Engineering the dynamics of topological spin textures by anisotropic spin-orbit torques

Integrating topologically stabilized magnetic textures such as skyrmions as nanoscale information carriers into future technologies requires the reliable control by electric currents. Here, we uncover that the relevant skyrmion Hall effect, which describes the deflection of moving skyrmions from the current flow direction, acquires important corrections owing to anisotropic spin-orbit torques that alter the dynamics of topological spin structures. Thereby, we propose a viable means for manipulating the current-induced motion of skyrmions and antiskyrmions. Based on these insights, we demonstrate by first-principles calculations and symmetry arguments that the motion of spin textures can be tailored by materials design in magnetic multilayers of Ir/Co/Pt and Au/Co/Pt. Our work advances the understanding of the current-induced dynamics of these magnetic textures, which underlies a plethora of memory and logic applications.

Figure 1

(a),(b) Top and side views of the considered magnetic films that consist of Pt(111) underlayers (dark gray), a magnetic Co monolayer (light gray), and heavy-metal overlayers (dark red) such as Ir or Au. Bold integers label the different overlayers to highlight the $C_{3v}$ symmetry around the $z$ axis. Red arrows in (b) indicate the direction of the magnetic moment of Co. (c) Variation of the spin-orbit torkances ${\tau }_{yx}$ (red circles) and $-{\tau }_{xx}$ (blue squares) when alloying the overlayers from Ir over Pt to Au in the perpendicularly magnetized system. The effect of alloying is treated within the virtual crystal approximation.

Figure 2

(a),(b) First-principles dependence of (a) antidamping and (b) fieldlike torkance tensors on the magnetization direction in the ferromagnetic trilayer Ir/Co/Pt. The angles $\theta$ and $\phi$ are used to characterize the magnetization direction $\mathit{m}=(cos\phi sin\theta ,sin\phi sin\theta ,cos\theta )$ in spherical coordinates. Filled circles, squares, and triangles denote data for the $xx,yx$, and $zx$ elements of the tensors, respectively, and solid lines represent corresponding fits according to the Neumann expansion. (c),(d) Same as in panels (a) and (b) but for Au/Co/Pt.

Figure 3

(a) General, radially symmetric Néel-type skyrmion profile with core size $c$ and width $w$ of the domain wall. Colors encode the out-of-plane component of the local magnetization. (b),(c) Skyrmions and antiskyrmions are subject to current-induced forces that act parallel $(F_{\parallel })$ or perpendicular $\left(F_{\perp }\right)$ to the current flow direction. Green and yellow colors correspond to clockwise and counterclockwise rotational senses along the radial direction, respectively, and the brightness denotes the out-of-plane spin component. (d),(e) Forces $F_{\parallel }$ (dashed blue line) and $F_{\perp }$ (solid red line), per electric field $E$, on the radially symmetric Néel skyrmion in the magnetic trilayers (d) Ir/Co/Pt and (e) Au/Co/Pt. The position of the Fermi energy is varied, the value of the force $F_{\perp }$ perpendicular to the electric-field direction is scaled by a factor of four, and $c=w/2$ is used in (d)–(g). (f),(g) Real-space distribution of the force density $f_{\perp }$ in (f) Ir/Co/Pt and (g) Au/Co/Pt for the self-consistent value of the Fermi energy. Bold integers as well as circles of increasing radius indicate the radial distance $\rho$ in units of $w$, the skyrmion core is located in the center, and the arrow points along the electric field $\mathit{E}$. The black circle denotes the radius for which the out-of-plane component of the magnetization is zero.

Figure 4

Deformations of the skyrmion profile activate additional contributions to the perpendicular force $F_{\perp }$ in (a) Ir/Co/Pt and (b) Au/Co/Pt. The shaded area denotes the force for the rigid symmetric texture, whereas solid blue $\left(\delta c=0.1w\right)$ and solid red $\left(\delta c=0.2w\right)$ lines represent the corresponding changes owing to deformations of the spin profile via the modulated core size $c+\delta c{sin}^2\varphi$. In the case of $\delta c=0.2w$, the dashed black line indicates the new force component due to the lowest-order fieldlike torque.

Figure 5

(a) Due to the skyrmion Hall effect, skyrmions and antiskyrmions move under the angle ${\theta }_{\mathrm{sk}}$ with respect to the applied electric field $\mathit{E}$. The color scale for the spin textures follows the one of Fig. 3. (b) Skyrmion Hall angle ${\theta }_{\mathrm{sk}}$ in the magnetic trilayers Ir/Co/Pt (solid red line) and Au/Co/Pt (dashed blue line) as the Fermi energy is varied. The results are obtained by considering the symmetry-allowed first-principles shape of the spin-orbit torques. For comparison, the thin gray line indicates the value of ${\theta }_{\mathrm{sk}}$ in the absence of anisotropic fieldlike torques. The chosen Gilbert damping parameter of ${\alpha }_{\mathrm{G}}=0.05$ is assumed to be independent of the energy.