Manipulating 1-dimensional skyrmion motion by the external magnetic field gradient

We have investigated an approximated analytic form of the one-dimensional motion of skyrmions accelerated by a gradient of the external magnetic field. We find excellent agreement between the analytical calculations and micromagnetic simulations when the skyrmion size is large. The skyrmion motion is related to not only the skyrmion size but also the skyrmion wall width. We also have performed the numerical calculation without approximation in comparison. The numerical calculation results are entirely in agreement with those of micromagnetic simulation for all the skyrmion size. These results introduce an efficient control of skyrmions to next-generation spintronic devices.


Introduction
In magnetic heterojunction systems, the crucial competition among Heisenberg exchange interaction, Dzyaloshinskii-Moriya (DM) interaction, and magnetocrystalline anisotropy can exhibit complex spin textures such as Skyrmions [1,2], chiral magnetic domain walls (DWs) [3][4][5], and Bloch lines [6,7]. A contribution of interfacial DM interaction is directly related to the spin-orbit coupling at the interfaces between heavy metals and ferromagnets combined with the broken inversion symmetry at the interfaces [8][9][10][11][12]. These exotic spin textures arising from DM interaction are mostly topologically stable and are appropriate for various applications as an information carrier and information-processing devices [13,14]. Indeed, a large majority of theoretical and numerical studies have shown the possibilities that magnetic skyrmions and chiral DWs could be essential ingredients of the next-generation spintronic devices for storage devices and logic application [8,14]. A single or a bunch of these topological objects are manipulated by laterally applied electrical currents due to spin-transfer torque [15,16] or spin-orbit torque [17,18]. Below the critical current density, the magnetic textures are fastened in a pinning potential. When the electric current is larger than the critical current density, the non-negligible displacements of topological objects occur. While the electrical current injection technique is a promising method to drive multiple skyrmions and chiral DWs synchronously, a large critical current density caused by low resistivities of magnetic materials and the extremely narrow and long nanoscale wire architectures makes an insurmountable obstacle which is the so-called 'Joule heating problem' owing to Ohmic losses [19,20].
Many studies of alternative propagation techniques have been carried out using not only to overcome the issue of the Joule heating problem but also application in skyrmions such as magnetic anisotropy gradient [21], electric field gradient [22,23], temperature gradient [24][25][26], spin waves and magnons [26][27][28][29]. At present, extensive experiments observed the skyrmion motion on the tracks with the magnetic anisotropy energy [30], controlled by external electric fields [31]. Moreover, there are several papers about the skyrmion motion under external magnetic field gradient [32][33][34][35][36]. Zhang et al [32] report that rotational motion under the radial field gradient. Casiraghi et al [33] reported experimental results that skyrmion manipulates individually through the local field gradient. Liang et al [34] demonstrate that the skyrmion motion under a magnetic field gradient at the limited region of damping. Komineas and Papanicolaou [35] report cyclotron-types rotational motion in the two-dimension. Wang et al [36] claim that radially spatial gradient magnetic field along the width direction on the nanowire. Utilizing magnetic anisotropy energy, electric fields, and gradient magnetic field, we have established ultra-low energy-consumption spintronics devices. The skyrmion tracks patterned with difference of the magnetic anisotropy energy, will be playing an important role to realize the Brownian computation [30].
The magnetic field-driven chiral DWs and magnetic solitons have been received a great amount of interests because the system is totally governed by the Landau-Lifshitz (LL) equation, which is equivalent to the Landau-Lifshitz-Gilbert (LLG) equation when the magnetic damping constant of the system is small enough. Moreover, various manipulation ideas such as direct current or alternative current magnetic field-driven magnetic solitons, the transverse magnetic field pulse induced DW, and skyrmion racetrack are demonstrated recently [37,38]. For practical applications for information storage devices or logic applications, the magnetic skyrmion or DW racetrack should be compatible with the complementary metal-oxide-semiconductor (CMOS) architectures, and the continuous miniaturization of CMOS architectures is essential for increasing the data capacity of the devices. For the magnetic field-driven skyrmion or DW motions in real spintronic devices, the external magnetic field is applied from the ultrashort electrical current pulses passing through the conduction lines adjacent skyrmion or DW racetracks to minimize the energy consumptions. Naturally, the applied magnetic field to the magnetic racetracks is not uniform due to Oersted law.
In this work, we investigate analytically and computationally the skyrmion motion in a one-dimensional nanowire driven by the magnetic field along the z-direction while the field gradient is taken in the x-direction. According to the analytic formula, the skyrmion velocity in the nanowire with a magnetic field gradient is proportional to the skyrmion radius and the wall width. The analytic calculation is in good agreement with the micromagnetic simulation results when the skyrmion size is large. Furthermore, we also perform the numerical calculation without approximation in comparison. The numerical calculation results are entirely in agreement with those of micromagnetic simulation in a whole range of the skyrmion size. The observed results can be introduced to next-generation spintronics devices.

