Detection of antiskyrmions by topological Hall effect in Heusler compounds

Heusler compounds having $\textit{D}$${}_{2d}$ crystal symmetry gained much attention recently due to the stabilization of a vortex-like spin texture called antiskyrmions in thin lamellae of Mn${}_{1.4}$Pt${}_{0.9}$Pd${}_{0.1}$Sn as reported in the work of Nayak $\textit{et al.}$ [Nature (London) 548, 561 (2017)]. Here we show that bulk Mn${}_{1.4}$Pt${}_{0.9}$Pd${}_{0.1}$Sn undergoes a spin-reorientation transition from a collinear ferromagnetic to a noncollinear configuration of Mn moments below 135 K, which is accompanied by the emergence of a topological Hall effect. We tune the topological Hall effect in Pd and Rh substituted Mn${}_{1.4}$PtSn Heusler compounds by changing the intrinsic magnetic properties and spin textures. A unique feature of the present system is the observation of a zero-field topological Hall resistivity with a sign change which indicates the robust formation of antiskyrmions.

A chemical substitution in place of the heavy element Pt in the non-centrosymmetric Heusler compound Mn1.4PtSn can allow for systematic control of the intrinsic properties that give rise to topological textures, namely spin orbit coupling (SOC), electron occupation and magnetization. These properties determine fundamental parameters such as the exchange interactions, the DMI, and the magnetocrystalline anisotropy (MCA). In turn, the competition of these parameters determines which magnetic texture is formed, i.e. a long wavelength spiral or a topological spin texture. The presence of more than one magnetic sublattice and the competition between parallel and antiparallel exchange interactions in the non-centrosymmetric crystal can lead to a spin reorientation transition [33]. In Mn1.4PtSn below 160 K the formation of a non-collinear spin structure was reported [15]. In this article, we investigate the detailed magnetic structures of Mn1.4Pt0.9Pd0.1Sn below and above the spin reorientation transition temperature TSR and demonstrate that below TSR a THE emerges with features which are evidence for the presence of aSKs in the bulk material. Fractional substitution of Pt by Pd (isoelectronic substitution) in Mn1.4PtSn is used to exclusively change the SOC, whereas substitution by Rh varies the electron occupancy of the bands and in consequence changes the SOC as well as the magnetization. Our results demonstrate the tunability of topological spin textures in the Heusler system which paves the way for systematic design of aSK phase containing compounds suitable for applications in spintronics. The THE can be used as a sensitive probe for characterizing the spin textures. b(Sn) = 6.228 fm [35]. The magnetic form factor of the Mn atoms was taken from Ref. [36].
Magnetization measurements were performed using a vibrating sample magnetometer (MPMS3, Quantum Design, µ 0 max = 7 T). The electrical transport properties were investigated using a physical property measurement system (PPMS9, Quantum Design, µ 0 max = 9 T). The samples were cut into rectangular bar shaped pieces with dimensions of approximately 3.5×1.5×0.5 mm 3 . Point contacts were made on the samples with 25 μm diameter Pt wires and silver paint. The five probe geometry was used for the Hall experiments. The Hall resistivity was measured at different temperatures between 2 and 300 K in the field range from 5 to 5 T in hysteresis mode by applying the field perpendicular to the rectangular surface of the bar. In order to avoid any artifact due to the demagnetization effect, both magnetization and resistivity measurements were performed on the same piece (length × breadth × height) where the magnetic field was applied perpendicular to the rectangular surface (length × breadth) under identical conditions. It should be noted that in a superconducting coil trapped magnetic flux may lead to deviations of reported and real magnetic field. Especially, when measuring the magnetization, magnetoresistance, or the Hall resistivity of a soft ferromagnet in a field sweep through zero, the reversal of the hysteresis (i.e. of the coercive fields) may be observed [37].
The presented data were measured in a MPMS3 magnetometer with a 7 T magnet and a PPMS with a 9 T magnet, respectively. In field sweeps both systems typically produce a maximum field error of 20 Oe around zero field [37].

