Observation of Topological Hall Effect and Signature of Room Temperature Antiskyrmions in Mn-Ni-Ga D2d Heusler magnets

TL;DR

This study reports the observation of room-temperature antiskyrmions in Mn-Ni-Ga Heusler alloys, revealing a large topological Hall effect and potential for racetrack memory applications.

Contribution

It demonstrates the stabilization of antiskyrmions at high temperatures in Mn-Ni-Ga alloys, linking D2d symmetry to topological spin structures and their Hall effects.

Findings

Large topological Hall effect observed beyond room temperature.

Antiskyrmions stabilized up to 550 K via composition tuning.

Micromagnetic simulations confirm stabilization mechanisms.

Abstract

Topologically stable nontrivial spin structures, such as skyrmions and antiskyrmions, display a large topological Hall effect owing to their quantized topological charge. Here, we present the finding of a large topological Hall effect beyond room temperature in the tetragonal phase of a Mn-Ni-Ga based ferrimagnetic Heusler shape memory alloy system. The origin of the field induced topological phase, which is also evidenced by the appearance of dips in the ac-susceptibility measurements, is attributed to the presence of magnetic antiskyrmions driven by D2d symmetry of the inverse Heusler tetragonal phase. Detailed micromagnetic simulations asserts that the antiskyrmionic phase is stabilized as a result of interplay among inhomogeneous Dzyaloshinskii-Moriya interaction, the Heisenberg exchange, and the magnetic anisotropy energy. The robustness of the present result is demonstrated by stabilizing the antiskyrmion hosting tetragonal phase up to a temperature as high as 550 K by marginally varying the chemical composition, thereby driving us a step closer to the realization of ferrimagnetic antiskyrmion based racetrack memory.

Full text

Topological Hall Effect, Skyrmions, Heusler compounds

In recent years, there is a significant interest towards non-collinear magnetism, where the local magnetic state can be periodically altered via spin transfer torque by passing a spin polarized current Parkin08 . The prospect of non-collinear magnetism can be greatly enhanced when the aforementioned magnetic structure is topologically stable in nature. One of such spin textures is the recently discovered magnetic skyrmion, which is a vortex-like object with a swirling spin configuration Pfleiderer09 ; Yu10 . The topological nature of the skyrmions helps them to get decoupled from the crystal lattice, thereby assisting to move at much lower current density in comparison to that of domain walls Schulz12 . The topologically stable spin texture of the skyrmions is accompanied by a topological charge $Q=\frac{1}{4\pi}\int{\bf m}\cdot(\frac{\partial{\bf m}}{\partial x}\times\frac{\partial{\bf m}}{\partial y})dxdy=\pm 1$, where ${\bf m}$ is the unit vector along the local magnetization Nagaosa13 . When a conduction electron approaches a skyrmion, the spin of the electron tries to align with the local magnetization of the skyrmion owing to a large Hund’s coupling. Consequently, the electrons experience a large fictitious magnetic field, resulting in an additional component to the observed Hall voltage, named as topological Hall effect (THE) Neubauer09 . Effective fictitious magnetic field of 4000 T can be realized for a skyrmion of size 1 nm Kanazawa15 . Depending upon the topological charge of the skyrmion ($\pm 1$), the topological Hall component adds or subtracts from the normal and anomalous Hall components to develop a hump or dip type of behavior in the total Hall voltage observed in various bulk materials and thin films Gallagher17 ; Matsuno16 ; Kanazawa15 ; Li13 ; Schulz12 ; Huang12 ; Kanazawa11 ; Neubauer09 .

