Abstract
Currentinduced motion of nonaxisymmetric skyrmions within tilted ferromagnetic phases of polar helimagnets with the easy plane anisotropy is studied by micromagnetic simulations. Such nonaxisymmetric skyrmions consist of a circular core and a crescentshaped domainwall region formed with respect to the tilted surrounding state. Currentdriven motion of nonaxisymmetric skyrmions exhibits two distinct time regimes: initially the skyrmions rotate towards the current flow direction and subsequently move along the current with the skyrmionic crescent first. According to the Thiele equation, the asymmetric distribution of the topological charge and the dissipative force tensor play an important role for giving the different velocities for the circular and the crescentshaped constituent parts of the skyrmion what underlies such a shuttlecocklike movement. Moreover, the currentvelocity relation depends on the angle of the tilted ferromagnetic phase what makes in particular the transverse velocity of skyrmions sensitive to their fielddriven configurational transformation. We also argue the possibility of magnetic racetrack waveguides based on complex interplay of robust asymmetric skyrmions with multiple twisted edge states.
Introduction
Magnetic chiral skyrmions are topological excitations with particlelike properties that have complex noncoplanar spin structure^{1,2,3,4,5}. Skyrmions were recently discovered in bulk noncentrosymmetric helimagnets^{6,7,8,9,10} and in nanostructures with confined geometries over larger temperature regions^{11,12,13}. According to general arguments by Hobart and Derrick^{14}, however, any multidimensional localized states are unstable in many physical field models: particlelike inhomogeneous states may arise only as dynamic excitations, but static configurations are unstable and collapse into topological singularities^{14}. Consequently, nonlinear field equations produce only onedimensional solitonlike solutions. Chiral DzyaloshinskiiMoriya interactions (DMI)^{15} represent a distinct stabilization mechanism working against the constraints of the HobartDerrick theorem^{14} and thus protects chiral skyrmions from radial instability^{1,16}. As a result, noncentrosymmetric magnets constitute a particular class of materials where skyrmions can exist and thus are of special interest in fundamental physics and mathematics^{17,18}. Nanometer size of chiral skyrmions, their topological stability and manipulation by electric currents^{19,20,21} enabled a new burgeoning field of research in nonvolatile memory and logic devices^{22,23}. In particular, in the skyrmion racetrack^{23,24,25} – a probable model for future information technology – the flow of information is encoded in the moving metastable skyrmionic bits^{26}.
The customary approach to enhance the functionality of the skyrmionbased racetrack memory is the mechanical patterning of underlying nanosamples. As an example, suggested devices may feature a regular arrangement of notches to divide the track into a sequence of parking lots for the skyrmions^{26,27}. An additional nanostrip on top of the racetrack may create an energy barrier along the middle and thus forms two channels for a skyrmion movement^{24}. This enables the information storage in the lane number of each skyrmion. Moreover by adding stripes with high magnetic crystalline anisotropy at the edges, one confines the skyrmions inside and prevents their escape from the nanotrack^{28}. One should also mention elaborate schemes that include, e.g., Yshaped junctions^{22,23,29}. By these, the skyrmions can be selectively driven into different nanotracks and form complementary data representation. Moreover, spin logic gates such as the “AND” and “OR” operations based on manipulations of skyrmions can be designed^{22,23,29}.
An alternative approach to enhance the functionality of the skyrmionbased racetrack memory is to utilize the unique properties of nonaxisymmetric isolated skyrmions (NISs) that in particular may emerge within tilted ferromagnetic (TFM) phases of polar magnets with the easyplane anisotropy^{30} (Fig. 1(a), right). TFM state represents a homogeneously magnetized state with a ferromagnetic moment tilting away from the polar axis due to the competition between the easyplane anisotropy and the field. As compared with the ordinary axisymmetric isolated skyrmions (AISs) within the fieldsaturated state^{16} (left side of Fig. 1(a,b)), NIS may acquire both polarities in their cores (although the vorticity bears the same sign) and thus are subject to the skyrmion Hall effect with opposite shift directions (and hence naturally form two channels, Fig. 1(d,e)). Based on the opposite sign of the topological charge, one may call two types of NISs – skyrmions (Fig. 1(d)) and antiskyrmions (Fig. 1(e)), and consider them as binary data bits for possible practical applications. Note that skyrmions and antiskyrmions with the same polarity but the opposite vorticity were recently investigated in frustrated magnets with competing exchange interactions^{31,32,33}.
