Critical Sample Aspect Ratio and Magnetic Field Dependence for Antiskyrmion Formation in Mn1.4PtSn Single-Crystals

Mn1.4PtSn is the first material in which antiskyrmions have been observed in ultra-thin single crystalline specimens. While bulk crystals exhibit fractal patterns of purely ferromagnetic domain ordering at room temperature, ultra-thin Mn1.4PtSn lamellae clearly show antiskyrmion lattices with lattice spacings up to several $\mu$m. In the work presented here, we systematically investigate the thickness region from 400 nm to 10 $\mu$m using 100 $\times$ 100 $\mu$m$^2$ -wide Mn1.4PtSn plates, and identify the critical thickness-to-width aspect ratio $\alpha_0 = 0.044$ for the ferromagnetic fractal domain to the non-collinear texture phase transition. Additionally, we also explore these non-collinear magnetic textures below the critical aspect ratio $\alpha_0$ above and below the spin-reorientation transition temperature $T_{SR}$ while applying variable external magnetic fields. What we find is a strong hysteresis for the occurrence of an antiskyrmion lattice, since the antiskyrmions preferentially nucleate by pinching them off from helical stripes in the transition to the field polarized state.


I. INTRODUCTION
Magnetic materials that exhibit topological spin arrangements are promising candidates for future applications, in particular for spintronics. One type of such topological arrangements are the so-called skyrmion lattices (SkLs) that may arise in the presence of an antisymmetric Dzyaloshinskii-Moriya interaction (DMI). Every skyrmion represents a spatial distribution of noncoplanarly arranged magnetic moments, whose orientations once cover all spatial directions. This mutual noncoplanar orientation of neighboring spins is generally described by the topological charge that, in the end, is able to differentiate between skyrmions (Skys) and antiskyrmions (A-Skys) [1][2][3][4][5][6].
Subsequent LTEM experiments revealed that for some particular temperature and field regions within the phase diagram as well as for selected sample thicknesses, A-Skys may also arrange in a squared lattice fashion [22,23]. Moreover, elliptically-distorted skyrmions of both handedness as well as the non-topological bubble lattice were shown to appear when applying the symmetry-breaking in-plane magnetic field in combina-tion with an out-of-plane field [22,23]. This makes the tetragonal Heusler compounds a unique class of materials hosting a rich variety of controllable magnetic textures.
Moreover, in bulk samples of Mn 1.4 PtSn an anisotropic fractal ferromagnetic closure domain pattern (FMD) has been reported in clear contrast to the LTEM results, which shows, that Mn 1.4 PtSn is an easy-axis ferromagnet at room temperature with the easy-axis aligned along the tetragonal c-axis and a pronounced quadratic anisotropy within the ab-plane, and highlights the importance of long-ranged dipolar magnetic interactions for the formation of antiskyrmions in these materials [27]. Also, a recent study of the magnetic textures in Mn 1.4 PtSn using wedge-shaped samples came to the same conclusion [24]. Furthermore, at T SR ∼170K Mn 1.4 PtSn undergoes a spin reorientation transition from the coplanar above T SR to the non-coplanar spin orientation below T SR [28,29].
Hence, to clearly determine the subtle differences in the phase diagram as well as to safely quantify the crossover from the reported fractal ferromagnetic domain (FMD) to the non-coplanar texture (NCT) phase in Mn 1.4 PtSn as needed for instance for realizing miniaturized room-temperature devices based on A-Skys, it is essential to characterize the different magnetic structures over a broad thickness range on samples with no or only weak thickness variations on the length scale of the antiskyrmion radius. We thus employ in the present study magnetic force microscopy (MFM) in order to clearly resolve the different magnetic textures of Mn 1.4 PtSn in Mn 1.4 PtSn plate samples cut out from bulk single crystals by means of focused ion beam (FIB) milling. What we observe is how the different magnetic patterns depend on sample thickness, and how and when these structures change when varying both temperature and the applied magnetic field.
Magnetic force microscopy (MFM) has proven to be a valuable tool when studying complex spin textures such as helices and skyrmions as well as complex domain patterns [11,16,17,30,31]. MFM can be effectuated with high precision not only under the various temperature and magnetic field conditions, but moreover, can be applied to investigate near-surface properties of any sample thickness, from monolayer to bulk systems. In this sense, MFM is not restricted to the analysis of ultra-thin lamellae, but will provide insight for any of these Mn 1.4 PtSn plate samples.
The paper is organized as follows: After describing the necessary experimental details, we (i) firstly discuss the thickness-dependent characteristic features of the realspace patterns obtained by MFM in zero field and above the spin-reorientation transition temperature T SR ; here we clearly identify the critical thickness of D 0 = 4.4 µm for Mn 1.4 PtSn plates, indicating the transition from the FMD to the NCT state. (ii) Secondly, we then focus on the field dependence and magnetic texture formation for a plate thickness of D = 2 µm well below the critical thickness D 0 at temperatures above T SR ; of interest here are field-induced phase transitions between the two states. (iii) Finally, we briefly substantiate and discuss the impact of temperature, especially when cooling below T SR .

