Manipulating and trapping skyrmions by magnetic field gradients

The effective manipulation of skyrmion motion introduces the technologically relevant possibility of skyrmion-based spintronics. Here we show how a magnetic skyrmion can be captured by a radially spatial gradient of the magnetic field. A theoretical model governed by the Thiele equation is employed to study the skyrmion motion. The analytical predictions are compared to micromagnetic simulations based on the Landau–Lifshitz–Gilbert equation, showing that the dynamic behavior of skyrmions strongly depends on the gradient strength of magnetic fields as well as the radius of the skyrmion and the Gilbert damping factor. Moreover, we find that the skyrmions can also be dragged by a moving magnetic tip.


Introduction
Magnetic skyrmions, topologically stable magnetization textures with particle-like properties, have recently attracted great attention. The magnetic skyrmions are fundamental magnetic structures that hold promise for applications in spintronic devices; for example, functional skyrmion racetrack memory [1][2][3][4][5][6][7]. In comparison with magnetic vortices [8][9][10] or bubbles [11], one of the most interesting characteristics of skyrmions is the topological protection, which maintains the robustness with respect to the perturbations and disorder [12]. Moreover, the critical currents required to move skyrmions are very small, typically a few 10 6 A m −2 , which is about five orders smaller than the threshold of current-induced domain wall motion [1]. To date, three different types of skyrmions have been reported in materials with perpendicular magnetic anisotropy (PMA): bubble skyrmions [13,14], nano-skyrmions [2,15,16], and dynamical skyrmions [17]. They are stabilized by different mechanisms, as follows. (i) The bubble skyrmions with a typical lateral size of the order of 1 μm are stabilized by the competition between the PMA and magnetostatics (i.e. long-range dipole-dipole interaction) [18]. (ii) The nano-skyrmions have typical dimensions of few tens of nanometers, and are stabilized by PMA and the Dzyaloshinskii-Moriya interaction (DMI) [19,20]. (iii) The dynamical skyrmions can be stabilized by the PMA and dynamics [17], without magnetostatics or DMI.
Reliable control of skyrmion motion is of great importance and interest. Previous work has shown that the motion of skyrmions can be driven by using spin-polarized current [21][22][23][24][25][26], magnetic field gradients [27,28], electric field gradients [29,30], temperature gradients [27,31,32], magnons [33,34], or spin waves [35]. Among these, the magnetic field gradients along the y-axis direction, as shown in figure 1(a), can induce a Hall-like motion of skyrmions, showing that in addition to the main velocity along the x-axis direction (v x ) induced by the magnetic field gradient, a slight transverse velocity can also be caused along the y-axis direction (v y ) due to the damping effect. Recently, a theoretical study on the dynamics of skyrmions driven by a magnetic field gradient has been published [36], in which the skyrmion is considered as a soliton and its dynamics is analyzed by the momentum calculation.
In this paper, we investigate the dynamics of skyrmions driven by a magnetic field gradient. The theoretical predictions are verified by micromagnetic simulations, showing how parameters such as gradient strength and Gilbert damping affect the dynamical characteristics of skyrmions. Interestingly, we show that a radially gradient magnetic field can be used to trap a series of skyrmions.

Analytical model
As a simulation model, we consider a Néel-type skyrmion formed in the center-line of a nanowire, as shown in figure 1(a). The spin dynamics of this system can be described by the Landau-Lifshitz-Gilbert (LLG) equation [17,37] including the DMI effect. The DMI energy density is written as m D m m m , where D i is the interfacial DMI constant. The applied spatially dependent magnetic field at the mass center X = (X, Y) of the skyrmion is where (r, j) are the local polar coordinates (see figure 1(b)), h 0 is the local magnetic field at the center, and g is the gradient parameter. Here, we consider the skyrmion as a rigid structure (without any or with very slight structure deformation). Thus the Thiele approach based on the course-grain approximation can be employed to analytically model the skyrmion dynamics [27,35,[38][39][40]: where Θ and Ф stand for the polar angle and azimuthal angle of the spin moment in the standard spherical coordinates, as shown in figure 1(c), and E is the Zeeman energy density. F edge in equation (2) is the repulsive force generated by the boundary of the sample. In this study, we consider that the skyrmion is far away from the boundary of sample. Thus, the edge effect is ignored and F edge ≈ 0. From equation (2) we can obtain an explicit solution for the skyrmion displacement driven by the magnetic field gradient: where F grad,y is the y-component of the driving force. The subscripted R SK indicates the area of the skyrmion. Note that the displacement velocity of the skyrmion shown in equation (4) contains two components (v x and v y ), where v x is perpendicular to the field gradient and caused by the gyromagnetic force; v y is parallel to the field gradient and associated with the Gilbert damping (that is, the dissipative force).

