Driven gyrotropic skyrmion motion through steps in magnetic anisotropy

The discovery of magnetic skyrmions in ultrathin heterostructures has led to great interest in possible applications in memory and logic devices. The non-trivial topology of magnetic skyrmions gives rise to a gyrotropic motion, where, under an applied energy gradient a skyrmion gains a component of motion perpendicular to the applied force. So far, device proposals have largely neglected this motion or treated it as a barrier to correct operation. Here, we show that skyrmions can be efficiently moved perpendicular to an energy step created by local changes in the perpendicular magnetic anisotropy. We propose an experimentally-realizable skyrmion racetrack device which uses voltage-controlled magnetic anisotropy to induce a step in magnetic anisotropy and drive a skyrmion unidirectionally using alternating voltage pulses.


Driven gyrotropic skyrmion motion through steps in magnetic anisotropy
Yifan Zhou, Rhodri Mansell & sebastiaan van Dijken the discovery of magnetic skyrmions in ultrathin heterostructures has led to great interest in possible applications in memory and logic devices. the non-trivial topology of magnetic skyrmions gives rise to a gyrotropic motion, where, under an applied energy gradient a skyrmion gains a component of motion perpendicular to the applied force. so far, device proposals have largely neglected this motion or treated it as a barrier to correct operation. Here, we show that skyrmions can be efficiently moved perpendicular to an energy step created by local changes in the perpendicular magnetic anisotropy. We propose an experimentally-realizable skyrmion racetrack device which uses voltage-controlled magnetic anisotropy to induce a step in magnetic anisotropy and drive a skyrmion unidirectionally using alternating voltage pulses.
Magnetic skyrmions are particle-like spin textures with a non-trivial topological number 1,2 . Several mechanisms can give rise to skyrmion formation in different materials, such as four-spin exchange 3 , dipole-dipole interactions 4 , frustrated exchange interactions 5 and the Dzyaloshinskii-Moriya interaction (DMI) [6][7][8] . Amongst these, skyrmions in magnetic heterostructures with interfacial DMI and perpendicular magnetic anisotropy (PMA) have attracted most interest for technological applications [8][9][10][11][12] . The current-induced motion of such skyrmions has been investigated intensively, and the required driving current has been shown to be much lower than that required for domain walls in similar systems 13 . However, the energy required by current-controlled devices is still relatively high per bit compared to approaches based on controlling magnetism through electric fields 14,15 . These devices exploit the ability to electrically control the magnetic anisotropy. For perpendicularly magnetized materials it has been shown that the anisotropy can be controlled by applying voltages across ferromagnet-metal oxide interfaces [16][17][18] . In most implementations, the electric field at the interface alters the 3d orbital occupation of a magnetic transition metal, modifying the perpendicular magnetic anisotropy 17 , although the largest effects have come from altering the oxidation state at the interface 18 . Voltage-controlled magnetic anisotropy (VCMA) has been combined with skyrmion devices in various proposals to create static and racetrack-like memories [19][20][21][22][23] . However, these proposals, similarly to proposals concerning current-driven skyrmions, do not exploit the gyrotropic motion created by the topology of skyrmions 9 , with few exceptions 24 .
In this work, we demonstrate how skyrmion motion can be induced by a step in magnetic anisotropy, which could be created by VCMA, and propose a device implementation exploiting this effect. We start by studying the motion of skyrmions in an anisotropy gradient 24 , comparing the results of micromagnetic simlations with predictions of the analytical Thiele equation 25 which describes the motion of a rigid skyrmion. We investigate how different material parameters affect the skyrmion motion. Finally, we demonstrate that a step in anisotropy can provide a similar effect to a gradient and introduce a device that could allow the experimental demonstration of an electric-field controlled skyrmion racetrack memory.
as it moves to lower anisotropy. We can compare the simulation with the Thiele equation, which decomposes the motion of a rigid skyrmion under an applied force into gyrotropic and dissipative terms. By assuming that the spin structure of a skyrmion is rigid, we can define m(r, t) = m(r − C(t)), where r is the position vector of the spins to the skyrmion center and C(t) is the centre-of-mass coordinate of the structure. The dynamics of a magnetic skyrmion structure can then be described by the Thiele equation 9,25-27 : Here, ∇U is the energy gradient within a skyrmion, and F is the equivalent driving force. G = (0, 0, G) is the gyromagnetic coupling vector where G = 4π for a skyrmion.  is called the dissipative force tensor. We model an energy gradient in a VCMA material as a gradient of the uniaxial anisotropy, K u , varying along y direction: Here, E K u is the skyrmion energy due to the uniaxial anisotropy and d is the thickness of the magnetic layer and ẑ is the unit vector in the growth direction. The two-dimensional velocity described in the Thiele equation can then be separated into x and y axis: where S x and S y give the skyrmion position perpendicular and parallel to the anisotropy gradient, respectively, and the dot indicates a time derivative. These equations can be numerically solved as a time-dependent ordinary differential equation. This also leads to the drift angle 22 : x y  where a high value of θ indicates motion nearly perpendicular to the gradient. From Eqs (4) and (5) it is clear that a force acting on a skyrmion in one direction will, in general, cause motion both parallel and perpendicular to the force 9 . Further, if the damping term α and the dissipative force tensor  are small then the motion will be predominantly perpendicular to the force. Such motion is seen in Fig. 1, where the drift angle is θ = 87.9°. We can use the above equations to estimate the drift angle expected from the Thiele equation to check that the theoretical description above accurately describes the simulations. Since the skyrmion radius changes with time, we simplify the estimate by using the average value of the radius during the simulation. We calculate a single value of  for use in the Thiele equation by integrating over the whole system according to Eq. 2 and time averaging over snapshots taken every 0.3 ns. Due to the finite width of the system and the presence of the anisotropy gradient,  is depend- www.nature.com/scientificreports www.nature.com/scientificreports/ ent on both the size and position of the skyrmion. Using this method we obtain a drift angle of θ = 87.8°, in good agreement with the simulation.
The slight difference can be explained through the fact that in simulations the radius of the skyrmion increases with time, and so changes the dissipative force tensor. Further, the skyrmion expansion also leads to acceleration both in the transverse direction as well as against the anisotropy gradient.
Influence of material parameters. Using the same simulation geometry from Fig. 1, we further investigate how different material parameters influence skyrmion motion under anisotropy gradients. It has be shown using the Thiele equation that an anisotropy energy gradient ΔE K u controls the distance travelled by the skyrmion in x, while the damping constant α and the dissipative force tensor  determine the drift angle (see Eqs 4 and 5). Figure 2(a) shows the distance of travel, S x , after 30 ns as a function of ΔK u with a comparison of the simulation and theoretical calculation.
In Fig. 2(b), the drift angle, θ is plotted as a function of the damping constant α. We find that larger ΔK u increases the travel distance, and larger α decreases the drift angle, which corresponds to the Thiele equation. However, the simulation and theoretical results differ in large ΔK u and α. This is due to the fact that the skyrmion energy is position-dependent in the simulations due to skyrmion expansion, while in the theoretical calculation it is simplified using a fixed skyrmion radius. If ΔK u > 3 × 10 12 J/m 4 , the inflation of the skyrmion radius breaks this assumption down 24 . Similarly, for large α, the skyrmion radius expands during the simulation. This increases the dissipative force tensor  and leads to a decreased drift angle in the simulations in accordance with Eq. 5. The theoretical drift angles, where  is fixed in each calculation, are consequently larger than in the simulations.
As discussed above, the skyrmion radius R plays an important role in determining the travel distance of a skyrmion after a fixed time. In Fig. 3, we show the travel distance and the final skyrmion radius as a function of (a) exchange energy A, (b) damping parameter α, (c) DMI energy D, and (d) perpendicular magnetic anisotropy K u . The anisotropy gradient ΔK u is held constant in these simulations. There is a strong correlation between the skyrmion radius at the end of the simulation and the travel distance along x.
From the four plots, we can estimate that the relationship between the change in skyrmion radius after 30 ns and each of the parameters roughly follows Δ ∝ α R D AK u . Moreover, larger skyrmions lead to a longer travel distance transverse to the anisotropy gradient, and the distance changes almost linearly with the skyrmion radius. These results show that careful selection of the material parameters will lead to well-defined gyrotropic motion. Moreover, skyrmions have reasonable scaling properties, so that if the device length becomes smaller, the skyrmion size can also be reduced.

