Introduction

The field of skyrmionics comprises new phases of magnetic systems, where the individual spins align as whirls1,2,3,4, that hold the potential to advance the understanding of topology in condensed-matter physics. Fundamentally, magnetic textures, such as skyrmions3,5, antiskyrmions6,7 (ASKs), and related skyrmionic systems8,9,10, 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 racetrack memory spintronic devices11,12. 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 compounds5,13, multilayer heterostructures14,15, thin films16, oxides17, and, more recently, tetragonal Heusler systems6,18. While experimental observations from, e.g., Lorentz transmission electron microscopy6,19,20, neutron scattering5, and magnetic force microscopy21,22 have shown great advancement over the last decades, detection by electrical means is required for the realization of energy-efficient spintronic devices11,12, probabilistic23, and neuromorphic computing24. Furthermore, a pivotal point for applications is the ability to control and scale skyrmionics25,26, or even to transform one species of magnetic texture into another in multiskyrmion systems18,27,28.

The tetragonal Heusler compound Mn1.4PtSn29 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 compounds22,27,30, 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 Mn1.4PtSn 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 TSR ≈ 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 dipole–dipole interaction due to large moments (>4μB). Above TSR, 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 found6.

These ASKs exhibit an extraordinary stability to sample-thickness variations to at least a few hundred nanometers confirmed by Lorentz transmission electron microscopy20 and X-ray studies31. 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 samples22,27. For very thin platelets, the phase diagram becomes more complex. Multiple skyrmion-like textures that are different above and below TSR can be stabilized by varying the field orientation18,30. Surprisingly, neutron-scattering experiments on bulk Mn1.4PtSn did not resolve any ASKs32. 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 TSR, exceeding expectations for skyrmions33. 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 effects34,35, where emergent fields due to a real-space variation of the local magnetism, describable by the Berry curvature36, 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 electrons37. 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 contributions37:

$${\rho }_{xy}={\rho }_{xy}^{{{{{{{{\rm{OHE}}}}}}}}}+{\rho }_{xy}^{{{{{{{{\rm{AHE}}}}}}}}}+{\rho }_{xy}^{{{{{{{{\rm{THE}}}}}}}}}.$$
(1)

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 scattering37,38; The THE, which can be described in terms of the chiral product:

$${{{{{{{{\boldsymbol{\chi }}}}}}}}}_{ij}={{{{{{{{\bf{S}}}}}}}}}_{i}\cdot ({{{{{{{{\bf{S}}}}}}}}}_{j}\times {{{{{{{{\bf{S}}}}}}}}}_{k}),$$
(2)

where Si 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.

Fig. 1: Transport signature of ASKs in micro-fabricated Mn1.4PtSn devices.
figure 1

a Schematic visualization of thickness-dependent ASKs induced by magnetic field, H∥c axis, in a step-like sample with applied current along the a axis. b False-color SEM images of FIB devices A and B with meander-shape (three thicknesses: d = 11.2, 8.4, and 2.7 μm) and Hall-bar geometry (d = 1.0 μm), respectively, contacted via sputter-deposited gold contacts. (Further details of all investigated devices, dimensions, and fabrication are presented in Supplementary Notes 1–3). c Hall-resistivity loop of samples A, B, and C with different thicknesses, d, along the c direction with H∥c at T = 300 K. d Greyscale MOKE images of Hall-bar transport device F (d = 0.8 μm) at four different fields for up and down sweep, respectively. e In situ Hall loop for the left and right contacts of device F marked by red and black color, respectively.

