Configurable pixelated skyrmions on nanoscale magnetic grids

Topological spin textures can serve as non-volatile information carriers. Here we study the current-induced dynamics of an isolated magnetic skyrmion on a nanoscale square-grid pinning pattern formed by orthogonal defect lines with reduced magnetic anisotropy. The skyrmion on the square grid can be pixelated with a quantized size of the grid. We demonstrate that the position, size, and shape of skyrmion on the square grid are electrically configurable. The skyrmion center is quantized to be on the grid and the skyrmion may show a hopping motion instead of a continuous motion. We find that the skyrmion Hall effect can be perfectly prohibited due to the pinning effect of the grid. The pixelated skyrmion can be harnessed to build future programmable racetrack memory, multistate memory, and logic computing device. Our results will be a basis for digital information storage and computation based on pixelated topological spin textures on artificial pinning patterns.

A square lattice of square-shaped antiskyrmions was also observed experimentally by Peng et al. [82] in a noncentrosymmetric magnet. Besides, it is found that skyrmions and antiskyrmions can show elliptical shapes in samples with anisotropic DM interactions [83][84][85]. The deformation of a skyrmion induced by external forces may also result in a non-circular shape [31,65]. All these findings on skyrmions showing different shapes reflect the importance of controlling the shape of a skyrmion, which may lead to novel spintronic applications based on topological spin textures with different shapes.
Recently, Juge et al. [87] and Ohara et al. [88] independently demonstrated the control of skyrmion position by locally modifying the magnetic properties. They experimentally realized the confinement of skyrmions in nanoscale tracks with modified magnetic properties on a large film.
In particular, the local modification of perpendicular magnetic anisotropy (PMA) can result in an energy barrier, which plays a key role on the confinement and pinning of skyrmions. Therefore, by locally modifying PMA or other magnetic properties it is envisioned that one can fabricate different types of artificial pinning patterns on magnetic materials, such as parallel defect lines, grids, and square patterns [18,86,. These artificial pinning patterns may lead to very special static and dynamic behaviors of topological spin textures interacting with them [18,86,.
For example, the skyrmion Hall effect can be controlled or reduced for skyrmions moving over two-dimensional periodic pinning arrays in certain cases [91,105]. Moreover, the artificial pinning patterns would also offer the possibility to study basic science issues since particles and quasiparticles (e.g., superconducting vortices and colloids) on periodic substrates is a wide-ranging problem [18,[110][111][112][113][114][115][116].
In this work, we report the properties of a skyrmion in a magnetic thin film with the square-grid pinning pattern formed by nanoscale orthogonal defect lines with reduced PMA. We show that the square grid leads to the formation of pixelated skyrmions, of which the position and area are quantized in the unit of the grid cell. We find that the position, size, and shape of a pixelated skyrmion can be manipulated precisely by a current pulse.

