Skyrmions and antiskyrmions in quasi-two-dimensional magnets

A stable skyrmion, representing the smallest realizable magnetic texture, could be an ideal element for ultra-dense magnetic memories. Here, we review recent progress in the field of skyrmionics, which is concerned with studies of tiny whirls of magnetic configurations for novel memory and logic applications, with a particular emphasis on antiskyrmions. Magnetic antiskyrmions represent analogs of skyrmions with opposite topological charge. Just like skyrmions, antiskyrmions can be stabilized by the Dzyaloshinskii-Moriya interaction, as has been demonstrated in a recent experiment. Here, we emphasize differences between skyrmions and antiskyrmions, e.g., in the context of the topological Hall effect, skyrmion Hall effect, as well as nucleation and stability. Recent progress suggests that anitskyrmions can be potentially useful for many device applications. Antiskyrmions offer advantages over skyrmions as they can be driven without the Hall-like motion, offer increased stability due to dipolar interactions, and can be realized above room temperature.


INTRODUCTION
Magnetic skyrmion is a non-collinear configuration of magnetic moments with a whirling magnetic structure (see Figure 1). Magnetic skyrmions represent (2+1)-dimensional analog of Skyrme model [1] which is (3+1)-dimensional theory. Thus, magnetic skyrmions are often referred to as baby-skyrmions. From the topology point of view, magnetic skyrmion is described by an integer invariant that arises in the map from the physical two-dimensional space to the target space S 2 . The invariant describes how many times magnetic moments wrap around a unit sphere in the mapping (see Figure 1). Erasing a skyrmion requires to globally modify the system and as a result skyrmions possess topological protection. The integer topological invariant of a skyrmion also referred to as topological charge is: where m is a unit vector pointing in the direction of the magnetization. Initially, magnetic skyrmions of Bloch type (see Figure 1) [2,3,4,5] were discovered in chiral B20 compounds such as MnSi, FeGe, and Fe 1−x Co x Si in which the absence of the center of inversion and spin-orbit interactions lead to appearance of the Dzyaloshinskii-Moriya interaction (DMI) [6,7]. Sufficiently strong DMI can then lead to formation of isolated skyrmions or even skyrmion lattices.
Skyrmions stabilized by DMI are commonly referred to as Bloch-and Néel-type skyrmions (see Figure 1). These two types of skyrmions differ by their helicity but have the same polarity and topological charge. On the other hand skyrmions stabilized by DMI can also have the same polarity and opposite topological charge. Skyrmions with the same polarity and opposite topological charge are often referred to as skyrmions and antiskyrmions. Antiskyrmions (see Figure 1) can be stabilized by bulk DMI with lower symmetry as has been first predicted [8,9] and later realized experimentally in Heusler compounds with D 2d symmetry [10]. It has also been predicted that interfacial DMI with C 2v symmetry can lead to formation of antiskyrmions in ultrathin magnetic films [11,12]. In general, various types of skyrmions and antiskyrmions shown in Figure 3 can be stabilized by changing the form of DMI tensor [11]. Anisotropic interfacial DMI with C 2v symmetry has recently been realized in epitaxial Au/Co/W magnetic films [13].
Microscopically, DMI arises when interaction of two magnetic atoms is mediated by a non-magnetic atom via the superexchange or double-exchange mechanisms (see Figure 2). In the absence of an inversion center, a non-collinear configuration of magnetic moments is preferred by DMI energy: where S 1 and S 2 describe the spins and DMI vector D 12 ∝ r 1 × r 2 (see Figure 2). The strength of DMI is proportional to the strength of the spin-orbit interaction which is expected to scale with the fourth power of the atomic number. However, in some cases the particular form of the band structure and/or effects related to charge transfer can influence the strength of the spin-orbit interaction. Particularly strong spin-orbit interaction and DMI can arise at interfaces between magnetic films and non-magnetic metals where the 3d orbitals of magnetic atoms interact via 5d orbitals of the heavy metal [14]. Néel type-skyrmions stabilized by interfacial DMI have been obtained at low temperatures in epitaxially grown Fe and PdFe magnetic layers on Ir [15,16]. Another approach is to stack magnetic and non-magnetic layers in such a way that additive interfacial DMI leads to formation of Néel skyrmions. Such approach leads to formation of room temperature skyrmions in magnetic layers (e.g. Co) sandwiched between two different non-magnetic layers (e.g. Ir and Pt) [17,18,19,20].
Non-collinear magnetic textures can also arise due to dipolar interactions in a form of magnetic bubbles. Compared to skyrmions, magnetic bubbles have larger size and no definite chirality. Thus antiskyrmions can be realized in systems with dipolar interaction [21]. Magnetic skyrmion bubbles are similar to magnetic bubbles but have definite chirality induced by DMI [22,23,24]. It has been predicted that skyrmions of both chiralities can be stabilized in lattices with frustrated exchange interactions [25,26].