A. Crystal structure and lattice parameters
We determined the crystal structure and lattice parameters of the prepared samples by Rietveld refinement of the powder x-ray diffraction patterns shown in Fig. S1  first increases a and decreases c up to y = 0.5 and thereafter a reverse trend is observed i.e., a decreases and c increases from y = 0.6 to 0.8. Consequently, the c/a ratio decreases up to y = 0.5 and then increases considerably. However, the cell volume decreases linearly up to y = 0.5 and thereafter the decrement is non-monotonic.
In this system, the tetragonal structure results from two cubic unit cells slightly displaced along the c axis which leads to a c/a ratio close to 2. Hence, the degree of tetragonality increases when the c/a ratio tends away from the value 2 or in other words the c/a ratio moves away from the pseudo-cubic condition (c = a‫,׳‬ where a‫׳‬ = 2a). The decrease in the c/a ratio for all Pd substitutions and moderate Rh substitutions up to y = 0.5 suggests an enhanced tetragonality.
Accordingly, higher Rh substitution, y > 0.5, leads to a decrease in tetragonality as reflected in the drastic rise in the c/a ratio.

B. Neutron powder diffraction
We investigated the detailed magnetic structure of the present system using neutronpowder-diffraction experiments. In Fig. 2 the neutron-powder-diffraction patterns of Mn1.4Pt0.9Pd0.1Sn are shown, the corresponding Rietveld refinement results are summarized in Table S1 of Supplemental Material [38]. The data collected at 449 K (well above the Curie temperature, TC = 390 K) was successfully refined in the space group I4 ̅ 2 (No. 122) as shown in Fig. 2 Fig. 2(b)), indicating a ferromagnetic ordering of the Mn atoms with a propagation vector k = 0. Due to the fact that magnetic intensity is observable on the reflections 200 and 220 it can be assumed that the magnetic moments are predominantly aligned parallel to the c axis. Therefore, we used a magnetic structure model where only the z components of the magnetic moments μz(Mn1) and μz(Mn2) were allowed to vary, see Table S1 in Supplemental Material [38]. The magnetic moments were found to be that the high-temperature magnetic configuration has a collinear ferromagnetic spin alignment along the tetragonal axis, in contrast to the previous suggestion of a ferrimagnetic spin structure [15]. The proposed magnetic structure was based on the general model of Mn2-based inverse Heusler compounds, which frequently have a ferrimagnetic configuration due to antiparallel spin alignment between Mn atoms at different sites [39,40]. A simulation of a ferrimagnetic model demonstrates that strong magnetic Bragg intensity is generated at the position of the relatively weak nuclear reflection 101 which is not observed.
Below 135 K, our neutron powder diffraction data showed a spontaneous increase of magnetic intensity on the reflections 101 and 004. The strong increase of the magnetic intensity of the 004 reflection indicates the presence of an additional spin alignment in the ab plane. We focused on the data set collected at 2.4 K (shown in Fig. 2  Our neutron investigations demonstrate that Mn1.4Pt0.9Pd0.1Sn displays a collinear ferromagnetic spin alignment along the tetragonal axis in the temperature range TSR < T < TC, whereas the magnetic structure transforms into a non-collinear spin configuration for T < TSR. In Fig. 3 the magnetic structures below (2.4 K) and above (200 K) TSR, as well as the angle between the Mn1 and Mn2 moments are shown. The spin reorientation transition involves spin canting of each Mn sublattice as well as an increase in the total moments. These findings are consistent with the temperature dependent magnetization data presented in Fig. S3 of Supplemental Material [38]. The low-temperature spin structure is non-collinear, but still coplanar. Recently, a non-coplanar spin arrangement has been discussed as a candidate structure for Mn1.4PtSn even at zero-field, where an additional antiferromagnetic component may be present parallel to y [41]. This order should generate magnetic intensity at the position of the reflection 002. From our data of Mn1.4Pt0.9Pd0.1Sn collected at 2.4 K we were not able to find significant magnetic intensity at this position. Simulations showed that any magnetic moment parallel to y is expected to be smaller than 0.2 μB.
Neutron patterns of Mn1.4Pt0.6Rh0.4Sn verify that for moderate degrees of Rh substitution a non-collinear spin structure and collinear ferromagnetic structure are retained below and above TSR, respectively, whereas for higher substitution levels y > 0.5 an essentially collinear ferrimagnetic (FiM) spin configuration with both Mn moments lying in the basal plane emerges at all temperatures below TC (see the supplemental Material [38]). The change in spin structure is associated with a decrease in the tetragonal distortion of the crystal structure for y > 0.5 as discussed in Section III.A. (c.f. Fig. 1). The neutron diffraction data, however, cannot exclude the persistence of a small spin canting which might lead to a topological Hall effect in an external magnetic field.