Analytical formula for the skyrmion velocity driven by the magnetic-field gradient
We briefly describe a theory for the skyrmion motion in our system. The motion of skyrmion in a two-dimensional film can be expressed by the Thiele equations [39] where R is the center coordinate, G the gyromagnetic coupling, α the Gilbert damping constant, D the dissipation dyadic, and F the external force, e.g., by electric currents, magnetic field gradients, and thermal fluctuations.
, M s is the saturation magnetization, γ the gyromagnetic ratio. The normalized magnetization at r = (r, φ) is described by polar and azimuthal angles Θ (r, φ) and Φ (r, φ) so that m = (sin Θ cos Φ, sin Θ sin Φ, cos Θ). A skyrmion centered at r = 0 can be described [40] by, with boundary conditions of Θ (0) = 0 and Θ (∞) = π for the axial symmetric skyrmion. The integer q is the vorticity (q = 1 for the skyrmion and q = −1 for an anti-skyrmion), and γ is the helicity classifying the type of skyrmions (γ = 0 or π for Néel skyrmions and γ = ±π/2 for Bloch skyrmions). For the Néel skyrmion, the magnetization profile looks like a 360 • Néel domain wall along the radial direction as described below [41,42] where R is the radius of the skyrmion, w the wall width of the skyrmion, and r the position of magnetization. The components of the dissipative force, which is the second term of equation (1), αD describes the friction of the skyrmion, We study the effects of a non-uniform perpendicular magnetic field with a longitudinal direction of a nanowire. We neglect thermal fluctuation. The force is given by The above equation shows the force attribute to the skyrmion edge where the magnetization is reversed. Because we apply the magnetic field along the z-direction while field gradient is in x-direction, force by magnetic field gradient can be expressed as [41]: here, h z g is perpendicular magnetic field gradient and f z R/w is only needed to calculate numerically and its asymptotic form is given by R/w 2 /2 + π 2 /24.
Then, we consider the one-dimensional equation of skyrmion motion in equation (1). Using magnetic field gradient force and dissipation above with approximation, we can calculate how the velocity of skyrmion depends on the magnetic field gradient in an analytical form for q = ±1 and R/w 1, the velocity of a skyrmion is simply calculated by The value of v x increases with increasing skyrmion radius. The magnetic field gradient force has a large effect on large skyrmion because of sizable magnetic field differences from the left and right sides of the skyrmion. Surprisingly, the skyrmion velocity is proportional to not only the radius of the skyrmion but also the width of the skyrmion. Therefore, we find the relationship between the skyrmion velocities divided by skyrmion width (v x /w) and the skyrmion radius as follows: The skyrmion velocity is related to not only skyrmion radius but also skyrmion width. It is might be hard to determine skyrmion width experimentally. Contrary, it can be determined the skyrmion width thorough equation (8), knowing the skyrmion radius and velocity.

