnetic resonance reveal a semi-hard magnetic phase, and, hence, the unique link of ASKs to anisotropies, not possible in the cubic B20 compounds with weak anisotropy. We find this semi-hard magnetic behavior to strengthen as we reduce the thickness of the sample, which is particularly interesting for scalable electronic devices [46][47][48] Fig. 1b).
Dimension of the devices were determined using a scanning electron microscope. The authors declare no competing interests.

DATA AVAILABILITY
All data supporting the findings of this study are included in the main text and Supplementary Information. Raw data in ASCII format will be provided by the corresponding author upon reasonable request.

SUPPLEMENTARY MATERIALS
In addition, we provide the following supplementary material:

This PDF file includes:
Supplementary Notes 1 to 9 Supplementary Figures 1 to 16 Other Supplementary Materials for this manuscript include the following:

Fig. S2. Different FIB-microstructure designs used in our study (see text below).
In this study, we investigated the transport properties of more than 5 different devices, covering various thicknesses and designs. In Fig. S2, we show SEM images of devices A to E (left) and sketches of the respective contact designs (right). Note: Device F, shown in Fig. 2d and in the Supplementary Movie sequence, is not shown here. Device A (also shown in Fig. 1b in the main text) is a meander-shaped Hall-bar device, divided into steps with different thicknesses, namely d = 11.2, 8.4, and 2.7 µm. We chose this design to allow for simultaneous measurements on different material thicknesses. The crystallographic c axis is oriented out of plane and the current is applied within the ab plane. Another characteristic of the structure is the particular arrangement of the electrical contacts. For each thickness, there are three contacts, two on one side and one centered on the opposite side. When measuring on opposite sides, the measured signal is a superposition of ρxx (H) and ρxy(H). The latter is antisymmetric to magnetic field, and, hence, can be separated numerically after measuring both components simultaneously via one diagonal contact pair. We have fabricated various microstructures from Mn1.4PtSn, individually tailored to serve specific tasks. Device C is a Hall-bar device with d = 2.4 µm. We did not cut into the lamella and only deposited gold contacts on top.
Device D is a series connected Hall-bar device with the b axis, i.e., the [010] direction, oriented perpendicular to the substrate surface. In the two series-connected sections with thickness d = 1.6 µm, the current is forced along the a axis and along the c axis, respectively. This enables a simultaneous Hall measurement of both current configurations.
Device E is a Hall-bar device with d = 0.7 µm. (A similar device F with d = 0.8 µm was produced in the same way for the results presented in Fig. 1e and d, and in the video). The lamella was transferred onto a glass substrate without additional glue or epoxy. Platinum contacts were deposited on the sides by the help of a FIB gas-injection system, providing connections to gold leads on the glass substrate that were fabricated by sputter deposition.

Supplementary Note 3. Basic transport characterization
When examining structures produced by FIB milling, it is important to ensure that the physical properties of the material stay unaltered. Typically, the modification process with Ga ions leaves an amorphized, only few nanometer-thick layer on the surface of the specimen. The implantation of Ga ions depends on the acceleration voltage. The typical value of 30 eV gives rise to primary and secondary effects penetrating the surface with a maximum depth of 10-20 nm. For metallic materials used in devices with micron thicknesses, the electrical properties are preserved, since the major part of the current flows through the damage-free part of the sample. In addition, inhomogeneous mechanical stress may affect the devices, specific for each design. As we will exemplify in the following, the main transport characteristics, such as the temperature and field dependence of the resistivity and the Hall effect are not altered as compared to the results previously reported for bulk single crystals, see Vir et al. [29]. Fig. S3 shows R(T) for j || a and j || c measured simultaneously in device D. In comparison to a bulk sample the overall T dependence of R is preserved. Furthermore, the resistivity anisotropy remains very weak. Only slight variations between the two current directions are observed that may indicate weak differences in the experienced stress induced by the coupling to the substrate. Nevertheless, the striking feature, i.e., the kink in the slope at the spin-reorientation transition temperature, TSR ≈ 170 K, is not altered (see also Fig. S4).
In Fig. S5, we exemplify the magnetoresistance (MR) in relative units and the Hall resistivity measured on device D for the two distinct transport directions, j || a and j || c. It matches previously reported bulk data [29,33]. Both transport channels exhibit an overall negative MR of a few percent in magnitude for the full T range between 2 and 300 K. At low fields, below the saturation field µ0Hs ≈ 1 T, variations of the slope are observable, likely related to the helical phase. Once we cool down to below TSR, this anomaly grows significantly. In contrast, for H || c (realized for device B) the MR changes to an almost linear field dependence once T < TSR. The low-temperature Hall resistivity (see 150 and 2 K in Fig. S5) exhibits a broad hump for fields below 3 T aligned within the ab plane, matching with previous reports for bulk single crystals [33]. This hump is associated with the topological Hall contribution due to the noncoplanar magnetic phase, established below TSR. For field aligned along the c direction, the Hall resistivity acquires a nonsaturating positive slope at high fields, above the sharp shoulder at Hs accompanied with a small hysteresis as we describe in the main text (see Fig. 1c). This hysteretic behavior only emerges in thin devices and intensifies the lower the thickness, as we discuss in the main text.