Simulation and results
The Bloch l = = A periodic boundary condition in the x-direction is used. The total simulation time is 50 ns. The displacement of the skyrmion is characterized by the position of skyrmion center X = (X, Y), which can be given by the distribution of the local magnetization [11]: and the gradient increases from 0 to 0.18 mT nm −1 . Apparently, the simulation results (symbols) agree well with the theoretical prediction of equation (4) (solid lines), showing a proportional relation between the velocity and field gradient. However, the skyrmion state will be broken if the magnetic field is too strong [14]. It can be seen in figure 2(a) that the v x (perpendicular to the magnetic field gradient direction) is positive while v y (along the gradient direction) is negative. v y is two orders smaller than v x when the Gilbert damping constant α is adopted to be 0.03. The influence of the Gilbert damping is shown in figure 2(b), where v x (black squares) is almost invariant for the small Gilbert damping factor (α < 0.1) while v y (red circles) increases linearly with the damping factor. This feature can be explained well by equation (4), in which v y is proportional to the dissipative factor D (related to the damping α) and v x is independent of D. The ratio of velocity along the x-and y-directions is coincident with the Hall-effect-like angle tanΘ H = v y /v x = −D/G, which stands for the deflection angle [40]. However, both v x and v y in the inset of figure 2(b) show a nonlinear dependence when α is further increased from 0.3 to 2.0. We attribute this nonlinear decrease to the over-damping effect, which induces an inhibition contribution to the skyrmion's movement.

Radially spatial gradient of magnetic field
The skyrmion motion triggered by the gradient magnetic fields provide a potential approach to addressing magnetic skyrmions. In the following, we will show quite an interesting dynamics: a radial direction of the magnetic field gradient can trap and capture skyrmions in a confined thin film structure. As an example, here we consider a square shaped confined structure with dimensions of 512 × 512 × 2 nm 2 . The discretized cell for simulations is set to be 2 × 2 × 2 nm 3 . In the simulations, a magnetic skyrmion is nucleated at the 'start' position marked in figure 3(a) using a local magnetic field pulse (along the −z-direction). After that, an inhomogeneous perpendicular magnetic field is applied along the +z-direction. The inhomogeneous field has a radial direction gradient with the minimum zero field at the sample center marked by a symbol 'O' and the maximum field is ∼57 mT at the boundary of the sample. Correspondingly, the radial gradient strength is 0.175 mT nm −1 . This is a reasonable value which can be created by a magnetic tip with typical parameters (not shown). Under the action of the magnetic field gradient, the skyrmion consequently moves toward the center of the sample along a clockwise spiral trajectory, as shown in figure 3(a). Figure 3(b) shows the evolution of the skyrmion velocity and the displacement d deviated from the sample center point O. It is interesting to note that the skyrmion is first accelerated and then decreases rapidly after its maximum velocity obtained at t = 2.4 μs. This acceleration behavior can be explained from the theory based on equations (4) and (5). It is found that the radius of the skyrmion depends on the external magnetic field, which can be derived from the minimum-energy principle: where R SK is the size of the skyrmion region, which increases with the decrease of magnetic field H app and anisotropy K 1 . This will enhance the integration value of equation (5) and further enlarge the velocity given in equation (4). As a result, the skyrmion slightly accelerates, as shown in figure 3(b). In this simulation, the gradient field is not strong enough to change the skyrmion shape. Interestingly, the sudden decrease in velocity is observed in figure 3(b) for t > 2.4 μs. At this moment, the periphery of the skyrmion starts crossing the sample center 'O' point. The velocity is mainly determined by the force distribution calculated by equation (5). At the moment 'A', the skyrmion is far away from the sample center (μ 0 H = 0) and the force F generated by the magnetic field is along the direction of the field gradient, as shown in figure 3(c). In contrast, at the moment 'B', the force F 1 generated in the left region of the solid line is pointing to the right while the force F 2 in the right region is pointing to the left, as illustrated in figure 3(d), here F 2 < F 1 . The sum of the two forces results in a reduced driving force acting on the skyrmion particle. Consequently, the skyrmion will slow down and eventually stop at the sample center. This simulation suggests an effective method to trap or capture the skyrmions.