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In this work, we successfully combined in situ transport with magneto-optical microscopy applied to mesoscale structures fabricated from single crystals by focused-ion-beam assisted patterning. This approach enables us to directly examine the Hall signature of the ASK phase in Hall-bar devices with thicknesses ranging from 500 nm to 10 μm, observable from TSR to room temperature and above. The antiskyrmionic Hall component \({\rho }_{xy}^{{{{{{{{\rm{ASK}}}}}}}}}={\rho }_{xy}^{{{{{{{{\rm{AHE}}}}}}}}}+{\rho }_{xy}^{{{{{{{{\rm{THE}}}}}}}}}\) is comprised of an anomalous and a topological component. We study its temperature, field orientation, and device-thickness dependence. As the expected THE is rather small, we conclude that the major contribution originates from the AHE of the ASKs. In contrast to the general approach of micromagnetic simulations, we apply atomistic spin-dynamic calculations that capture the underlying complex magnetism of Mn1.4PtSn, inevitably linked to the emergence of ASKs at high temperatures. We show experimentally how the periodicity of magnetic textures and the \({\rho }_{xy}^{{{{{{{{\rm{ASK}}}}}}}}}\) of ASKs scales with thickness. Our ferromagnetic resonance (FMR) studies reveal the distinct magnetic anisotropies above and below TSR that lay the foundation for the emergence of multiple skyrmionic textures reported previously.

Results and discussion

Room-temperature Hall signature of the antiskyrmion phase

In Fig. 1a, we show a schematic of a step-like device, where an ASK lattice is formed in an external magnetic field applied perpendicular to the ab plane of the step device (for further details see Supplementary Notes 1–3). ASKs possess an anisotropic winding with a topological charge of opposite sign as compared to isotropic Bloch and Néel skyrmion systems7. As the thickness, d, of a device made from Mn1.4PtSn is reduced the periodicity of the ASK lattice decreases and, hence, the ASK size, too. This in turn increases the density and should, therefore, directly affect the magnitude of the topological charge that contributes to the Hall voltage, Vxy, transverse to the electric current, i.e., the THE. Furthermore, the local net magnetization of an individual ASK is roughly zero. Consequently, the volume magnetization should exhibit a reduction as ASKs emerge. This would affect the anomalous Hall effect, which is proportional to the magnetization39,40. However, direct measurements of the magnetization for sub-micron-thin, small-in-volume samples of Mn1.4PtSn remains a technical challenge.

Figure 1b displays false-color scanning electron microscope images of two different devices fabricated by focussed-ion-beam (FIB) assisted patterning, namely, a step-like device with thicknesses d = 11.2, 8.4, and 2.7 μm and a standalone device with d = 1.0 μm. In Fig. 1c, we display the total measured Hall effect for all thicknesses from the demagnetized state (μ0H = 0) up to a nearly saturated state (HS). For d < 8.4 μm, a noticeable hysteresis appears in the Hall effect before saturation, which details a noncoplanar magnetic and ASK spin texture, when the field is applied parallel to the c axis. Device F with d = 0.8 μm was set up as a Hall bar to simultaneously measure the Hall voltage while observing the magnetic state in situ via polar magneto-optical Kerr effect (MOKE) microscopy41. Selected MOKE images of the device are presented in Fig. 1d. The Hall data in Fig. 1e were simultaneously measured with polar MOKE microscopy, shown in Fig. 1d as a function of external field. Figure 1e displays the Hall-resistivity hysteresis loops recorded for the left (red) and the right (black) contacts (shown in Fig. 2f) of device F with the sweep direction indicated by black arrows. The down sweep stays in the saturated state down to ~460 mT due to the anisotropy in the magnetic exchange interactions and the critical aspect ratio of the sample that adds an additional shape-anisotropy contribution22. However, on the up sweep there are two clearly distinguishable slopes: The first originates from distinct conical magnetic spin spirals formed by the two non-equivalent magnetic Mn sublattices33. This hysteresis is also discernible in the magnetization, as we show in Supplementary Note 4. The second, highlighted by the yellow-shaded area, shows a slope change and coincides with the formation of the ASK lattice. The MOKE images of Fig. 1d, recorded simultaneously with the Hall resistivity, show a saturated magnetized state down to above 450 mT, where band domains associated with magnetic spin spirals suddenly emerge. When increasing field from zero, the conical spin-spiral state starts to slowly disintegrate into individual ASKs and ASK strings. Then suddenly at 535 mT, an ASK lattice emerges, exactly where the shoulder in the Hall resistivity develops. A more refined video sequence that shows a full field scan between saturation and zero field can be found in Supplementary Movie 1. In Supplementary Note 5, we present an additional sequence of MOKE images recorded for device C with d = 2.4 μm before it was structured into the Hall-bar geometry, where we observed much larger band domains and consequently a less dense ASK lattice.