In the present paper, we explore the currentinduced dynamics of introduced nonaxisymmetric skyrmions. We show that depending on the direction of the spinpolarized current (SPC) with respect to the skyrmion orientation, NIS undergoes a rotation towards the SPC with its subsequent currentaligned movement. The velocity of NIS is effectively regulated by the field magnitude (and thus by the tilt angle of the surrounding angular phase). We also underline prospects of using NISs in racetrack memory devices. Anisotropic skyrmionskyrmion interaction that depends on their mutual orientation^{30} alongside with the three types of edge states naturally formed at the lateral edges of a racetrack, make NISs effective candidates to be employed in nanoelectronic devices of the next generation in which nanopatterning is boiled down to a minimum.
Micromagnetic Model
The equilibrium solutions for NIS are derived within the standard discrete model of a polar helimagnet^{9,10,30} where the total energy is given by:
M_{i} is the unit vector in the direction of the magnetization at the site i of a twodimensional square lattice and <i, j> denote pairs of nearestneighbor spins. \(\hat{x}\) and \(\hat{y}\) are unit vectors along x and y directions, respectively. The first term describes the ferromagnetic nearestneighbor exchange with J < 0, the second term is the Zeeman interaction with the magnetic field parallel to the z axis, and the third term is the easyplane anisotropy with K < 0. Throughout the paper, we use the value of K that enables only TFM formation, i.e. we omit the regions of the H − K phase diagram that host modulated skyrmion lattice (SkL), spiral and elliptical cone phases^{34} (Fig. 2). DzyaloshinskiiMoriya interaction (DMI) stabilizes NISs with the Néel type of the magnetization rotation. The DMI constant D = Jtan(2π/p) defines the characteristic size of skyrmions with the period of modulated structures p. In the following simulations, D is set to 0.5J. Within model (1), AISs exist as metastable excitations of the saturated state for H > H_{cr} = 2K (Fig. 1(b)), while NISs (Fig. 1(c)) are present for lower fields. We use K = −2.6D^{2}/J to consider metastable NISs, and hence H_{cr} = 5.2D^{2}/J (Fig. 2).
AIS are characterized by azimuthal (θ) and polar (ψ) angles of the spins according to \(\theta =\theta (\rho ),\,\psi =\varphi \). Here the boundary conditions are θ(0) = π, θ(∞) = 0, while φ and ρ are cylindrical coordinates of the spatial variable (Fig. 1(a), left). On the other hand, NISs are confined by the following inplane boundary conditions: θ(0) = π, θ(∞) = θ_{TFM} = arccos(H/2K). These boundary conditions violate the rotational symmetry, forcing the skyrmions to develop an asymmetric shape (Fig. 1(a), right).
The complete phase diagram (Fig. 2) of states of the model (1) has been reproduced from refs. ^{30,34} and includes stability regions of modulated phases and regions of metastable skyrmions. The phase diagram also allows to generalize the processes of skyrmion lattice formation. Along the line a − b the skyrmion lattice appears as a result of condensation of isolated skyrmions (building blocks of the hexagonal skyrmion lattice), as found for axisymmetric skyrmions in the easyaxis case^{1,4,16}. Along the line c − d, hexagonal skyrmion lattice may appear as a result of local cutting of the cycloid (in this sense, two merons may be considered as nuclei of the skyrmion lattice. Along the firstorder phase transition line a − d, however, none of the aforementioned mechanisms is appropriate. Presumably, domains of the skyrmion lattice and the elliptical cone state coexist with nontrivial domain boundary between them.
Figure 3 exhibits the magnetic structure of all the states from the phase diagram in Fig. 1. The states in Fig. 2 include in particular skyrmion chains, disordered glassy states of NISs, as well as square arrangements of AISs. We, however, state that additional minimization with respect to the size of considered numerical grids preserves only the phases from the phase diagram in Fig. 2. All other phases are the result of imposed confinement and thus could be realized in nanostructures with confined geometries.