DESCRIPTION OF SKYRMIONS AND ANTISKYRMIONS
Magnetic skyrmions in thin magnetic films can be well understood by considering a continuous model with the free energy density written for a two-dimensional ferromagnet well below the Curie temperature: where the free energy is F = d 2 rF, we assume summation over repeated index i = x, y, and m is a unit vector along the magnetization direction. The first term in Eq. (3) describes isotropic exchange with the exchange stiffness A, the second term describes uniaxial anisotropy with the strength K, the third term describes the Zeeman energy due to the external magnetic field H e , H ≡ µ 0 H e M where M is the magnetization, and the last term corresponds to DMI described by a general tensor D ij = (D j ) i [11] where for a two-dimensional magnet j is limited to x and y. DMI is the most important term in Eq. (3) for formation of magnetic skyrmions. Modest strength of DMI favors isolated metastable skyrmions [27] while strong DMI leads to condensation into a skyrmion lattice [28].
The form of DMI tensor D ij is determined by the crystallographic symmetry of the system [11]. In particular, non-zero elements of DMI tensor are determined by relations: where R (α) are generators of the point group corresponding to the crystallographic symmetry, α = 1, 2, . . ., and the summation over repeated indices l and m is assumed. Note that the constraints on DMI tensor in Eq. (4) can be equivalently expressed via Lifshitz invariants [29,8,30].
A system invariant under SO(3) rotations then results in D ij = Dδ ij where δ ij is the Kronecker delta. Such DMI stabilizes Bloch-type skyrmions (see Figure 1). A system with C ∞v symmetry is invariant under proper and improper rotations around the z axis and allows only two nonzero tensor coefficients D 12 = −D 21 = D. Such DMI stabilizes Néel-type skyrmions (see Figure 1). Another important example arises for a system invariant under D 2d symmetry for which again only two nonzero tensor coefficients are allowed, i.e., D 12 = D 21 = D. The latter case realizes a system with antiskyrmions (see Figure 1). Within a simple model given by Eq. (3), all three examples given above are mathematically equivalent as they can be mapped to each other by a global spin rotation/reflection accompanied by an appropriate transformation of DMI tensor [11]. Since the free energy does not change in such a mapping one can expect that the same stability diagram will describe the above skyrmions and antiskyrmions (other examples of equivalent skyrmions and antiskyrmions are shown in Figure 3) [11]. This equivalence is no longer valid in the presence of dipolar interactions [31] or more complicated magnetocrystalline anisotropy.
The parameters A, K, H, and D in the above examples enter the free energy density (3) and they determine whether magnetic skyrmions or antiskyrmions can be present in the system. Minimization of the free energy corresponding to Eq. (3) leads to the phase diagram shown in Figure 4 where the phase boundaries separate the cycloid or spiral phase (SP), the hexagonal skyrmion lattice (SkX), the square cell skyrmion lattice (SC), and the ferromagnetic phase (FM) [11]. It is convenient to introduce a critical DMI: corresponding to the strength of DMI at which the formation of Dzyaloshinskii domain walls becomes energetically favorable [32]. In an infinite sample, the transition from the isolated skyrmions to the skyrmion lattice or cycloid phase happens in the vicinity of this critical DMI strength [28,33,11] (see Figure 4). The magnetic skyrmion size changes substantially as one varies the strength of DMI. At D < D c the skyrmion size has been calculated analytically R s ≈ ∆/ 2(1 − D/D c ) where ∆ = A/K [32]. The effects related to finite temperature and dipolar interaction modify this behavior, especially close to the divergence when R s → ∞ at D = D c [19]. At D > D c inside the skyrmion phase the skyrmion lattice period can be estimated by the period of the equilibrium helix, L D = 4πA/D [28,34,11].
Here, we discuss (anti)skyrmion dynamics due to STT and SOT induced by the in-plane charge currents. These two mechanisms differ as the former vanishes in the absence of magnetic textures while the latter does not. Both mechanisms can be included in the Landau-Lifshitz-Gilbert equation: where s = M s /γ is the spin density, H eff = −δF/δm is the effective field, γ is (minus) the gyromagnetic ratio (γ > 0 for electrons), β is the factor describing non-adiabaticity, j s is the in-plane spin current proportional to the charge current j, and τ so is the spin-orbit torque [47]. For the spin Hall contribution where t f is the thickness of ferromagnetic layer, θ SH is the spin Hall angle, and e the electron charge. The Thiele approach [48] applied to Eq. (6) leads to equations of motion describing (anti)skyrmion dynamics: whereη is the damping dyadic tensor and F = F so + F st is the total force acting on (anti)skyrmion due to the SOT and STT. The SOT contribution is F so =B · j where the tensorB is proportional to the spin Hall angle and is determined by the configuration of the (anti)skyrmion [49,50]. The STT contribution is F st = (Qẑ × +βη)j s [39]. We obtain the (anti)skyrmion velocity in an infinite sample: where for typical (anti)skyrmions η is comparible to 1. The velocity in Eq. (8) scales as 1/Q. An interesting situation happens in a nanotrack geometry close to the edge. Due to the edge repulsion additional force term appears in Eq. (8), v y = 0, and the (anti)skyrmion velocity becomes v x = F x /(αη). As a result a large (anti)skyrmion velocity is possible close to the edge due to SOT in systems with low Gilbert damping [38,51].
From Eq. (8) it is clear that (anti)skyrmions will move along the current with an additional side motion, resulting in the (anti)skyrmion Hall effect with the Hall angle θ H = tan −1 (v y /v x ). Antiskyrmions exhibit the anisotropic Hall angle dependent on the current direction due to SOT as the tensorB becomes anisotropic for antiskyrmion profile which is in contrast to isotropic behavior of skyrmions (the principal values are given by B xx = −B yy for antiskyrmion with γ = 0 in Figure 3, and B xx = B yy for Néel skyrmion) [50]. It is also possible to completely suppress the antiskyrmion Hall effect by properly choosing the direction of the charge current [50]. Note that antiferromagnetic skyrmions [52] or synthetic antiferromagnetic skyrmions in antiferromagnetically coupled layers [53,54] also exhibit vanishing Hall angle. This behavior can be useful for realizations of magnetic memories.
Skyrmion dynamics consistent with Eq. (8) has been observed in magnetic multilayers, e.g., in Pt/Co/Ta [18], Ta/CoFeB/TaO x [22,23], (Pt/CoFeB/MgO) 15 [55], and (Pt/Co/Ir) 10 [56]. From theoretical modeling [38], SOT has been identified as a dominant driving mechanism, i.e., only F x is important in Eq. (8) and the sign of the transverse response is determined by the sign of the spin Hall angle and the topological charge. Some aspects of skyrmion dynamics are still puzzling, i.e., the dependence of the skyrmion Hall angle on the velocity and the dependence of the velocity on the skyrmion size. Variations in the skyrmion Hall angle have been attributed to the presence of pinning sites [57]. In principle, the presence of pinning sites could be problematic for realizations of magnetic memories [58]. Nevertheless, the presence of the skyrmion Magnus force induced by pinning sites results in significant reduction of the depinning threshold current density [59,57], e.g., by several order of magnitude if compared to domain wall dynamics.

