Chiral droplets and current-driven motion in ferromagnets

We find numerically skyrmionic textures with skyrmion number $Q=0$ in ferromagnets with the Dzyaloshinskii-Moriya interaction, perpendicular anisotropy, and the magnetostatic field. These have a skyrmion part and an antiskyrmion part, and they may be called chiral droplets. They are stable in an infinite film as well as in disk-shaped magnetic elements. Droplets are found in films for values of the parameters close to the transition from the ferromagnetic to the spiral phase. Under spin-transfer torque, they move in the direction of the spin flow and behave as solitary waves of Newtonian character, in stark contrast to the Hall dynamics of the standard $Q=1$ skyrmion.

Figure 1

The magnetization configuration produced by the form (13) representing a skyrmion-antiskyrmion pair. The lower half of this configuration has the features of a skyrmion, and the upper half has the features of an antiskyrmion. This is used as the initial condition in the energy relaxation algorithm for finding a static solution of the equation.

Figure 2

(a) A static skyrmion-antiskyrmion droplet with $Q=0$ found numerically for the parameter values given in Table 1 and for film thickness $d_{\mathrm{f}}=0.5$ nm. For comparison, a $Q=1$ skyrmion in the same system has a radius of $\sim 16$ nm. (b) Blowup of the antiskyrmion part of the configuration, where we show the full resolution of the simulation. The simulated domain in the plane of the film is $400\times 400{\mathrm{nm}}^2$, and periodic boundary conditions are used. The cell size in the simulations is approximately $0.195\times 0.195\times 0.5{\mathrm{nm}}^3$ ($2048\times 2048\times 1$ cells). We have performed simulations with different cell sizes to confirm that the presented droplet configuration does not depend on the cell size.

Figure 3

The topological density $q$ for the chiral droplet in Fig. 2. The part where $q<0$ (blue) occupies a very small region compared to the part where $q>0$ (red). The density $q$ takes very large negative values in a small region, while in the region with positive values, $q$ is small. The integrated topological density, or $Q$, is zero.

Figure 4

(a) Snapshots of an initially static droplet during a simulation where a current $J_{\mathrm{e}}$ is applied in the $x$ direction for times $0\le t\le 75\mathrm{ns}$. The parameters are $U=3.86\mathrm{m}/\mathrm{s},\beta =0.075$, and the damping parameter is $\alpha =0.03$. (b) The velocity of the droplet as a function of time. Upon switching on the current the velocity is $V=4.02\mathrm{m}/\mathrm{s}$ in the current direction (denoted $V_x$). The droplet is then accelerated up to $V=9.45\mathrm{m}/\mathrm{s}$, which is close to the value $2.5U$. After switching off the current, the velocity drops instantly by approximately $4\mathrm{m}/\mathrm{s}$ (which is close to $U$), to $V=5.37\mathrm{m}/\mathrm{s}$. The droplet continues to travel, and the damping term decelerates the motion until it stops. The component of the velocity perpendicular to the current (denoted $V_y$) is very small and goes to zero for steady-state motion.

Figure 5

Snapshots of the droplet in Fig. 2 when it is placed under spin torque at time $t=0$ and is set in motion. The flow velocity is (a) $(U,0)$ and (b) $(0,U)$ with $U=3.72\mathrm{m}/\mathrm{s}$. The damping parameter is $\alpha =0.1$, and the nonadiabaticity parameter is $\beta =0.2$. The simulation domain is $400\times 400{\mathrm{nm}}^2$, and we apply periodic boundary conditions.

Figure 6

Static skyrmion-antiskyrmion droplets in disk elements with thickness $d_{\mathrm{f}}=5\mathrm{nm}$. (a) The disk diameter is $d=70\mathrm{nm}$. The parameter values are as in Table 1. (b) The disk diameter is $d=120\mathrm{nm}$. The parameter values are $D=1\mathrm{mJ}/{\mathrm{m}}^2,K=4.43505\times {10}^5\mathrm{J}/{\mathrm{m}}^3$ (other parameters are as in Table 1). In this case, we have a more extended antiskyrmion part. For comparison, we note that a usual skyrmion (with $Q=1$) has an approximate radius of $15\mathrm{nm}$ for the dot in (a) and $30.5\mathrm{nm}$ for the dot in (b). They are thus somewhat larger than the corresponding $Q=0$ configurations shown here.