Curvature-induced Local & Nonlocal Chiral Effects

in Curvilinear Ferromagnetic Shells and Wires

Conventional magnetic nanoscale devices are based on planar thin films and straight racetracks hosting magnetic topological solitons. Recent progress in fabrication and characterization methods allows to realise and study of complex-shaped planar and three-dimensional (3D) architectures. In the planar case, boundaries of nanodots lead to the formation of inhomogeneous textures, such as vortices and antivortices. In 3D, the magnetostatic interaction favours a spatially inhomogeneous shape anisotropy, which acts as easy-axis anisotropy along wires or hard axis of anisotropy perpendicular to the film surface. These interactions track the sample geometry and enable curvature-induced symmetry-breaking effects, such as topology-induced magnetization patterning and emergent anisotropic and chiral responses of the Dzyaloshinskii-Moriya interaction (DMI) type [1,2].

Curvature-induced magnetic responses can be classified as being local or nonlocal. In ferromagnets, local effects stem from the exchange interaction and DMI. The curvature-induced DMI originates from exchange: it is linear in curvatures and has the symmetry of the interfacial DMI. Its strength can be comparable with typical values of the intrinsic DMI. This is experimentally confirmed by the stabilization of chiral domain walls (CDW) on the apex of a Permalloy parabola-shaped stripe [3]. The strength of the CDW depinning field gives an estimation for the curvature-induced DMI constant and can be tuned by the geometry. In contrast to curvature itself, also curvature gradients offer a possibility to pin CDW, which was studied with an example of a circular indentation with a conic cross-section profile. This geometry supports circular CDWs described by the forced skyrmion equation, where the effective force acts as the stabilizing factor for large-radius skyrmion and skyrmionium states [4].

The magnetostatic interaction is a source of novel curvature-induced chiral effects, which are essentially nonlocal, in contrast to the conventional DMI [5]. The effect emerges in shells with non-zero mean curvature due to the non-equivalence between the top and bottom surfaces of a geometrically curved shell. It is possible to show that the analysis of nonlocal effects in curvilinear shells can be more intuitive with a split of a conventional volume magnetostatic charge into two terms: (i) tangential charge, governed by the tangent to the sample's surface, and (ii) geometrical charge, given by the normal component of magnetization and the mean curvature. In addition to the shape anisotropy (local effect), four additional nonlocal terms appear, determined by the surface curvature. Three of them are zero for any magnetic texture in shells with the geometry of minimal surfaces. The fourth term becomes zero only for the special symmetries of magnetic textures.

The impact of local and nonlocal chiral effects on magnetic textures in curvilinear architectures will be discussed in this presentation.

Nanomagnetism, spin texture, domain walls

[1] D. Makarov, O. M. Volkov, A. Kákay, O. V. Pylypovskyi, B. Budinská and O. V. Dobrovolskiy, "New dimension in magnetism and superconductivity: 3D and curvilinear nanoarchitectures", Adv. Mater., **34**, 2101758 (2022)

[2] D. D. Sheka, O. V. Pylypovskyi, O. M. Volkov, K. V. Yershov, V. P. Kravchuk and D. Makarov, "Fundamentals of Curvilinear Ferromagnetism: Statics and Dynamics of Geometrically Curved Wires and Narrow Ribbons", Small, 2105219 (2022)

[3] O. M. Volkov, A. Kákay, F. Kronast, I. Mönch, M. Mawass, J. Fassbender and D. Makarov, "Experimental Observation of Exchange-Driven Chiral Effects in Curvilinear Magnetism", Phys. Rev. Lett., **123**, 077201 (2019)

[4] O. V. Pylypovskyi, D. Makarov, V. P. Kravchuk, Y. Gaididei, A. Saxena and D. D. Sheka, "Chiral Skyrmion and Skyrmionium States Engineered by the Gradient of Curvature", Phys. Rev. Applied, **10**, 064057 (2018)

[5] D. D. Sheka, O. V. Pylypovskyi, P. Landeros, Y. Gaididei, A. Kákay and D. Makarov, "Nonlocal chiral symmetry breaking in curvilinear magnetic shells", Comm. Phys., **3**, 128 (2020)