Topological constraints on positions of magnetic solitons in multiply connected planar magnetic nanoelements

Here we consider an interplay between the topology of the magnetization texture (which is a topological soliton, or skyrmion) in a planar magnetic nanoelement and the topology of the element itself (its connectivity). We establish the existence of a set of constraints, coupling these topologies, which are specific for multiply connected elements and are absent in the simply connected case. As an example, a specific constraint is derived for a case of a planar ring magnet, relating the angular positions of magnetic vortices and antivortices inside. We analyze the recent experimental data on the vortex magnetic domain walls in the ring to validate our findings.

Figure 1

The configuration of local magnetic moments (magnetization texture) in a ring, containing two ${180}^{\circ }$ head-to-head vortex domain walls. The arrows correspond to the dimensionless in-plane magnetization vectors $\{m_X,m_Y\}$ plotted against the dimensionless coordinates $X$ and $Y$ normalized by the outer radius of the ring. The magnetization components are calculated according to (1) and (2) with $e_1=0.2$ and $e_2=2$ and the analytical expression for $f\left(z\right)$ from Ref. [13].

Figure 2

Integration contour in Eq. (3) for the case of two head-to-head ${180}^{\circ }$ vortex walls in a ring, shown in Fig. 1. Two bold arrows show the general direction of the magnetization inside two magnetic domains. The topological singularities (vortices and antivortices) are shown schematically with their centers marked by bold dots (and half-dots if they are on the boundary). The vortices are labeled with 1 and the half-antivortices at the boundary with $\overline{1/2}$.

Figure 3

The dots show the difference of the sums of the azimuth of vortices and antivortices $\delta$ as a function of time during the complex dynamical evolution of two vortex walls in a ring, extracted from the experimental data of Ref. [12]. The solid line shows the magnitude of the external field during the evolution in arbitrary units taken with the minus sign for easier comparison. The field is increasing, performing a ${360}^{\circ }$ rotation in the ring plane, then decreasing. The error bars (vertical lines) are computed in the text, taking into account both systematic and statistical errors.