Fig. 2: Magnetic structure of Mn1.4PtSn.
figure 2

a Sketch of the relevant interaction and coupling constants linking the two Mn sublattices (red and blue). b, c Atomistic spin-dynamic simulations for H∥c: Color maps for an array of (372 × 372) unit cells showing the perpendicular z component in the spin-spiral phase (left) and ASK phase (right), respectively. Black arrows represent the in-plane component. The color scale is given in radiant, where 1/−1 corresponds to a spin alignment parallel/antiparallel to the normal. Lower panels show the in-plane spin-winding component. d Upper panel: Calculated magnetization loop for fixed temperature given in meV; Lower panel: Calculated topological charge for opposite field-sweep directions (orange and black line). e Full up and down Hall-resistivity traces for device C (d = 2.4 μm) recorded at 300 K. Inset: Reduced-area plot. Arrows indicate the field-sweep direction. Red dashed line is a linear fit. f Magnetization for a bulk sample and device C, recorded before it was structured into a Hall bar by FIB. Inset: Reduced-area plot. g Differences between up and down sweeps for the Hall and magnetization data shown in e and f, respectively. Insets: Background-subtracted component related to the field region where ASKs are detectable.

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The magnetic structure of Mn1.4PtSn and the expected transport signature in the ASK phase

In Fig. 2a, we sketch the spin configuration and the respective interactions present in Mn1.4PtSn. The combination of DMI, induced by the D2d symmetry, FM and AFM interaction, is an ideal foundation for diverse magnetic textures33. We find that the physics of topological textures in Mn1.4PtSn can be accurately modeled purely from the exchange interactions and magnetocrystalline anisotropy in three atomic layers of the two magnetic sublattices10,42,43. More significantly, the ratios of the exchange constants determine the underlying physics; therefore, we tuned the classical parameters in atomistic spin-dynamic simulations to clearly show the effects seen in the experiments. There are three sets of exchange interactions and DMI crucial for the stabilization of in-plane ASKs in Mn1.4PtSn. The first interaction, J1 between the Mn(1) and Mn(2) sublattice, is ferromagnetic. It reaches along the shortest magnetic distance, where the DMI is strongest. The second ferromagnetic interaction, J2 between magnetic sublattices of the same type and perpendicular to the c axis, is the main interaction responsible for the Curie temperature. The interaction J3 for the Mn sublattice Mn(2) at the Wyckoff position 4d is of AFM nature, and assists in the formation of the noncoplanar structure below TSR. These AFM and FM interactions compete to form double-chiral spin spirals in order to minimize the energy. The chirality degeneracy is lifted by the DMI.

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 c, 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, χij leads to a topological winding number, \(w=1/4\pi \int\hat{{{{{{{{\bf{M}}}}}}}}}\cdot \left({\delta }_{x}\hat{{{{{{{{\bf{M}}}}}}}}}\times {\delta }_{y}\hat{{{{{{{{\bf{M}}}}}}}}}\right)dxdy\), where δi = δ/δi, and \(\hat{{{{{{{{\bf{M}}}}}}}}}(r,t)={{{{{{{\bf{M}}}}}}}}(r,t)/| {{{{{{{\bf{M}}}}}}}}|\) is the direction of the magnetization at each spatial position. The presence of w causes an emergent magnetic field, \({H}^{e}=\frac{\hslash }{2}{\epsilon }_{zxy}\hat{{{{{{{{\bf{M}}}}}}}}}\cdot \left({\delta }_{x}\hat{{{{{{{{\bf{M}}}}}}}}}\times {\delta }_{y}\hat{{{{{{{{\bf{M}}}}}}}}}\right)\), originating from a real-space Berry curvature34. The so-called topological charge, qT, is negative for skyrmions and positive for ASKs. Thereby, the THE is a direct link to the topology of the magnetic texture:

