Tunable Magnetic Antiskyrmion Size and Helical Period from Nanometers to Micrometers in a D2d Heusler Compound

Skyrmions and antiskyrmions are magnetic nano‐objects with distinct chiral, noncollinear spin textures that are found in various magnetic systems with crystal symmetries that give rise to specific Dzyaloshinskii–Moriya exchange vectors. These magnetic nano‐objects are associated with closely related helical spin textures that can form in the same material. The skyrmion size and the period of the helix are generally considered as being determined, in large part, by the ratio of the magnitude of the Heisenberg to that of the Dzyaloshinskii–Moriya exchange interaction. In this work, it is shown by real‐space magnetic imaging that the helix period λ and the size of the antiskyrmion daSk in the D2d compound Mn1.4PtSn can be systematically tuned by more than an order of magnitude from ≈100 nm to more than 1.1 µm by varying the thickness of the lamella in which they are observed. The chiral spin texture is verified to be preserved even up to micrometer‐thick layers. This extreme size tunability is shown to arise from long‐range magnetodipolar interactions, which typically play a much less important role for B20 skyrmions. This tunability in size makes antiskyrmions very attractive for technological applications.

DOI: 10.1002/adma.202002043 boundary than aSk, was recently discovered. [2] The boundaries in an aSk are composed of successive left-hand Bloch, left-hand Néel, right-hand Bloch, and right-hand Néel wall segments. [2] The intrinsic stability of aSks over a large range of field and temperature was recently demonstrated. [3] The detailed magnetic structures of both Sks and aSks are controlled by the magnitude and symmetry of the Dzyaloshinskii-Moriya exchange interaction (DMI). [4] The dependence of Sks and helices on the thickness of the host material has been studied in several B20 materials including FeGe, [5] Mn 1-x Fe x Ge of various compositions, [6] and Fe 0.5 Co 0.5 Si. [7] In all these cases no significant dependence of the size of the Sk or the helix period on the thickness of the host compound was found. It was concluded that the fundamental length scale is determined by a competition between the Heisenberg exchange and the DMI. By contrast, it was shown some time ago that the size and arrangement of achiral magnetic bubbles is strongly influenced by magnetostatic interactions. [8] What have been called "biskyrmions" and "skyrmions" in centrosymmetric materials fall into this latter category. [9] We show here that antiskyrmion systems are unique in that both DMI and magnetostatic interactions play a significant role and that, thereby, give rise to extensive size tunability as well as intrinsic stability. We demonstrate, using a combination of Lorentz transmission electron microscopy (LTEM) and magnetic force microscopy (MFM) that the size of the antiskyrmion scales with the period of the helical structure and that both can be tuned over a wide range of more than an order of magnitude by adjusting the thickness of the host layer in which they are imaged.
In these studies, we focus on exploring spin textures in thin lamellae formed from a single crystal of Mn 1.4 PtSn which is an inverse tetragonal Heusler compound with D 2d symmetry (see Supporting Information). High-quality single crystals were prepared using a flux method. One single crystal was fashioned into a uniform thickness lamella (L1) and several wedgeshaped lamellae (L2-L4) by Ga + focused ion beam milling. The spin textures reported here were imaged using two different techniques. LTEM was used to both confirm the existence of aSks and to explore their size and the helix period for thinner wedges. Thicker wedges were studied using MFM. A schematic drawing of the aSk spin texture, that consists of chiral boundaries formed from helicoids (Bloch-like) and cycloidal Skyrmions and antiskyrmions are magnetic nano-objects with distinct chiral, noncollinear spin textures that are found in various magnetic systems with crystal symmetries that give rise to specific Dzyaloshinskii-Moriya exchange vectors. These magnetic nano-objects are associated with closely related helical spin textures that can form in the same material. The skyrmion size and the period of the helix are generally considered as being determined, in large part, by the ratio of the magnitude of the Heisenberg to that of the Dzyaloshinskii-Moriya exchange interaction. In this work, it is shown by real-space magnetic imaging that the helix period λ and the size of the antiskyrmion d aSk in the D 2d compound Mn 1.4 PtSn can be systematically tuned by more than an order of magnitude from ≈100 nm to more than 1.1 µm by varying the thickness of the lamella in which they are observed. The chiral spin texture is verified to be preserved even up to micrometer-thick layers. This extreme size tunability is shown to arise from long-range magnetodipolar interactions, which typically play a much less important role for B20 skyrmions. This tunability in size makes antiskyrmions very attractive for technological applications.