Supplementary Note 4. Magnetization of a thin, lamella-shaped sample of Mn 1.4 PtSn
A reliable measurement of the magnetization is not feasible for the microstructured devices due to their small volume. Nevertheless, the basic magnetic behavior of the material is important for understanding the magnetotransport experiments. In particular, the analysis of the anomalous Hall effect, AHE , which is proportional to the magnetization, is needed for disentangling the various Hall effects. In Fig. S6, we present magnetization data recorded for a FIB-cut lamella with (a x b x c) = (60 x 160 x 2.4) µm 3 , which was later used to fabricate device C. We performed the measurements in a Quantum Design 7 T SQUID magnetometer. For H || c, we observe a hysteresis close to the saturation field (see Fig. S6a). This feature is similar to what we observed in the magnetotransport of the device shown in the main text in Fig. 1c and e. It, however, does not occur in bulk samples [29]. Below TSR, we also observed a hysteresis for a high tilt angle of 85° off the c axis towards the a axis (see Fig. S6b). Note: The 310 K difference plot in Fig. S6c exhibits two maxima, which indicate a nonuniform thickness of the thin (but large in area) lamella sample. This may affect the nucleation process of the chiral domains in the magnetizing loop, on the way back to zero. Indeed, we were able to confirm this feature by Kerr-effect measurements on the same lamella (see Supplementary Note 5).

Supplementary Note 5. Polar magneto-optical Kerr effect (MOKE) details
For our MOKE experiments, we transferred thin platelets prepared by FIB into a glue droplet on a glass substrate. We reduced the amorphized surface-layer thickness by an additional Ar-ion etching step afterwards. Low Ar-plasma voltages of the order of 1 kV keep the impact-layer thickness to a few nanometers. We exemplify the MOKE imaging process recorded for the 2.4 µm thick sample that was used later on for device C. First, we recorded a background image in the field-polarized state, where no magnetic textures are visible any more. The background is then subtracted from each image taken at lower fields in order to better visualize the field-induced contrast changes.
In Fig. S7, we present a series of MOKE images collected for device C. A water-cooled electromagnet was used. The sample was sitting at a distance of approximately 5 mm above the pole of the magnet. Therefore, the absolute field at the sample level should be scaled by a factor 0.8. Starting from zero, the field was increased to +700 mT, in steps of 10 mT, where we took an image that was used as reference background. Next, we went back to zero, further to -700 mT, and back to +500 mT keeping the same step size. At 500 mT, we continued at a slower pace with 1 mT steps until +700 mT and returned to zero. During this slow field-stepping process the focus and contrast was manually adjusted to compensate for slight drifts. Figure S7a shows how the band domains slowly disintegrate into dot-like objects that assemble in a hexagonal structure at around 571 mT. This is exactly the point when the shoulder feature in the Hall effect emerges. As we further increase the field, these points remain fixed and gradually reduce in size until they vanish within the background noise. This suggests that these objects are no simple bubbles. The clear negative sign of the AHE provides strong evidence for ASKs as the local magnetization of these objects should decrease due to the zero net magnetization of ASKs. Once we turn back to lower fields, the sample remains in the field-polarized state until suddenly (within less than a millitesla) band domains and only few dots reestablish. As the field approaches zero, these patterns turn into longer continuous bands and the distance/width is shrinking with increasing field. Figure S8a-e shows Kerr-effect images of the whole lamella during the demagnetization, coming from the field-polarized regime. It can be clearly seen that the lamella is divided into two regions. While at 410 mT one region already shows the helical phase with domain bands along the ab direction, typical for Mn1.4PtSn, the other half is still in the field-polarized state. At 390 mT, the hysteresis loop is complete and magnetic spirals have formed throughout the sample. This proofs that the origin of the major hysteresis is related to the establishment of the helical spin-spiral phase. It also explains the double-step of the magnetization hysteresis for this particular sample, shown already above in Fig. S6c. In Fig. S8f, we show a zoomed-in version of the room temperature magnetization difference from Fig. S6c.  Fig. S6(c).

