Skyrmion dynamics in a frustrated ferromagnetic film and current-induced helicity locking-unlocking transition

The helicity-orbital coupling is an intriguing feature of magnetic skyrmions in frustrated magnets. Here we explore the skyrmion dynamics in a frustrated magnet based on the J 1-J 2-J 3 classical Heisenberg model explicitly by including the dipole-dipole interaction. The skyrmion energy acquires a helicity dependence due to the dipole-dipole interaction, resulting in the current-induced translational motion with a fixed helicity. The lowest-energy states are the degenerate Bloch-type states, which can be used for building the binary memory. By increasing the driving current, the helicity locking-unlocking transition occurs, where the translational motion changes to the rotational motion. Furthermore, we demonstrate that two skyrmions can spontaneously form a bound state. The separation of the bound state forced by a driving current is also studied. In addition, we show the annihilation of a pair of skyrmion and antiskyrmion. Our results reveal the distinctive frustrated skyrmions may enable viable applications in topological magnetism.

Supplementary Figure 11. Relaxed spin configurations of a magnetic thin film (40 × 40 spins) with the OBC at (a) a certain in-plane applied field H y and (b) a certain out-of-plane applied field H z .
The out-of-plane spin component is color-coded: blue is into the plane, red is out of the plane, white is in-plane. The fixed parameters (in units of J 1 = 1) are J 2 = −0.8 and J 3 = −1.2, and K = 0. The simulation is carried out by the OOMMF CG minimizer. The edge spin configuration differs from the bulk spin configuration at certain applied fields due to the effects of OBC and DDI. We construct a radial symmetric antiskyrmion as the initial state. We check whether the initial geometry is preserved or broken after the relaxation. For the initial states of antiskyrmions with Q = −1, −2, −3, the numbers (Q, η) do not change after the relaxation although the radius shrinks. However, the antiskyrmion with Q = −4 is unstable and split into two antiskyrmions with Q = −2. The helicity η changes during this splitting process from that of the initial state with η = ±π/2. Micromagnetic energy difference between the system with and without a skyrmion as a function of the saturation magnetization M S of the magnetic thin film, of which the length, width, and thickness are fixed at 40a, 40a, and a, respectively. (b) Micromagnetic energy difference between the system with and without a skyrmion as a function of the length-to-width ratio of the magnetic thin film, of which the width and thickness are fixed at 40a and a, respectively. (c) Micromagnetic energy difference between the system with and without a skyrmion as a function of the thickness of the magnetic thin film, of which the length and width are fixed at 40a. (d) Energy to total skyrmion energy ratio as a function of the thickness of the magnetic thin film, of which the length and width are fixed at 40a. In all simulations, the fixed parameters (in units of J 1 = 1) are J 2 = −0.8, J 3 = −1.2, K = 0.1, and H z = 0.1. For the system with the skyrmion, a ground-state skyrmion with Q = 1 and η = π/2 is placed at the film center and the given system is relaxed. For the system without the skyrmion, the relaxed state of the film is a ferromagnetic state with spins pointing along the +z direction. The energy difference between the systems with and without a skyrmion can be seen as the energy of the skyrmion. The black, red, blue, green, purple, yellow, and turquoise curves denote the differences of the total energy, NN exchange energy, NNN exchange energy, NNNN exchange energy, anisotropy energy, Zeeman energy, and DDI energy, respectively. It shows that the skyrmion energy does not change with the length-to-width ratio. On the other hand, the NN exchange energy of a skyrmion increases with increasing thickness, while the NNN exchange, NNNN exchange, and DDI energies of a skyrmion decrease with increasing thickness. The Zeeman, anisotropy, and total energies of a skyrmion only slightly increases with increasing thickness. Besides, it can be seen the skyrmion energy varies with increasing M S , especially when M S is larger than a certain value. The skyrmion has an initial helicity number of η = π/2. The driving current density ranges from j = 70 × 10 10 A m −2 to j = 100 × 10 10 A m −2 . (a) At small driving current densities (e.g., j = 70 × 10 10 A m −2 ), the skyrmion moves toward the right. (c) When the driving current density