Topological control of magnetic textures

A micromagnetic study is carried out on the role of using topology to stabilize different magnetic textures, such as a vortex or an antivortex state, in a magnetic heterostructure consisting of a permalloy disk coupled to a set of nanomagnetic bars. The topological boundary condition is set by the stray field contributions of the nanomagnet bars and thus by their magnetization configuration, and can be described by a discretized winding number that will be matched by the winding number of the topological state set in the disk. The lowest number of nanomagnets that defines a suitable boundary is 4, and we identify a critical internanomagnet angle of 225 ° between two nanomagnets, at which the boundary fails because the winding number of the nanomagnet configuration no longer controls that of the disk magnetization. The boundary also fails when the disk-nanomagnets separation is >50 nm and for disk diameters >480 nm. Finally, we provide preliminary experimental evidence from magnetic force microscopy studies in which we demonstrate that an energetically unstable, antivortex-like structure can indeed be stabilized in a permalloy disk, provided that the appropriate topological conditions are set.

Figure 1

A discrete form of the topological winding number is introduced. Schematics show the winding numbers associated with different Ising spin configurations (yellow arrows).

Figure 2

(a)–(d) Micromagnetic simulations of the relaxed magnetization state in a Py disk for winding numbers of the boundary of −1 (a), (d), 0 (b), and −1 (c). (e)–(g) Topologically equivalent magnetic textures (all with a winding number +1) belonging to different topological sectors, i.e., different values of $c$.

Figure 3

(a)–(d) Micromagnetic simulation of a Py disk with surrounding elliptical nanomagnets, as a function of nanomagnet position. The associated winding number of the nanomagnets is shown in top left. (a)–(c) An antivortex forms in the disk for values of (${\vartheta }_{\max }$) of 190 °, 200 °, and 210 °. (d) Uniform magnetization observed for $\left({\vartheta }_{\max }\right)={225}^{\circ }$.

Figure 4

(a) Micromagnetic simulation of an antivortex state with a winding number of −1, in which the boundary state also has a winding number of −1. (b) MFM image of an antivortexlike magnetic texture in a Py disk, for which the surrounding Py nanomagnets create a boundary with winding number of −1.

Figure 5

mumax simulations of coupled disks, in which two permalloy disks are interacting with each other across a connecting nanomagnet. (a) The magnetization in the disks corresponds to an antivortex (disk 1) and a soliton free state (disk 2) corresponding to their respective boundary $s$ with $W=–1$ and $W=0$ respectively. (b) Boundary for disk 2 changed to have $W=+1$—a vortex is now stabilized.