In the present manuscript, however, we consider only metastable NISs surrounded by the TFM state (orange shaded region in Fig. 2) what is required for racetrack memory devices. We avoid regions of the phase diagram where skyrmions form skyrmion lattices or undergo elliptical instability, as well as the regions of onedimensional spiral states.
The currentdriven dynamics of NISs and AISs was simulated using LandauLifshitzGilbert (LLG) equation^{35,36}:
Here γ is the gyromagnetic ratio and α = 0.01 is the Gilbert damping constant. This value of α is common and widely used in skyrmionics. The value of the same order, α = 0.04 or even smaller α = 0.004, have been used to theoretically study spinwave modes and their intense excitations activated by microwave magnetic fields in the SkL phases of insulating magnets^{37}. α = 0.04 is a typical value for the ferromagnetic metal and the dilute magnetic semiconductors^{38}. α = 0.01 was used to address the dynamics of skyrmions in frustrated magnets^{31,33}. Since the velocity of AISs is inversely proportional to α and the SPCvelocity characteristics in the SkL phase remain universal, independent on α, nonadiabatic effect β and impurities^{38}, the substantial room for improvement of skyrmion velocity is provided, because many magnetic materials show α much smaller than that for cobalt^{26}, α = 0.3.
H_{eff} is a local effective magnetic field, which at the site i is given by H_{eff} = −∂w/∂M_{i}. The spin transfer torque (STT) consists of the adiabatic part, τ = A(j ⋅ ∇)M_{i}, and the nonadiabatic term, τ_{β} = AβM_{i} × (j ⋅ ∇)M_{i}, where A = Jgμ_{B}/2eM_{s} is the coefficient proportional to the SPC density. The SPC j comprises an angle θ with xaxis (Fig. 1(c)). An orientation of NIS is characterized by an angle φ between a vector, which connects centers of the circular core and the crescent (skyrmion dipole q), and xaxis (Fig. 1(c)).
CurrentInduced Motion (A ShuttlecockLike Movement)
To systematically investigate the currentdriven dynamics of NISs, we applied the SPC with different angles with respect to the skyrmion dipole initially oriented with \(\varphi =0\) (Fig. 4(a)). It was found that after an initial rotation towards the SPC, NISdipoles are always currentaligned with \(\varphi =\theta \) (Fig. 4(a)). A relatively small increment of the SPC angle θ allowed to exclude local minima of the skyrmion orientation with respect to the SPC. In particular, a NIS motion with its core along the SPC was excluded (although in a numerical experiment of Fig. 4(a) such a movement opposite to the SPC appeared to be feasible).
Rotation of NIS towards the SPC direction
Figure 4(a) shows the time dependence of the dipole angle \(\varphi \) for different SPC directions θ. As expected, more time is needed to orient NIS along the SPC with increasing angle θ. The skyrmion rotation originates from the nonaxisymmetric internal structure. By applying the charge current, the magnetization structure is modulated by the spin transfer torque. This modulation changes the total energy of the magnetic system. The energy gain due to the spin transfer torque E^{STT} can be expressed as \({E}^{{\rm{STT}}}={\int }_{{\rm{sk}}}\,d{\bf{r}}{{\bf{M}}}_{i}\cdot {{\bf{H}}}_{{\rm{eff}}}\). Figure 4(b) shows the E^{STT} as a function of θ featuring minima only along the SPC. We use the magnetic configuration of t = 10^{−13} s after the current is applied. The minima, however, become shallow with the increasing magnetic field and completely disappear for H > H_{cr} with the onset of the fieldsaturated state.
Translational movement of NISs
The currentinduced translational motion of NIS is well understood in terms of the Thiele equation^{39}
where v is the velocity of the skyrmion and V is the pinning potential. The gyromagnetic coupling vector G = (0, 0, G_{z}) equals the topological charge
\(\boldsymbol{\mathscr{D}}\) is the dissipative force tensor:
which is not symmetric for NISs \(({\boldsymbol{\mathscr{D}}}_{xx}\ne {\boldsymbol{\mathscr{D}}}_{yy})\), even the offdiagonal element has a finite value \(({\boldsymbol{\mathscr{D}}}_{xy}\ne \mathrm{0)}\). Thus, when the charge current j is applied, the skyrmion feels both the longitudinal and the Magnus forces^{38}.