C. Magnetic properties
The evolution of the magnetic properties for both substitution series is depicted in Fig An approximately linear decrease of the magnetization is observed with increasing y, i.e.
decreasing number of valence electrons, for Rh contents up to y = 0.5, which demonstrates a Slater-Pauling-like reduction [42].

D. Topological Hall effect
In Fig below the field, above which the magnetization tends to saturate and (2) A large coercivity ( −450 Oe) in the Hall resistivity hysteresis (within ±0.5 T) which is reversed to that of the magnetic isotherm. As mentioned in the experimental section the maximum error in the magnetic field in our superconducting magnets is 20 Oe, such small field errors cannot be responsible for the hysteresis loop reversals observed for the present material. Indeed, the observation of enhanced feature below the saturation field in ( ) indicates that actually a non-coplanar spin configuration emerges from the canted spins below TSR which is induced by application of a magnetic field in the Hall experiment and gives rise to a THE. On the other hand, the inverse hysteretic observation at lower field in ( ) suggests the presence of aSKs whose cores are antiparallel to the field direction and which are stabilized by the MCA. In B20 compounds this effect is observed when the anisotropy is induced by the thin film limit [7,22,43], while in the compounds with D2d symmetry the anisotropy is inherent to the crystal lattice. In addition, microstructural defects like twins and grain boundaries of the polycrystalline samples may also act as pinning centers to stabilize the aSKs. Previous reports on analogous Heusler compounds e.g. bulk Mn2PtSn [44] and thin films of Mn2PtSn [45], Mn2-xPtSn [46] and Mn2RhSn [47] also revealed a THE but did not show an inverse hysteresis in the loop. In Fig. 5(b) the topological Hall resistivity, T at T = 2, 120 and 150 K is shown (the method for the extraction of T is shown in Supplemental Material [38]). T displays an inverse hysteresis as compared to the magnetization at 2 and 120 K (< TSR), whereas no hysteresis is where T is expected to be independent of the variation in longitudinal resistivity [19]. Further evidence of aSKs at H = 0 is obtained by the opposite signs of max T and 0 T , which can only be caused by a sign change in the topological texture on going from a NCP structure to an aSK phase. In the work of Nayak et al. [15], the aSK phase was even stable above TSR in thin lamellae of Mn1.4Pt0.9Pd0.1Sn, which is probably related to the reduced dimensionality. By contrast, the present study of Hall experiments demonstrates that bulk polycrystalline materials show aSKs only in the non-collinear magnetic phase (< TSR).
Upon increasing the Pd content up to x = 0.3, the largest possible Pd substitution level, only slight changes in magnetic and transport properties are observed (see the Supplemental Material [38]). This implies that a moderate change in SOC due to variation of the Pd substitution does not change much the physical situation reflected by Fig. 5. By contrast, substitution of Rh for Pt has more pronounced implications for the properties. In particular, the magnetization features a strong Slater-Pauling-like reduction with decreasing electron count as well as a decrease in TC (see Fig. 4). Analysis of the field dependence of the magnetization indicates that the MCA in the present system is strengthened by increasing the Rh content (see  Table 1). This is further shown as a schematic of arrows in Fig. 6 Finally, the aSK phase (blue background) occurs in both Rh and Pd substituted samples in the composition ranges 0 < y < 0.25 and 0 ≤ x ≤ 0.3, respectively. The aSK phase is stabilized at lower fields (μ0H ≤ |0.5| T) and remained stable at zero field. The max T has a nearly constant value of  0.2 μΩ cm. Furthermore, C M is smallest in this range but its sign is also opposite to that of C T . This is a key feature of the formation of aSKs. In this composition range, due to the enhanced magnetization, the DMI is sufficiently large to induce transformation of the NCP structure to the aSK phase, whereas the MCA is primarily important for the survival of the aSK at zero field. The opposite sign for 0 T and max T is due to the change in topology of the magnetic spin structure.
The max T found in the present system is due to the NCP spin structure, which displays the largest value of  0.5 μΩ cm, and is comparable to that observed in other similar Heusler compounds as bulk Mn2PtSn (1.53 μΩ cm) [44], thin films of Mn2PtSn (0.5 μΩ cm) [45] and Mn2-xPtSn (1.2 μΩ cm) [46]. However, these cases did not show the reversal of hysteresis in the Hall resistivity compared to that of the magnetization, which is attributed to the aSK phase.
The presence of aSKs results in a finite value of 0 T with a negative sign, which has the largest magnitude of  0.08 μΩ cm for x = 0.1. The THE in the B20 compounds due to the stabilization of skyrmions has been well described by the size of their spin texture. The emergent magnetic field ( em ) shows quadratic decrement with the size ( sk ) of the skyrmion in the adiabatic limit, em ∝ 1/ sk 2 [18]. For instance, MnGe [19] and FeGe [22] display T values of 0.16 and 0.08 μΩ cm at a finite field for sk = 3 and 70 nm, respectively. Intriguingly, despite the comparatively large size of the aSKs in Mn1.4Pt0.9Pd0.1Sn (150 nm) [15], we find a large topological Hall resistivity even at zero fields. This is likely due to the role of the electronic structure in the topological Hall constant T = T / em [24], and provides the possibility to engineer even a larger THE in aSK phases with smaller periods.

