Antiferromagnetic Skyrmion: Stability, Creation and Manipulation

Magnetic skyrmions are particle-like topological excitations in ferromagnets, which have the topo-logical number Q = ± 1, and hence show the skyrmion Hall effect (SkHE) due to the Magnus force effect originating from the topology. Here, we propose the counterpart of the magnetic skyrmion in the antiferromagnetic (AFM) system, that is, the AFM skyrmion, which is topologically protected but without showing the SkHE. Two approaches for creating the AFM skyrmion have been described based on micromagnetic lattice simulations: (i) by injecting a vertical spin-polarized current to a nanodisk with the AFM ground state; (ii) by converting an AFM domain-wall pair in a nanowire junction. It is demonstrated that the AFM skyrmion, driven by the spin-polarized current, can move straightly over long distance, benefiting from the absence of the SkHE. Our results will open a new strategy on designing the novel spintronic devices based on AFM materials.


Results
AFM system. We investigate the AFM system with the lattice Hamiltonian, , , 2 where m i represents the local magnetic moment orientation normalized as |m i | = 1, and 〈 i, j〉 runs over all the nearest neighbor sites. The first term represents the AFM exchange interaction with the AFM exchange stiffness J > 0. The second term represents the DMI with the DMI vector D. The third term represents the perpendicular magnetic anisotropy (PMA) with the anisotropic constant K. The dynamics of the magnetization m i is controlled by applying a spin current in the current-perpendicular-to-plane (CPP) configuration 14,23 . We numerically solve the Landau-Lifshitz- Gilbert with μ 0 the vacuum magnetic permittivity, d the film thickness, M s the saturation magnetization, and j the current density. We take the p = −z for creating the AFM skyrmion, while p = −y for moving the AFM skyrmion. Although an antiferromagnet comprises complex two sublattices of reversely-aligned spins, the STT can be applicable also for the AFM system provided the lattice discreteness effect is taken into account with an ultra-small mesh size in the micromagnetic simulations 17,18 . The STT is induced either through spin-polarized current injection from a magnetic tunnel junction polarizer or by the spin Hall effect 14,24 . We can safely apply this equation for the AFM system since there is no spatial derivative terms. A comment is in order. We cannot straightforwardly use the current-in-plane (CIP) configuration to control the dynamics of the magnetization as it stands, since spatial derivative terms are involved in the LLGS equation, that is, Topological stability. The skyrmion carries the topological number. In the continuum theory it is given by x y 2 However, the AFM system has a two-sublattice structure made of the A and B sublattices. In our numerical computation we employ the discretized version of the topological charge equation (3), ijk i j k for each sublattice (τ = A, B). Hence, we propose to assign a set of two topological numbers (Q A , Q B ) to one skyrmion. We obtain Q A = − Q B = 1 for a skyrmion in a sufficiently large area. Even if the skyrmion spin texture is deformed, its topological number cannot change. A skyrmion can be neither destroyed nor separated into pieces, that is, it is topologically protected. The color scale represents the magnetization direction, which has been used throughout this paper: orange is into the plane, green is out of the plane, white is in-plane.
Scientific RepoRts | 6:24795 | DOI: 10.1038/srep24795 Creation of an AFM skyrmion by a vertical spin current. We employ a CPP injection with a circular geometry in a nanodisk. The CPP injection induces spin flipping at the current-injected region. When we continue to apply the current, the spins continue to flip. As soon as we stop the current, an AFM skyrmion is nucleated to lower the DMI and AFM exchange energies (see Supplementary Movie 1). It is relaxed to the optimized radius irrespective of the injected region, as shown in Fig. 2a (see Supplementary Movie 2). Once it is relaxed, it stays as it is for long, demonstrating its static stability. We show the spin configuration of an AFM skyrmion obtained numerically in Fig. 2b. It is made of a toroidal DW with fixed radius and width determined by the material parameters. There exists a threshold current density to create an AFM skyrmion, as shown in Fig. 2c. It is natural that the spins cannot be flipped if the injected current is not strong enough.
The time-evolution of the topological charges of the AFM system is shown in Fig. 3. Note that there is a non-zero topological number τ Q 0 in the AFM background state, which is created by the tilting magnetization at edges due to the DMI. It is for the A sites. The topological charge oscillates during the CPP injection. As soon as the CPP injection is off, the topological charge develops suddenly to a fixed values. By subtracting Q A 0 from that of the AFM skyrmion in the A sites, we find Q A = 0.9865, which is almost 1. Similarly, we find Q B is almost −1. The AFM skyrmion can be created equally by a vertical current injection polarized along the + z-direction or the −z-direction (see Supplementary Movie 3).