Anisotropy step and device implementation.
Having studied the effects of the underlying materials parameters on skyrmion motion in an anisotropy gradient, we now move to a device implementation. For a device, rather than creating a gradient, a step-like profile in anisotropy would be easier to obtain. A schematic of a possible device, where a magnetic nanotrack is half-covered by a top-gate, is shown in the insets of Fig. 4. By applying a voltage on the gate metal (white) across a metal oxide (red) on top of the magnetic layer (yellow) the anisotropy can be varied through VCMA. A skyrmion at the anisotropy step will move along the step (i.e. transverse to the direction of changing anisotropy). Whilst this mechanism can also be described by the Thiele equation, an analytical description of this motion is non-trivial. In our simulations we assume an abrupt anisotropy step. In a real device the step will have some width, at least on the order of the thickness of the oxide layer, but this width is likely to be smaller than the radius of the skyrmion.
There are two approaches to creating skyrmion motion by tuning the anisotropy strength on one side of the nanotrack: increasing the anisotropy by applying a negative voltage, as shown in Fig. 4(a), or decreasing the anisotropy by applying a positive voltage, as shown in Fig. 4(b) 28 . In both cases, the velocity of the skyrmion www.nature.com/scientificreports www.nature.com/scientificreports/ reduces with time as the skyrmion moves at a small angle away from the anisotropy step. Further, the skyrmions in both anisotropy configurations travel similar distances with comparable drift angles. However, there are also differences. For the anisotropy step in Fig. 4(a), the skyrmion maintains its size during motion, as the skyrmion  www.nature.com/scientificreports www.nature.com/scientificreports/ remains in the initial low anisotropy region. For the step in Fig. 4(b), the skyrmion enlarges continuously while moving into the area with a lower anisotropy value. This difference in behavior suggests that a controlled increase of magnetic anisotropy would create more consistent skyrmion motion by maintaining the skyrmion size.
Based on the anisotropy step simulations, a device implementation for a skyrmion racetrack through VCMA is shown in Fig. 5(a). The magnetic nanotrack is covered by an array of ten gate electrodes on one side of the track. The size of each electrode is 50 nm × 50 nm, forming a total length of 500 nm. Negative voltage pulses with a duration of 10 ns are applied sequentially on two sets of alternating and parallel-connected electrode blocks (1 and 2). The simulation is initialized with a skyrmion placed at the left edge of the first electrode as shown in Fig. 5(a). By applying a negative voltage on electrode block 1, regions with higher magnetic anisotropy are periodically created along the nanotrack. This causes the skyrmion to move along the voltage-induced anisotropy step to the first boundary between regions 1 and 2. Then, the voltage on electrode block 1 is turned off, and the voltage on electrodes 2 is turned on. This reverses the modulation of magnetic anisotropy, as shown in Fig. 5(b), and forces the skyrmion to move along the newly formed anisotropy step to the next boundary between electrodes 2 and 1. Because the voltage pulse is longer than the time it takes for the skyrmion to travel 50 nm, the skyrmion moves around the corner of the anisotropy step, at the end of each pulse. When the voltage is switched between the electrode blocks, the skyrmion moves back to the middle and continues its travel along the x axis. By alternating the voltage pulses on the two electrode blocks, the skyrmion moves forward one block per pulse, thus giving rise to a synchronous unidirectional movement along the nanotrack, as shown in Fig. 5(c). With these parameters, the skyrmion traverses the 500 nm long device in 100 ns, and the skyrmion is always at least 20 nm from the edge of the nanotrack to avoid any edge effects. A supplementary video showing the device operation is available online.