The topologically stable spin texture of the skyrmions arises from the spin-orbit interaction mediated Dzyaloshinskii-Moriya interaction (DMI), which competes with the Heisenberg exchange ($J$) and magnetic anisotropy to form a stable skyrmion lattice. The DMI energy that can be expressed as $\bf D_{ij}\cdot(\bf S_{i}\times\bf S_{j})$, where $D$ is the DMI vector and $S_{i}$ and $S_{j}$ are spins at the $i^{th}$ and $j^{th}$ sites, respectively, exists in systems with broken inversion symmetry and large spin-orbit coupling Dzyaloshinskii57 ; Moriya60 . The magnetic materials with B20 and related crystal classes that possess intrinsic bulk DMI display Bloch-type skyrmions Pfleiderer09 ; Yu10 ; Yu11 ; Seki12 ; Tokunaga15 , whereas, most of the layered thin films with interfacial DMI and some bulk materials with suitable crystal symmetry host Néel skyrmions Heinze11 ; Sampaio13 ; Jiang15 ; Woo16 ; Soumyanarayanan17 ; Kezsmarki15 ; Kurumaji17 . Artificial magnetic skyrmions with nano-patterning is also realized without DMI Li14 ; Sun13 ; Miao14 . A latest addition to the skyrmion family is the recently observed antiskyrmions in $D_{2d}$ crystal symmetry based inverse tetragonal Heusler compounds Nayak17 . The special crystal symmetry of these materials ensures an inhomogenious DMI vector ($D_{x}=-D_{y}$) in contrast to the homogeneous DMI observed in materials exhibiting Bloch and Néel skyrmions ($D_{x}=D_{y}$) Leonov17 ; Hoffmann17 ; Camosi18 .

It has been established that Mn${}_{2}YZ$- based Heusler compounds display a non-centrosymmetric crystal structure Graf11 . The DMI in these materials can be set up in case of Mn${}_{2}YZ$ tetragonal Heusler compounds Meshcheriakova14 ; Nayak17 . The Heusler shape memory alloys (SMA) that undergo a martensite transtion from high temperature cubic to low temperature tetragonal phase, possess a great potential to host nontrivial spin texture like skyrmions. However, most of these alloys exhibit a modulated and centrosymmetric tetragonal structure that preclude DMI in the system Nayak14 ; Planes09 ; Yu15 . An asymmetric tetragonal structure with $D_{2d}$ symmetry can be stabilized in case of Mn2NiGa when a single Mn atom in the Mn-Mn plane is replaced by Ni atom in Mn3Ga, as shown in Fig. 1(a) Balke07 ; Nayak15 ; Liu05 ; Liu06 . The Mn sitting in Mn-Mn/Mn-Ni plane and the Mn at Mn-Ga plane align antiferromagnetically, account for the ferrimagnetic ordering in the system Barman08 . The existence of a more complex non-collinear spin structure and/or the presence of slight antisite disorder intrinsic to most of the Heusler materials can also result in the mismatch between experimentally observed moment with that of theoretical prediction Barman08 . In this letter we show that Mn-rich Mn-Ni-Ga based inverse Heusler system indeed displays a large topological Hall effect in the tetragonal phase, suggesting the presence of antiskyrmions in the system.

Like most of the Mn2- based Heusler compounds, Mn2NiGa exhibits a ferrimagnetic ordering with a Curie temperature of $\sim$ 650 K. The $D_{2d}$ symmetry of the tetragonal phase can ensure a competing iteraction between the DMI and the Heisenberg exchange that can result in an one-dimensional helix in the [100] direction and a cycloid along the [110] direction, as shown in Fig. 1(b). In this scenario, application of magnetic field can generate ferrimagnetic antiskyrmions as schematically demonstrsted in Fig. 1(c). The thermomagnetic $M(T)$ curves measured in field-cooled (FC) and field-heating (FH) modes up to 400 K for Mn2NiGa and Mn2.1NiGa0.9 are depicted in Fig. 1 (d). The signature of structural transition in Mn2NiGa can be seen from the presence of large hysteresis in cooling and heating $M(T)$ curves around 300 K, whereas, no such transition is found in Mn2.1NiGa0.9. Both the samples exhibit the Curie temperature ($T_{C}$) well above the room temperature [inset of Fig. 1(d)]. An experimental signature that hints at the presence of an additional magnetic phase in both the samples was obtained from the isothermal magnetization $M(H)$ data which display a small kink marked by arrows in Fig. 1(e). Since Mn2NiGa transforms to the cubic phase above 300 K, this transition disappears for $M(H)$ loops measured at higher temperatures. The magnetic moment of the present Mn2NiGa matches well with the previous report Liu06 .

