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Magnetic skyrmions are magnetic quasi-particles with enhanced stability and different manipulation mechanisms using external fields and currents making them promising candidates for future applications for instance in neuromorphic computing. Recently, several measurements and simulations have shown that thermally activated skyrmions in confined geometries, as they are necessary for device applications, arrange themselves predominantly based on commensurability effects. In this simulational study, based on the Thiele model, we investigate the enhanced dynamics and degenerate non-equilibrium steady state of a system in which the intrinsic skyrmion-skyrmion and skyrmion-boundary interaction compete with thermal fluctuations as well as current-induced spin-orbit torques. The investigated system is a triangular-shaped confinement geometry hosting four skyrmions, where we inject spin-polarized currents between two corners of the structure. We coarse-grain the skyrmion states in the system to analyze the intricacies of skyrmion arrangements of the skyrmion ensemble. In the context of neuromorphic computing, such methods address the key challenge of optimizing read-out positions in confined geometries and form the basis to understand collective skyrmion dynamics in systems with competing interactions on different scales.

One of the most important magnetic spin structure is the topologically stabilised skyrmion quasi-particle. Its interesting physical properties make them candidates for memory and efficient neuromorphic computation schemes. For the device operation, detection of the position, shape, and size of skyrmions is required and magnetic imaging is typically employed. A frequently used technique is magneto-optical Kerr microscopy where depending on the samples material composition, temperature, material growing procedures, etc., the measurements suffer from noise, low-contrast, intensity gradients, or other optical artifacts. Conventional image analysis packages require manual treatment, and a more automatic solution is required. We report a convolutional neural network specifically designed for segmentation problems to detect the position and shape of skyrmions in our measurements. The network is tuned using selected techniques to optimize predictions and in particular the number of detected classes is found to govern the performance. The results of this study shows that a well-trained network is a viable method of automating data pre-processing in magnetic microscopy. The approach is easily extendable to other spin structures and other magnetic imaging methods.

Magnetic Bloch points (BPs) are highly confined magnetization configurations,that often occur in transient spin dynamics processes. However, opposingchiralities of adjacent layers for instance in a FeGe bilayer stack canstabilize such magnetic BPs at the layer interface. These BPs configurationsare metastable and consist of two coupled vortices (one in each layer) withsame circularity and opposite polarity. Each vortex is stabilized by oppositesign Dzyaloshinskii-Moriya interactions. This stabilization mechanismpotentially opens the door towards BP-based spintronic applications. An openquestion, from a methodological point of view, is whether the Heisenberg (HB)model approach (atomistic model) as to be used to study such systems or if the-- computationally more efficient -- micromagnetic (MM) models can be used andstill obtain robust results. We are modelling and comparing the energetics anddynamics of a stable BP obtained using both HB and MM approaches. We find thatan MM description of a stable BP leads qualitatively to the same results as theHB description, and that an appropriate mesh discretization plays a moreimportant role than the chosen model. Further, we study the dynamics byshifting the BP with an applied in-plane field and investigating the relaxationafter switching the filed off abruptly. The precessional motion of coupledvortices in a BP state can be drastically reduced compared to a classicalvortex, which may be also an interesting feature for fast and efficientdevices. A recent study has shown that a bilayer stack hosting BPs can be usedto retain information [1].

Computational modeling and numerical simulations have become indispensable tools in science, technology, engineering and mathematics (STEM), and in industrial research and development. Consequently, there is growing demand for computational skills in undergraduate and postgraduate students and a need to revise existing curricula. We report from a case study where an existing materials science module was modified to contain numerical simulation of the materials: students use simulation software to explore the material's behavior in simulated experiments. In particular, the Ubermag micromagnetic simulation software package is used by the students in order to solve problems computationally that are linked to current research in the field of magnetism. The simulation software is controlled through Python code in Jupyter notebooks. This setup provides a computational environment in which simulations can be started and analyzed in the same notebook. A key learning activity is a project in which students tackle a given task over a period of approximately 2 months in a small group. Our experience is that the self-paced problem-solving nature of the project work -- combined with the freedom to explore the system's behavior through the simulation -- can facilitate a better in-depth exploration of the course contents. We report feedback from students and educators both on the training in material science and the Jupyter notebook as a computational environment for education. Finally, we discuss which aspects of the Ubermag and the Jupyter notebook have been beneficial for the students' learning experience and which could be transferred to similar teaching activities in other subject areas.