Discussion
In the above simulations we have used an anisotropy step of 5 × 10 4 J/m 3 , corresponding to a change in interfacial anisotropy of 2 × 10 −5 J/m 2 at the ferromagnet/metal oxide interface. Assuming a moderate electric field across the interface of 200 MV/m, easily achievable in real devices, this leads to a minimum required VCMA effect of 100 fJ/Vm, a magnitude which has been demonstrated in a variety of systems 18,29 . Whilst we do not specify the skyrmion creation or readout here, all electrical creation and detection methods have already been proposed elsewhere 29,30 .
A key materials feature for the implementation of such a device is low damping. The deviation away from the anisotropy step on each voltage pulse is largely controlled by this parameter. In this paper we have used a value of 0.03 for the device simulations, which is relatively low for perpendicularly magnetized heterostructures, but is similar to that found in CoFeB based systems 31,32 .
We have shown here that a skyrmion can efficiently move transverse to an anisotropy step due to the topology-induced gyrotropic motion. The underlying material parameters affect the transverse motion, in, particular by determining the skyrmion radius, which is closely linked to the velocity of the skyrmion. The gyrotropic motion can be exploited to create a technologically feasible device, where a skyrmion is moves synchronously www.nature.com/scientificreports www.nature.com/scientificreports/ along a series of magnetic anisotropy steps in a fully electrically controlled skyrmion racetrack. Implementation of the proposed device could lead to a new class of low energy data storage devices.

Methods
Analytical expressions. The Hamiltonian of the magnetic skyrmion system includes Heisenberg exchange, the Dzyaloshinskii-Moriya interaction, the Zeeman interaction and an uniaxial anisotropy term, given by: where m i is the three dimensional unit spin vector, defined as M s /|M s |. J ij is the exchange constant, D ij the DMI vector, μ s the local atomic magnetic moment, B app the applied magnetic field, k u the uniaxial anisotropy constant and e the direction of easy axis. The time evolution of a magnetic moment at zero temperature is described by the Laudau-Lifshitz-Gilbert (LLG) equation, written as:

Data Availability
The scripts and datasets generated during the current study are available from the corresponding author on reasonable request.