RKKY-like interactions between two magnetic skyrmions

Abstract

Understanding skyrmion-skyrmion interactions is crucial for effectively manipulating the motion of multiple skyrmions in racetrack and logic devices. However, the fundamental nature and microscopic origins of these interactions remain poorly understood. In this study, we investigate skyrmion-skyrmion interactions in chiral magnetic films and reveal that they possess intrinsic, anisotropic, and oscillatory characteristics. Specifically, we demonstrate that the attractive and repulsive forces between skyrmions oscillate with a well-defined period, akin to the Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling observed between two magnetic moments in metals. Our analysis uncovers the essential physics behind a previously unrecognized universal wavy tail in the skyrmion spin texture. Notably, the resulting RKKY-like interaction between skyrmions is universal for all tilted skyrmions, irrespective of whether the titled easy-axis is from an external field or a crystalline magnetic anisotropy. These findings introduce a novel physical principle for the design of skyrmion molecules or skyrmion superstructures, which hold significant potential for applications in skyrmion-based spintronics and neuromorphic computing.

Figures (4)

• Figure 1: (a) Energy surface $E(x,y)$ of two skyrmions for $H=0.164$T along the $y$-axis. $E=0$ is chosen for two isolated skyrmions infinitely apart. (b) $E(x,y)$ as a function of $x$ along $y = 0$ (black line), or $y$ along $x=0$ (red line), or $\sqrt{x^2+y^2}$ along $y=x$ (blue line). The vertical blue dash-lines indicate local minima.
• Figure 2: (a) A perpendicular skyrmion with $H = 0.164$T along the $-z$-axis. (b) $\Theta(\rho)$ (red) for $\phi=0$ (stars), $40^0$ (squares), $70^0$ (triangles) and $\Phi(\phi)$ (blue) for $\rho=10$nm (stars), $40$nm (squares), $70$nm (triangles). (c) An in-plane skyrmion $H=0.164$T along the $-y$-axis. Inset: density plots of $m_z>0$ (red) and $m_z<0$ (blue). The skyrmion center is indicated by a dashed circle. (d) $m_x$ (red) and $m_z$ (blue) as a function of $y$ for $x=5$nm (solid curve) and -15 nm (dashed lines).
• Figure 3: $\Phi(l)$ of skyrmion centred at $(0,0)$ (red symbols) and $(x_0, y_0)$ (blue symbols), and $\Phi(l)$ of the two coupled skyrmions (black dashed line) for $(x_0, y_0)= (0,66\mathrm{nm} )$, $(100\mathrm{nm},66\mathrm{nm})$, and $(-200\mathrm{nm}, 67\mathrm{nm})$ (a) and for $(x_0, y_0)=(0,113\mathrm{nm} )$, $(100\mathrm{nm}, 113\mathrm{nm})$, and $(-200\mathrm{nm},113\mathrm{nm})$ (b).
• Figure 4: Phase diagrams of RKKY-like interaction (the red), repulsive interaction (the green) , and non-existence of isolated skyrmion (the orange) in $\kappa'-\theta_H$ plane for $\mathbf{H}$ controlled asymmetric anisotropy (a) and in $\kappa-\theta$ planes for $K-\theta$ controlled symmetric magnetic anisotropy (b). (c) Trajectories of a bi-skyrmion molecule under a spin–orbit torque. Torque is applied only on the top skyrmion whose motion drags the other skyrmion to move together.