Results and Discussion
Static properties of the square-shaped skyrmion. Figure 1(a) depicts the simulation geometry.
We consider a ferromagnetic (FM) thin layer attached to a heavy-metal layer, where the FM layer has certain PMA and interface-induced DM interaction. The FM layer thickness is fixed at 1 nm in all simulations. We assume that the square-grid pinning pattern in the FM layer is formed by orthogonal defect lines with reduced PMA, which can be realized in experiments by locally modifying the magnetic properties (i.e., using additional sputtered layers or ion irradiation) [87,88,117]. The width of each defect line is defined as w. The distance between two nearestneighboring parallel defect lines is defined as l. The number of unit square patterns along the x or y directions is defined as n. Hence, the total side length of the FM layer is equal to nl + (n + 1)w [see Fig. 1(a)]. The magnetic parameters and other modeling details are given in the Methods.
We first study a static single isolated skyrmion in the sample with the square grid. At the initial state, a Néel-type skyrmion with a theoretical topological charge Q = 1 is placed at the sample center, which is relaxed to a stable or metastable state by the OOMMF conjugate gradient minimizer [118]. The topological charge Q is defined as Q = − 1 4π m · ( ∂m ∂x × ∂m ∂y )dxdy with m being the reduced magnetization. In this work, we assume that the initial skyrmion diameter is smaller than the defect-line spacing l. For example, the relaxed ordinary skyrmion has a diameter As shown in Fig. 2(a), an ordinary skyrmion shows directional motion in the sample with K d /K = 1 when a single current pulse is applied. The direction of motion depends on the spin polarization direction p, which is controlled by the current injection direction in experiments.
When p = +x, the ordinary skyrmion moves smoothly toward the +y direction and shows an obvious transverse shift in the −x direction due to the skyrmion Hall effect [64,65]. The skyrmion stops when the current pulse is off at t = τ = 400 ps, and the final state obtained at t = 1000 ps shows that the skyrmion is closer to the upper left corner of the sample (see Supplementary Movie 1). The skyrmion moves closer to the lower left, lower right, and upper right corners driven by the current pulses with p = +ŷ, p = −x, and p = −ŷ, respectively.
When K d /K < 1, we find that the square-shaped skyrmion shows very different currentinduced dynamic behaviors, which depend on the value of K d /K. In the sample with K d /K = 0.5, First, we review the results on an ordinary skyrmion with K d /K = 1. m z slightly decreases to a stable value during the application of the current pulse, indicating the steady motion of the skyrmion with a slightly reduced size during the pulse application as shown in Fig. 3(a). The timedependent total energy E is given in Fig. 3(d), which rapidly recovers to the initial-state value after the pulse application, suggesting that the initial and final states are the same. Figure 3(g) shows the time-dependent numerical topological charge Q, which slightly increases during the skyrmion motion. We note that the numerical Q is not exactly equal to 1 at the initial and final state. The non-integer value is caused by the discretized meshes and current-induced deformation of the skyrmion. We have excluded the effect of tilted edge spins on the calculation of Q.
Next, we study the case with K d /K = 0.5, where a square-shaped skyrmion shows a hopping motion. m z oscillates during the pulse application, which implies the oscillating changes of the skyrmion size and shape during its hops across square grid cells as shown in Fig. 3(b). The detailed process is shown in Fig. 4(c). After the pulse application, both m z and E are recovered to their initial-state values at t = 0 ps [see Fig. 3(e)], justifying that the initial and final skyrmion states are the same because of the translational symmetry of the square-shaped skyrmion. We note that the numerical Q oscillates during the pulse application [see Fig. 3(h)], which is caused by the oscillation of the square-shaped skyrmion.
Finally, we study the case with K d /K = 0.2. As the square-shaped skyrmion is enlarged and deformed during the pulse application, m z significantly decreases during t = 0 − 400 ps and reaches a stable value at t = 500 ps. See Fig. 3(c). The detailed process is shown in Fig. 4(d). We also study the effects of j and α on the dynamics of a square-shaped skyrmion (see Supplementary Note 3). The effect of j is similar to that of τ . A large α will reduce the skyrmion Hall effect for the skyrmion hop and deformation, which leads to the current-induced formation of the rectangle-shaped skyrmion. A basic phase diagram of the system transitions from the single skyrmion hopping to the skyrmion deformation is given in Fig. 6. We point out four possible cases induced by the current pulses with different pulse lengths. First, for the samples with relatively stronger pinning strengths (K d /K < 0.5), a weak current pulse cannot drive the square-shaped skyrmion. Namely, the square-shaped skyrmion is pinned at its initial position during and after the pulse application. For a strong current pulse, the square-shaped skyrmion will be transformed to a rectangle-shaped skyrmion by the current pulse. Second, for the sample with a moderate pinning strength (K d /K = 0.5), a weak current pulse cannot drive the square-shaped skyrmion, but a stronger current pulse may drive the square-shaped skyrmion into a hopping motion or shrinking.
Third, for the sample with relatively weaker pinning strength (K d /K > 0.5), the square-shaped skyrmion may not be stable on the square pinning pattern. Hence, once a current pulse is applied, the square-shaped skyrmion will first depin and then shrink to a smaller round-shaped skyrmion.
Such a smaller round-shaped skyrmion could be pinned again on the defect line after the pulse application.