$${\rho }_{xy}^{{{{{{{{\rm{THE}}}}}}}}}={R}_{xy}^{{{{{{{{\rm{THE}}}}}}}}}{H}^{e},$$
(3)

where the Hall coefficient, \({R}_{xy}^{{{{{{{{\rm{THE}}}}}}}}}\), is tied to the complex multiorbital electronic structure44. The composite nature of \({R}_{xy}^{{{{{{{{\rm{THE}}}}}}}}}\) and He complicates the differentiation of skyrmionic configurations with distinct winding numbers across multiple materials. However, in Mn1.4PtSn distinct magnetic textures are induced by the external field.

In the lower panel of Fig. 2d, we plot qT against the external field. In the field-up sweep (orange curve), qT 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 qT in the down sweep. The down sweep (black curve) remains flat and only exhibits a positive contribution at lower fields, originating from the re-establishing 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 Mn1.4PtSn, ASK sizes can reach a few hundred nanometers in diameter. For these huge objects the expected THE is rather small39. Figure 2e, f, and g provides a comparison of the signatures observed in the Hall resistivity as well as in the magnetization for a 2.4 μm thick lamella sample. The hysteresis shows up for both quantities and a linear fit of the slope in the differences Δρxy and ΔM 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 Mn1.4PtSn, it is limited in spatial resolution by the wavelength of the optical light used41. 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 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 thickness22,32.

Fig. 3: Thickness and temperature dependence of the magnetic textures and Hall component associated with ASKs.
figure 3

a MFM images (local magnetic z component represented by Δϕ showing chiral domains and an ASK lattice for two samples prepared by FIB patterning with thicknesses of 5 and 1.5 μm. The data were recorded at room temperature in zero field and at 550 mT applied parallel to the c axis, i.e., parallel to the line of sight. The lower panels show the respective FFTs (in arb. unit with logarithmic color scale). b ASK Hall component, \({\rho }_{xy}^{{{{{{{{\rm{ASK}}}}}}}}}\), plotted against magnetic field for devices B, C, E, and G with thicknesses of 1, 2.4, 0.7, and 0.8 μm, respectively. c Thickness dependence of \({\rho }_{xy}^{{{{{{{{\rm{ASK}}}}}}}}}\) compared to the domain periodicity, w, determined by recent MFM measurements22,27. The error bars represent the standard deviation of the values for variations in the details of the background subtraction. d \({\rho }_{xy}^{{{{{{{{\rm{ASK}}}}}}}}}\) for temperatures between 10 and 300 K for device B with d = 1 μm. e Maximum amplitude of \({\rho }_{xy}^{{{{{{{{\rm{ASK}}}}}}}}}\) in device B (d = 1 μm) plotted against temperature. Red dots are a guide to the eye. The error bars are defined as in c. f Temperature dependence of the maximum difference in the Hall-resistivity loop, Δρxy between up and down sweeps for devices A, B, and E.

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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 \({\rho }_{xy}^{{{{{{{{\rm{ASK}}}}}}}}}\) 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 component39. 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 \({\rho }_{xy}^{{{{{{{{\rm{ASK}}}}}}}}}\). In Supplementary Note 7 we provide a theoretical estimate following the previous approaches5,44 for a known ASK density. For our microscale devices the ASK size is ~100 nm, which would lead to a theoretically expected THE component of ∣ρTHE∣ ~ 50 nΩcm.

