Skyrmion states, engineered by curvature gradients

Skyrmions attract a special attention for spintronic and spinorbitronic devices by their unique static and dynamic properties [1]. The interplay between geometry and magnetization texture gives additional degrees of freedom in control of such topologically nontrivial patterns via geometry-induced anisotropy and Dzyaloshinskii-Moria interaction (DMI) in materials with easy-normal anisotropy [2-5]. It is sufficient for appearance of skyrmions and skyrmion lattices as a ground state on small curvilinear defects [6].

Here, we propose a new way to stabilize skyrmions and control their size via curvature gradient in a nanoindentation even without intrinsic DMI [7]. Our mathematical formalism also allows to describe planar films with inhomogeneous distribution of material parameters. We consider a thin membrane with easy-normal anisotropy and circular nanoindentation of a conic frustrum shape with inner and outer radii R and R , respectively. This geometry can be described by two principal curvatures, k and k , providing the whole information about geometry in the given point of the membrane. While k (r) is inversely proportional to the distance from origin r, k (r) has sharp peaks in points of the bend of the membrane. To compare textures in curved membranes and flat films we propose a projection of a surface of revolution to a plane, which reconstructs a skyrmion equation. The energy of the magnetic texture in projected coordinates obtains a curvature-modified anisotropy and two DMI terms, related to principal curvatures. In contrast to the planar case, the corresponding skyrmion equation is characterized not only by DMI coefficients itself, but also their spatial derivatives playing a role of the external driving force, proportional to d(k + k )/dr. The main consequences of the driving force are: (i) the ground state cannot be strictly normal to the membrane with nonconstant curvature; (ii) a gradient of k can result in stabilization of the Neel skyrmion of radius R or R (Fig. 1). Skyrmions, stabilized at the inner and outer bends, have the opposite chiralities. If the difference between R and R is large enough, both skyrmions can coexist forming a skyrmionium state with zero total winding number. In the limiting case of sharp bends, the minimal angle α of the indentation side for stabilizing topologically nontrivial textures equals 4L/R radians, with R being either R or R (α = 0 corresponds to a flat film) and L being a magnetic length. Numerical analysis of skyrmion stability is performed in a wide range of geometrical parameters (Fig. 2). It is shown that the strength and spatial localization of the DMI coefficient, associated with k , plays the main role in the pinning of topologically nontrivial textures and pinning strength is estimated to be hundredths of Kelvin for typical parameters of Co/Pt multilayers.

In conclusion, we propose a mathematical framework which allows us to describe magnetic nanomembranes with rotational symmetry and planar films with circular distribution of material parameters using the same apparatus. It uncovers two mechanisms of skyrmion stabilization, namely DMI-driven [8] and DMI gradient-driven [7]. The first one does not require the curvature gradients and lead to formation of small-radius skyrmions, while the second allows stabilization of large-radius skyrmions and skyrmionium states of the geometrically defined size.

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