Magnetic skyrmions are topologically stabilized quasi-particles and are promising candidates for energy-efficient applications, such as storage but also logic and sensing. Here we present a new concept for a multi-turn sensor-counter device based on skyrmions, where the number of sensed rotations is encoded in the number of nucleated skyrmions. The skyrmion-boundary force in the confined geometry of the device in combination with the topology-dependent dynamics leads to the effect of automotion for certain geometries. For our case, we describe and investigate this effect with micromagnetic simulations and the coarse-grained Thiele equation in a triangular geometry with an attached reservoir as part of the sensor-counter device.

Periodically patterned metamaterials are known for exhibiting wave properties similar to the ones observed in electronic band structures in crystal lattices. In particular, periodic ferromagnetic (FM) materials, also known as magnonic crystals (MCs), are characterized by the presence of bands and bandgaps (BGs) at tunable frequencies in their spin-wave (SW) spectrum. While those frequencies typically cover the GHz-range, no fundamental reason prevents one from extending this range towards THz-frequencies, a regime of high importance in communication technologies. Recently, the fabrication of magnets hosting Dzyaloshinskii-Moriya interactions (DMIs) has been pursued with high interest since properties such as the stabilization of chiral spin textures and nonreciprocal SW propagation originate from this antisymmetric exchange interaction. In this context, to further engineer the band structure of MCs, we propose the implementation of MCs with periodic DMIs, which can be obtained, for instance, patterning periodic arrays of heavy metals (HMs) on top of an ultrathin FM film. We demonstrate through theoretical calculations and micromagnetic simulations that such systems exhibit a unique evolution of the standing SWs around the BGs in areas of the FM film that are in contact with the HM wires. We also predict the emergence of at SW bands and indirect magnonic gaps, and we show that these effects depend on the strength of the DMI. This study opens further pathways towards engineered metamaterials for SW-based devices.