Moreover, we find that \({\rho }_{xy}^{{{{{{{{\rm{ASK}}}}}}}}}\) remains as pronounced as at room temperature all the way down to temperatures close to TSR (see Fig. 3d and Supplementary Note 7). Below TSR, 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 TSR, where noncoplanar magnetism starts to take over. Its overall amplitude starts to subside around TSR as can be seen from Fig. 3e. Below 150 K, we cannot unambiguously link the observed deviations to skyrmions. To visualize the crossover around TSR we show the maximum Δρxy 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 TSR, 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 TSR and the field being oriented away from the c direction, we observe a prominent hump-like enhancement of ρxy(H) close to the saturation field (see for example the maximum in ρxy(H) at 180 K and H∥b in Fig. 4b). This feature was already explored for bulk samples29,33. 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, ρxy(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 ρxy(H) occur (see for example the 150 K curve in Fig. 4b). Hence, our data indicate 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 \(\cos \theta\) dependence, observable at room temperature, as temperature is tuned to TSR 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).

Fig. 4: Angle-dependent Hall and FMR at various temperatures.
figure 4

a, b ρxy(H) of device B with 1 μm thickness recorded for H∥c and H∥a, respectively, at various temperatures. c Extracted ASK Hall signature at three fixed temperatures for various angles. The error bars are defined as in Fig. 3c. d Anomalous Hall coefficient extracted from linear fits to the high-field part above 5 T of ρxy(H) plotted against the tilt angle θ. Red dashed lines are fits with \({\rho }_{xy}(H)\propto \cos \theta\). e, f FMR forward-transmission parameter, S21. e FMR spectra for H∥c and H∥b recorded for a 0.8 μm thick lamella at 10 and 260 K. f Angular dependence of the FMR field. Error bars represent the resonance linewidth values.

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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 TSR, 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 Fe1.9Ni0.9Pd0.2P28. At temperatures well below TSR, 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 TSR. These significant changes in the magnetic anisotropies can explain the previously reported vanishing of ASKs and emergence of Bloch-type skyrmions at low temperatures30. 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 bimerons45.

Conclusion

In summary, we demonstrated the direct detection of the Hall resistivity due to the emergence of ASKs in the Heusler compound Mn1.4PtSn 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 devices46,47,48. Many hard-magnetic materials are strongly susceptible to finite sizes and may display nontrivial magnetic textures detectable by Hall transport24,49. As the thickness is decreased, the density of ASKs increases, which we observe directly using MFM, leading to an increase in the Hall-effect signature. Our theoretical prediction of the ASK THE magnitude matches the observed Hall signature values, thereby displaying the importance of the electronic structure for the unexpectedly large Hall signature. However, the disentanglement of the anomalous and topological Hall contributions remains a challenge, without spin-resolved visualization capabilities. In addition, the temperature dependence of the magnetic anisotropy, reflected in our Hall and FMR results, reveals the intimate link to the crystalline, electronic, and magnetic structure. Our atomistic model, solely based on the competing magnetic exchange interactions, captures the mechanism behind the emergence of ASKs in Mn1.4PtSn. The novelty of tunability of texture sizes and the importance of the exchange interactions offers additional ways to manipulate and detect ASKs. Yet, smaller sizes and new materials with enhanced THE are desirable. Our observations offers a route to room-temperature skyrmionic applications based on multiple distinct types of skyrmionic textures.

Methods

Experimental design

The objectives of this study were to visualize the emergence of ASKs in mesoscale devices that allow a simultaneous investigation of the electrical transport under the influence of an external magnetic field and disentangle other contributions to the electrical-transport signature related to the complex magnetism of the multiskyrmion host compound Mn1.4PtSn. We conducted polar magneto-optical Kerr microscopy as well as magnetic force microscopy on micron-sized samples fabricated by focussed-ion-beam assisted patterning. The former microscopy technique was successfully applied for the in situ detection of the Hall effect at room temperature. Furthermore, a detailed study of the ASK Hall signature depending on sample thickness, field orientation, and temperature was conducted. Chip-based FMR measurements depending on temperature, field, and angle were performed to obtain information on the ferromagnetic component of the complex magnetic background around temperatures at which the THE, associated with ASKs, is subsiding.

Crystal growth

Single crystals were prepared by flux-growth technique showing no microtwinning confirmed by Laue diffraction. Details can be found elsewhere29.

