# Trochoidal motion and pair generation in skyrmion and antiskyrmion dynamics under spin–orbit torques

## Abstract

Magnetic skyrmions are swirling magnetic spin structures that could be used to build next-generation memory and logic devices. They can be characterized by a topological charge that describes how the spin winds around the core. The dynamics of skyrmions and antiskyrmions, which have opposite topological charges, are typically described by assuming a rigid core. However, this reduces the set of variables that describe skyrmion motion. Here we theoretically explore the dynamics of skyrmions and antiskyrmions in ultrathin ferromagnetic films and show that current-induced spin–orbit torques can lead to trochoidal motion and skyrmion–antiskyrmion pair generation, which occurs only for either the skyrmion or antiskyrmion, depending on the symmetry of the underlying Dzyaloshinskii–Moriya interaction. Such dynamics are induced by core deformations, leading to a time-dependent helicity that governs the motion of the skyrmion and antiskyrmion core. We compute the dynamical phase diagram through a combination of atomistic spin simulations, reduced-variable modelling and machine learning algorithms. It predicts how spin–orbit torques can control the type of motion and the possibility to generate skyrmion lattices by antiskyrmion seeding.

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## Acknowledgements

This work was partially supported by the Horizon2020 Framework Programme of the European Commission under grant no. 665095 (MAGicSky). J.-V.K. acknowledges support from the Deutscher Akademischer Austauschdienst under award no. 57314019. U.R. acknowledges support from the Deutsche Forschungsgemeinschaft (grant RI2891/1-1). U.R., B.D. and J.S. acknowledge the Alexander von Humboldt Foundation, the Deutsche Forschungsgemeinschaft (grant DU1489/2-1), the Graduate School of Excellence Materials Science in Mainz (MAINZ), the ERC Synergy Grant SC2 (no. 610115), the Transregional Collaborative Research Center (SFB/TRR) 173 SPIN+X and the Grant Agency of the Czech Republic (grant no. 14-37427G).

## Author information

Authors

### Contributions

B.D. and S.H. initiated the project. U.R. and S.v.M. developed the atomistic spin dynamics code and U.R. performed the atomistic spin dynamics simulations. U.R. and J.-V.K. interpreted the simulation results and developed the analytical model. S.H., B.D., J.-V.K. and U.R. wrote the manuscript. All of the authors discussed the data.

### Corresponding author

Correspondence to Ulrike Ritzmann.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

Supplementary Figures 1–11 and Supplementary Notes 1–5

### Supplementary Video 1

Atomistic spin dynamics simulation showing the linear motion of a skyrmion under spin–orbit torques of $$\hbar \beta _{{\mathrm{FL}}} = \hbar \beta _{{\mathrm{DL}}} = 0.04$$ meV. The animation shows the time evolution over 1 ns of the z-component of the magnetization in the system with periodic boundary conditions.

### Supplementary Video 2

Atomistic spin dynamics simulation showing the linear motion of an antiskyrmion under spin–orbit torques of $$\hbar \beta _{{\mathrm{FL}}} = \hbar \beta _{{\mathrm{DL}}} = 0.04$$ meV. The animation shows the time evolution over 1 ns of the z-component of the magnetization in the system with periodic boundary conditions.

### Supplementary Video 3

Atomistic spin dynamics simulation showing the deflected motion of an antiskyrmion under spin–orbit torques of $$\hbar \beta _{{\mathrm{FL}}} = \hbar \beta _{{\mathrm{DL}}} = 0.06$$ meV. The animation shows the time evolution over 1 ns of the z-component of the magnetization in the system with periodic boundary conditions.

### Supplementary Video 4

Atomistic spin dynamics simulation showing the trochoidal motion of an antiskyrmion under spin–orbit torques of $$\hbar \beta _{{\mathrm{FL}}} = \hbar \beta _{{\mathrm{DL}}} = 0.09$$ meV. The animation shows the time evolution over 1 ns of the z-component of the magnetization in the system with periodic boundary conditions.

### Supplementary Video 5

Atomistic spin dynamics simulation showing skyrmion–antiskyrmion pair generation from a single antiskyrmion seed under spin–orbit torques of $$\hbar \beta _{{\mathrm{FL}}} = 0.01$$ meV and $$\hbar \beta _{{\mathrm{DL}}} = 1.35$$ meV. The animation shows the time evolution over 0.1 ns of the topological charge density in a system with periodic boundary conditions.

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Ritzmann, U., von Malottki, S., Kim, JV. et al. Trochoidal motion and pair generation in skyrmion and antiskyrmion dynamics under spin–orbit torques. Nat Electron 1, 451–457 (2018). https://doi.org/10.1038/s41928-018-0114-0

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