Phase diagrams.
A skyrmion is topologically protected. Nevertheless, it may shrink or expand with the topological charge unchanged. We present a phase diagram in Fig. 4. It is convenient to understand it in terms of the DMI constant D. The DMI prevents a skyrmion from shrinking in antiferromagnets as in the case of ferromagnets. (1) Near D = 0, a skyrmion shrinks and disappears (blue region). (2) There are two cases when a skyrmion exists as a static stable object (yellow region): see also Fig. 2d. In one case (smaller D), its energy is more than that of the AFM ground state. It is an energetically metastable state, but it is topologically stable. In the other case (larger D), its energy is less than that of the AFM ground state. It would undergo condensation if it were not

Creation of an AFM skyrmion from an AFM DW wall pair. A FM skyrmion can be created from a FM
DW pair using a junction geometry 11 . We show that a similar mechanism works in creating the AFM skyrmion as shown in Fig. 5. We first make an AFM DW pair through the CPP injection with p = −z. The AFM DW pair is shifted by applying a spin-polarized current through the STT on AFM DW 25 as shown in Fig. 5 (see the process from t = 10 ps to t = 20 ps in Fig. 5). Here we consider the vertical injection of a spin current polarized along the −y-direction. The CPP injection moves the AFM DW in the rightward direction (+ x). When the AFM DW arrives at the junction interface (t = 20 ps), both the end spins of the DW are pinned at the junction, whereas the central part of the DW continues to move due to STT in the wide part of the nanotrack. Therefore, the structure is deformed into a curved shape and an AFM skyrmion texture forms at t = 30 ps. This process is analogous to blowing air through soapy water using bubble pipes or plastic wands to create soap bubbles. The skyrmion will break away from the interface when the bulk of its structure continues to move rightward as shown at t = 40 ps. By continuously "blowing" AFM DWs into the junction, a train of AFM skyrmions is generated.

Current-driven motion of an AFM skyrmion in a nanotrack.
We can move the AFM skyrmion by the CPP injection as in the case of the FM skyrmion. We show the relation between the magnitude of the injected current and the velocity in Fig. 6a, where the velocity is proportional to the injected current.
We recall that the FM skyrmion is easily destroyed by touching the sample edges due to the SkHE. At the same time, the maximum velocity of the FM skyrmion in a FM nanotrack is typically much less than 10 2 m s −1 , limited by the confining force of ∼(D 2 /J) 26 .
Conversely, there is no SkHE for the AFM skyrmion. Hence, it can move straightly in an AFM nanotrack without touching the edge. It is shown in Supplementary Movie 5, where a chain of encoded AFM skyrmions moves in a nanotrack with a speed of ∼1700 m s −1 driven by a vertical current without touching edges.
In Fig. 6b we compare the AFM and FM skyrmions. The velocity of AFM skyrmions can be very large compared to FM skyrmion, which is suitable for ultrafast information processing and communications. The steady motion of AFM skyrmions is demonstrated in Supplementary Movie 6, where they move in a thin film without boundary effect driven by the vertical spin current. This highly contrasts with the case of FM skyrmions demonstrated in Supplementary Movie 7, where skyrmions do not move either parallel or perpendicular to the film edges.

Discussion
We have proposed magnetic skyrmions in the AFM system. The dynamics of AFM skyrmion is very different from those in the FM system, since they are topologically protected and are free from the SkHE. We have first checked that our simulation software reproduces a linear dispersion relation inherent to the two-sublattice structure of the AFM system, and then employ it to explore various properties of the AFM skyrmion. It is worth mentioning there are two recent preprints 27,28 on AFM skyrmions, including the preliminary version of the present work 27 . Our work is focused on the injection and vertical spin current-driven dynamics of AFM skyrmions. Regarding the other work in ref. [28], the thermal properties as well as in-plane current-induced dynamics of an AFM skyrmion have been studied and a high-speed motion (ν ∼ 10 3 m s −1 ) of an AFM skyrmion has also been shown in the absence of the SkHE, consistent with the present work. We believe that the AFM skyrmions will play a very significant role in the emerging field of AFM spintronics. Methods Modeling and simulation. We perform the micromagnetic simulations using the Object Oriented MicroMagnetic Framework (OOMMF) together with the DMI extension module 14,[29][30][31][32] . The time-dependent magnetization dynamics is governed by the LLGS equation [33][34][35][36][37] . The OOMMF has been developed originally and used extensively for the simulation of FM systems, and we have checked that one may use it to analyze the nanotexture in the AFM system as well. Indeed, we have successfully reproduced a linear dispersion relation inherent to the two-sublattice structure, as shown in Supplementary Figure 1.
For micromagnetic simulations, we consider 0.4-nm-thick magnetic nanodisks and nanotracks on the substrate. With respect to the material parameters, we recall 38 that an antiferromagnet is a special case of a ferrimagnet for which both sublattices A and B have equal saturation magnetization. Both the DMI and the PMA arise from the spin orbit coupling, albeit in different forms. We have checked that our results hold for a wide range of material parameters (cf. Fig. 4). Here, we use the parameters of the same order as those given in Ref. 39 for AFM materials. We thus adopt the magnetic parameters from Refs 6 and 14: the Gilbert damping coefficient α = 0.3, the gyromagnetic ratio γ = − 2.211 × 10 5 m A −1 s −1 , the sublattice saturation magnetization M s = 290 kA m −1 , the exchange constant J = 0∼ 20 × 10 −21 J, the DMI constant D = 0∼ 10 × 10 −21 J, and the PMA constant K = 0∼ 2 × 10 −21 J unless otherwise specified. All samples are discretized into tetragonal cells of 1 nm × 1 nm× 0.4 nm in the simulation, which ensures reasonable numerical accuracy as well as run time. The output time step is fixed at 1 ps for the simulation of the dispersion relation, which is increased to 10 ps for the simulation of the skyrmion dynamics. The polarization rate of the spin-polarized current is defined as P = 0.4 in all simulations. The Zeeman field is set as zero because the AFM skyrmion, having no net magnetization, is insensitive to it (see Supplementary Figure 2).