Dynamical switching of confined magnetic skyrmions under circular magnetic fields

Understanding the dynamical properties of magnetic skyrmions is of prime importance for realization of skyrmion based devices. Here we report on the dynamics of a confined magnetic skyrmion under application of circular magnetic fields (CMFs). We provide the complex map of the dynamical states and demonstrate that under some conditions CMFs can change the skyrmion number and lead to the switching of the core polarity. Combing the results of micromagnetic simulations with those of analytical calculations, based on the extended Thiele equation, we unravel the physical mechanism behind this dynamical switching. The core switching is associated with the dynamical motions of the skyrmion driven by a CMF. The formation of domain walls and their movement from the border towards the center leads to shrinking of the skyrmion and the creation of a skyrmion with opposite core polarity. The results provide a guideline for checking the skyrmion's stability as well as an efficient way for switching the core polarity using CMFs.

Figure 1

(a) Trajectory of a confined skyrmion, recorded 0.63 ns after application of a Gaussian pulse field with a width of $85\mathrm{ps}$ and a magnitude of about $104\mathrm{kA}/\mathrm{m}$. (b) The FFT of the $x$ and $y$ components of the position of the skyrmion's guiding center. The skyrmion trajectory when a sine field with frequencies of (c) ${\nu }_1=1$, (d) ${\nu }_2=6.8$, and (e) ${\nu }_3=12.5\mathrm{GHz}$ is applied to the system. The color bar shows the time evolution in the units of ns.

Figure 2

The map of dynamical states of a skyrmion subjected to a clockwise CMF. The vertical axis represents the amplitude, and the horizontal axis represents the frequency of the CMF. In region I the skyrmion rotates in the clockwise direction without changes in its skyrmion number $Q$. It does not collide with the border within 20 ns. In region II the skyrmion rotates in the clockwise direction without changes in its $Q$, but it collides with the border within this time. In regions III and IV the sign of $Q$ changes while applying the CMF. Region III is the switching region in which a skyrmion with an opposite core polarity can be achieved. In some cases a Zeeman field along the core direction of the initial skyrmion is needed to stabilize this skyrmion. The magnitude of this field is given as the color bar. Note that the color bar applies only to the region III. In region IV the skyrmion collides with the border, and chiral spin textures are created. The yellow, dark yellow, and dark blue represent regions I, II, and IV, respectively. In these regions no stabilizer field was applied at all.

Figure 3

The skyrmion is initially stabilized in the disk for 4 ns. It is then subjected to a clockwise CMF with (a) $H_0=30\mathrm{kA}/\mathrm{m}$ and ${\nu }_h=5\mathrm{GHz}$ and (b) $H_0=190\mathrm{kA}/\mathrm{m}$ and ${\nu }_h=25\mathrm{GHz}$. The CMF is turned off at $t=4.54$ ns in (a) and at $t=4.4$ ns in (b). After turning the CMF off this skyrmion gyrates in a clockwise direction and is finally stopped at the center. The color scale indicates the direction of the local magnetic moments along the $z$ direction, perpendicular to the disk (red: up, blue: down). Arrows represent the transverse components of local moments. (c) The time evolution of the skyrmion number during the CMF. (d) The trajectory of the skyrmion before the CMF is applied. The initial position of the skyrmion is first defined to be at point O $\left(x_0=1\mathrm{nm},y_0=2\mathrm{nm}\right)$. In order to avoid effects caused by deformations and breathing, the data are recorded from point A where the skyrmion's shape is rather circular. The skyrmion rotates counterclockwise and finally stops near the disk center (point B). (e) The trajectory of the skyrmion during and after application of the CMF $\left(H_0=30\mathrm{kA}/\mathrm{m},{\nu }_h=5\mathrm{GHz}\right)$. During the field the skyrmion rotates in clockwise direction from point B to C. At point C the up-core skyrmion switches to a down-core one. At this point the CMF is turned off. The down-core skyrmion rotates in a clockwise direction and finally stops at the center. Arrows indicate the direction.

Figure 4

The spin texture formed after application of the CMF with amplitude and frequency of (a) $H_0=170\mathrm{kA}/\mathrm{m}$ and ${\nu }_h=8\mathrm{GHz}$ and (b) $H_0=230\mathrm{kA}/\mathrm{m}$ and ${\nu }_h=12\mathrm{GHz}$. These points are located in region IV of Fig. 2. The field is turned on at $t=4$ ns and is turned off at $t=4.25$ ns. After application of such CMFs the skyrmion moves rather fast, and it quickly transforms into a magnetic texture composed of three or four magnetic domains separated with domain walls of Bloch type. The sign of $Q$ of such spin textures in most of the cases is negative (opposite to the one of the initial skyrmion).