is increased to a larger value (e.g., j = 84 × 10 10 A m −2 ), the helicity unlocking event occurs once, and (d) the skyrmion moves toward the left after the flip of the helicity. (f) When the driving current density is further increased to a more larger value (e.g., j = 94 × 10 10 A m −2 ), the helicity is totally unlocked, and the skyrmion moves in an orbital circle. The model is a square element with 100 × 100 spins. The parameters (in units of The simulation is carried out by the OOMMF solver integrating the LLG equation including the spin torque. The unit of the current density is 10 10 A m −2 . The antiskyrmion has an initial helicity number of η = π/2. The driving current density ranges from j = 10 × 10 10 A m −2 to j = 94 × 10 10 A m −2 . (a) At small driving current densities (e.g., j = 10 × 10 10 A m −2 ), the antiskyrmion moves toward the right. (b) When the driving current density is increased to a larger value (e.g., j = 20 × 10 10 A m −2 ), the helicity unlocking event occurs once, and the antiskyrmion moves toward the left after the flip of the helicity. (c)-(f) When the driving current density is further increased to a more larger value (e.g., j = 73 × 10 10 A m −2 ), the helicity is totally unlocked, and the antiskyrmion moves in an orbital circle. The model is a square element with 100 × 100 spins. The parameters (in units of The simulation is carried out by the OOMMF solver integrating the LLG equation including the spin torque. The unit of the current density is 10 10 A m −2 . x , y ( n m ) t ( p s ) x , y ( n m ) t ( p s ) In the presence of DDI and thermal effect, the skyrmion driven by a small current with locked helicity moves toward a certain direction (depending on η), however, the trajectory is fluctuated and the skyrmion shows Brownian motion behavior. Besides, the skyrmion driven by a large current with unlocked helicity shows rotational motion associated with Brownian motion behavior. When the DDI is removed, the trajectory of the skyrmion at finite temperature is rather fluctuated, which also shows a mixed dynamics of rotational motion and Brownian motion.  Figure 19. Initial spin configuration (m z component) for the study of the skyrmionskyrmion, antiskyrmion-antiskyrmion, and skyrmion-antiskyrmion interactions. The model is a magnetic rectangular element with 18 × 9 spins with the OBC. We construct two skyrmions (antiskyrmions) with Q = ±1 and different η placed at the left and right sides of the sample, respectively, with three types of initial spin configuration. The initial spin configuration of profile A assumes that the two skyrmions have no overlap at the initial state, which is used to study the natural interaction between the two skyrmions. The initial spin configuration of profile B assumes that the two skyrmions have a moderate overlap at the initial state, which is used to study the interaction between the two skyrmions when they are colliding each other driving by moderate external driving forces. The initial spin configuration of profile C assumes that the two skyrmions have a strong overlap at the initial state, which is used to study the interaction between the two skyrmions when they are colliding each other driving by strong external driving forces.
3π/ 2 3π/ 2 3π/ 2 π π π 3π/ 2 π π/ 2 π/ 2 π/ 2 π/ 2 3π/ 2 π π/ 2 0 0  Supplementary Fig. 19). When the initial spin configuration of profile A is employed, the skyrmion with η = π/2 and the skyrmion with η = 3π/2 merge into one skyrmion with Q = 2, indicating the natural attractions between these two skyrmions. When the initial spin configuration of profile B is employed, the two skyrmions with unequal η merge into a skyrmion with Q = 2, while the two skyrmions with identical η cannot merge into one skyrmion with |Q| > 1, indicating the natural repulsions between the two skyrmions with identical η. When the initial spin configuration of profile C is employed, the skyrmion with η = 0 and the skyrmion with η = π merge into one skyrmion with Q = 2. The skyrmion with η = π/2 and the skyrmion with η = 3π/2 merge into one skyrmion with Q = 2. However, for other cases, the relaxed state is only one skyrmion with Q = 1. This indicates that a strong collision between two skyrmions may lead to the annihilation of one skyrmion, unless one has η = 0 and the other has η = π or one has η = π/2 and the other has η = 3π/2. The model is a magnetic rectangular element with 18 × 9 spins. The parameters (in units of J 1 = 1) are J 2 = −0.8, J 3 = −1.2, K = 0.1, and H z = 0.1. The simulation is carried out by the OOMMF CG minimizer.