For θ = 0, the longitudinal (v_{x}) and the transverse (v_{y}) components of the velocity are represented as
where ξ = G_{z} − αD_{xy}.
We plot the spatial configuration of the dissipative force tensor in Fig. 5(b–d). With a sufficiently large value of d_{xx}d_{yy} = (∂_{x}M⋅∂_{x}M)(∂_{y}M⋅∂_{y}M) (plotted in Fig. 5(b)), v_{x} = β/α, which is the same as for AISs^{38} and is consistent with the universal j − v_{x} relation independent of β (Fig. 6(a)). The transverse velocity, on the contrary, is proportional to d_{xx} (Fig. 5(c)) and is strongly fielddependent (Fig. 6(b)), since the field affects the spin configuration of NISs, which also may be reflected in the skyrmion Hall angle of the NIS. One can also see that d_{xx}d_{yy} of the crescent part is larger than that of the circular part (Fig. 5(a,b)). This asymmetry of d_{xx}d_{yy} underlies the faster velocity of the crescent resulting in the rotational motion and is the reason why we dubbed such a motion “a currentinduced shuttlecocklike movement”. We neglect the effect of the offdiagonal element of the dissipative force tensor d_{xy} = ∂_{x}M⋅∂_{y}M, since it is much smaller than d_{xx} and has opposite sign for upper and lower parts of NISs (Fig. 5(d)).
Edge States
The practical use of NISs in racetracks hinges on their interaction with edge states. For racetracks with the fieldsaturated magnetization, the edge states manifest themselves as remnants of the helical spiral and repulse AISs^{40}. For racetracks with the TFM state, however, three different types of edge states can arise (in Fig. 7 the edge states are marked with capital letters A, B, and C).
Transitions between edge states
The type A edge state with collinear inplane spin components (azimuthal angle of the magnetization is ψ = π/2) is induced for lower values of the applied magnetic field. zcomponent of the magnetization acquires opposite signs at the opposite edges which leads to an asymmetric NISedge interaction potential (Fig. 7(d)). With increasing magnetic field, the Atype edge state undergoes the firstorder transition into the type B edge state with the rotating magnetization across the racetrack width L. Once m_{z} reaches unit value in the racetrack middle, the Btype edge state by the secondorder transition transforms into the type C edge state with ψ = π/2 (Fig. 7(c)).
Orientational confinement of NISs
Three different types of edge states formed at the lateral boundaries of the racetrack also impose an orientational confinement on nonaxisymmetric skyrmions. The Atype state implies perpendicular orientation of a skyrmion dipole q with respect to the racetrack edges (white arrow in Fig. 7(a) and magnetic configurations with NISs located near both stripe edges, Fig. 7(d)). At the same time due to the asymmetry of the magnetization distribution within the opposite edges, a NIS will be located closer to one edge than to the other (Fig. 7(d)). The SPCj_{x}, by inducing the skyrmion rotation, may also initiate a transition AB between edge states in spite of the Btype state is a metastable solution. The Btype state, on the contrary, may accommodate NISs with qx. Thus, a moving NIS due to the skyrmion Hall effect will shift the whole stripe with the maximal M_{z}component (marked by the dashed lines in Fig. 7(b)) towards one of the edges. The Ctype edge state allows two opposite NIS orientations degeneracy of which could be removed by the SPC j_{x} (Fig. 7(c)). In the present calculations, we avoid such extreme regimes under the larger current densities when (i) the skyrmions overcome the repulsive potential barrier from edge states what leads to skyrmion annihilation; (ii) the skyrmions are strongly deformed or even decay into magnons.
We also show that NISs rotate the surrounding homogeneous state, which otherwise is insensitive to SPC. Thus, NISs could be utilized as tumblers in nanostructures that rotate the surrounding oblique phases. In Supplementary Video, the current is applied perpendicular to the NISdipoles. After NISs have been rotated along the current, the current is switched off (j = 0 for t > 35 ns). This leads to the repulsive interaction between NISs^{30}.