Supplementary Note 6. Atomistic spin-dynamic simulations
The origin of the noncollinear magnetic textures is due to a competition of classical exchange interactions between the inter-site interactions of the magnetic atoms Mn1 and Mn2 (with Wykoff positions 4a and 8d, respectively) and the intra-site interactions. This can be modeled by a classical exchange Hamiltonian Here, Si is the unit vector of spin for each sublattice (l), J1 is the inter-sublattice interaction, and J2 and J3 are the 8d Mn2 and 4a Mn1 intra-site interactions for each sublattice, respectively. All interactions together are necessary for a noncollinear spin texture. The positive 8d Mn2 interactions, J2, align the spins parallel and are the largest contribution to the ordering temperature. The 4a Mn1 interactions, J3, are negative, which favors an antiparallel alignment. Lastly, the intersite interaction, J1, is positive and causes a noncoplanar magnetic structure depending on the relation with the other interactions. When J1 is weak (J1 < ¼ J2), the coupling between the 8d Mn2 and 4a Mn1 sublattice is weak, and the noncoplanar structure stabilizes as the ground state. When J1 is strengthened, the 4a Mn1 sublattice follows that of the 8d Mn2 sublattice for a collinear alignment and is, therefore, the origin of the spin reorientation. The DMI Hamiltonian is where D2 and D3 are nominally equal and opposite. Lastly, the effective uniaxial-anisotropy field and the external magnetic field are modeled by ku is the uniaxial anisotropy constant and Hext is the external field. The total Hamiltonian is the sum of the previously mentioned contributions tot = exch + DMI + field . We map these interactions onto the atomistic Landau-Lifshitz-Gilbert equation The gyromagnetic ratio is and is the microscopic damping. eff = − 1 tot + th is the effective field with thermodynamic fluctuations ( th ) modeled by Langevin dynamics and tot , which includes the exchange interactions, a uniaxial anisotropy and the external magnetic field. Within the vampire code [33,53], we simulate a 50 x 50 nm trilayer of Mn1.4PtSn for periodic boundary conditions in the a direction and an open boundary in the b and c crystal axis. The lattice constant is chosen to be 0.2715 nm. In Fig. S9, we plot the M(H) loops of the spin texture as a function of the field along the c direction for both polar angles (top) and azimuthal angles (bottom). In high field, the ASK state begins to form. In Fig. S10, we show similar plots of the skyrmion state as a function of temperature and field. The color red marks spins that align with the magnetic field, whose direction points out of the plane and blue marks spins pointing into the plane. In a field range between 1.6 and 2.6 T the ASK state forms from the low-field helical phase and is destroyed at higher field. The ASK state is confirmed by the pin-wheeled shape of the azimuthal plots, where red and blue mark positive and negative in-plane components.  1.7, 3.4, 4.3 meV). For the polar angle, θ we plot the contour density of each spin as cos(θ), correspondingly for the azimuthal angle, φ, we plot sin(φ). For larger anisotropy the ASK state becomes more stable.

Supplementary Note 7. Extraction of the ASK Hall-resistivity component
In order to extract the antiskyrmionic contribution to the Hall effect, ASK , we need to determine the ordinary (orbital) and anomalous (magnetic) components, OHE (M) and AHE , respectively (see Eq. 1 in the main text). The first can be obtained from the high-field slope in the saturated (polarized) state. AHE (M) is proportional to the magnetization, ( ). However, due to the tiny volume of the microstructured devices, and thus the hardly resolvable magnetic moment, we had to choose another approach. As we have shown in Supplementary Note S4, we observe a hysteresis in the magnetization in a thin platelet of Mn1.4PtSn. Apparently, there is a direct relation to the Hall response. In Fig. S11, we show four Hall-response examples recorded at 300, 180, 150, and 2 K. Interestingly, there is a shoulder-like feature in the up sweeps right before saturation is reached. Supported by polar MOKE microscopy and MFM measurements we attribute this to the formation of the ASK lattice and the associated ASK .
With increasing field, the ASK lattice establishes from the helical phase at field c1 . It is stable up to the transition to the field-polarized state at c2 . As we decrease the field, no spin textures are observed in the MOKE measurements until the hysteresis closes. Thus, the difference ∆ directly reflects the influence of the spin textures compared to the field-polarized state (see Fig. S11c, d). Our in-situ measurements in combination with MOKE on devices C and E provide unambiguous evidence for the ASK lattice as the origin of this sudden change in the Hall signal. We associate this feature with the ASK Hall-resistivity component. As the topological charge for these magnetic excitations is opposite to that of Bloch-and Néel-type skyrmions a negative THE is expected. However, its magnitude is expected to be very small because of the huge dimensions of the ASKs for micron-thick devices. We also show in the main text for the example of a 2.4 µm thin sample that there also seems to be a signature in the magnetization (Fig. 2g). Unfortunately, it is already within the noise for this unstructured relatively large lamella-shaped sample, and, hence, was not resolvable for thinner structured devices. We extract ASK from ∆ by subtracting off a linear fit as demonstrated in Figs. S11c, d, e, and f. Here, we also present the magnetoresistivity data. Although the overall magnitude with 10 -3 % is rather small, we observe a hysteretic feature as well. Similar to the magnetization, no separate feature is discernible in the respective field range where ASKs are detected, marked by the yellow regions in Fig. S12. For temperatures below TSR, where the noncoplanar phase is completely dominant, the overall hysteretic behavior reduces and additional step-like changes occur. Figure S12 shows the extracted ASK over the entire temperature range studied. Above 170 K, the picture is consistent and the shoulder-like feature can be distinctively traced. However, below and down to base temperature we cannot directly link our data to ASKs. It becomes clear that TSR has a significant effect on the ASK signature. It recently has been demonstrated for electron-transparent lamellas that at low temperature Bloch-type skyrmions emerge [18]. The presence of skyrmionic textures with varying net magnetizations or topological winding would result in compensating/varying AHE and THE components. The schematic temperature dependence of the ASK region is highlighted in Fig. S13a. MFM confirmed ASKs for temperatures as indicated (blue diamonds) above TSR. Further detailed magneto-optical measurements are necessary to reveal the origin of the observed low-temperature behavior.
In Fig. S13b, we show ρ xy ASK extracted for device B with 1 µm thickness at various angles and at T = 180 and 150 K. Apparently, the ASK Hall-resistivity contribution is maintained also for tilts away from H || c. Even at 150 K, we were able to trace it until 70° tilted off the c direction.

