Fractional Vortices and Composite Domain Walls in Flat Nanomagnets

We provide a simple explanation of complex magnetic patterns observed in ferromagnetic nanostructures. To this end we identify elementary topological defects in the field of magnetization: ordinary vortices in the bulk and vortices with half-integer winding numbers confined to the edge. Domain walls found in experiments and numerical simulations in strips and rings are composite objects containing two or more of the elementary defects.

Figure 1

(a) and (b) Edge defects with winding numbers $n=+\frac{1}{2}$ and $-\frac{1}{2}$. (c) A vortex ($n=+1$) offsets the influence of two $-\frac{1}{2}$ edge defects. (d) A domain wall composed of two edge defects $\pm \frac{1}{2}$. (e) The $-\frac{1}{2}$ defect in the limit where magnetostatic energy dominates.

Figure 2

Numerical simulation of magnetization dynamics in a permalloy disk. Color encodes the direction of the field $\widehat{\mathbf{m}}$ (also shown by arrows). Convergence of different colors at a point signals the presence of a topological defect. In the first three panels the disk contains two $+1$ vortices and two $-\frac{1}{2}$ edge defects. After the edge defects and a vortex mutually annihilate (last panel), a single vortex remains.

Figure 3

Stable and metastable states of a magnetic nanoring in the thin-film limit. (a) One of the two ground states without topological defects. (b) State with a remnant magnetization contains two composite domain walls with $+\frac{1}{2}$ defects at the outer edge and $-\frac{1}{2}$ defects at the inner edge.