FIB microfabrication

We fabricated transport devices from high-quality single crystals of Mn1.4PtSn by the application of Ga or Xe FIB microstructuring, which enable high-resolution investigations of anisotropic transport. FIB micromachining has already proven extremely powerful in various other metallic compounds. A detailed description of the fabrication is provided in Supplementary Notes 1 and 2. Further details on FIB-patterned transport devices can be found in previous works of some of the authors50,51. Thin (few microns thick) lamella-shaped slices of Mn1.4PtSn were separated with FIB and manually transferred ex situ onto a sapphire substrate into a thin layer of insulating two-component epoxy. We then deposited a 100 nm thick gold (Au) layer on top in order to electrically connect the leads with the crystal. With the help of FIB, we thereafter patterned the Au interfaces into separate terminals. In a next step, we removed the Au from the central area of the slice before cutting the lamellae into Hall-bar-shaped transport devices, highlighted by purple color in Fig. 1b. For device A, shown in Fig. 1b, the thin slice of the crystal was polished stepwise to three different thicknesses by FIB before the transfer onto the substrate. This way, a microstructure device was created with three 3-point hall-bar devices connected in series.

Magnetotransport measurements

We conducted electrical-transport measurements in a 14 T Quantum Design physical properties measurement system by applying a standard a.c. lock-in technique. A SynkTek multichannel lock-in amplifier as well as Zürich Instruments lock-in amplifiers were used. The current is directed through the FIB cut microstructures (see purple colored parts in Fig. 1b). Dimension of the devices were determined using a scanning electron microscope.

Magnetic force microscopy

MFM measurements were performed in two instruments. For room-temperature measurements without external fields we used a Park Systems NX10 with MFM probes from Nanosensors at lift heights between 100 and 150 nm. For measurements under magnetic fields, we used a AIST-NT SmartSPM 1000 with MFM probes from Nanosensors at lift heights between 250 and 350 nm together with a Nd-based permanent magnet. The magnetic flux strength was measured at the position of the sample with a MAGSYS HGM09s magnetometer. All data analysis was performed with the Gwyddion software.

Polar magneto-optical Kerr microscopy

Experiments were conducted with an AxioScope-type Carl Zeiss wide-field polarization microscope at room temperature. The sample was placed onto the pole of a solenoid-type, water-cooled electromagnet with a maximum magnetic field of 1.5 T aligned along the c axis. Special care was taken to remove parasitic Faraday contributions by the application of a motorized analyzer. Further details of the method can be found elsewhere41.

Ferromagnetic resonance

A broadband frequency vector network analyzer (VNA)52 was used to probe the resonance properties of a Mn1.4PtSn lamella (20 × 100 × 0.8) μm3 prepared by FIB. A 20 μm wide coplanar wave guide was used for the transmission in an impedance-matched (50 Ω) chip-carrier design. The detection of the FMR modes was carried out via the change in the forward-transmission parameter S21 using a Keysight N5225A VNA. Field- and temperature-dependent measurements were performed in an Attocube DRY 1100 cryostat with a split-coil magnet at constant temperature and fixed frequency of 20 GHz using 10 dBm microwave input power. At each field step, the real and imaginary components of the S21 parameter were recorded, using 100 times averaging.

Numerical calculations

First-principle calculations were performed on Mn1.4PtSn by using the full-potential linearized augmented plane-wave code FLEUR following the work of Vir et al.33. From this, exchange parameters were extracted, where the largest interactions were J1, J2, J3, and D. The atomistic spin calculations were performed in the code VAMPIRE for a lattice with 372 × 372 unit cells53. For computation efficiency the Heisenberg exchange parameters are decreased (J1 = 10 meV, J2 = 12.0 meV, J3 = −4 meV) and the DMI magnitude increased (D1 = 2.0 meV). The magnetocrystalline anisotropy is single axis and was chosen to be 0.5 μeV. To simulate the effect of the spin-reorientation transition, we set J3 = −2 meV. We used the Landau-Lifshitz-Gilbert equation to study the spin dynamics of the system.