Skyrmion elongation, duplication, and rotation by spin-transfer torque under spatially varying spin current

The effect of the spatially varying spin current on a skyrmion is numerically investigated. It is shown that an inhomogeneous current density induces an elongation of the skyrmion. This elongation can be controlled using current pulses of different strength and duration. Long current pulses lead to a splitting that forms two replicas of the initial skyrmion while for short pulses the elongated skyrmion relaxes back to its initial circular state through rotation in the MHz-GHz frequency range. The frequency is dependent on the strength of the damping coefficient.

Figure 1

(a) Device model schematic. We consider a device made of a ferromagnet built over two heavy metal layers separated by a gap where the spin current flows in opposite directions. (b) Elongated skyrmion through the effect of counterpropagating spin currents. Spin current directions for $y\ge 0$ and $y<0$ are shown schematically with white arrows. The current pulse used is $3\times {10}^{12} \mathrm{A}/{\mathrm{m}}^2$ for 0.67 ns with device parameters $A=2.5$ pJ/m, $D_{\perp }=1 \mathrm{mJ}/{\mathrm{m}}^2, {\mu }_{0}H=0.1\mathrm{T}, P=0.4, M_s=1000$ kA/m, and $\alpha =0.015$. To determine the length of the skyrmion elongation in the wider axis (dark gray arrow in the figure), we consider the skyrmion boundaries to be located at $M_z/\left|M\right|=-0.8$.

Figure 2

(a) Skyrmion elongation evolution in time as a function of a single inhomogeneous current pulse of 0.5 ns. (b) Skyrmion elongation in time as a function of an infinite pulse for different LLG parameters, where $J$ and/or $D$ is halved or doubled. LLG parameters are $J=2.5$ pJ/m, $D=1 \mathrm{mJ}/{\mathrm{m}}^2, {\mu }_{0}H=0.1\mathrm{T}$, and $\alpha$ = 0.015.

Figure 3

(a) Skyrmion elongation evolution in time as function of different spin current pulse lengths. The applied current is fixed at $5\times {10}^{12} \mathrm{A}/{\mathrm{m}}^2$. The half-open dots indicate the skyrmion splitting point. If the skyrmion is elongated too much, it is unable to relax to its original configuration and splits in two. (b) Skyrmion elongation evolution in time as a function of current pulses strength, but differently from (a) with current pulses selected for each case to be large enough to split the skyrmion. For each current the breaking point happens between 123–133 nm, indicated by the gray area. LLG parameters are $J=2.5$ pJ/m, $D=1 \mathrm{mJ}/{\mathrm{m}}^2, {\mu }_{0}H=0.1\mathrm{T}$, and $\alpha$ = 0.015.

Figure 4

(a) Skyrmion elongation evolution in time after different spin current pulses (0.4 ns current pulse for the black, red, and blue curves and 0.7 ns for the green curve). (b) Frequency of the elongated skyrmion rotation versus the time for the same cases as in (a). The dotted line shows where the length and frequency are equal. The LLG parameters are $J=2.5pJ/m, D=1mJ/m^2, \alpha =0.015$.

Figure 5

(a) Skyrmion elongation relaxation from the initial conditions for different damping parameters on the LLG equation. Note how the skyrmion elongation decreases linearly but with a different slope depending on the damping. (b) Rotation frequency of the skyrmion in time for the same damping-dependent relaxation processes as in (a). The frequency increases in a parabolic manner while the radius decreases. The LLG parameters are $J=2.5\text{pJ/m}, D=1{\text{mJ/m}}^2, \alpha =0.015$.

Figure 6

(a) Shows $(\omega /{\omega }_0)$ vs ${(r/r_0)}^{-2}$ and the different slopes for the currents 3, 4, and $5\times {10}^{12}$ with a damping coefficient of 0.015 and one for $\alpha =0.1$ (green curve). (b) The slope vs the damping coefficient is plotted for the current $4\times {10}^{12}$. The slope saturates for higher $\alpha$ to just below 2. (c) Higher damping coefficient will damping out faster to their original size with a straight line according to $1/\alpha$.

Figure 7

The evolution of the skyrmion in time under a spatial current density of $j_e=4\times {10}^{12} \mathrm{A}/{\mathrm{m}}^2$ for a current pulse of 0.6 ns, which is larger than in Fig. 1, and above the boundary between them. The elongation will result in a rupture, creating two new skyrmions. The LLG parameters are $A=2.5$ pJ/m, $D_{\perp }=1 \mathrm{mJ}/{\mathrm{m}}^2, {\mu }_{0}H=0.1\mathrm{T}, P=0.4, M_s=1000$ kA/m, and $\alpha =0.015$.

Figure 8

The evolution of the skyrmion in time under a spatial current density of $j_e=4\times {10}^{12} \mathrm{A}/{\mathrm{m}}^2$ for a current pulse of 0.5 ns. The elongated skyrmion starts rotating counterclockwise as indicated by the black arrow. The black straight line indicates the angle of the skyrmion. The LLG parameters are $A=2.5$ pJ/m, $D_{\perp }=1 \mathrm{mJ}/{\mathrm{m}}^2, {\mu }_{0}H=0.1\mathrm{T}, P=0.4, M_s=1000$ kA/m, and $\alpha =0.015$.