3π/ 2 3π/ 2 3π/ 2 π π π 3π/ 2 π π/ 2 π/ 2 π/ 2 π/ 2 3π/ 2 π π/ 2 0 0  Supplementary Fig. 20. Two antiskyrmions and different η are placed at the left and right sides of the magnetic rectangular element with three types of initial spin configuration (cf. Supplementary Fig. 19). When the initial spin configuration of profile A is employed, the antiskyrmion with η = 0 and the antiskyrmion with η = π merge into one antiskyrmion with Q = −2, indicating the natural attractions between these two antiskyrmions. When the initial spin configuration of profile B is employed, the two antiskyrmions with unequal η merge into an antiskyrmion with Q = −2, while the two antiskyrmions with identical η cannot merge into one antiskyrmion with |Q| > 1, indicating the natural repulsions between the two antiskyrmions with identical η. When the initial spin configuration of profile C is employed, the antiskyrmion with η = 0 and the antiskyrmion with η = π merge into one antiskyrmion with Q = −2. The antiskyrmion with η = π/2 and the antiskyrmion with η = 3π/2 merge into one antiskyrmion with Q = −2. However, for other cases, the relaxed state is only one antiskyrmion with Q = −1. This indicates that a strong collision between two antiskyrmions may lead to the annihilation of one antiskyrmion, unless one has η = 0 and the other has η = π, or one has η = π/2 and the other has η = 3π/2. 3 π/ 2 π π π π/ 2 π/ 2 π/ 2 0 0 0 3 π/ 2 3 π/ 2 3 π/ 2 3 π/ 2 π π π π π/ 2 π/ 2 π/ 2 π/ 2 3 π/ 2 0 π π/ 2 0 0 0 with different η are placed, respectively, at the left and right sides of the magnetic rectangular element with three types of initial spin configuration (cf. Supplementary Fig. 19). When the initial spin configuration of profile A is employed, there is no attraction between the skyrmion and the antiskyrmion. The η of the relaxed skyrmion and the antiskyrmion will be altered to the case with lower energy, unless the skyrmion and the antiskyrmion: i. have identical η = π/2 or η = 3π/2; ii. one has η = 0 and the other has η = π; iii. one has η = π/2 and the other has η = 3π/2. When the initial spin configuration of profile B is employed, the skyrmion and the antiskyrmion will process a pair annihilation unless they have identical η. The skyrmion and the antiskyrmion have identical η will repel each other. Similarly, when the initial spin configuration of profile C is employed, the skyrmion and the antiskyrmion will process a pair annihilation unless they have identical η. If the skyrmion and the antiskyrmion have identical η = 0 or η = π, the skyrmion will be annihilated, while if the skyrmion and the antiskyrmion have identical η = π/2 or η = 3π/2, the antiskyrmion will be annihilated. The results indicate that a moderate or strong collision between a skyrmion and an antiskyrmion will lead to the pair annihilation of the skyrmion and the antiskyrmion, unless they have identical η where the annihilation of only the skyrmion or only the antiskyrmion will be processed under a strong collision. . We first place a relaxed bi-antiskyrmion at the center of the sample. Then, a driving current is vertically injected to the sample, which can be realized by the spin Hall effect in a heavy-metal substrate. It is found that the bi-antiskyrmion starts to rotate counterclockwise, and forms a clear peanut-like shape, as shown in (d). When it has rotated almost 90 degrees, two antiskyrmions are almost generated, as shown in (h). As shown in (l), the bi-antiskyrmion is successfully split into two isolated antiskyrmions. This forced separation is possible since the bi-antiskyrmion is composed of two antiskyrmions with opposite helicities, which rotate in opposite directions. the Hamiltonian is rewritten as + n 2 π (4 + 3n 2 ) 2r 4 sin 2 θ (r) + n 2 π sin 2θ (r) 2r 2 (7θ (r) + θ (r)) + n 2 π cos 2θ (r) 2r 2 θ (r) 2 −J 2 π 4 θ (r) 2 + θ (r) 4 + n 2 + 1 r 2 θ (r) 2 − π 2r θ (r) θ (r) + n 2 (4 + n 2 ) 4r 4 sin 2 θ (r) + n 2 π sin 2θ (r) 4r 2 (5θ (r) + θ (r)) + 3n 2 π cos 2θ (r) 4r 2 θ (r) 2 .
This is independent of η, which implies that the energy is independent of the helicity. Furthermore, the Hamiltonian is invariant for n −→ −n, which implies the energy between the skyrmion and the antiskyrmion is identical irrespective to the helicity η.

Supplementary Note 2 | Thiele equation for the collective coordinate
The collective coordinate ξ i of a skyrmion is well described by the Thiele equation [1] G ijξi + F j − Γ ijξi = uΥ j , with where F i represents the force with U being the associated potential, Γ ij the damping tensor, G ij the gyromagnetic tensor, and Υ i the current induced torque. Note that u, defined by Equation (4), is proportional to the applied current density j.
The helicity dependence of the velocity is given by this set of equations, which accounts for Fig. 6c of the main text. On the other hand, the helicity-dependent skyrmion Hall angle is given as Φ SkHE = tan −1 ∓ 4π cos η + Dα sin η 4π sin η − Dα cos η .
The skyrmion Hall angle is Φ SkHE = 172.6 • in Fig. 6c of the main text for the stable Bloch skyrmion (e.g., η = π/2), from which we obtain D = 16.34. On the other hand, we find D = 15.58 by the numerical integration of the skyrmion profile based on Supplementary Equations (10) and (14). These values coincide very well.
The condition of the transition is that the sign of dV /dη is always the same, which determines the critical value of u as uΥ η = 2U 0 / , yielding Equation (8) with the use of Equation (4).