II. EXPERIMENTAL DETAILS
Single crystals of Mn 1.4 PtSn were grown by flux method using Sn as flux. For details please refer to Vir et. al [28]. After selecting crystals with appropriately oriented surfaces, a series of thin-plate samples with thicknesses ranging from 400 nm to 10 µm suitable for MFM were FIB milled by cutting a 100 × 100 µm 2 groove into the side of the single crystal using Xe ions at a current of 4 -60 nA. All samples hence are supported from the three (bulk) sides resulting in an extremely high mechanical stability for MFM inspection, as displayed in Fig. 1(A). Note that all sample thicknesses were checked by scanning electron microscopy with an error of less than 50 nm. We also checked that gentle sample surface polishing using Xe or Ga ions at currents below 10 nA, did not impact the magnetic textures of any sample at all.
MFM measurements were performed with two different instruments: Room-temperature inspection without applying any external magnetic fields was carried out on the Park Systems NX10 [32] using standard PPP-MFMR probes from Nanosensors [33] at lift heights between 100 and 150 nm. Low-temperature measurements with external fields applied, were run in our Omicron cryogenic ultra-high vacuum STM/AFM instrument [34] operated with the R9 electronics from RHK [35]. Here we employed PPP-QMFMR probes from Nanosensors driven at mechanical oscillation amplitudes of ≈ 20 nm at lift heights between 400 and 800 nm.

A. Thickness dependence
The domain structure strongly depends on the sample thickness. In Fig. 1(B) we show the magnetic pattern in zero field at room temperature when inspecting the border between the 10 µm-thick Mn 1.4 PtSn plate and the bulk. While the bulk part exhibits the typical anisotropic fractal domain pattern, the magnetic pattern seen in the plate area is much simpler showing one row of nested domains, only. Note, however, that magnetic domain walls in both areas are aligned along the same preferred crystallographic directions, i.e. along [100] and [010].
A series of MFM images obtained on thin-plates with different thicknesses is analyzed in Fig. 1(C). Plates with thicknesses between 10 µm and the treshhold value D 0 = 4.4 µm clearly exhibit stripy patterns with one row of nested domains [see Figs. 1(C3,C4)]. The corresponding 2-dimensional fast Fourier transforms (FFTs) (D3), (D4) display two distinct features, (i) a set of blurred streaks (highlighted by the white ellipse) originating from the nested domains, and (ii) a series of linearly aligned peaks within each streak (highlighted by arrows) that correspond to the higher Fourier components of the nonsinusoidal stripe profile. This behavior changes drastically for thicknesses below D 0 , where the stripe profiles now become more sinusoidal showing no nested magnetic domains in real-space images. The FFTs in turn, show only a one dimensional pattern with less or no higher order peaks. Thus, D 0 clearly marks the critical thickness of the first branching, at which the magnetic pattern fundamentally changes.
From the MFM images we extract line profiles and measure the domain width W or the helix period λ = 2W , respectively. For patterns with nested domains, we measure the width of both the base domain W b and of the smallest closure domains W s . All measurements are summarized in Fig. 1(E).
The dependence of magnetic domain structure on sample geometry in the case of thin infinitely extended plates has been described in detail for uniaxial ferromagnetic systems [31]. In general a power law dependence of the domain width W on plate thickness D is expected in the form W (D) = a · D b , where constants a and b are tied to the exchange constant A, the uniaxial anisotropy constant K u as well as the saturation magnetization M s . For our experiment, the aspect ratio varies as a function of thickness, and therefore, we account for varying demagnetization factor N resulting in a slight thickness dependency of factors a in our analysis. A more detailed summary can be found in the supplemental material.
In the thickness regime D < D 0 , the domain width W can be well fitted as a function of plate thickness D by W = a · D 1/2 , as expected for stripe domains [36][37][38] [orange line in Fig. 1(E)]. Yet, leaving the exponent b as a free fitting parameter W ∝ D b , we find b = 0.65 ± 0.03, as shown in Fig. 1(E) by the blue solid line. Since in this regime the observed texture is quite sinusoidal, the stray field is weaker than for uniform domains of the same width. In contrast to our observations, for only dipolar interactions this would result in a larger W for constant D. The small deviations from the Kittel-law therefore may be attributed to the DMI that favors perpendicular spin arrangements and is responsible for the sinusoidal profile.
For thicknesses D > D 0 , free-exponent fitting for the base domain width W b results in b = 0.66 ± 0.08 [green solid line in Fig. 1(E)] in excellent agreement with the theoretical b = 2/3-behavior [31]. Meanwhile, the smallest surface domain width W s remains nearly constant as expected [31,39]. Thus, we conclude, that for thicknesses above D 0 , Mn 1.4 PtSn is well described as a uniaxial ferromagnet with pronounced anisotropy within the ab-plane. Below D 0 , DMI impacts dominantly such that spin helices instead of stripe domains nucleate.
From D 0 = 4.4 µm, the saturation magnetization M s ≈ 4 × 10 5 A/m at room temperature [28], and the ratio K u /K d = 1.71 determined by the fit in the regime D > D 0 we find K u ≈ 1.71 · µ 0 · M 2 s /2 ≈ 1.71 × 10 5 J/m 3 , which is small compared to common permanent magnets like Nd 2 Fe 14 B or SmCo 5 with K u = 5 × 10 6 J/m 3 and K u = 1.7 × 10 7 J/m 3 respectively [40] and on the same order of magnitude as the dipolar energy.
Thus, we may argue here, that magneto-crystalline and shape anisotropy compete and provide the effective, thickness-tunable c-axis anisotropy. While Mn 1.4 PtSn constitutes an easy-axis magnet along the c-axis, the shape anisotropy of a thin plate is thickness dependent and makes the surface normal of the extended surface to become the magnetic hard axis. For our plate samples investigated here, the c-axis is at the same time the surface normal, and hence the two anisotropies counteract. For cubic helimagnets with uniaxial anisotropy, it was predicted that stabilizing non-collinear spin textures might be possible for selected values of uniaxial anisotropies only [41,42]. We thus may speculate that similarly the lowering of the effective anisotropy is the reason for the thickness-dependent fundamental change of the reported magnetic textures in Mn 1.4 PtSn. Therefore, instead of emphasizing on the relevance of an absolute thickness D 0 , we may rather consider the critical thickness-to-width aspect ratio α 0 for our quadratically-shaped plates as α 0 = D 0 /100µm = 0.044. Nevertheless, we have to add that the special sample shape, where the plate is still connected to the bulk, may also have an impact on this value.