Micromagnetic simulations
The micromagnetic simulations were performed by using the MuMax3, which is numerically solved the LLG equation [43]. Here, we consider a nanowire shaped with 1000 nm length, 100 nm width, and 1.2 nm thick. The discretized cell for simulations is set to be 2.0 × 2.0 × 1.2 nm 3 , which is smaller than the exchange length (∼7.5 nm). The skyrmion has not much to do with the simulation cell size, which again manifests that the system discretization does not affect the skyrmion properties [44]. In the simulation, a magnetic skyrmion is nucleated at the center of the nanowire, as shown in figure 1(a). As an example, here the magnetic parameters are saturation magnetization, M s = 560 kA m −1 , exchange stiffness constant, A ex = 12 pJ m −1 , perpendicular magnetic anisotropy energy K = 1.1 MJ m −3 , interfacial DM interaction energy density D = 4.0 mJ m −2 [13]. To characterize the size and the shape of the skyrmion, we take normalized magnetization, n z (n z = cos Θ) profiles across the center of skyrmion. It can be rewritten the   line profile across a skyrmion and the skyrmion position along the longitudinal direction of nanowire using equation (3) as where r is the position of magnetization, c the skyrmion center position, w the width of the skyrmion, and R the radius of skyrmion. The open black circles in figure 1(b) calculated values with the fitting parameter using equation (9). The obtained R and w are 22.49 nm and 3.57 nm, as shown in figure 1(b), respectively. The material parameters in our simulations are chosen as table 1. The magnetic parameters were selected for stable skyrmion conditions. In order to investigate skyrmion motion under applied magnetic field gradient, h z g = (H final − H initial ) /L, is applied to the whole nanowire along the x-direction. Magnetic field gradient could be generated by a hybrid nanostructure of diluted magnetic semiconductor quantum well [45][46][47] or using the Oersted field at the edge [48,49]. Furthermore, using magnetic force microscopy, the radial gradient of the magnetic field [36] can be generated. These gradient fields can drive the skyrmions. We estimated the displacement in terms of the difference between the skyrmion center position at the initial time and at the final time. The initial time is defined at 10 ns after the simulation starts to remove the time period of the transient processes to steady-states [50]. The final time is determined as the moment when the skyrmion reaches the nanowire edge. The skyrmion has a constant velocity during the motion. The skyrmion velocity is obtained from the skyrmion displacement and the time required. Changes of R and w during the motion are less than 5% and 1%, respectively. The contribution of these changes to the velocities is negligible.

Magnetic field gradient-driven skyrmions in a nanowire
The skyrmion width and radius are determined by magnetic parameters such as M s , A ex , K, and D. Since equation (7) implies the skyrmion velocity proportional to the skyrmion radius, however, the skyrmion  Based on the theoretical calculation, other magnetic parameters such as the damping constant and field gradient can also affect the skyrmion motion. In order to reveal the effects of skyrmion motion with various damping constant (α) and field gradient (h z g ), we perform micromagnetic simulations. Figures 3(a)   The black open circles are micromagnetic simulation results, and the solid red line is calculated values using equation (8). The results show the inverse proportionality between v x /w and α which is consistent with equation (8). We find that the decrease in alpha from 0.5 to 0.0125 increased in v x /w from 0.07 to 1.79 GHz. The micromagnetic simulation results are in good agreement with the theoretical calculations. Whereas the small damping region, the results from micromagnetic simulations are less than the theoretical calculations. For the damping constant less than 0.02, the terminal time of skyrmion at the edge of the nanowire is incapable since there is not clear to determine the skyrmion velocities. The chiral DW has existed with smaller interfacial Dzyaloshinskii-Moriya interaction energy density cases. We also observe the chiral DW motion in comparison, see appendix A.

