Magnetic skyrmions are magnetic quasi-particles with a number of interesting properties for possible future applications in, memory storage devices, or non-conventional computing [1-7]. We have shown that in low energetic landscapes of thin film magnetic multilayers, skyrmions show Brownian-like diffusion [7], which can be influenced by magnetic fields, temperature, and the magnetic properties of the pinning site [8,9]. While the diffusive dynamics have experimentally been studied in the continuous film [7,8,10], the device application requires a fabrication process in general, which means that skyrmions move rather in confined geometry. Here we show that the diffusive dynamics of skyrmions can be significantly modified for the geometry of the confined system and the skyrmion density. It is experimentally shown that the critical factor to unravel the underlying physics is the commensurability effect which is corroborated with Molecular dynamics simulations, as well as the pinning effect. The combination of experiment and analysis enables us to reveal the intriguing behavior that pinning sites for magnetic skyrmion depends on the magnetic field strength of the order of μT. Our findings could pave the way to control the thermal dynamics of skyrmions and the pinning effect to realize novel skyrmionic device applications. Stable skyrmions are nucleated by applying a magnetic out-of-plane (OOP) field while pulsing once an in-plane field [10]. We study the skyrmion diffusion in confined geometries varying both the symmetry of the geometry as well as the number of skyrmions from a sparse population to a fully lattice-like situation [11]. Basic geometries like triangles and circles are used. By measuring the mean squared displacement (MSD) of the trajectories of the skyrmions in the structures, the diffusion coefficient is calculated. Figure 1 shows the thermal behavior of skyrmions as a function of the skyrmion density in triangular geometry [11]. As can be seen in Fig. 1(a), The number of skyrmions can be precisely controlled by varying the OOP field to allow us to investigate the skyrmion density dependence of the skyrmion diffusion in confined geometries. Figure 1(b) presents the time-averaged configurations over 9600 Kerr images, where some configurations exhibit blur skyrmions which move with arbitrary time scale. It is shown that the diffusion qualitatively depends on the commensurability of the skyrmion number. Also, we find that the states for skyrmion numbers of 1, 3, 6, and 10 commensurate with the geometry and lead to a saturation of the MSDs as a function of time. The MSDs for an incommensurate state exhibit a non-monotonic relation and do not saturate over the experimental timescale. As shown in Fig. 1(c), molecular simulations agree qualitatively with the experimental findings that corroborating the basic dependence of skyrmion numbers on the motion in triangular confinement [11]. The quantitative difference from the simulation could be related to the pinning effect on the magnetic skyrmion. To explore the behavior closely, we investigate the magnetic field dependence of the skyrmion diffusion under a simplified environment. Figure 2 shows the OOP magnetic field dependence of histograms of skyrmion center positions in the circular geometry. For eliminating the commensurability effect, and the symmetry breaking of magnetic potential, we nucleate a single skyrmion in a circular geometry. We can see not only the modification of the depth of the pinning potential but also the pinning site transition with a small magnetic field of the order of μT, where the size of the skyrmion is modified. More intriguingly, the behavior shows hysteresis to a magnetic field, on the other hand, the reproducibility confirmed after applying a large magnetic field of 8 T suggests that the pinning sites themselves are not linked with the magnetic origin of pinned spins. Thus, the possible origin could arise from the topological spin texture itself. In conclusion, we reveal that the skyrmion diffusion is governed by the commensurability in confined geometry, as well as the magnetic field-dependent skyrmion pinning. The results show that for any potential skyrmionic device based on thin magnetic layers the numbers of skyrmions in the structured geometry as well as the geometry itself influence not only the arrangement but also the thermal diffusive motion of the skyrmions. Also, the pinning effect remains significant however, our findings indicate that magnetic skyrmions could be a new useful tool for experimentally exploring the pinning effect, which is a critical factor for any magnetic objects, and the possibility that the pinning sites are able to be tailored even after film deposition and fabrication process.

**References:**

[1] A. Fert et al., Nat. Nanotechnol.8, 152 (2013). [2] X. Zhang et al., Sci. Rep. 5, 7643 (2015). [3] X. Zhang et al., Sci. Rep. 5, 9400 (2015). [4] F. Garcia-Sanchez et al., New J. Phys. 18, 075011 (2016). [5] S. Zhang et al., New J. Phys. 17, 023061 (2015). [6] D. Prychynenko et al., Phys. Rev. Appl. 9, 014034 (2018). [7] J. Zázvorka et al., Nat. Nanotechnol. 14, 658 (2019). [8] T. Nozaki et al., Appl. Phys. Lett. 114, 012402 (2019). [9] N. Kerber et al., preprint arXiv:2004.07976 (2020). [10] J. Zázvorka et al., Adv. Funct. Mater. 30, 2004037 (2020). [11] C. Song et al., preprint arXiv:2009.03995 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3767ae61748fcafeb8f7fd6ebb2d7005.jpg)
Fig. 1 Thermal behavior of skyrmions as a function of the skyrmion density in triangular geometry. (a) The initial state of the skyrmion configurations. (b) The time-averaged configurations over 9600 Kerr images. (c) Simulations of the histograms of skyrmion center positions for configurations of 1 to 10 skyrmions inside the triangle containing positions from 1 million independent frames.

![](https://assets.underline.io/uploads/markdown_image/1/image/4f7ce9fe1971db64496cd0232ff4f441.jpg)
Fig. 2 The OOP magnetic field dependence of histograms of skyrmion center positions at 341.5 K in the circular geometry, where we plot and integrate the position where skyrmion exists at each frame in a 60 minutes video (57600 frames) on a two-dimensional map. A brighter contrast means that the skyrmion dwell time is longer at these points corresponding to an energy minimum.

 Skyrmion diffusion in a confined system

*Session Chair: Chun-Yeol You, Daegu Gyeongbuk Institute of Science and Technology*

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