One of the major topics in spintronics today is the study of the steady state and dynamical properties of spin textures with various topologies. [1] Amongst these are magnetic skyrmions (Sks) which are magnetic nano-objects with chiral magnetic boundaries, namely, an in-plane magnetized region that separates the interior and exterior of the object where the magnetization points up/down or vice versa. Another distinct magnetic nano-object, an antiskyrmion (aSks), that has a more complex spin propagations (Néel-like), is shown in Figure 1a. Néel domain walls of right-and left-hand propagate along the [110] and [110] directions, respectively, and Bloch domain walls of right-and left-hand propagate along the [100] and [010] directions, respectively. The helix in the D 2d system propagates along these latter directions (in the absence of an in-plane magnetic field) because its energy is lower for the Bloch structure (see Supporting Information).
Antiskyrmions were observed over a wide range of temperature and field in all samples. Only room temperature data are reported here because we find that the characteristic size of the spin textures in Mn 1.4 PtSn increases only a little as the temperature is reduced below room temperature, for otherwise the same thickness. Typical LTEM images are given in Figure 1 for a lamella with a uniform thickness of t ≈ 200 nm (sample L1) and a wedged lamella whose thickness varies from ≈116 to ≈206 nm (sample L2). Figure 1b shows a magnified image of a single antiskyrmion in L1 that has a distinctive four-spot pattern with alternating black and white contrast, as previously observed for the closely related compound Mn 1.4 Pt 0.9 Pd 0.1 Sn. [2] A scanning electron microscopy (SEM) image of the wedge L2 is shown in Figure 1c, and a schematic diagram showing the definition of the coordinate axes and tilting angles used in the discussion below is given in Figure 1d. Electron energy loss spectroscopy was used to determine the thickness t of the wedge. [10] The [001] zone axis is oriented along the transmission electron microscopy (TEM) column using a double-tilt sample holder. An LTEM image recorded in zero magnetic field is shown in Figure 1e, in which the helices have been oriented along the [100] direction by using the in-plane component of an applied magnetic field. The most important finding is that the helix period varies significantly and monotonically from ≈96 to ≈185 nm as the sample thickness is varied from ≈116 to ≈206 nm. Magnified images of two representative regions A and B, each 1 µm × 1 µm in area, with thicknesses of ≈142 and ≈206 nm are shown in Figure 1h,i. The helix period is ≈108 ± 13 nm in region A ( Figure 1h) and ≈185 ± 2 nm in region B (Figure 1i). Note that across the width of the wedge, where the thickness is unchanged, the helix period does not change significantly.
The helical state evolves into an aSk phase when a perpendicular magnetic field is introduced. A magnetic field oriented along the microscope column is generated by applying current in the objective lens of the TEM. The following protocol was used: after an LTEM image is taken, the sample's [001] axis is tilted away from the TEM column axis by α = 30° and the magnetic field is then increased in steps of 32 mT. Note that the presence of the in-plane field provided by the tilt allows for the aSk phase to be more easily stabilized. [2] The sample is then tilted back to 0° and the next LTEM image is recorded. This tilting procedure and step-wise increase in magnetic field is repeated until the magnetic field is so large that the sample reaches the fully magnetized ferromagnetic state. Typical LTEM images of L2 corresponding to fields of 192 and 320 mT are shown in Figure 1f  components, as the field is increased till the fully polarized state is reached, is shown in Figure S1 of the Supporting Information. These data illustrate the sensitivity of the field dependence of the spin texture to thickness. Moreover, it is clear that the size of the aSk, just like the helix period, depends strongly on the wedge thickness. This is more visible in Figure 1j,k that show magnified images of regions A (192 mT) and B (320 mT), in which the aSk size is respectively, ≈128 ± 5 and ≈200 ± 4 nm. As shown previously for Mn 1.4 Pt 0.9 Pd 0.1 Sn, [2] the aSk size changes little with magnetic field. Thus, the aSk size variation arises predominantly from the change in thickness of the lamella rather than the change in magnetic field.
To explore possible variations in the aSk size and the helix period at thicknesses beyond those measurable by LTEM, we performed MFM measurements on two wedged lamellae with thicknesses varying from ≈630 to ≈4260 nm (L3) and ≈60 to ≈1600 nm (L4). The helical phase is clearly revealed in the MFM images. Typical results in zero magnetic field are summarized in Figure 2. An SEM image of the wedge L3 is shown in Figure 2a where the inset shows a schematic of the coordinate axis and the in-plane field direction ϕ H . A saturation field of 1 T is applied in the plane of L3 at an angle ϕ H and is then reduced to zero where the MFM image is taken: the MFM contrast corresponds to the stray field produced mainly by the out of plane magnetization component, M z . By varying ϕ H the helix propagation axis can be switched from predominantly The inset in Figure 2g shows the helical period variation for the wedge L2 obtained from LTEM studies and comparison with similar data on a wedge of comparable geometry for the sister compound Mn 1.4 Pt 0.9 Pd 0.1 Sn. [2] The data closely resemble one another.
