Control of skyrmion magnetic bubble gyration

The skyrmion magnetic bubble in a ferromagnetic disk exhibits hypocycloidal gyrations contrary to the vortex gyration, showing a simple circular trajectory. To describe the hypocycloidal bubble gyration, a mass term is needed in Thiele's equation. In this study, we analytically derived both mass and spring constant term, which are crucial parameters for describing the bubble gyration. Values obtained by these analytic expressions were consistent with those obtained by simulations. We could find the dependences of these two terms on several external parameters, including the bubble radius. Especially using the radius's dependence, we could obtain regular polygonlike trajectories such as a square and a triangle confirmed by the numerical simulations. Based on this effective method to control the bubble gyration, the regular polygonlike trajectories of this skyrmion magnetic bubble make it possible to study the bubble gyration without time-resolved experiments.

Figure 1

(a) A skyrmion magnetic bubble domain state in a perpendicular magnetic anisotropy disk. The magnetization of the bubble is aligned in the −$z$ direction (colored red). The blue color represents alignment of the magnetization in the +$z$ direction. Black arrows in the disk denote the in-plane magnetization direction. A white dashed circle represents the domain wall position where the total energy is minimized. The bubble position shifted −6.4 nm from the white dashed circle to the $x$ direction for gyration. (b) A regular pentagonlike trajectory of the bubble domain gyration with the external field $H_z$ = −7 mT. The line represents the trajectory of the center of the bubble. The trajectory has five vertices and thus resembles a regular pentagon. (c) Several examples of regular polygonlike trajectory with respect to the external field. The time scale is the same as in (b).

Figure 2

Frequencies obtained by gyration results with respect to the external magnetic field $H_z$ applied to the perpendicular direction to the disk; ${\omega }_{+}$ (□) is obtained by a clockwise rotation of the bubble, and ${\omega }_{-}$ (○) represents counterclockwise rotation of the bubble. The thick green line () denotes the frequency derived from the massless Thiele's equation. The black (─) and red () lines are obtained by analytic equation. The inset shows the averaged radius $\overline{r}$ as a function of $H_z$.

Figure 3

Mass constant $\mathcal{M}$ and spring constant $\mathcal{K}$ for the skyrmion bubble gyration. (a) Mass of the bubble with respect to $\overline{r}$. The line is obtained by the analytic Eq. (5). (b) Origin of the skyrmion bubble mass. The energy difference between the Bloch and Néel-type wall generates the mass of the bubble. The Bloch wall does not make any surface magnetic charge on the domain wall, but the Néel wall does. (c) The dependence of the spring constant on the bubble radius. The line is obtained by the analytic Eq. (7). (d) The schematic diagram describing the origin of the spring constant. When the disk moves from the initial position (dashed circle) to the $x$ direction, excessive magnetic surface charges are induced.

Figure 4

Time-averaged images of bubble gyrations. We assumed the zero wall width and neglected the damping effect. Different colors denote the time-averaged perpendicular magnetic component $\langle m_z\rangle$${}_t$ during the bubble gyration. The angular frequency ratio ${\omega }_{-}$/${\omega }_{+}$ is integers (a–c) and not integer (d).