Motivated by the signature of magnetic phase transition in the $M(H)$ data, we have performed Hall effect measurements at different temperatures as depicted in Fig. 2. For Mn2NiGa, the total Hall resistivity $\rho_{xy}$ exhibits a dip around 0.5 T for all temperatures $T\leq 300$ K [Fig. 2(a)&(b)]. This peculiar behavior disappears for the $\rho_{xy}$ data collected above the martensite transition at $T=385$ K in the cubic phase [Fig. 2(c)]. Although the martensite transition sets the DMI in the tetragonal phase, it has no role in the observed anomaly in the $\rho_{xy}$ data. This is demonstrated in another sample Mn2.1NiGa0.9 that exhibits tetragonal phase in the whole temperature range up to the $T_{\text{C}}$, without any structural transition. For this sample the $\rho_{xy}$ data acquired up to 380 K, the highest possible measured temperature, display a similar dip kind of behavior around 1 T [Fig. 2(d)-(f)].

It is well known that the total Hall resistivity can be expressed as $\rho_{xy}=\rho_{N}+\rho_{AH}+\rho^{T}_{xy}$, where $\rho_{N}$, $\rho_{AH}$ and $\rho^{T}_{xy}$ are normal, anomalous, and topological Hall resistivities, respectively. Normal Hall resistivity can be written as $\rho_{N}=R_{0}H$, where $R_{0}$ is the normal Hall coefficient. Anomalous Hall resistivity, which is in general directly proportional to the magnetization in a ferri-/ferromagnet, can be expressed in terms of the longitudinal resistivity ($\rho_{xx}$) and magnetization ($M$) as $\rho_{AH}=b\rho_{xx}^{2}M$, where $b$ is a constant. The effect of skew scattering and side-jump on AHE in the present sample are not taken into consideration due to the fact that the longitudinal resistivity in the present bulk materials is too high and acan be neglected completely at high temperatures. In case of Mn2NiGa, the anomaly in the $\rho_{xy}$ data is only found for fields less than 1 T, whereas, Mn2.1NiGa0.9 displays such behavior for fields up to 2 T . Hence it is assumed that the high field $\rho_{xy}$ data do not consist of any $\rho^{T}_{xy}$ component. At high fields, $\rho_{xy}$ can be further simplified to $\rho_{xy}=R_{0}H+b\rho_{xx}^{2}M$. The linear fit between $\frac{\rho_{xy}}{H}$ and $\frac{\rho_{xx}^{2}M}{H}$ gives us slope $b$ and intercept $R_{0}$. In the present case, the values of b and $R_{0}$ are calculated by using $\rho_{xy}$ data for $\mu_{0}H>\pm~{}3~{}T$. Afterwards, $\rho_{xy}$ sans $\rho^{T}_{xy}$ was calculated using $\rho_{xy}=R_{0}H+b\rho_{xx}^{2}M$, as shown by red lines on the top of the experimental $\rho_{xy}$ curves in Fig. 2. It can be clearly seen that the experimental and the calculated $\rho_{xy}$ curves display a substantial difference at the field where both the magnetization and Hall resistivity exhibit dips, whereas, perfect matching is obtained for higher field regions. The calculated $\rho_{xy}$ was subtracted from the experimental $\rho_{xy}$ to obtain $\rho^{T}_{xy}$ as plotted in Fig. 4(a)&(b). The validity of the extraction of THE by the present method is well verified in case of Mn1.8Ni1.2Ga, where both the experimental and calculated curves match at all field regime, as this sample does not exhibit any anomaly in Hall effect measurements.