The helical phase transforms into an aSk phase in the presence of a magnetic field, as shown in Figure 3 for wedge L3. Typical MFM images are shown for an initial field of zero, followed by magnetic fields of 300, 420, and 480 mT applied at 30° from the z-axis (Figure 3a  fully polarized state that depends on sample thickness. The critical fields at which these transformations take place depend sensitively on the in-plane component of the field (see Supporting Information). A large thickness range where the aSks can be seen simultaneously was found for a tilt angle of 30 ≈° which also matches the tilt angle used for the LTEM data. At a field of ≈550 mT the complete wedge was fully magnetized.
The aSk phase could be stabilized at all thicknesses which enabled the dependence of the aSk size on the lamella thickness to be determined. These results are summarized in Figure 3e together with results from the LTEM data in Figure 1 that are shown in the inset to Figure 3e. The protocol used to determine the aSk size is discussed in the Supporting Information. The combined MFM and LTEM data clearly show that the aSk size has a weak dependence on magnetic field but a very strong dependence on the lamella thickness. Indeed, the aSk size increases by one order of magnitude from ≈128 nm to nearly 1.2 µm as the lamella thickness is varied from 142 nm to ≈4.2 µm. The aSk size and helix period are found to be close to one another for the same lamella thickness (see Figure S3, Supporting Information).
For the thinner wedge sample that is measured by LTEM, one can see from Figure 1 that the antiskyrmions appear at the thinner side first under a smaller field and then appear at the thicker side under a larger field. However, the result is opposite for the thicker wedge sample that is measured by MFM in Figure 3. These differences could arise from the different field tilting angles used in the measurements of these images, which is ≈ 0° for LTEM and ≈ 30° for MFM, or possibly from the different shapes (inclination angles) of the wedges used. However, the measured helix period and aSk size are hardly influenced by the measurement protocols or the wedge shapes.
In order to study the large-scale magnetic textures that we find in thicker lamellae, MFM measurements were performed in the presence of an in-plane magnetic field applied along distinct directions. We first apply an in-plane field of H = 1.5 T along the direction ϕ H = 0° to polarize the magnetization along +x, after which H is gradually decreased to 0.3 T. Then we observe that the magnetic texture has become triangularly shaped throughout much of the lamella (L3), as shown in Figure 4a. All the triangles point along the same direction: we note that their size corresponds well to that of the original helix or aSk. Similar measurements for ϕ H = -90°, ϕ H = -180°, and ϕ H = -270° were performed, as shown in Figure 4b-d. In each case similar triangular magnetic textures are found but, most interestingly, they clearly point along different directions.
Let us consider the evolution of an aSk structure under an in-plane field applied along 〈100〉. Along these directions, the boundaries are Bloch like so that one expects that the area of the regions of in-plane magnetization within these boundaries pointing along the field direction will increase, whereas those pointing in the opposite direction will decrease in area. This will give rise to a distorted triangularly shaped object that is observed. Furthermore, when the field direction is rotated in a clockwise fashion, we find experimentally that the triangular shape rather rotates in the counterclockwise direction. This is in agreement with the expected distortion of the aSk structure under an in-plane magnetic field, as illustrated schematically in Figure 4e. Note that for a skyrmion structure, whether Bloch or Néel, the distorted spin textures would rotate in the same direction as the field. Another explanation for the triangularly shaped magnetic structures, which has recently been proposed, are "nontopological bubbles," [11] in which half of the object is a square antiskyrmion and the other half is a round skyrmion. This "nontopological bubble" will also rotate in the opposite direction to the in-plane magnetic field. Since MFM cannot readily distinguish between an antiskyrmion and such a "nontopological bubble," the detailed magnetic structures, especially in the large thickness regime, require further studies. There is, however, strong experimental evidence, by comparing Figure 3 with Figure 4, that the round shape magnetic object in Figure 3 is indeed an antiskyrmion.