WRITING, ERASING, AND DETECTING
Interfacial skyrmions in Fe on Ir have been controllably written and erased by STT induced by an STM tip at 4.2K in the presence of high magnetic fields [16]. For practical applications it is necessary to write and erase skyrmions at room temperature in the presence of weak or, preferably, no magnetic fields. STT has been demonstrated to generate skyrmions in a confined geometry of a Pt/Co nanodot [38]. At room temerature, SOT has been used to create elongated chiral domains that under application of inhomogeneous current break into skyrmion bubbles [22]. This process has been reproduced by micromagnetic simulations [60]. A slightly different mechanism in which a pair of domain walls is converted into a skyrmion has been demonstrated numerically [61]. Detailed analysis of topological charge density revealed that in the process of creating skyrmion by SOT an unstable antiskyrmion can also be created for a short period of time [62]. As antiskyrmion is not stable in the presence of interfacial DMI (C ∞v case) it eventually annihilates due to the presence of the Gilbert damping [63]. Thus, detailed studies of antiskyrmions can help in realizing new ways of writing and erasing skyrmions. Generation of skyrmions by charge currents have also been demonstrated for magnetic multilayer stacks [56] as well as for symmetric bilayers hosting pairs of skyrmions with opposite chirality [51].
For device application, it is preferable to detect the presence of a skyrmion electrically. Several transport techniques have been suggested. The non-collinear magnetoresistance can be used to detect a skyrmion [64]. However, only non-collinearity is being detected and no information about the topological structure of a skyrmion is obtained. Measuring the z-component of magnetization is possible by employing the anomalous Hall effect [65]. This technique has been used to detect a single skyrmion [66]. Measurement of the topological Hall effect can directly reveal the topological nature of skyrmion as the effect originates in the fictitious magnetic field proportional to the topological charge. The presence of skyrmions has been detected by measurements of the topological Hall effect in non-cetrosymmetric bulk materials in B20 group [67,68]. In principle, antiskyrmions should also exhibit the topological Hall effect but with the reversed sign due to the opposite topological charge [10]. On the other hand, in magnetic mulitlayers the topological Hall effect is expected to be much smaller than the anomalous Hall contribution [66].

CONCLUSIONS
In this article, we review recent progress in the field of skyrmionics -a field that concerns with studies of tiny whirls of magnetic configurations for novel memory and logic applications. A particular emphasis Frontiers has been given to antiskyrmions. These, similar to skyrmions, are particle-like structures with topological protection. Compared to skyrmions, antiskyrmions have opposite charge and are anisotropic. Recent experimental observation of antiskyrmions [10] encourages further studies of transport and dynamical properties such as the topological Hall effect, antiskyrmion Hall effect, antiskyrmion nucleation and stability, and others. On the other hand, realization of antiskyrmions requires careful material engineering which should also be addressed in future studies. It is expected that similar to skyrmions, antiskyrmions can potentially result in new device concepts for memory and computing applications [69].

CONFLICT OF INTEREST STATEMENT
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

AUTHOR CONTRIBUTIONS
The author confirms being the sole contributor of this work and approved it for publication.

FUNDING
This work was supported by the DOE Early Career Award DE-SC0014189.