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Antiferromagnetic Skyrmion: Stability, Creation and Manipulation

Xichao Zhang,1, 2 Yan Zhou,1, 2, and Motohiko Ezawa3, 1Department of Physics, University of Hong Kong, Hong Kong, China

2School of Electronics Science and Engineering, Nanjing University, Nanjing 210093, China3Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan

(Dated: July 17, 2018)

Magnetic skyrmions are particle-like topological excitations in ferromagnets, which have the topologi-cal number Q = 1, and hence show the skyrmion Hall effect (SkHE) due to the Magnus force effectoriginating from the topology. Here, we propose the counterpart of the magnetic skyrmion in the anti-ferromagnetic (AFM) system, that is, the AFM skyrmion, which is topologically protected but withoutshowing the SkHE. Two approaches for creating the AFM skyrmion have been described based on micro-magnetic lattice simulations: (i) by injecting a vertical spin-polarized current to a nanodisk with the AFMground state; (ii) by converting an AFM domain-wall pair in a nanowire junction. It is demonstrated thatthe AFM skyrmion, driven by the spin-polarized current, can move straightly over long distance, benefit-ing from the absence of the SkHE. Our results will open a new strategy on designing the novel spintronicdevices based on AFM materials.

PACS numbers: 75.50.Ee, 75.78.Cd, 75.78.Fg, 12.39.Dc

Skyrmion is a topologically protected soliton in continuousfield theory, which is recently realized in both bulk non-centrosymmetric magnetic materials [1, 2] and thin films [3],where the ferromagnetic (FM) background is described by thenon-linear sigma model with the Dzyaloshinskii-Moriya in-teraction (DMI) [4]. The study of the magnetic skyrmion isone of the hottest topics in condensed matter physics, due toits potential applications in information processing and com-puting [5, 6]. There are several ways to create magneticskyrmions, e.g., by applying spin-polarized current to a nan-odisk [7, 8], by applying the laser [9], from a notch [10] andby the conversion from a domain wall (DW) pair [11, 12]. Amagnetic skyrmion can be driven by the spin-polarized cur-rent [13, 14]. However, it does not move parallel to the in-jected current due to the skyrmion Hall effect (SkHE), sinceits topological number is 1. This will pose a severe chal-lenge for realistic applications which require a straight mo-tion of magnetic skyrmions along the direction of the appliedcurrent [13].

In this work, we demonstrate that a skyrmion can be nu-cleated in antiferromagnets, as illustrated in Fig. 1, based onmicromagnetic lattice simulations. We refer to it as an antifer-romagnetic (AFM) skyrmion. We further show that the AFMskyrmion can move parallel to the applied current since theSkHE is completely suppressed, which is very promising forspintronic applications.

Recently, antiferromagnets emerge as a new field of spin-tronics [1518]. A one-dimensional topological soliton inantiferromagnets is an AFM DW [19]. An AFM DW canbe moved by spin transfer torque (STT) induced by spin-polarized currents or spin waves [20, 21]. Besides, a two-dimensional (2D) topological soliton, that is, the magneticvortex, has been studied in 2D AFM materials [22]. The AFMsystem has an intrinsic two-sublattice structure. The spins of

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Figure 1. Illustration of an AFM skyrmion spin texture in a2D AFM monolayer with the two square antiparallel magneticsublattices. 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.

the ground state are perfectly polarized in each sublattice. Wemay calculate the topological number of the spin texture pro-jected to each sublattice. Hence, we propose to assign a set oftopological numbers (+1,1) to one AFM skyrmion, whichshows no SkHE since it has no net magnetization.

We also present two approaches to create an AFMskyrmion. One is applying a spin-polarized current perpen-dicularly to a disk region, which flips the spin in the appliedregion. The other is a conversion from an AFM DW pairin junction geometry as in the case of the conversion of aFM skyrmion from a FM DW pair [11]. Furthermore, weshow that it is possible to move an AFM skyrmion by apply-ing a spin-polarized current. The AFM skyrmion can travelvery long distance without touching the sample edges. It isalso insensible to the external magnetic field. These resultswill be important from the applied perspective of magneticskyrmions.

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ResultsAFM system. We investigate the AFM system with the latticeHamiltonian,

HAFM = Ji,j

mi mj+i,j

D (mimj)Ki

(mzi )2,

(1)where mi represents the local magnetic moment orientationnormalized as |mi| = 1, and i, j runs over all the nearestneighbor sites. The first term represents the AFM exchangeinteraction with the AFM exchange stiffness J > 0. The sec-ond term represents the DMI with the DMI vector D. Thethird term represents the perpendicular magnetic anisotropy(PMA) with the anisotropic constant K.

