Controlling the deformation of antiferromagnetic skyrmions in the high-velocity regime

While antiferromagnetic skyrmions display appealing properties, their lateral expansion in the high-velocity regime hinders their potential for applications. In this work, we study the impact of spin Hall torque, spin transfer torque, and topological torque on the velocity-current relation of antiferromagnetic skyrmions with the aim of reducing this deformation. Using a combination of micromagnetic simulations and analytical derivations, we demonstrate that the lateral expansion of the antiferromagnetic skyrmion is reminiscent of the well-known Lorentz contraction identified in one-dimensional antiferromagnetic domain walls. We also show that in the flow regime the lateral expansion is accompanied by a progressive saturation of the skyrmion velocity when driven by spin Hall and topological torques. This saturation occurs at much smaller velocities when driven by the topological torque, while the lateral expansion is reduced, preventing the skyrmion size from diverging at large current densities. We extend this study toward synthetic antiferromagnets, where the weaker antiferromagnetic exchange leads to much larger lateral expansion at smaller current densities in all cases. This study suggests that a compromise must be made between skyrmion velocity and lateral expansion during the device design. In this respect, exploiting the topological torque could lead to better control of the skyrmion velocity in antiferromagnetic racetracks.

Figure 1

Schematics of the current-driven motion of an antiferromagnetic skyrmion upon (a) spin Hall torque arising from an adjacent metal and (b) topological torque arising from electron flowing along the magnetic texture. $j_e^{\mathrm{HM}}$ and $j_e^{\mathrm{AFM}}$ denote the current density flowing in the metallic layer and in the antiferromagnetic layer, respectively. The antiferromagnetic skyrmion is represented by the dark blue circle and its longitudinal and lateral widths are defined in (a). The curved red and blue arrows denote the bent trajectory of up and down spins upon the spin Hall effect.

Figure 2

Skyrmion motion in an antiferromagnet under spin Hall torque. (a) Simulated snapshots of spin Hall torque driven skyrmion motion showing the skyrmion deformation at high current densities. (b) Simulated skyrmion velocity-current characteristics (black dots). The dashed line shows the fit using Eq. (19). (c) Simulated skyrmion size variation, ${\mathrm{\Delta }}_x$ (open circles) and ${\mathrm{\Delta }}_y$ (open squares), as a function of applied current density. The dashed lines correspond to Eqs. (14) and (15). The scattered values of ${\mathrm{\Delta }}_{x,y}$ are due to the difficulty in accurately extracting the skyrmion size in the simulations due to the large micromagnetic unit cell (2 nm).

Figure 3

Skyrmion motion in an antiferromagnet under spin transfer torque. (a) Snapshots of spin transfer torque driven skyrmion motion showing the texture deformation at high current densities for $\beta =100\alpha$. (b) Simulated skyrmion velocity-current characteristics for $\beta =\alpha$ (black), $\beta =10\alpha$ (red), and $\beta =100\alpha$ (blue). The dashed line is a fit using Eq. (20). (c) Simulated skyrmion size variation, ${\mathrm{\Delta }}_x$ (open circles) and ${\mathrm{\Delta }}_y$ (open squares), as a function of applied current density for $\beta =100\alpha$. The dashed lines correspond to Eqs. (14) and (15).

Figure 4

Skyrmion motion in an antiferromagnet under topological torque. (a) Simulated snapshots of topological torque driven skyrmion motion at high current densities. (b) Simulated skyrmion velocity-current characteristics for $\beta =\alpha$ and ${\lambda }^2=0$ (black), ${\lambda }^2=1.6{\mathrm{nm}}^2$ (blue), and ${\lambda }^2=13{\mathrm{nm}}^2$ (red). The dashed line is a fit using Eq. (21). (c) Simulated skyrmion size variation, ${\mathrm{\Delta }}_x$ (open circles) and ${\mathrm{\Delta }}_y$ (open squares), as a function of applied current density for $\beta =\alpha$ and ${\lambda }^2=13{\mathrm{nm}}^2$. The dashed lines correspond to Eqs. (14) and (15). The small variation of the skyrmion shape and the relatively large size of the micromagnetic unit cell (2 nm) make the accurate estimation of ${\mathrm{\Delta }}_{x,y}$ difficult.

Figure 5

Skyrmion motion in a synthetic antiferromagnet under spin Hall torque with ${\theta }_{\mathrm{sh}}=0.12$ (red), spin transfer torque with $\beta =200\alpha$ (green), and topological torques, with $\lambda =1.6{\mathrm{nm}}^2$ (blue) and $\lambda =50{\mathrm{nm}}^2$ (black). In the latter two cases, we also set $\beta =\alpha$. (a) Simulated skyrmion velocity-current characteristics and (b) size variation as a function of applied current density. In (b), the solid lines are guides for the eye.

Figure 6

(a) Skyrmion diameter as a function of $A_{\mathrm{FM}}$, for $A_{\mathrm{AFM}}=-6.59$ pJ/m. Skyrmion velocity as a function of the current density when driven by the (b) spin Hall torque, (c) nonadiabatic torque with $\beta =100\alpha$, and (d) topological torque with ${\lambda }^2=13{\mathrm{nm}}^2$, for three values of $A_{\mathrm{FM}}$ and $A_{\mathrm{AFM}}=-6.59$ pJ/m.

Figure 7

Skyrmion diameter as a function of the (a) perpendicular anisotropy constant $K_z$, and (b) DMI parameter $D$, for $A_{\mathrm{FM}}=-A_{\mathrm{AFM}}=6.59$ pJ/m.

Figure 8

(a) Skyrmion diameter as a function of $A_{\mathrm{AFM}}$, when $A_{\mathrm{FM}}$ is fixed to 6.59 pJ/m. Skyrmion velocity as a function of the (b) spin Hall torque current density, (c) nonadiabatic torque current density when $\beta =100\alpha$, (d) including the topological torque when ${\lambda }^2=13{\mathrm{nm}}^2$, for three values of $A_{\mathrm{AFM}}$ and $A_{\mathrm{FM}}=6.59$ pJ/m.

Figure 9

(a) Skyrmion diameter as a function of $M_{\mathrm{s}}$, when $A_{\mathrm{FM}}=-A_{\mathrm{AFM}}=6.59$ pJ/m. Skyrmion velocity as a function of the current density when driven by the (b) spin Hall torque, (c) nonadiabatic torque with $\beta =100\alpha$, and (d) topological torque with ${\lambda }^2=13{\mathrm{nm}}^2$, for three values of $M_{\mathrm{s}}$ and $A_{\mathrm{FM}}=-A_{\mathrm{AFM}}=6.59$ pJ/m.

Figure 10

Skyrmion velocity as a function of the current density when driven by the spin Hall torque for three values of $\alpha$, with $A_{\mathrm{FM}}=-A_{\mathrm{AFM}}=6.59$ pJ/m and $M_{\mathrm{s}}=376$ kA/m.