A broken chiral symmetry in a magnetic system manifests itself as the appearance of either periodical (e.g. helical or cycloid modulations [1-3]) or localized magnetization structures (e.g. chiral domain walls and skyrmions [3-6]). The origin of these magnetic textures is spin-orbit Dzyaloshinskii-Moriya interaction (DMI), which is observed in bulk magnetic crystals with low symmetry [7-8] or at interfaces between a ferromagnet and a nonmagnetic material with strong spin-orbit coupling [9]. This DMI is intrinsic to the crystal or layer stack. Recently, it was reported that geometrically-broken symmetry in curvilinear magnetic systems leads to the appearance of curvature-driven DMI-like chiral contribution in the energy functional [10-14]. This chiral term is determined by the sample geometry, e.g local curvature and torsion, and is therefore extrinsic to the crystal or layer stack. It reveals itself in the domain wall pinning at a localized wire bend [15] and is responsible for the existence of magnetochiral effects in curvilinear magnetic systems, e.g. in ferromagnetic Möbius rings [16], nanorings [11] and helix wires [12, 13, 17]. The intrinsic and extrinsic DMI act at different length scales and, hence, their combination can be reffered to as a mesoscale DMI. The symmetry and strength of this term are determined by the geometrical and material properties of a three-dimensional (3D) object. Although, intrinsic and extrinsic terms separately are broadly investigated, their synergistic impact is not known yet. Here, we study the properties of the mesoscale DMI in a 1D curvilinear wire and in 2D curvilinear shells. We derive the general expressions for the mesoscale DMI term and analyse the magnetization states which arise in a helix wire and in a thin spherical shell with intrinsic DMI. The clear cut comparison a helix wire with a straight wire with homogeneous tangential intrinsic DMI reveals: (i) The magnetic states of a curved wire is governed by a single vector of magnetochirality — a vector of the mesoscale DMI — originating from the vector sum of the intrinsic and extrinsic DMI vectors; (ii) The symmetry and period of the chiral structures are determined by the strength and direction of the vector of the mesoscale DMI, which depends on both material and geometrical parameters of a curvilinear wire (Figure, panel a); (iii) Similarly to the case of the straight wire [18] both types of phase transitions (of the first and the second order) are found in the helix. The appearance of each state can be determined by measuring of the average values of the magnetization components and/or by establishing space Fourier spectra of the coordinate-dependent magnetic signals from the helices. In the case of 2D curvilinear magnetic shells, it’s shown the existence of a skyrmion solution on a thin magnetic spherical shell even without any additional intrinsic DMI [19]. Such skyrmions can be stabilized by curvature effects only, namely by the curvature-induced, extrinsic DMI (Figure, panel b). In addition to the striking difference to the case of a planar skyrmion, magnetic skyrmions on a spherical shell are topologically trivial. This is due to a shift of the topological index of the magnetization field caused by topology of the surface itself. As a result, a skyrmion on a spherical shell can be induced by a uniform external magnetic field. Further, the curvature stabilized skyrmions are very small, with a lateral extension of several nm only (Figure, panel b). The size of the skyrmion core can be tailored e.g. by an additional intrinsic DMI (Figure, panel c). One can note here, that the curvature stabilized skyrmions are always of Neel type (at least, for a surface of rotation). Due to their small sizes and ease in manipulating using homogeneous magnetic fields, we envision those topological objects to be relevant for the realization of on-demand tunable topological logic. Indeed, topological Hall effect can be digitally switched on or off by exposing a sample withferromagnetic spherical shells submerged by a nonmagnetic conductor. References: [1] I. E. Dzyaloshinskii, “Theory of helicoidal structures in antiferromagnets. i. nonmetals,” Sov. Phys. JETP, vol. 19, pp. 964–971 (1964). [2] A. Bogdanov, U. Rössler, and C. Pfleiderer, “Modulated and localized structures in cubic helimagnets,” Physica B: Condensed Matter, vol. 359-361, pp. 1162–1164 (2005). [3] S. Rohart and A. Thiaville, “Skyrmion confinement in ultrathin film nanostructures in the presence of dzyaloshinskii-moriya interaction,” Physical Review B, vol. 88, p. 184422 (2013). [4] A. Thiaville, S. Rohart, É. Jué, V. Cros, and A. Fert, “Dynamics of Dzyaloshinskii domain walls in ultrathin magnetic films,” EPL (Europhysics Letters), vol. 100, p. 57002 (2012). [5] A. Bogdanov and A. Hubert, “The stability of vortex-like structures in uniaxial ferromagnets,” Journal of Magnetism and Magnetic Materials, vol. 195, pp. 182–192 (1999). [6] N. Nagaosa and Y. Tokura, “Topological properties and dynamics of magnetic skyrmions,” Nature Nanotechnology, vol. 8, pp. 899–911 (2013). [7] I. E. Dzialoshinskii, “Thermodynamic theory of “weak” ferromagnetism in antiferromagnetic substances,” Sov. Phys. JETP, vol. 5, pp. 1259–1272 (1957). [8] T. Moriya, “New mechanism of anisotropic superexchange interaction,” Phys. Rev. Lett., vol. 4, pp. 228–230 (1960). [9] A. Fert, “Magnetic and transport properties of metallic multilayers,” Materials Science Forum, vol. 59-60, pp. 439–480 (1990). [10] Y. Gaididei, V. P. Kravchuk, and D. D. Sheka, “Curvature effects in thin magnetic shells,” Phys. Rev. Lett., vol. 112, p. 257203 (2014). [11] D. D. Sheka, V. P. Kravchuk, and Y. Gaididei, “Curvature effects in statics and dynamics of low dimensional magnets,” Journal of Physics A: Mathematical and Theoretical, vol. 48, p. 125202 (2015). [12] D. D. Sheka, V. P. Kravchuk, K. V. Yershov, and Y. Gaididei, “Torsion-induced effects in magnetic nanowires,” Phys. Rev. B, vol. 92, p. 054417 (2015). [13] O. V. Pylypovskyi, D. D. Sheka, V. P. Kravchuk, K. V. Yershov, D. Makarov, and Y. Gaididei, “Rashba torque driven domain wall motion in magnetic helices,” Scientific Reports, vol. 6, p. 23316 (2016). [14] R. Streubel, P. Fischer, F. Kronast, V. P. Kravchuk, D. D. Sheka, Y. Gaididei, O. G. Schmidt, and D. Makarov, “Magnetism in curved geometries (topical review),” Journal of Physics D: Applied Physics, vol. 49, p. 363001 (2016). [15] K. V. Yershov, V. P. Kravchuk, D. D. Sheka, and Y. Gaididei, “Curvature-induced domain wall pinning,” Phys. Rev. B, vol. 92, p. 104412 (2015). [16] O. V. Pylypovskyi, V. P. Kravchuk, D. D. Sheka, D. Makarov, O. G. Schmidt, and Y. Gaididei, “Coupling of chiralities in spin and physical spaces: The Möbius ring as a case study,” Phys. Rev. Lett., vol. 114, p. 197204 (2015). [17] K. V. Yershov, V. P. Kravchuk, D. D. Sheka, and Y. Gaididei, “Curvature and torsion effects in spin-current driven domain wall motion,” Phys. Rev. B, vol. 93, p. 094418 (2016). [18] M. Heide, G. Bihlmayer, and S. Blügel, “Non-planar dzyaloshinskii spirals and magnetic domain walls in noncentrosymmetric systems with orthorhombic anisotropy,” Journal of Nanoscience and Nanotechnology, vol. 11, pp. 3005–3015 (2011). [19] V. P. Kravchuk, U. K. Rößler, O. M. Volkov, D. D. Sheka, J. van den Brink, D. Makarov, H. Fuchs, H. Fangohr, and Y. Gaididei, “Topologically stable magnetization states on a spherical shell: Curvature-stabilized skyrmions,” Phys. Rev. B, vol. 94, p. 144402 (2016).

