Asymmetric skyrmion-antiskyrmion production in ultrathin ferromagnetic films

Ultrathin ferromagnets with frustrated exchange and the Dzyaloshinskii-Moriya interaction can support topological solitons such as skyrmions and antiskyrmions, which are metastable and can be considered particle-antiparticle counterparts. When spin-orbit torques are applied, the motion of an isolated antiskyrmion driven beyond its Walker limit can generate skyrmion-antiskyrmion pairs. Here, we use atomistic spin dynamics simulations to shed light on the scattering processes involved in this pair generation. Under certain conditions a proliferation of these particles and antiparticles can appear with a growth rate and production asymmetry that depend on the strength of the chiral interactions and the dissipative component of the spin-orbit torques. These features are largely determined by scattering processes between antiskyrmions, which can be elastic or result in bound states or annihilation.

Figure 1

(a) Film geometry illustrating the Pd/Fe bilayer on an Ir(111) substrate, with schematic illustrations of a skyrmion $s$ and antiskyrmion $\overline{s}$. $\mathbf{B}$ is the applied field, and ${\theta }_p$ is the angle associated with the spin polarization vector $\mathbf{p}$. (b) Phase diagram of antiskyrmion dynamics under fieldlike (FL) and dampinglike (DL) spin-orbit torques [25].

Figure 2

Main scattering processes following pair generation from the seed $\overline{s}$ under SOT. (a) Maximal production, minimal asymmetry process $(N=2,\eta =0)$ leading to proliferation in which the generated $s\overline{s}$ pair splits and collision between the seed and generated $\overline{s}$ conserves skyrmion number. (b) $(N=2,\eta =0)$ process leading to premature saturation or stasis, where collision between the seed and generated $\overline{s}$ proceeds through a transient $Q=-2$ state ($2\overline{s}$) before decaying to an $\overline{s}\overline{s}$ bound pair, preventing further generation. (c) Minimal production, maximal asymmetry process ($N=1,\eta =1$) in which the generated $s\overline{s}$ pair splits and collision between the seed and generated $\overline{s}$ is inelastic, leading to annihilation of seed $\overline{s}$. Crosses denote the point of reference in the film plane, and the color map indicates the charge density $q$ of a unit cell. Arrows are shown for moments for which $\sqrt{m_{i,x}^2+m_{i,y}^2>0.9}$, and the open circles denote the approximate position of the core.

Figure 3

Representative examples of different growth regimes of the total skyrmion charge $Q$ for three values of $(\hslash {\beta }_{\mathrm{FL}},\hslash {\beta }_{\mathrm{DL}})$. (a) Proliferation, (0.02, 1.5) meV. (b) Stasis or premature saturation, (0.13, 1.1) meV. (c) Linear growth, (0.1, 1.35) meV.

Figure 4

(a) (${\beta }_{\mathrm{FL}},{\beta }_{\mathrm{DL}}$) phase diagram illustrating the total number of skyrmions and antiskyrmions produced over 0.1 ns, where $N$ is represented by the circle size on a logarithmic scale and the asymmetry parameter $\eta$ is shown on a linear color scale. (b) $N$ and (c) $\eta$ as a function of DL torques for the proliferation regime (for different FL torques). (d) $N$ as a function of DL torques for linear growth ($\eta =1$).

Figure 5

(${\theta }_p,D_{ij}$) phase diagram illustrating the total number of skyrmions and antiskyrmions produced over 0.1 ns, where $N$ is represented by the circle size on a logarithmic scale and $\eta$ is shown on a linear color scale for (a) $({\beta }_{\mathrm{FL}},{\beta }_{\mathrm{DL}})=$ (0.02, 1.5) meV, (b) (0.1, 1.35) meV, and (c) (0.13, 1.1). (d) $\eta$ and $N$ as a function of $D_{ij}$ for the case in (a).

Figure 6

Minimum-energy paths for the merging of the $\overline{s}\overline{s}$ pair into (a) a $2\overline{s}$ state and (b) an $\overline{s}$ state. The image indices are given in the bottom left corner. The associated energy profile along the (normalized) reaction coordinate, where (c) corresponds to the paths that results in the $2\overline{s}$ state and (d) corresponds to the path that results in the $\overline{s}$ state. The total topological charge remains constant at $Q=-2$ in (c), while its variation with the reaction coordinate is shown in (d). The inset in (c) shows the saddle point configuration (image 7), where the dashed arrows indicate the reference axis along which the clockwise (CW) or counterclockwise (CCW) Bloch states are defined and through which the merging of $\overline{s}$ occurs. The inset in (d) represents an expanded view of the region around the energy barrier.