Moving field trap
Furthermore, we find that the skyrmion can be dragged by a moving magnetic field trap which creates a radially gradient magnetic field. For that, we performed a series of simulations for the skyrmion driven by a moving trap along the x-axis direction with different velocities V trap . The results are shown in figure 4, and supplemental movie I is available at stacks.iop.org/NJP/19/083008/mmedia. In these simulations, we assume that the magnetic field trap only acts on the region with a 30 nm radius (marked by the dotted circles) and the gradient strength is 0.15 mT nm −1 . The skyrmion center and the trapping center start from a same position. Due to the active area of the trap being much larger than the skyrmion size (R SK ∼ 10 nm), the skyrmion is completely captured by the trap at the initial moment. When the trap of the magnetic field gradient moves along the xdirection with V trap = 1.1 m s −1 , the result shows that the skyrmion can be dragged perfectly but its trajectory behaves as a periodic motion of parabola curves, as shown in figure 4(a). To gain insight into this parabolic motion, figure 4(b) shows the relative position between the skyrmion and the magnetic field trap center at three different moments (A, B, and C). In the beginning (moment 'A'), the skyrmion is behind the trap because v x smaller than V trap . But the skyrmion speed (i.e. the norm of its velocity vector, v v v 1.5 m s x y larger than that of the trap. Due to the spiral motion of the skyrmion driven by a radially gradient fields, the v x of skyrmion in region II of figure 3(a) is enhanced. As a result, the skyrmion is accelerated in the x-direction and catches up with the trap somewhere between A and B. After this, the skyrmion is located in region I of figure 3(a), and v x will decrease. Thus, the moving trap will catch up and pass the skyrmion again, see moment 'C' in figure 4(b). Consequently, the skyrmion follows the periodic motion of a parabolic curve, as shown in figure 4(a) and supplemental movie I. However, when the magnetic field trap speed is increased to 1.5 m s −1 , the skyrmion cannot catch up with the trap and it will stop somewhere, as shown in figure 4(c). In this case, the approach using the moving magnetic field trap to control skyrmion motion will be invalid. Figures 4(d) and (e) summarize the trajectories and the  Figure 5 shows a series of skyrmions with different numbers of 2 × 2, 3 × 3, and 4 × 4 trapped by the radially gradient magnetic field. In this case, all skyrmions are propelled by the magnetic field gradient towards the center of the sample. Different from the case of an isolated skyrmion, a collision effect among the skyrmions will occur, but the particle-like skyrmions cannot be fused together due to the repulsive interaction between them. As a result, a stable hexagonal distribution is eventually formed for these skyrmions, see figures 5(a)-(f) [44]. The separation between two skyrmions is determined by the competition between the repulsive force and the driving force from the gradient field [45]. The supplemental movie II shows the trapping process of a group of 3 × 3 skyrmions.

Trapping multiple skyrmions
Finally, we would like to point out that the trapping of magnetic skyrmions has attracted much attention recently. Different techniques have been proposed, for example, using a hole (defect effect) [42,46,47] or electric field gradient [30] to guide the skyrmion motion. Here we would like to point out that a radially gradient magnetic field may be generated by different experimental techniques, such as a magnetic tip or a magnetic dot deposited on top of a thin film. For example, for magnetic force microscopy, if the magnetic tip is made of CoCr material (M S = 80 kA/m) with a size of 40 × 40 × 80 nm 3 , our calculation shows that a radial gradient of the magnetic field with maximum value of 0.5 mT nm −1 can be created (not shown), in which we assume that the separation between the tip and the sample surface is 20 nm. Obviously, this gradient field can be used to trap the skyrmions. In addition, the gradient magnetic field may also be created by a superconducting vortex system [48,49] or circularly polarized laser technique [18,32,50,51].

Conclusion
In summary, we have investigated the skyrmion motion triggered by a magnetic field gradient. The velocity of skyrmion motion is predicted analytically through the Thiele approach, which agrees well with micromagnetic simulations. The skyrmion velocity increases almost linearly with increased magnetic field gradient. The component of velocity along the x-axis direction is almost unchanged for the small Gilbert damping factor (α < 0.1) while the y-component of the velocity slightly increases with the damping factor. More interesting is that we have demonstrated a potential approach for trapping skyrmions using a radially spatial gradient of magnetic fields. The skyrmions can also be dragged by a moving magnetic tip which is used to create a radially spatial gradient of magnetic fields.