For a deep understanding of the origin of the observed topological Hall effect, we have carried out ac-susceptibility measurements which have been extensively used to characterize skyrmions in several materials Kurumaji17 ; Wilhelm11 ; Bauer12 . In the present case, the real part of the ac-susceptibility, $\chi^{{}^{\prime}}(H)$, for Mn2NiGa exhibits a dip/peak type behavior around the fields where a large topological Hall effect is found. The magnitude of this dip/peak behavior initially increases with increasing temperature before getting slowly suppressed for $T\geq 250$ K due to the presence of a small amount of cubic phase with higher magnetic suceptibility. The dip/peak completely vanishes at 370 K in the cubic phase. For Mn2.1NiGa0.9, a pronounced and well defined dip/peak can be found up to 380 K, suggesting the presence of magnetic antiskyrmions in the present system. It is worth to mention that the tetragonal Heusler compound Mn1.4Pt0.9Pd0.1Sn that displays antiskyrmions up to room temperature Nayak17 also exhibits a similar behavior in the ac-susceptibility data Manna18 . The robostness of the ac-susceptibility measurements is vindicated in case of Mn1.8Ni1.2Ga that does not exhibit any anomaly in $\chi^{{}^{\prime}}(H)$ as no THE is found in this sample.

The occurrence of large topological Hall effect that is underpinned by the observation of dips/peaks in the ac-susceptibility data in the present material lends firm support for the existence of some non-trivial spin texture, such as, skyrmions. The $D_{2d}$ crystal symmetry of the present system ensures an anisotropic DMI with $D_{x}=-D_{y}$, thereby leading to antiskyrmions. In order to gain more insights into the magnetic field and temperature dependence of the antiskyrmions in the present system, we have plotted $\rho^{T}_{xy}$ at different magnetic fields as shown in Fig. 4(a)&(b). In general, the topological Hall effect scales (i) directly with the density and (ii) inversely with the size of the skyrmions/antiskyrmions. Since for a given system the size of the skyrmions/antiskyrmions remains almost constant, the enhanced $\rho^{T}_{xy}$ at room temperatures for Mn2NiGa can be attributed to a significant increase in antiskyrmion density due to the higher nucleation ability of antiskyrmions at the tetragonal to cubic phase transition. This can be understood from the fact that in case of a bulk system the nucleation probability of skyrmions/antiskyrmions increases around the magnetic ordering temperature Nayak17 ; Pfleiderer09 ; Yu11 ; Tokunaga15 . In case of Mn2.1NiGa0.9, the magnitude of $\rho^{T}_{xy}$ almost remains constant up to the highest measured temperature of 380 K. Figure 4(c)&(d) represent the $H-T$ phase diagrams for Mn2NiGa and Mn2.1NiGa0.9. For Mn2NiGa the THE almost vanishes for a field of about 1.5 T, whereas, Mn2.1NiGa0.9 exhibits THE for field as high as 3 T.

A factor that significantly contributes to the size and stability of the antiskyrmions in the present tetragonal materials is the anisotropy energy. The presence of considerable amount of magnetic anisotropy in the present system can be seen from the out-of-plane type hysteretic behavior of the $M(H)$ loop in Fig. 1(d). A slight change in the Mn/Ga ratio significantly changes the coercive field and the magnetic ordering temperature. This signifies a considerable change in the magnetic anisotropy as well as the exchange constant $J$, which can modify the size and stability of the antiskyrmion phase. For a detailed understanding of the stability of antiskyrmion phase at different anisotropy constant $K\textsubscript{u}$, DMI constant $D$ ($D_{x}=-D_{y}$), and exchange stiffness constant $A$, we have carried out a detailed micromagnetic simulation using public domain software package Object Oriented MicroMagnetic Framework (OOMMF) oommf with DMI extension module dmi . Initially, a $1000\times 1000\times 2$ nm3 thin film was relaxed from a random magnetization state in presence of perpendicular magnetic field for different values of $D$ and $A$ with zero anisotrpy. $A$ was calculated from the $T\textsubscript{C}$ and saturation magnetization $M\textsubscript{s}$ from $M(H)$ loop at 2 K. After the initial relaxation, the anisotropy constant $K\textsubscript{u}$ was increased to various values to check the stability of antiskyrmion lattice at the corresponding values of $D$, $A$ and $K\textsubscript{u}$. Figure 4(e) shows the $K\textsubscript{u}$-$D$ phase diagram corresponding to the experimental parameters $A=3.0\times 10\textsuperscript{-11}$ J/m , $M\textsubscript{s}=2.37\times 10\textsuperscript{5}$ A/m, and $\mu_{0}H$ = 500 mT. A stable antiskyrmion lattice can be found for $D\approx 2.0-4.0$ mJ/m2 and $K\textsubscript{u}\approx 0-5\times 10\textsuperscript{5}$ J/m3. A rough estimation of the anisotropy constant from the $M(H)$ loops yields $K\textsubscript{u}\approx 3.0-5.0\times 10\textsuperscript{5}$ J/m3. A higher $D$ and $K\textsubscript{u}$ results in a mixed or scattered antiskyrmion phase. For a fixed $D=4.0$ mJ/m2, a stable antiskyrmion lattice can be found for $A\approx 3.0\times 10\textsuperscript{-11}$ J/m [Fig. 4(f)]. At lower values of $A$, which is expected for Mn2.1NiGa0.9, mixed phase and scattered antiskyrmions were stabilized at higher values of $K\textsubscript{u}$. A decrease in the size of antiskyrmions at lower $A$ and higher $K\textsubscript{u}$ leads to a significant increase in the density ($n$) even in the mixed and scattered antiskyrmion state.