B. Field dependence
Next, we focus on the field dependence and evolution of magnetic patterns for plates with thickness D < D 0 at T > T SR , where ASkLs have been reported. The field dependent behavior in bulk samples has been discussed in a recent publication [27]. A series of images measured for D = 2 µm at T = 200 K is illustrated in Fig. 2(A). The complete dataset and datasets obtained at various other temperatures above T SR are available in the supplemental material.
Starting from zero external field as shown in Fig. 2(A1), we find a spin helix up to the critical field strength B c1 = 550 mT. With increasing field strength a systematic increase of the helix period λ is observed [see Figs. 2(A2&A3)], closely resembling recent findings in Cu 2 OSeO 3 in the helical phase [43]. Above B c1 , helical stripes break up into antiskyrmions [see Fig. 2(A3)], that start to assemble in a hexagonal ASkL with a periodicity of approximately twice the helix period in zero field, as shown in Fig. 2(A4). When further increasing the magnetic field, individual A-Skys start to unwind until the transition to the field-polarized state is reached at B c2,up = 650 mT [see Fig. 2(A5)]. In contrast for decreasing field strengths, the field-polarized state survives until B c2,down = 600 mT, where eventually single A-Skys nucleate as seeds for the helical stripe phase [compare Fig. 2(A6)]. Note, that no ASkL lattice is formed when lowering the external field from saturation, clearly indicating the hysteretic behavior at this phase transition. For B < B c2 , the helical phase is well stable down to zero magnetic field [see Figs. 2(A7&A8)]. Also note, that reversing the magnetic field excellently reproduces the above behavior, i.e. with B c2,up = −650 mT and B c2,down = −600 mT. In Fig. 2(B) we analyze the field dependence of the helix period λ. Apparently, the helix period changes as a function of external magnetic field for all domains. Moreover within the helical phase and as seen for instance in Fig. 2(A1), we find magnetic domains within one and the same image that have different λ. In principle, there are two possible explanations for such a behavior: Firstly, when assuming that the helix wave-vector q is always aligned in-plane, i.e. always lies in the image plane, then the helix transforms into an anharmonic helicoid and the magnitude of the period λ must effectively change in magnetic field [41]. Yet, this explanation conflicts with our observations, as firstly domains with different period are clearly seen, and secondly the expected field dependency with an approximately constant period up to a critical field, where the period starts to diverge, does not fit to our data.
Secondly, a magnitude λ that remains constant directly implies tilting of q with respect to the sample surface normal n, i.e. the external magnetic field direction. What MFM then measures at the sample surface, is always the local projection onto the image plane, as has similarly been reported for the helical state in Cu 2 OSeO 3 [43]. As a result, only the in-plane component of q causes the contrast in MFM. Thus, an apparently longer period λ = 2π/[q · sin(φ)] = λ 0 / sin(φ) with φ = (q, n) the inclination angle and λ 0 = 2π/q the zero field period is measured. Vice versa, from the values of the apparent helix period λ(B), we can compute φ(B) = arcsin λ0 λ(B) . By doing so, we find an approximately linear dependence φ(B) for the lower domain [see Fig. 2(C)]. Extrapolating this linear fit to φ(B s ) = 0 (i.e. the fully aligned case q||B) yields the critical field B s ≈ 700 mT, a value that is slightly larger than the onset of the field-polarized state deduced experimentally here. In contrast to Cu 2 OSeO 3 , the tilted helix becomes unstable already at an angle of ≈ 15°, long before q||B.
In small external fields, the upper domain deviates slightly from the linear behavior, which hints towards an incomplete q-vector rotation, due to the very slow dynamics connected to the size of the domain and the large number of topological defects at the domain boundary. Similar long relaxation times were also reported for FeGe and MnSi [44,45]. In total, the field dependence of the apparent wavelength can be well explained by rotation of the helix wave-vector q into the external field with linear dependence ϕ(B).