Estimate of the THE component:
The Hall resistivity linked to the emergent field H e induced by the ASK phase may be estimated by [13]: where is the spin polarization of the charge carriers, 0′ is the Hall coefficient representing the effective charge density contributing to the THE of spin-up and spin-down carriers, and is the antiskyrmion density and Φ0 = ℎ/ is the magnetic flux quantum. R0 is derived from the difference of spin-up and spin-down contributions. A rough estimate was already provided by calculations [33]. R0 for Mn1.4PtSn was calculated to 0.5•10 -8 Ωcm/T. The experimental value, determined from the slope of the Hall resistivity in fields above 2 T at room temperature, is an order of magnitude larger with 4•10 -8 cm/T. The ASK density for a 1 m thick device can be approximated to 1/(100 nm) 2 . Assuming a low spin-polarization value of ~0.1 we would be able to attempt a rough estimate using the equation above. Hence, the expected THE component may be of the order of 1 ncm, which is similar to what was observed for MnSi [13].

Supplementary Note 8. Angle-resolved transport measurement
We investigated the Hall resistivity for various devices. In Fig. S14a, we show a detailed study of the angle-and field-dependent Hall resistivity conducted on device B with 1 µm thickness. The measurements were realized as illustrated in Fig. S14b by performing a field sweep at an angular configuration and then increasing the tilt angle. As expected, the general intensity of the Hall signal decreases with higher tilt angles. However, the characteristic transport properties of the material are clearly visible. Particularly striking is the THE of the noncoplanar spin structure, which starts to develop at 180 K and becomes even stronger at lower temperatures. This manifests itself in an increasingly pronounced maximum at higher tilt angles. The overall Hall signal exhibits a peculiar temperature dependence: Its high-field slope changes sign for < SR as the field orientation is approaching the in-plane direction. In Fig. 4d of the main text, we show the AHE component plotted against . The deviation from the conventional cos dependence is a consequence of the slope change at low temperature.
For < SR , a hysteresis in the Hall effect is observable at almost any field orientation, even close to H || b ( = 90°). The hysteretic field range shifts to higher field values with increasing tilt angle away from the c direction. Figure S14c shows field sweeps with H close to the b direction. At 2 K, the hysteresis is particularly pronounced and small steps can be resolved in both up and down sweeps.
To highlight the distinct behavior of the hysteresis above and below TSR, Fig. S15 shows plots of ∆ for 180 and 2 K. At 180 K, the peak due to the hysteresis is clearly visible (see Fig. S15a). To higher angles, there is a drastic reduction in the intensity of this peak until it disappears completely near 80°. To illustrate the disappearance of the peak and, hence, of the hysteresis at high tilt angle, Fig. S15b shows ∆ for field orientations close to H || b. The behavior at 2 K shows distinct differences (see Fig. S15c). First, the magnitude of the peak in ∆ reduced. Second, a sign change of the peak occurs above 80° (see Fig. 15d). As the angle approaches the b axis, the intensity of the negative peak and the step field increase drastically. Once θ passes 90°, the sign is reversed, which is expected from the sign change in the out-of-plane component of the field with respect to the current.