In order to probe the microscopic mechanism of the thickness dependence of the aSk size and helix period, an analytic calculation was carried out for a model D 2d system. The DMI energy density in a D 2d material is given by where D is the DMI strength (J m -2 ). As shown in Figure 5a, we assume that the helix magnetization changes only along the helix direction with a Bloch sine function and a period of λ. The model assumes a slab of infinite extent in the xy plane and a finite thickness t along z. We assume the magnetization is constant along z which is a good approximation in a D 2d system where the DMI has no component along z. We note that this same assumption may not be appropriate for B20 systems in which the DMI imposes a twisted structure along z. The resultant total energy density is given by where A is the exchange stiffness (J m -1 ), K is the anisotropy energy (J m -3 ), E Exc is the volume exchange energy (J m -3 ), E DMI is the volume DMI energy, E Ani is the volume magnetic anisotropy energy, E Dip is the volume magnetic dipolar energy, and E Tot is the volume total energy. Since the exchange interaction, DMI and anisotropy in this system are of bulk origin, there are no thickness dependent prefactors in the energy functionals. The DMI energy term does not depend on the helix/cycloid propagation direction. [12] The detailed calculations are summarized in the Supporting Information. Most importantly the long-range magnetostatic energy E Dip can be given as an exact analytical expression. Atomistic numerical calculations were also carried out which agree well with this expression (see Supporting Information). The dipolar energy as a function of helix period for various t are plotted in Figure 5b. When the film thickness is very , which has a linear dependence on λ with a very small slope. In these two regimes, the dipolar energy has a weak dependence on λ, as shown in Figure 5b  . When the DMI strength is weaker, λ drops rapidly to a minimum value as t is increased and then slowly increases back to 4 A D π for very thick layers. As shown in Figure 5d, when M s is increased, λ shrinks. The dipole energy is dominantly located at the surfaces of the lamella except for very thin t. As shown in Figures S10 and S11 of the Supporting Information, E Dip is greatest at the surfaces, decaying exponentially into the interior of the lamella with a characteristic length-scale that is set by λ. Thus, in the limit of large t, the helix wavelength is no longer influenced by the dipole energy but is rather determined by the ratio of the Heisenberg and DMI energies. However, for the range of t of interest here the dipole energy strongly influences λ because, in very simplistic terms, the twisted helical structure reduces the dipole charge at the top and bottom surfaces. Since the total dipolar energy decreases, in this range, as ≈1/t, thus λ monotonically increases as t is increased, thereby accounting for our experimental observation that λ approximately linearly increases with thickness. In Figure S13 of the Supporting Information a comparison between our experimental results and the model discussed above is made. Qualitatively the model accounts well for the observed trend of increasing λ with thickness. But it is difficult to find within the limitations of our model an exact agreement. We attribute this to deviations from the assumed pure helical Bloch-like form of the magnetic structure throughout the thickness. Our MFM data show that the helix structure is nearly sinusoidal and, since the magnitude of the MFM signal increases linearly with thickness, that the helix exists throughout the thickness. Nevertheless, there can clearly be deviations from this simple structure due to the complex interplay between all the relevant energy terms including the dipolar energy.
The calculation of the magnetodipolar interaction for the aSk phase is much more complicated and needs further theoretical studies. However, since, as shown in our experiments (Figures 2g and 3e), the helix period and aSk size are similar to each other, it is reasonable to assume that the same mechanism, i.e., the thickness dependence of the dipolar energy, accounts for the size variation of the aSk, as well as the helix period.
The thickness dependent behavior of the characteristic size of the spin texture reported in this work is distinct from other material systems due to important differences in the underlying physical mechanisms. In magnetic bubble systems, [8] the magnetic structure is achiral since there is no DMI interaction. For a given thickness, the size of the bubble is not fixed and can vary strongly, and, moreover, has a significant dependence on the external magnetic field. Recently, our group reported the experimental discovery of Néel skyrmions in lamellae formed from PtMnGa which has a C 3v structure. [13] There, the Néel skyrmion size was also found to have a large thickness dependence. However, the width of the boundary of the Néel skyrmion is small compared to the size of the object itself, so that this system is more akin to conventional magnetic "bubbles." In an interfacial DMI system, [14] although it has been reported theoretically that dipolar interactions could play some role, [15] the effective strength of the DMI decreases with magnetic layer thickness, so limiting these systems to the nanometer thickness range.
In B20 bulk materials [5][6][7] the DMI has nonzero components along all three spatial directions, which is distinct from the D 2d system where the DMI is zero along the [001] direction. Thus, skyrmion tubes in a B20 material will have an additional twist in their magnetic structure along the tube direction. Such a twisted structure might weaken the contribution of the dipolar