C. Temperature dependence
Finally, we extend our analysis to compare the magnetic pattern evolution in thin Mn 1.4 PtSn plates both above and below the spin-reorientation transition temperature T SR at 170 K. In bulk samples, for T < T SR , the quadratic ab-plane anisotropy is lost, resulting in a much less rectangular domain pattern arrangement [27].
When inspecting the plate with D = 2 µm, the effects on the magnetic pattern can be described as follows: Upon zero field cooling to T = 80 K, the spin helices evolving above T SR , remain. Only when magnetically saturating the sample and ramping the field back to zero, can we notice some subtle changes in the magnetic pattern. In more detail, the period slightly increases and single phase fronts appear less straight, i.e. the helix wave-vector locally rotates slightly within the ab-plane [see Fig. S3(A1 to A3) and Fig. S1 in the Supplemental Material]. As for bulk samples, the quadratic ab-plane anisotropy characteristic above T SR vanishes. Even when ramping the field up into the field-polarized state showed no evidence of the occurrence of an ASkL, but only of single antiskyrmions or bubble domains [see Fig.3(B1 to B3)], in agreement with recent reports on Hall-effect measurements in bulk samples showing distinct behaviour above and below T SR [29].

IV. DISCUSSION AND CONCLUSIONS
In summary, we performed a thickness-dependent study of the magnetic patterns observed in Mn 1.4 PtSn, both under variable magnetic fields and over a wide temperature range. We found for our quadratic 100 × 100 µm 2 -wide plates the critical thickness D 0 = 4.4 ± 0.4 µm corresponding to a critical thickness-to-width aspect ratio α 0 = 0.044 ± 0.004 above which the material can be well described as a uniaxial ferromagnet with a domain pattern fully built-up from stripe domains and nested closure domains.
Below D 0 , a spin helix in low fields and an ASkL at fields close to saturation field were found, in agreement with literature. The phase diagram shows a strong hysteretic dependence, which manifests in the absence of the ASkL when the field is decreased from the saturated state. This might be connected to the observed A-Sky formation by pinching them off from helical stripes. Note that the reversed process had formerly been identified to unwind the SkL state towards the topological trivial helical state in Fe 0.5 Co 0.5 Si [11]. The nucleation of further A-Skys in the field-polarized state happens to be less favorable than the area growth of already nucleated A-Skys into helical stripes, and hence, no ASkL but a stripe pattern forms. Thus, for selected magnetic field ranges, stable antiskyrmions may be easily nucleated by means of an external stimulus without formation of an ASkL at the same field and temperature. This is quite unusual for single crystalline systems, but offers the possibility to study single antiskyrmions at will in Mn 1.4 PtSn.
The apparent period of the spin helix shows a field dependence that can be well described by the rotation of the spin helix into the field direction, with a linear dependency of the inclination angle on the applied field. For temperatures below the spin-reorientation transition temperature the anisotropy within the ab-plane is lost, and the helix wave-vector may locally rotate within the ab-plane.
To conclude, in thin-plate samples of Mn 1.4 PtSn both the DMI as well as dipolar interactions are equally important for the stability of antiskyrmions. By varying the sample thickness and hence the effective c-axis anisotropy, the helical pitch and the ASkL size can be tuned to very large values as compared to other skyrmion hosting materials. Together with the likely ease to nucleate single antiskyrmions at will and at any predefined position by means of an external stimulus without formation of an ASkL, an even wider set of experimental techniques may be available such as optical detection with highest temporal resolution for studies of the antiskyrmion formation and dynamics.