A broken chiral symmetry in a magnetic system manifests itself as the appearance of either periodical (e.g. helical or cycloid modulations [1]) or localized magnetization structures (e.g. chiral domain walls and skyrmions [2]). The origin of these magnetic textures is spin-orbit Dzyaloshinskii-Moriya interaction (DMI), which is observed in bulk magnetic crystals with low symmetry [3-4] or at interfaces between a ferromagnet and a nonmagnetic material with strong spin-orbit coupling [5]. This DMI is intrinsic to the crystal or layer stack. Recently, it was reported that geometrically-broken symmetry in curvilinear magnetic systems leads to the appearance of curvature-driven DMI-like chiral contribution in the energy functional [6]. This chiral term is determined by the sample geometry, e.g local curvature and torsion, and is therefore extrinsic to the crystal or layer stack. It reveals itself in the domain wall pinning at a localized wire bend and is responsible for the existence of magnetochiral effects in curvilinear magnetic systems, e.g. in ferromagnetic Möbius rings, nanorings and helix wires [7]. The intrinsic and extrinsic DMI act at different length scales and, hence, their combination can be referred to as a mesoscale DMI. The symmetry and strength of this term are determined by the geometrical and material properties of the three-dimensional (3D) object. Although, intrinsic and extrinsic terms separately are broadly investigated, their synergistic impact is not known yet. Here, we study the properties of the mesoscale DMI in a 1D curvilinear wire and in 2D curvilinear shells. We derive the general expressions for the mesoscale DMI term and analyze the magnetization states which arise in a helix wire and in a thin spherical shell with intrinsic DMI. [1] I. E. Dzyaloshinskii, Sov. Phys. JETP, vol. 19, pp. 964–971 (1964). [2] N. Nagaosa and Y. Tokura, Nature Nanotechnology, vol. 8, pp. 899–911 (2013). [3] I. E. Dzialoshinskii, Sov. Phys. JETP, vol. 5, pp. 1259–1272 (1957). [4] T. Moriya, Phys. Rev. Lett., vol. 4, pp. 228–230 (1960). [5] A. Fert, Materials Science Forum, vol. 59-60, pp. 439–480 (1990). [6] Y. Gaididei, V. P. Kravchuk, and D. D. Sheka, Phys. Rev. Lett., vol. 112, p. 257203 (2014). [7] R. Streubel, P. Fischer, F. Kronast, V. P. Kravchuk, D. D. Sheka, Y. Gaididei, O. G. Schmidt, and D. Makarov, Journal of Physics D: Applied Physics, vol. 49, p. 363001 (2016)

Topologically stable structures, e.g. vortices in a wide variety of matter, skyrmions in ferro- and antiferromagnets, hedgehog point defects in liquid crystals and ferromagnets, are characterized by integer valued topological quantum numbers. In this context the closed surfaces are a prominent subject of study, because they realize a link between fundamental mathematical theorems and real physical systems. Here we perform a topological analysis of equilibrium magnetization states for a thin spherical shell with easy-normal anisotropy. Skyrmion solutions are found for a range of parameters. These magnetic skyrmions on a spherical shell have two principal differences compared to the planar case: (i) they become topologically trivial, and (ii) can be stabilized by curvature effects only, also when Dzyaloshinskii-Moriya interactions are absent. Due to its specific topological nature a skyrmion on a spherical shell can be simply induced by an uniform external magnetic field.