Mesoscale Dzyaloshinskii-Moriya interaction: 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 [1-3]) 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.
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