Skyrmion dynamics in chiral ferromagnets

We study the dynamics of skyrmions in Dzyaloshinskii-Moriya materials with easy-axis anisotropy. An important link between topology and dynamics is established through the construction of unambiguous conservation laws obtained earlier in connection with magnetic bubbles and vortices. In particular, we study the motion of a topological skyrmion with skyrmion number $Q=1$ and a nontopological skyrmionium with $Q=0$ under the influence of an applied field gradient. The $Q=1$ skyrmion undergoes Hall motion perpendicular to the direction of the field gradient with a drift velocity proportional to the gradient. In contrast, the nontopological $Q=0$ skyrmionium is accelerated in the direction of the field gradient, thus exhibiting ordinary Newtonian motion. When the applied field is switched off the $Q=1$ skyrmion is spontaneously pinned around a fixed guiding center, whereas the $Q=0$ skyrmionium moves with constant velocity $v$. We give a systematic calculation of a skyrmionium traveling with any constant velocity $v$ that is smaller than a critical velocity $v_c$.

Figure 1

The static axially symmetric ($Q=1$) skyrmion represented through the projection $(m_1,m_2)$ of the magnetization vector on the plane for anisotropy $\kappa =3$.

Figure 2

The profiles $m_3=cos\theta =m_3\left(\rho \right)$ of skyrmions ($Q=1$) for three values of the anisotropy parameter $\kappa =2.6,2.8,3.0$.

Figure 3

The static axially symmetric skyrmionium ($Q=0$) represented through the projection $(m_1,m_2)$ of the magnetization vector on the plane for anisotropy $\kappa =3$.

Figure 4

The profiles $m_3=cos\theta =m_3\left(\rho \right)$ of a skyrmion ($Q=1$) and a skyrmionium ($Q=0$) for anisotropy $\kappa =3$.

Figure 5

Contour plots for $m_3$ for a skyrmion under the field gradient (26) with $g=-0.1,a=10$. Left: The static $Q=1$ skyrmion of Fig. 1 is placed at the origin at initial time $t=0$. Right: The skyrmion at simulation time $t=30$. It moves along the $y$ axis perpendicular to the field gradient. The contour levels plotted are $m_3=0.9,0.6,0.3,0.0$ (solid lines) and $m_3=-0.3,-0.6,-0.9$ (dashed lines).

Figure 6

Left: The trajectory of a $Q=1$ skyrmion under the influence of the field gradient (26) with $g=-0.1,a=10$. The guiding center propagates along the $y$ axis, thus in a direction perpendicular to the applied field gradient (straight black line). The trajectory for the $(X,Y)$ of Eq. (29) is the cycloid shown by a solid red line. After the applied field is suddenly switched off the guiding center ceases to move further and the skyrmion trajectory organizes itself in a rotational cyclotron-type motion around the fixed guiding center. Right: A magnification of the latest stages of the process is shown.

Figure 7

Contour plot for $m_3$ for the static skyrmionium (left) and the propagating skyrmionium with velocity $v=0.07$ (right). The contour levels plotted with solid lines are $m_3=0.9,0.6,0.3,0.0$ and with dashed lines $m_3=-0.3,-0.6,-0.9$.

Figure 8

Contour plot for the topological density $q$ of the static skyrmionium (left) and the propagating skyrmionium with velocity $v=0.07$ (right). The contour levels plotted are chosen arbitrarily. Solid lines mean $q>0$ and dashed lines $q<0$.

Figure 9

Left: Solid lines show the energy $W$ of Eq. (2) and linear momentum $P$ of Eq. (30) for a steady-state propagating skyrmionium as a function of its velocity $v$. The open circles show results from a direct simulation discussed in the text. Right: The energy $W$ versus momentum $P$ for a steady-state propagating skyrmionium. The curve is parabolic for low momenta in accordance with Eq. (41) but linear for large momenta in agreement with Eq. (42).

Figure 10

Linear momentum $P$ as a function of time for a simulation where a field gradient (26) with $g=-0.001,a=10$ is applied to an initially static skyrmionium. The field is applied for $0\le t\le 100$, and is then suddenly switched off.

Figure 11

Contour plots for $m_3$ for a skyrmionium under the field gradient (26) with $g=-0.001,a=10$. Left: The static $Q=0$ skyrmionium of Fig. 3 is placed at the origin at initial time $t=0$. Right: The skyrmionium at simulation time $t=160$. It propagates along the $x$ axis in the direction of the field gradient. The contour levels plotted are $m_3=0.9,0.6,0.3,0.0$ (solid lines) and $m_3=-0.3,-0.6,-0.9$ (dashed lines).