The dynamics of the magnetization mi is controlled byapplying a spin current in the current-perpendicular-to-plane(CPP) configuration [14, 23]. We numerically solve theLandau-Lifshitz-Gilbert-Slonczewski (LLGS) equation,

dmidt

= ||mi Heffi + mi dmidt

+ ||(mi pmi), (2)

where Heffi = HAFM/m is the effective magnetic fieldinduced by the Hamiltonian equation (1), is the gyromag-netic ratio, is the Gilbert damping coefficient originatingfrom spin relaxation, is the Slonczewski-like STT coef-ficient, and p represents the electron polarization direction.Here, = | ~0e |

jP2dMS

with 0 the vacuum magnetic permit-tivity, d the film thickness, MS the saturation magnetization,and j the current density. We take the p = z for creat-ing the AFM skyrmion, while p = y for moving the AFMskyrmion. Although an antiferromagnet comprises complextwo sublattices of reversely-aligned spins, the STT can beapplicable also for the AFM system provided the lattice dis-creteness effect is taken into account with an ultra-small meshsize in the micromagnetic simulations [17, 18]. The STT isinduced either through spin-polarized current injection froma magnetic tunnel junction polarizer or by the spin Hall ef-fect [14, 24]. We can safely apply this equation for the AFMsystem since there is no spatial derivative terms.

A comment is in order. We cannot straightforwardly usethe current-in-plane (CIP) configuration to control the dynam-ics of the magnetization as it stands, since spatial derivativeterms are involved in the LLGS equation, that is, umi (mi xmi) + umi xmi.

Topological stability. The skyrmion carries the topologicalnumber. In the continuum theory it is given by

Q = 14

d2xm(x) (xm(x) ym(x)) . (3)

However, the AFM system has a two-sublattice structuremade of the A and B sublattices. In our numerical compu-tation we employ the discretized version of the topologicalcharge equation (3),

Q = 1

4

ijk

mi (mj mk

), (4)

0 1 2 3 4 5 6 7 8

-23.85

-23.82

-23.79

-23.76

-23.73

-23.70

50 nm

40 nm

30 nm

Etotal

D

AFM skyrmion AFM ground state

80 nm

ds

mz-1 +1

80 nm

40 nm

6

5

4

3

D

j (1013 A m-2)1 3 5

(a) (b)

(c) (d)

Figure 2. Creation of an isolated AFM skyrmion in a 2D AFMnanodisk using a vertically injected spin-polarized current. (a)AFM skyrmion in the nanodisk with a diameter of 80 nm createdby a 2-ns-long spin-polarized current pulse (j = 5 1013 A m2)perpendicularly injected into the nanodisk in a circle region with dif-ferent size followed by a 1-ns-long relaxation (see SupplementaryMovie 1). The spin-polarized current injection region is denoted bythe blue circle, of which the diameter equals 30 nm, 40 nm, 50 nm,respectively. The size of all AFM skyrmions is found to be identicalirrelevant of the current injection region. (b) Magnetization distri-bution of the AFM skyrmion in an AFM nanodisk. It is made of atoroidal DW with fixed radius and width determined by the materialparameters. The AFM skyrmion size ds is defined by the diameter ofthe white circle, where mz = 0. (c) A 2-ns-long spin-polarized cur-rent pulse with different current density j is perpendicularly injectedinto the nanodisk with a diameter of 80 nm followed by a 1-ns-longrelaxation. The initial state of the nanodisk is the AFM ground state.The spin-polarized current injection region is denoted by the bluecircle, of which the diameter equals 40 nm. (d) Total micromagneticenergy Etotal for an isolated AFM skyrmion and the AFM groundstate as a function of the DMI constant D. The D and Etotal are inunit of 1021 J and 1017 J, respectively.

for each sublattice ( = A,B). Hence, we propose to assigna set of two topological numbers (QA, QB) to one skyrmion.We obtain QA = QB = 1 for a skyrmion in a sufficientlylarge area. Even if the skyrmion spin texture is deformed, itstopological number cannot change. A skyrmion can be neitherdestroyed nor separated into pieces, that is, it is topologicallyprotected.

Creation of an AFM skyrmion by a vertical spin current.We employ a CPP injection with a circular geometry in a nan-odisk. The CPP injection induces spin flipping at the current-injected region. When we continue to apply the current, the

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