As it can be seen, a stable antiskyrmion phase can be formed for $D=2.0-4.0$ mJ/m2 and $K\textsubscript{u}=0-5\times 10\textsuperscript{5}$ J/m3. The size (diameter) of the antiskyrmions corresponding to these values of $D$ and $K\textsubscript{u}$ is about 40-60 nm. It is known that the magnitude of topological Hall voltage greatly depends upon the size and density of skyrmions. We have estimated the size of the antiskyrmions from the measured topological Hall effect using the relation $\rho^{T}_{xy}=PR_{0}B_{eff}$, where $P$ is the conduction electron spin polarization and $B_{eff}$ is the effective (fictitious) magnetic field Neubauer09 . Further $B_{eff}$ can be expressed as $B_{eff}=-\phi_{0}/a_{sk}$, with $\phi_{0}=h/e$ is the magnetic flux generated by a single skyrmion and $a_{sk}$ is the size of the skyrmion Kanazawa15 . The conduction electron polarization can be roughly estimated as $P=M_{sp}/M_{s}$, where $M_{sp}$ is the ordered moment in the antiskyrmion phase and $M_{s}$ is the saturation magnetic moment in the system Neubauer09 . In the present case $P$ comes about 0.7. By taking the highest THE at room temperature for Mn2NiGa, the effective magnetic field is calculated as 8.8 T and the size of the antiskyrmions is found to be about 22 nm. A small mismatch of the antiskyrmion size might be arising from the fact that the simulations were carried for a thin film, whereas, experiments were performed on the bulk materials. It is worth to mention here that the size of the antiskyrmions in Mn-Ni-Ga system is much smaller in comparison to the recently observed antiskyrmion size of 150 nm in Mn-Pt(Pd)-Sn based Heusler materials Nayak17 . In the present case, by slightly changing the composition, both the magnetic anisotropy and the exchange interaction can be tuned significantly. This is evident from the increase in the coercive field and decrease in the $T_{\text{C}}$ as well as the saturation magnetic moment; which in principle could give rise to a reduced antiskyrmion size in Mn2.1NiGa0.9. Due to the ferrimagnetic ordering in the present system one can expect a reduced skyrmion Hall effect Woo18 , that might help the present ferrimagnetic antiskyrmions to move along the direction of applied currents.

In summary, we have established the presence of a large topological Hall effect that withstands above room temperature in the Mn-Ni-Ga based magnetic shape memory alloys. The topological Hall effect that exists in the $D_{2d}$ symmetry based tetragonal phase vanishes when the system undergoes a structural transition to the cubic phase. The origin of the observed THE is attributed to the presence of magnetic antiskyrmions. Owing to the large out-of-plane magnetic anisotropy in the present tetragonal phase, a detailed micromagnetic simulation was carried out to understand the stability of antiskyrmion phase in presence of different exchange interaction strength, anisotropy, and DMI. The present ferrimagnetic antiskyrmions with very high ordering temperatures possess a great potential for their application in racetrack memory devices as they are expected to display reduced skyrmion Hall effect in comparison to the ferromagnetic ones.