IV. CONCLUSIONS
We have shown the THE due to the robust formation of antiskyrmions below TSR in the bulk Mn1.

Detection of antiskyrmions by topological Hall effect in Heusler compounds
Vivek Kumar, 1

I. Powder x-ray diffraction (XRD)
We show room temperature XRD patterns in Fig. S1.   which again matches well with the value from the magnetization measurements.

Symmetry analysis
In order to proof the consistency of the proposed magnetic structures with the symmetry requirements of space group I4 ̅ 2 a symmetry analysis of the magnetic structure has been

III. Magnetic characterization
Temperature and field dependent magnetization measurements were performed using a vibrating sample magnetometer (MPMS3, Quantum Design). The DC M(T) measurements were carried out from 2 to 400 K in the zero field cooling (ZFC), field cooled cooling (FCC) and field cooled warming (FCW) modes using a field of 0.01 and 0.1 T. The AC susceptibility measurements were taken from 2 to 400 K using a small AC driving field of 5 Oe and a frequency of 18 Hz in the absence of a DC bias field. Magnetic isotherms M(H) were measured in the field range of 7 to 7 T at different temperatures between 2 and 300 K. The field dependent AC susceptibility measurements were carried out using a small AC driving field of 5 Oe and a frequency of 486 Hz in the field range of ±5 T.
We present the magnetic properties of Mn1.4Pt0.9Pd0.1Sn in Fig. S3. The TC (392 K) and the field range of ±0.5 T. It decreases with temperature and disappears above TSR. The bifurcation of the Hall resistivity ( Fig. 5(a)) and the irreversibility of χ' at different field sweep directions match well. This further supports the formation of a skyrmion-like spin texture [1,2] which gives rise to the finite value of the Hall resistivity at zero field despite of the zero magnetization.

IV. Transport measurements
The electrical resistivity , measured by lowering the temperature from 300 to 2 K is shown in Fig. S7(a).  Fig. S7(b). The MR at 2 K is positive for fields up to 0.5 T and becomes negative for higher fields. This anomaly in MR decreases with rise in temperature and vanishes above TSR which seems to be correlated with the Hall and the Therefore, we can take A = , where = + 2 is considered as a constant. Now we extract topological Hall resistivity assuming that is a constant (described in Fig. S9).
The constants 0 and are derived by the straight line fitting of the equation = 0 0 + . The extraction of topological Hall resistivity does not show any significant difference on the form of anomalous Hall resistivity due to the small variation of .
In Fig. S10 we present the log-log plot of (   However, there is no indication of THE for y = 0.8. The anomalous Hall conductivity A (shown in Fig. S15) as a function of temperature demonstrates a significant enhancement below TSR.
The compound with y = 0.6 has a slight increase in conductivity, however no enhancement is observed for y = 0.7 and 0.8. This is consistent with the scaling of A by the magnetization M.