Numerically calculation on the skyrmion velocity driven by the magnetic-field gradient for small R/w
The analytical formula of gradient field-driven skyrmion motion is presented so far. From micromagnetic simulations and analytic formula, equation (8). We found v x /w proportional to R in the condition of R/w >> 1. The analytic formula for the magnetic bubble dynamics has been reported and Malozemoff and Slonczewski [51] show the velocity of magnetic bubbles with Bloch lines under the external magnetic field in z-direction and its gradient is in x-direction. Without dipole interaction and in the limit of R/w >> 1, they obtain the identical equation to equation (6) which approximates equation (4). We also found the velocity of skyrmions along the z-direction magnetic field gradient identical between Bloch and Neel skyrmions. However, under the x-and y-direction magnetic field gradient, skyrmion velocities depend on the skyrmion number and helicities. We recall that our approximated analytical form on the velocity attribute to the approximation for the dissipation dyadic [52]. Although the approximated dissipation dyadic deviates from the real values when R/w < 3, only small deviations are introduced in resulting velocities given by equations (4) and (6). In this session, we numerically integrate equation (4) and obtain exact skyrmion motion. Figure 5 shows the numerically and approximately calculated v x /w 2 with respect to the ratio of R/w comparing to micromagnetic simulation results. Micromagnetic simulations are performed at h z g = 2.5 mT μm −1 and α = 0.05. The calculation results cannot be normalized by velocity only since the

Summary
In summary, we have investigated the skyrmion motion forced by a magnetic field gradient. The velocity of a skyrmion is predicted analytically through the Thiele approach, which agrees well with micromagnetic simulation results for large skyrmions. The skyrmion motion is related to skyrmion shape, Gilbert damping, magnetic field gradient. Interestingly, skyrmion velocities divided by skyrmion width is proportional to the skyrmion radius, magnetic field gradient, and inverse Gilbert damping constant. The numerical calculation results are entirely in agreement with the whole skyrmion size with those of micromagnetic simulation. We now discuss magnetic field gradient driven chiral DW motions. First, the DWs are nucleated at the center of the nanowire. The initial spin configuration of the DWs is parallel to the y-direction and the relaxation process longer than 10 ns is performed. The final spin configuration of the DWs is parallel to +x-direction, which is the so-called Néel type DWs. To manipulate the chiral DWs, the magnetic field gradient, h z g as 10.0 mT μm −1 is applied. The D values coincide with the experimentally determined values [3,4,[9][10][11][12]. The snapshots of the magnetizations as a function of simulation time with M s = 560 kA m −1 , A ex = 12 pJ m −1 , K = 1.1 MJ m −3 , α = 0.1, and D = 0.1 mJ m −2 , are depicted in figure 6(a). Due to the magnetic field gradient, the chiral DWs move along the +x-direction. As shown in colored lines in figure 6(b), the DW displacements are increased and these are saturated after t = 35 ns for D is 0.1 mJ m −2 and t = 33 ns for larger than 0.5 mJ m −2 . Surprisingly, the domain wall displacements are almost identical for various D and the chiral DW velocities are approximately 14.2 m s −1 , which are much faster than the cases of the skyrmion motions stimulated by the magnetic field gradients. The reason is that the efficiency of the magnetic soliton motions is indeed affected by the geometry of the DWs. When a homogeneous magnetic field is applied to the skyrmion on the nanowire, the Zeeman energy is canceled out for one edge and an opposite edge of skyrmion. That is no driven force to skyrmion. However, the x-directional gradient magnetic field applied to the skyrmion, the Zeeman energy is non-zero for both the edge region of the skyrmion. The force by magnetic field gradient conducts skyrmion motion as shown in equation (5). Therefore, the straight-type chiral DWs have faster DW velocities. Compare with other methods (electrical currents or conventional magnetic fields) to manipulate chiral DWs [54][55][56][57], the DW tilting angle is almost negligible for the cases of the magnetic field gradient driven DW motions as shown in figure 6(a).