Conclusions
In conclusion, we have studied the statics and dynamics of configurable skyrmions on the grid formed by orthogonal defect lines with identical spacings and reduced PMA. We find that the grid results in the pixelation of skyrmions, leading to the square-shaped, rectangle-shaped, and Lshaped skyrmions. The position and area of the square-shaped skyrmion are quantized in the unit of the grid cell, which are different to ordinary skyrmions, of which the position and size change continuously.
We numerically demonstrate that the position, size, and shape of a square-shaped skyrmion on the grid are manipulated electrically, which depend on the pinning strength, the applied current pulse, and the damping parameter. In particular, we show that the square-shaped skyrmion hops on the grid with weak pinning, and its skyrmion Hall effect can be controlled by the current pulse.
Especially, the skyrmion Hall effect of the square-shaped skyrmion is perfectly prohibited by appropriately tuning parameters. The straight hopping motion of skyrmion is vital for racetracktype memory devices. The control of the skyrmion Hall effect using a preset sequence of current pulses provides the possibility to build a logic computing device based on the transport route of skyrmions. In addition, it is possible to reduce the width of a nanotrack as wide as three grid cells since the skyrmion Hall effect is suppressed. It is highly contrasted with the case of an ordinary skyrmion, where we need to use a wider nanotrack in order to keep away a skyrmion from an edge.
It is possible to shift a skyrmion by N grid cells by applying N pulses since the skyrmion relaxes to the same structure only by changing its position after the pulse is over.
Besides, we find that the current pulse drives the square-shaped skyrmion to deform on the square grid with strong pinning, which transforms the square-shaped skyrmion to a rectangleshaped or L-shaped skyrmion in a controlled manner. It can be utilized to build a multistate memory [41] or an artificial synapse [59] based on different metastable topological spin textures in one sample, where topological spin textures with different m z stand for different states that can be detected by measuring magnetoresistance. It is worth mentioning that a reset function, that is, a method to transform an L-shaped skyrmion to an original square-shaped skyrmion may be required for the multistate memory and artificial synapse applications. Such a reset function can be achieved by applying an out-of-plane magnetic field pulse in our system (see Supplementary Note 4). Indeed, one can also reset the system by erasing the entire state and then nucleate a square-shaped skyrmion.
Our results give a deeper understanding of the complex dynamics of a skyrmion on a nanoscale grid formed by defect lines with modified magnetic anisotropy. It will be straightforward to generalize our results to the systems with artificial nanoscale triangular and honeycomb grids. However, the square-grid pinning pattern is most efficient to prohibit the skyrmion Hall effect and easily manufacturable, which is due to the fact that a typical lithography scanner system works in a way that favors horizontal and vertical scanning directions. For this reason, the fabrication of the triangular or irregular shape may result in obvious polygon edges. Such an effect may significantly reduce the pinning pattern quality when the resolution goes down to a few nanometers. Hence, the square-grid pinning pattern and rectangle-grid pinning pattern (see Supplementary Note 5) may be the most reliable choices. Besides, the advantage of using the square-grid pinning pattern to guide the skyrmion motion is that the skyrmion can be delivered toward different directions by controlling the driving current direction, current density, and pulse length. Such a feature may not be possible on other pinning patterns such as the parallel defect lines.
On the other hand, we would like to point out that the square-grid pinning pattern could also serve as a platform for the study of multiple skyrmions interacting with a pinning landscape (see Supplementary Note 6), and a great many directions one could go with this system such as different kinds of driving forces [18]. Last, from the point of view of electronic device applications, future works on this topic may focus on the performance analysis, such as the energy expenditures of skyrmion hopping and square-to-L deformation. Our results may provide guidelines for building spintronic applications utilizing the interaction between topological spin textures and artificial pinning patterns.

Methods
where the damping-like spin torque is generated through the spin Hall effect in the heavy-metal layer when an electric current is injected [35]. In Eq. 1, M is the magnetization, M S = |M | is the saturation magnetization, t is the time, γ 0 is the absolute value of gyromagnetic ratio, α is the Gilbert damping parameter, and H eff = −µ −1 0 ∂ε/∂M is the effective field. u = |(γ 0 )/(µ 0 e)| · (jθ SH )/(2aM S ) is the spin torque coefficient, p stands for the unit spin polarization direction, µ 0 is the vacuum permeability constant, is the reduced Planck constant, e is the electron charge, j is the driving current density, and θ SH is the spin Hall angle.
The average energy density ε contains the PMA, FM exchange, demagnetization, applied magnetic field, and interface-induced DM interaction energy terms, given as     is relaxed at the sample center as the initial state at t = 0 ps. A current pulse of j = 100 MA cm −2 and p = −ŷ is applied, and then the system is relaxed until t = 1000 ps. The final states are confirmed at t = 1000 ps.