Interaction of Individual Skyrmions in a Nanostructured Cubic Chiral Magnet

Haifeng Du, Xuebing Zhao, Filipp N. Rybakov, Aleksandr B. Borisov, Shasha Wang, Jin Tang, Chiming Jin, Chao Wang, Wensheng Wei, Nikolai S. Kiselev, Yuheng Zhang, Renchao Che, Stefan Blügel, and Mingliang Tian

Phys. Rev. Lett. 120, 197203 – Published 11 May 2018

Abstract

We report direct evidence of the field-dependent character of the interaction between individual magnetic skyrmions as well as between skyrmions and edges in $B 20$ -type FeGe nanostripes observed by means of high-resolution Lorentz transmission electron microscopy. It is shown that above certain critical values of an external magnetic field the character of such long-range skyrmion interactions changes from attraction to repulsion. Experimentally measured equilibrium inter-skyrmion and skyrmion-edge distances as a function of the applied magnetic field shows quantitative agreement with the results of micromagnetic simulations. The important role of demagnetizing fields and the internal symmetry of three-dimensional magnetic skyrmions are discussed in detail.

Received 31 October 2017
Revised 9 March 2018

DOI:https://doi.org/10.1103/PhysRevLett.120.197203

Physics Subject Headings (PhySH)

Dipolar interaction Exchange interaction Ferromagnetism Magnetic phase transitions Micromagnetism Spin relaxation

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Haifeng Du^1,2, Xuebing Zhao³, Filipp N. Rybakov⁴, Aleksandr B. Borisov⁵, Shasha Wang¹, Jin Tang¹, Chiming Jin¹, Chao Wang³, Wensheng Wei¹, Nikolai S. Kiselev^6,*, Yuheng Zhang^1,7,8, Renchao Che^3,†, Stefan Blügel⁶, and Mingliang Tian^1,2,8,‡

¹The Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory, Chinese Academy of Sciences and University of Science and Technology of China, Hefei 230026, China
²Department of Physics, School of Physics and Materials Science, Anhui University, Hefei 230601, China
³Laboratory of Advanced Materials, Department of Materials Science, Collaborative Innovation Center of Chemistry for Energy Materials, Fudan University, Shanghai 200438, China
⁴Department of Physics, KTH-Royal Institute of Technology, Stockholm, SE-10691 Sweden
⁵M.N. Miheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, Ekaterinburg 620990, Russia
⁶Peter Grünberg Institute and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
⁷Department of Physics, University of Science and Technology of China, Hefei 230031, China
⁸Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

^*n.kiselev@fz-juelich.de
^†rcche@fudan.edu.cn
^‡tianml@hmfl.ac.cn

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Issue

Vol. 120, Iss. 19 — 11 May 2018

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Images

Figure 1
Lorentz TEM images showing the evolution of skyrmion arrangement in a wide, $\sim 500 nm$ , FeGe nanostripe with a thickness of $\sim 120 nm$ with an increasing magnetic field applied normally to the plate. (a) A skyrmion chain and skyrmion cluster with long-range order formed near the edges of the sample at a low field. (b) A skyrmion cluster migrating to the central part of the sample with an increasing field. A long-range order in skyrmion arrangement and the distance between skyrmions is conserved. (c) A disordered skyrmion cluster in the middle part of the nanostripe at a high magnetic field. The distance between skyrmions is not conserved. All Lorentz-TEM images are taken at $T = 100 K$ in underfocus conditions with a focused value of $\sim 300 μ m$ .
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Figure 2
Field-driven evolution of interacting pairs of skyrmions in a nearly rectangular FeGe plate [the boundaries are marked in (a) with white dotted lines] with a width, length, and thickness of 430, 1590, and 120 nm, respectively. Color represents the direction of the in-plane component of magnetization; see the inset in (l). Attractive skyrmion-skyrmion and skyrmion-edge interactions are observed at a low magnetic field, while at a high magnetic field the character of the interactions changes to repulsive. Increasing (a)–(f) and decreasing (g)–(l) field magnetization processes illustrate the reversibility of the states with a very weak hysteresis effect. The lower-left isolated skyrmion and corresponding skyrmion pair are chosen to estimate the inter-skyrmion and skyrmion-edge distances, respectively, with the data presented in Figs. 3 and 3.
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Figure 3
(a) and (b) represent the isosurfaces $θ = 90 °$ and $θ = 5 °$ for a typical segment of a single SkT and a coupled pair of SkTs, respectively, while for a finite thickness film the corresponding SkTs are characterized by distortion near the ends of the tube due to the chiral surface twist [20]. The standard color code indicates the direction of the magnetic moments on a unit sphere, where black and white correspond to $m_{z} = - 1$ ( $θ = 180 °$ ) and $m_{z} = + 1$ ( $θ = 0 °$ ), respectively. (c) and (d) represent a cross section of a SkT and a pair of SkTs shown in (a) and (b) with $z = const$ plane. Closed curves indicate isolines for the $θ$ angle. (e) Skyrmion-edge distance as a function of the field. (f) Skyrmion-skyrmion distance. Open and solid (black) circles in (c) and (d) correspond to an increasing and a decreasing field, respectively. Red and blue circles represent theoretical dependencies. (g) The theoretical dependencies for $d_{se}$ and $d_{ss}$ calculated in the assumption of a demagnetizing field inside the sample, $B_{d} = 0$ .
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