Elliptical skyrmion moving along a track without transverse speed

It was recently demonstrated that introducing the shape anisotropy of a skyrmion is an alternative strategy for controlling the skyrmion Hall effect. For example, an elliptical skyrmion shows anisotropic motion under the applied driving force. Here, we study the fundamental properties of an elliptical skyrmion induced by uniaxial in-plane magnetic anisotropy. The dynamics of a skyrmion under various parameters, such as uniaxial in-plane magnetic anisotropy, interfacial Dzyaloshinskii-Moriya interaction, and the damping constant, are investigated, and we employ Thiele's theory and derive expressions for the velocity of the skyrmion that match very well with micromagnetic simulation results. We find that the skyrmion Hall angle can reach ${90}^{\circ }$ with the optimized angle between the in-plane easy axis and the current. Our results offer a potential application of racetrack memory based on skyrmions, and a type of racetrack configuration was designed.

Figure 1

Schematic diagram of current-driven skyrmion dynamics. (a) The bilayer geometry considered in this work. (b) Motion of ordinary skyrmions driven by current. The motion of an elliptical skyrmion driven by current with (c) ${\theta }_{\mathrm{Div}}>0$, (d) ${\theta }_{\mathrm{Div}}<0$, and (e) ${\theta }_{\mathrm{Div}}=0$ (${\theta }_{\mathrm{Div}}=\frac{\pi }{2}-{\theta }_{\mathrm{SKH}}$). J is the applied current in the heavy-metal layer, and v is the velocity of the skyrmion. The red dashed line represents the easy axis of the uniaxial in-plane magnetic anisotropy; for the convenience of representation, we use $x^{\prime }$ as the in-plane easy axis to establish the $x^{\prime }\text{−}{y}^{\prime }$ coordinate system. ${\psi }_J$ is the angle between the applied current and the easy axis of the uniaxial in-plane magnetic anisotropy.

Figure 2

Skyrmion velocity and Hall angle at different ${\psi }_J$. (a) The absolute value of the velocity of the skyrmion, (b) the velocity parallel to the track, (c) the velocity perpendicular to the track, and (d) ${\theta }_{\mathrm{Div}}$ at different ${\psi }_J$ and $K_{\text{IP}}$.

Figure 3

DMI-$K_{\text{IP}}$ phase diagrams of an elliptic skyrmion. Phase diagram of (a) the morphology of the skyrmion, (b) the area of the skyrmions, (c) the different diagonal elements of the dissipation tensor $d_D$, (d) the oscillation amplitude of the skyrmion Hall angle $\mathrm{\Delta }{\theta }_{\mathrm{Div}}$, (e) the average value of the skyrmion Hall angle ${\overline{\theta }}_{\mathrm{Div}}$, and (f) the first zero point ${\psi }_{\mathrm{ZP}1}$.

Figure 4

${\theta }_{\mathrm{Div}}$ and zero point under the influence of $\alpha$. (a)–(c) ${\theta }_{\mathrm{Div}}\text{−}{\psi }_J$ for various $\alpha$ and $K_{\text{IP}}$ and (d)–(f) the DMI-$K_{\text{IP}}$ phase diagrams of ${\psi }_{\mathrm{ZP}1}$ for various $\alpha$. The symbols represent the results of the simulations, and the solid lines shows the results of the theory.

Figure 5

DMI-$K_{\text{IP}}$ phase diagrams of ${\theta }_{\mathrm{SKH}}^{\min }$ with different $\alpha$ in elliptic skyrmions. $\alpha$ increases from 0.1 to 0.6 in (a) to (e).

Figure 6

Minimum value of the skyrmion Hall angle at different $\alpha$ under different combinations of DMI and $K_{\text{IP}}$.