Out-of-surface vortices in spherical shells

The interplay of topological defects with curvature is studied for out-of-surface magnetic vortices in thin spherical nanoshells. In the case of an easy-surface Heisenberg magnet it is shown that the curvature of the underlying surface leads to a coupling between the localized out-of-surface component of the vortex with its delocalized in-surface structure, i.e., polarity-chirality coupling.

Figure 1

Possible vortex phases $\Phi$ for the case of same polarities $p_1=p_2=p$ (right inset). Solid lines correspond to the exact numerical solution of Eq. (3b), where the out-of-surface component is chosen in the form of Eq. (4) with ${\vartheta }_c=0.05$. The corresponding approximate solutions (7) are indicated by dashed lines. The corresponding magnetization distribution on the sphere surface is schematically shown in the left inset using arrows and stream lines.

Figure 2

All possible vortex states of the spherical surface. The left column demonstrates the model out-of-surface magnetization distribution given by Eq. (4) for all possible combinations of polarities. The right graph shows the corresponding distributions of the phase $\Phi \left(\vartheta \right)$. For the correspondence, we use the notation $p_1{p}_2$, e.g., “+1$-$1” denotes $p_1=+1$ and $p_2=-1$.

Figure 3

Structure of the vortex state for the case of same polarities ($p_1=p_2=1$) obtained by different methods. Line 1: exact numerical solution of Eq. (3) with boundary conditions $\Theta \left(0\right)=\Theta \left(\pi \right)=0$ and ${\Phi }^{\prime }\left(0\right)={\Phi }^{\prime }\left(\pi \right)=0$ and $\ell /L=0.05$. Line 2: (left) the ansatz (4) and (right) corresponding solution of Eq. (3b) where the function $\Theta \left(\vartheta \right)$ is determined by Eq. (4) with an angular vortex core size ${\vartheta }_c=\ell /L=0.05$. This line coincides with line “$p=1$” in Fig. 1. Line 3 and line 4 correspond to micromagnetic simulations of types (i) and (ii), respectively (see text for details). Line 5 in the left graph shows the out-of-surface structure of the vortex core for the case of opposite polarities, when $p_1=-p_2=1$ and $\Phi \equiv \pi$.

Figure 4

Function $g\left(u\right)$ from Eq. (5) for ${\vartheta }_c=0.05$.