The considered effect is based on the anisotropic NISNIS interaction potential^{30}: NISs attract each other being oriented headtohead (initial configuration in Supplementary Video) and repulse being oriented sidetoside (the configuration after t = 35 ns). Thus, we stress that the SPC may disassemble the coupled pair of NISs or vice versa couples remote skyrmions into skyrmion chains. Recently, the chains of NISs were observed in chiral LC by the group of Ivan Smalyukh^{41,42,43}.
Conclusions
In conclusion, we examined the currentinduced dynamics of nonaxisymmetric skyrmions that exist within TFM states of bulk polar helimagnets with the easy plane anisotropy. In particular, uniaxial anisotropy of easyaxis and easyplane type is attributed to the bulk polar magnetic semiconductors GaV_{4}S_{8}^{9} and GaV_{4}Se_{8}^{10} (the C_{3v} symmetry), respectively. Since the value of this effective anisotropy in these lacunar spinels is temperaturedependent, these material family establishes an ideal ground for the thorough study of anisotropic effects on modulated magnetic states^{10}. The results obtained within the model (1) are also valid for thin films with interface induced DMI^{44,45}.
We considered a shuttlecocklike movement of NISs that consists in their rotation to coalign with the SPC. We succeeded in modifying the currentvelocity relation by the fielddriven control of the angle in a surrounding homogeneous state. In ref. ^{29}, conversion between skyrmions with axisymmetric and nonaxisymmetric shape was achieved in a setup where the left input and right output wide regions with different material parameters are connected by a narrow nanowire. We remark that in ref. ^{29}, the nonaxisymmetric skyrmions are called bimerons. Such a terminology is also widely used to describe NISs in frustrated magnets^{46,47}. Since frustrated magnets endow isolated skyrmions with the additional degrees of freedom  vorticity and helicity, easyplane anisotropy forces AIS to transform into a bound pair of energetically equivalent merons with opposite vorticities, each carrying topological charge 1/2. In ref. ^{46}, a meron cluster with a square lattice of vortices and antivortices was realized. Realspace observations of a twodimensional square lattice of merons and antimerons emerging from a helical state was also reported in a thin plate of the chirallattice magnet Co _{8} Zn _{9} Mn _{3}, which exhibits inplane magnetic anisotropy^{48}. In polar helimagnets, the pattern of DMI vectors stabilizes only one of the formed merons leading to a crescentshaped deformation of the other meron. Nevertheless, such a bimeron preserves its summary topological charge 1. The NISNIS interaction, however, has an anisotropic character and can be either attractive or repulsive depending on the relative orientation of the NIS pair. Thus, instead of square arrangement of merons^{46,48}, chiral NIS develop disordered metastable meron spin textures^{49}.
We also speculate that a NIS placed into the racetrack memory with three different types of the edge states not only undergoes an orientational confinement, but can also be used as a currentactivated tumbler between edge states. Our results are not only relevant to the application of magnetic skyrmions in memory technology but also elucidate the fundamental properties of skyrmions and the edge states formed in the TFM states of polar helimagnets^{10,50}.
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Acknowledgements
The authors are grateful to Istvan Kezsmarki, Katia Pappas, Ivan Smalyukh, and Maxim Mostovoy for useful discussions. This work was funded by JSPS CoretoCore Program, Advanced Research Networks (Japan), JSPS GrantinAid for Research Activity Startup 17H06889, GrantinAid for Scientic Research (A) 17H01052 from MEXT, Japan and CREST, JST. A.O.L. thanks Ulrike Nitzsche for technical assistance.
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R.M., A.O.L. and J.I.O. performed the calculations. A.O.L. and J.I.O. wrote the manuscript; A.O.L., K.I. and J.I.O. planned the project.
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Murooka, R., Leonov, A.O., Inoue, K. et al. Currentinduced shuttlecocklike movement of nonaxisymmetric chiral skyrmions. Sci Rep 10, 396 (2020). https://doi.org/10.1038/s41598019567913
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