Dynamics and switching processes for magnetic bubbles in nanoelements

We study numerically the dynamics of a magnetic bubble in a disk-shaped magnetic element which is probed by a pulse of a magnetic field gradient. Magnetic bubbles are nontrivial magnetic configurations which are characterized by a topological (skyrmion) number $\mathcal{N}$ and they have been observed in mesoscopic magnetic elements with strong perpendicular anisotropy. For weak fields we find a skew deflection of the axially symmetric $\mathcal{N}=1$ bubble and a subsequent periodic motion around the center of the dot. This gyrotropic motion of the magnetic bubble is shown here for the first time. Stronger fields induce switching of the $\mathcal{N}=1$ bubble to a bubble which contains a pair of Bloch lines and has $\mathcal{N}=0$. The $\mathcal{N}=0$ bubble can be switched back to a $\mathcal{N}=1$ bubble by applying again an external field gradient. Detailed features of the unusual bubble dynamics are described by employing the skyrmion number and the moments of the associated topological density.

Figure 1

Bubbles with (a) $\mathcal{N}=1$ and (b) $\mathcal{N}=0$. Only the domain wall is shown. We suppose that the magnetization points down inside the wall while it points up outside it.

Figure 2

(Color online) The orbit of the bubble under an external field gradient [Eq. (6)] with $g=-0.0025$. The solid line shows the coordinates $\left(R_x,R_y\right)$ of Eq. (8). The dashed line shows the coordinates $\left(X,Y\right)$ of Eq. (7). The circles mark the bubble position at times which are multiples of $5.33{\tau }_0$ (15 ps). The arrows indicate the point where the field is switched off.

Figure 3

(Color online) Snapshots from the simulation for a bubble with $\mathcal{N}=1$ under external field gradient [Eq. (6)] with $g=-0.0025$. They show the bubble (a) at the dot center (at time $\tau =0$), (b) when the external field is switched off $\left[\tau =44.5{\tau }_0\left(200\text{ps}\right)\right]$, and (c) when this has completed a cycle around the dot center $\left[\tau =267{\tau }_0\left(1200\text{ps}\right)\right]$.

Figure 4

The trajectory of the bubble under external field gradient [Eq. (6)] with $g=-0.025$. The solid line shows the coordinates $\left(R_x,R_y\right)$ of Eq. (8). They have been traced until $\tau =85.5{\tau }_0$ when we have switching. The dashed line shows the coordinates $\left(X,Y\right)$ of Eq. (7), which have been traced until $\tau =432{\tau }_0$.

Figure 5

(Color online) Snapshots from the simulation for a bubble under external field gradient [Eq. (6)] with $g=-0.025$. (a) A remanent $\mathcal{N}=1$ bubble in the dot center $\left(\tau =0\right)$, (b) the instant just before the wall unwinding $\left[\tau =83{\tau }_0\left(375\text{ps}\right)\right]$, where the arrow indicates the area where the VBLs have developed, (c) the instant just after the wall unwinding $\left[\tau =85.5{\tau }_0\left(385\text{ps}\right)\right]$, where the arrow indicates the same area as in the previous entry, and (d) a $\mathcal{N}=0$ bubble as a remanent state (at the end of the simulation).

Figure 6

(Color online) Blow ups of a part of the bubble which contains Bloch lines for (a) Fig. 5b and (b) Fig. 5c (the arrows correspond to those in Fig. 5).

Figure 7

(Color online) The trajectory of a $\mathcal{N}=0$ bubble which is subject to an external field [Eq. (6)] with $g=-0.025$. The field is switched off at $\tau =55{\tau }_0$ (250 ps). The bubble switches to a $\mathcal{N}=1$ bubble at $\tau =98{\tau }_0$ (440 ps), which is indicated by the arrows. We plot both the coordinates [Eq. (8)] (solid line), which are defined only after the switching for $\mathcal{N}=1$, and the coordinates [Eq. (7)] (dashed line). The total simulation time is $\tau =555.5{\tau }_0$ (2.5 ns).

Figure 8

(Color online) Snapshots from the simulation for a bubble under external field gradient [Eq. (6)] with $g=-0.025$. (a) A remanent $\mathcal{N}=0$ bubble in the dot center (at $\tau =0$), (b) the instant just before the wall unwinding where the arrow indicates the area where the VBLs have developed $\left[\tau =95.5{\tau }_0\left(430\text{ps}\right)\right]$, (c) the instant just after the wall unwinding where the arrow indicates the same area as in the previous entry $\left[\tau =98{\tau }_0\left(440\text{ps}\right)\right]$, and (d) the final results of the simulation, i.e., a static $\mathcal{N}=1$ bubble.

Figure 9

(Color online) Blow ups of a part of the bubble corresponding to (a) Fig. 8b and (b) Fig. 8c (the arrows correspond to those in Fig. 8).