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

Abstract

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.

Fig. 1

Skyrmions and antiskyrmions in a frustrated J ₁-J ₂-J ₃ ferromagnetic thin film. a Typical metastable states obtained by relaxing the magnetic thin film with a random initial spin configuration. b The spin configuration of the relaxed sample with K = H _z = 0.1. The skyrmion number Q is indicated. The arrow denotes the in-plane spin components (m _x, m _y). The color scale represents the out-of-plane spin component (m _z), which has been used throughout this paper. The model is a square element (40 × 40 spins) with the OBC. The fixed parameters (in units of J ₁ = 1) are J ₂ = −0.8 and J ₃ = −1.2. K and H _z are varied between 0 and 0.5, respectively

Fig. 2

Relaxed states of skyrmions with different values of Q and η. We construct a radial symmetric skyrmion as the initial state and check whether the initial geometry is preserved or destroyed after the relaxation. The model is a square element (40 × 40 spins) with the OBC. The parameters are J ₂ = −0.8, J ₃ = −1.2, K = 0.1, and H _z = 0.1

Fig. 3

Details of the micromagnetic energy of a relaxed (1, π/2)-skyrmion in the presence and absence of the DDI. The contribution from the DDI energy is found to be comparable to the anisotropy or Zeeman energy. The skyrmion energy is determined by the energy difference between the sample with a skyrmion and the sample without a skyrmion (i.e., with the FM state)

Fig. 4

Total micromagnetic energy E _total for a relaxed skyrmions and b relaxed antiskyrmions as functions of Q and η. The lowest energy states are found to be two degenerate Bloch-type states both for the skyrmion (Q = 1) and the antiskyrmion (Q = −1) due to the DDI. The corresponding spin configurations for relaxed skyrmions and antiskyrmions are given in Fig. 2 and Supplementary Fig. 12, respectively

Fig. 5

Helicity η as functions of the iteration for a skyrmions and b antiskyrmions with varied initial η. All of them are relaxed to the Bloch-type state unless they are initially in the Néel-type state. The model is a square element (11 × 11 spins) with the OBC. The parameters are J ₂ = −0.8, J ₃ = −1.2, K = 0.1, and H _z = 0.1

Fig. 6

Trajectories of skyrmions and antiskyrmions in a frustrated J ₁-J ₂-J ₃ ferromagnetic thin film. Trajectories of a skyrmions and b antiskyrmions with different initial η in the absence of the DDI. Trajectories induced by c, d a small driving current (j = 10 × 10¹⁰ A m⁻²) and by e, f a large driving current (j = 100 × 10¹⁰ A m⁻²) in the presence of the DDI. In the presence of the DDI, the skyrmion motion is along a straight line with the helicity fixed for the small driving current, but it is along a circle together with the helicity rotation for the large driving current. This is the helicity locking-unlocking transition. The model is a square element (100 × 100 spins) with the OBC. The parameters are J ₂ = −0.8, J ₃ = −1.2, K = 0.1, H _z = 0.1, and α = 0.1

Fig. 7

Flip of the skyrmion (antiskyrmion) helicity induced by a current pulse. a Skyrmion helicity as a function of time. A skyrmion is at rest with the helicity being η = -π/2. The helicity is flipped by a strong 80-ps-long current pulse and becomes η = π/2. Such a flip does not occur by applying a small current pulse with the same duration. b A similar flip occurs also for an antiskyrmion. c–f Snapshots of the skyrmion during the helicity flip process. g–j Snapshots of the antiskyrmion during the helicity flip process. The model is a square element (39 × 39 spins) with the OBC. The parameters are J ₂ = −0.8, J ₃ = −1.2, K = 0.1, H _z = 0.1, and α = 0.1

Fig. 8

Spontaneous formations of a bi-skyrmion and a bi-antiskyrmion. a Spontaneous formation of a bi-skyrmion by merging two skyrmions with different η. b Spontaneous formation of a bi-antiskyrmion by merging two antiskyrmions with different η. The model is a square element (18 × 9 spins) with the OBC. The parameters are J ₂ = −0.8, J ₃ = −1.2, K = 0.1, H _z = 0.1, and α = 0.1

Fig. 9

Forced separation of two skyrmions from a bi-skyrmion with Q = 2. a We first place a relaxed bi-skyrmion. b It starts to rotate counterclockwise under an injected driving current. After forming a clear peanut-like shape, it is separated into two skyrmions as in l. The model is a square element (40 × 40 spins) with the OBC. The parameters are J ₂ = −0.8, J ₃ = −1.2, K = 0.1, H _z = 0.1, α = 0.1, and j = 8 × 10¹¹ A m⁻². Q is indicated in a and l

Fig. 10

Pair annihilation of a skyrmion and an antiskyrmion. a We first place a skyrmion and an antiskyrmion. d They are spontaneously combined into a magnetic bubble (Q = 0). Then, by emitting spin waves, it disappears. The spin wave can be discerned from g–k. The positive and negative skyrmion number densities are denoted by red and blue colors, respectively. Q is indicated in a and d. The maximum value of the Q density is indicated in the lower right corner of each snapshot. The model is a square element (60 × 60 spins) with the OBC. The parameters are J ₂ = −0.8, J ₃ = −1.2, K = 0.1, H _z = 0.1, α = 0.1, and j = 8 × 10¹¹ A m⁻²