Spin-Transfer Torque on Curved Surfaces: A Generalized Thiele Formalism

Abstract

Curvature is a highly relevant parameter when considering nanostructures, favoring the stability and affecting the dynamics of magnetic textures. In this work, we address the spin-transfer torque phenomenon by deriving an expanded Thiele equation with the Zhang-Li term for curved surfaces. Our results show a coupling between current and curvature, which is perceived as a gyrovector and an additional dissipative tensor associated with this coupling. Using this model, we determine the dynamics of a skyrmion in a nanotube with Gaussian and variable mean curvature. The new terms included in the Thiele equation are responsible for an additional Hall effect in the skyrmion dynamics and for the generalization of the Walker limit condition.

Figures (4)

• Figure 1: Geometrical framework of the curved magnetic system. (a) Curvilinear coordinate system $(\xi^{1}, \xi^2)$ defining the manifold, with an inset showing the local magnetization parametrization via spherical angles. Coordinates $X^1$ and $X^2$ represent the position of the center of the skyrmion nucleated on the surface. (b) Diagram of a planar skyrmion profile mapped onto the curved surface. (c) Specific geometry of a bent nanotube. (d) Illustration of several nanotube opening angles and their resulting shape. (e,f) Spatial distribution of the Gaussian $\mathscr{K}$ and mean $\mathscr{H}$ curvatures for distinct opening angles with $r = 50 \ \text{nm}$.
• Figure 2: Skyrmion energy as a function of its position. (a) and (b) Show the energy of the Néel skyrmion for different opening angles and $L=1130$ nm, (c) shows the energy of the Bloch skyrmion for a fixed opening angle $\varphi = 20\pi/11$ and $L=565$ nm. (d) Shows the skyrmion profile reconstructed form micromagnetic simulations in its equilibrium position: top and bottom rows correspond to the Néel and Bloch skyrmions, respectively, for negative and positive DMI values. Symbols in (a)-(c) correspond to the data obtained by means of numerical simulations, and lines in (a) and (b) correspond to the analytical predictions.
• Figure 3: Skyrmion trajectory for $\alpha=\beta = 0.5$, $|J|=10\times10^{12}$ A/m $^2$, $\varphi=2\pi$, and $L=1130$ nm. (a) Skyrmion position as a function of time. Purple line and circle depict the trajectory predicted by the analytical model (line) and micromagnetic simulations (circles). The black-dashed line shows the skyrmion trajectory in the absence of the CCD. (b) depicts the skyrmion trajectory along the bent tube. The dashed line shows the trajectory during the first 50 ns.
• Figure 4: Comparison of the translational and oscillatory skyrmion motions is shown in panels (a) and (b), respectively. Here, we consider a torus with $R=10\ell$ and $r=2\ell$, and a skyrmion with $N_{\text{top}}=-1$ with radius $R_s=0.5\ell$. These parameters correspond to $V_g\approx-0.027v_0$ with $v_0=\gamma\sqrt{A_{ex}K}/M_s$, and $\Omega_0\approx-4.3\times10^{-4}\gamma K/M_s$. For $\alpha=0.1$ and $\beta=0.3$ we obtain $\Upsilon\approx0.062$ and estimate from \ref{['eq:uw']}$u_{\text{w}}\approx0.0271v_0$. The exact value numerically obtained from \ref{['eq:Thiele']} is $u_{\text{w}}\approx0.0286v_0$. For the considered values of $R$, $r$, and $R_s$, the condition $\Upsilon>1$ is shown on panels (c) and (d) by the orange shadowing. The dependence $u_{\mathrm{w}}(\alpha)$ determined by Eq. \ref{['eq:uw']} is shown on panel (d) for fixed $\beta=0.6$.