Mesoscale Dzyaloshinskii-Moriya interaction in curvilinear geometries: one-dimensional and two-dimensional cases

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)