Figure 5

The skyrmion trajectory calculated based in the generalized Thiele equation in the presence and absence of a CMF. (a) The trajectory of an up-core skyrmion when it is displaced to $\left(x_0=2\mathrm{nm},y_0=2\mathrm{nm}\right)$ and its motion is followed. The skyrmion starts to rotate in counterclockwise direction, and finally after a few nanosecond it stops at the center of the disk (not shown here). (b) The skyrmion trajectory in the presence of a clockwise CMF. The skyrmion rotates in the clockwise direction. The trajectory is plotted for the first 0.58 ns. The amplitude and frequency of the field are $H_0=30\mathrm{kA}/\mathrm{m}$ and ${\nu }_h=5\mathrm{GHz}$, similar to the simulations. (c) The trajectory of the skyrmion after switching its core polarity. The skyrmion has a down-core polarity, and it starts to rotate in a clockwise direction, as seen in the simulations. This figure will be compared to Figs. 3 and 3, where the results of the simulations for the same skyrmion are presented.

Figure 6

The displacement of the skyrmion from the disk center during application of different CMFs. The results of low- and mid-frequency fields are shown in (a) and (b). The results of high-frequency fields are shown in (c) and (d). The results of simulation are presented in (a) and (c). The results of calculations, based on the extended Thiele equation, are shown in (b) and (d). $H_0$ denotes the amplitude of CMF, and ${\nu }_h={\omega }_h/2\pi$ denotes its frequency. The values of $H_0$ and ${\nu }_h$ are given in kA/m and GHz, respectively.

Figure 7

(a) Trajectory of the skyrmion stabilized with $K=0.27\times {10}^6\mathrm{J}/{\mathrm{m}}^3$ and $D=3.3\times {10}^{-3}\mathrm{J}/{\mathrm{m}}^2$ driven by a Gaussian pulse field with the width of $65\mathrm{ps}$ and the magnitude of about $143\mathrm{kA}/\mathrm{m}$. In order to avoid the effects caused by the field, the data are recorded 0.62 ns after application of the field. The trajectory is obtained from the micromagnetic simulations and is recorded for 0.9 ns. (b) The FFT results of the $x$ and $y$ coordinates of the skyrmion's guiding center.

Figure 8

Switching of a confined skyrmion with $Q=+1$ to a skyrmion with $Q=-1$ and to a skyrmionium with $Q=0$ by a CMF. The skyrmion was stabilized with $K=0.27\times {10}^6\mathrm{J}/{\mathrm{m}}^3$ and $D=3.3\times {10}^{-3}\mathrm{J}/{\mathrm{m}}^2$. (a) Snapshots during the switching of $Q=+1$ to $Q=-1$ state. The skyrmion was first stabilized in the disk for 4 ns. It was then subjected to a clockwise CMF with $H_0=60\mathrm{kA}/\mathrm{m}$ and ${\nu }_h=7\mathrm{GHz}$. The CMF was turned off at $t=4.3$ ns. After turning the CMF off a down-core skyrmion with $Q=-1$ was stabilized. (b) The time evolution of the skyrmion number during the CMF with $H_0=60\mathrm{kA}/\mathrm{m}$ and ${\nu }_h=7\mathrm{GHz}$. (c) Snapshots during the switching of $Q=+1$ to $Q=0$ state. Again the skyrmion was initially stabilized for 4 ns. It was then subjected to a clockwise CMF with $H_0=60\mathrm{kA}/\mathrm{m}$ and ${\nu }_h=6\mathrm{GHz}$. The CMF was turned off at $t=4.3$ ns. After turning the CMF off a skyrmionium was created. (d) The time evolution of the skyrmion number during the CMF with $H_0=60\mathrm{kA}/\mathrm{m}$ and ${\nu }_h=6\mathrm{GHz}$. The color scale in (a) and (c) indicates the direction of the local magnetic moments along the $z$ direction, perpendicular to the disk (red: up, blue: down). The small arrows represent the transverse components of local moments. The CMF was applied for 2 ns in (b) and (d).