Current-induced dynamics of skyrmion tubes in synthetic antiferromagnetic multilayers

Topological spin textures can be found in both two-dimensional and three-dimensional nanostructures, which are of great importance to advanced spintronic applications. Here we report the current-induced skyrmion tube dynamics in three-dimensional synthetic antiferromagnetic (SyAF) bilayer and multilayer nanostructures. It is found that the SyAF skyrmion tube made of thinner sublayer skyrmions is more stable during its motion, which ensures that a higher speed of the skyrmion tube can be reached effectively at larger driving current. In the SyAF multilayer with a given total thickness, the current-induced deformation of the SyAF skyrmion tube decreases with an increasing number of interfaces; namely, the rigidity of the SyAF skyrmion tube with a given thickness increases with the number of ferromagnetic (FM) layers. For the SyAF multilayer with an even number of FM layers, the skyrmion Hall effect can be eliminated when the thicknesses of all FM layers are identical. Larger damping parameter leads to smaller deformation and slower speed of the SyAF skyrmion tube. Larger fieldlike torque leads to larger deformation and a higher speed of the SyAF skyrmion tube. Our results are useful for understanding the dynamic behaviors of three-dimensional topological spin textures and may provide guidelines for building SyAF spintronic devices.

Figure 1

(a) Schematics of the simulation models. The total sample thickness is fixed at 12 nm. $N$ denotes the number of FM layers in a sample. For $N=2$, the thickness of each FM layer equals 6 nm. For $N=12$, the thickness of each FM layer equals 1 nm. In each sample, the adjacent FM layers are antiferromagnetically exchange coupled, forming a SyAF structure. (b) Illustration of a SyAF 2-layer skyrmion tube (i.e., $N=2$). Black arrows indicate the Magnus force acting on each FM layer. (c) Illustration of a SyAF 6-layer skyrmion tube (i.e., $N=6$). (d) Definitions of $R_x$ and $R_y$, which are used to describe the size and shape of the skyrmion in the $x\text{−}y$ plane of each FM layer.

Figure 2

(a) Skyrmion tube velocity $v$ as a function of driving current density $j$ for a SyAF $N$-layer skyrmion. (b) Horizontal distance between the top layer and bottom layer skyrmion centers in the $y$ direction $\mathrm{\Delta }y$ as a function of driving current density $j$. Note that when the SyAF $N$-layer skyrmion is driven into motion in the $x$ direction, the velocities of skyrmions in each layer are the same. Thus, the skyrmion center positions in the $x$ direction are the same in all FM layers, i.e., $\mathrm{\Delta }x=0$. (c) $R_x$ as a function of driving current density $j$ for the skyrmion in the bottom FM layer. (d) $R_y$ as a function of driving current density $j$ for the skyrmion in the bottom FM layer. (e) $R_y-R_x$ as a function of driving current density $j$ for the skyrmion in the bottom FM layer. (f) $\mathrm{\Delta }{R}_x$ (i.e., $R_x^{\mathrm{bottom}}-R_x^{\mathrm{top}}$) as a function of $N$ when $j=20\times {10}^{10}$ A/m. The inset shows the corresponding $\mathrm{\Delta }{R}_y$ (i.e., $R_y^{\mathrm{bottom}}-R_y^{\mathrm{top}}$).

Figure 3

(a) Schematic illustrations of deformed moving SyAF skyrmion tubes. The total thickness is 12 nm, i.e., 12 spins in the thickness direction. $N$ denotes the number of FM layers in a sample. For $N=2$, the thickness of each FM layer equals 6 nm. For $N=12$, the thickness of each FM layer equals 1 nm. At the same driving current density, the Magnus-force-induced deformation of the SyAF 2-layer skyrmion tube is larger than that of the SyAF 12-layer skyrmion. (b) Sublayer skyrmion center locations (in the $y$ direction) of deformed SyAF $N$-layer skyrmion tubes driven by a current density of $j=40\times {10}^{10}$ A/m. The layer index indicates the single-spin-thick sublayer position, for example, 1 and 12 denote the bottommost and topmost layers of the SyAF structure. (c) Sublayer skyrmion areas of a deformed SyAF 12-layer skyrmion tube driven by a current density of $j=40\times {10}^{10}$ A/m.

Figure 4

(a) Damping dependence of the SyAF 12-layer skyrmion tube velocity $v$ at $j=100\times {10}^{10}$ A/m. (b) Damping dependence of $\mathrm{\Delta }y$ of the SyAF 12-layer skyrmion tube at $j=100\times {10}^{10}$ A/m. (c) Damping dependences of $R_x$ and $R_y$ of the SyAF 12-layer skyrmion tube at $j=100\times {10}^{10}$ A/m.

Figure 5

Effect of the fieldlike torque strength $\xi$ on the current-induced motion of a SyAF 12-layer skyrmion at $j=100\times {10}^{10}$ A/m. (a) Velocity, (b) $\mathrm{\Delta }y$, and (c) $R_x$ and $R_y$ as a function of $\xi$.

Figure 6

The current-induced motion of a SyAF bilayer skyrmion (i.e., $N=2$). Here the total thickness of the sample is fixed at 6 nm. The thicknesses of the top and bottom FM layers are defined as $T_{\mathrm{top}}$ and $T_{\mathrm{bottom}}$, respectively. Namely, $T_{\mathrm{top}}+T_{\mathrm{bottom}}=6$ nm. (a) Velocity, (b) skyrmion Hall angle ${\theta }_{\mathrm{SkHE}}$, and (c) $R_x$ and $R_y$ as a function of $T_{\mathrm{top}}$ at $j=20\times {10}^{10}$ A/m.