Role of nonlinear anisotropic damping in the magnetization dynamics of topological solitons

The consequences of nonlinear anisotropic damping, driven by the presence of Rashba spin-orbit coupling in thin ferromagnetic metals, are examined for the dynamics of topological magnetic solitons such as domain walls, vortices, and skyrmions. The damping is found to affect Bloch and Néel walls differently in the steady-state regime below Walker breakdown and leads to a monotonic increase in the wall velocity above this transition for large values of the Rashba coefficient. For vortices and skyrmions, a generalization of the damping tensor within the Thiele formalism is presented. It is found that chiral components of the damping affect vortexlike and hedgehoglike skyrmions in different ways, but the dominant effect is an overall increase in the viscouslike damping.

Figure 1

Bloch wall dynamics. (a) Steady-state domain-wall velocity $\langle {\dot{X}}_0\rangle$ as a function of perpendicular applied magnetic field ${\mu }_0{H}_z$ for several values of the Rashba coefficient ${\alpha }_{\mathrm{R}}$. The horizontal dashed line indicates the linear Walker velocity and the arrows indicate the Walker transition. (b) The ratio between the Walker velocity $v_{\mathrm{W}}$ to its linear damping value $v_{\mathrm{W},0}$ as a function of ${\alpha }_{\mathrm{R}}$. (c) Deviation in the wall angle from rest at the Walker velocity $\delta {\varphi }_{\mathrm{W}}$ as a function of ${\alpha }_R$.

Figure 2

Dzyaloshinskii (Néel) wall dynamics. (a) Steady-state domain-wall velocity $\langle {\dot{X}}_0\rangle$ as a function of perpendicular applied magnetic field ${\mu }_0{H}_z$ for several values of the Rashba coefficient ${\alpha }_{\mathrm{R}}$. The horizontal dashed line indicates the linear Walker velocity and the arrows indicate the Walker transition. (b) The ratio between the Walker velocity $v_{\mathrm{W}}$ to its linear damping value $v_{\mathrm{W},0}$ as a function of ${\alpha }_R$. (c) The wall angle at the Walker velocity ${\varphi }_{\mathrm{W}}$ as a function of ${\alpha }_R$.

Figure 3

In-plane magnetization fields for vortices and skyrmions. (a) Vector fields given by $\varphi (x,y)$ in (18) for different values of $q$ and $c$. (b) Vortex and (c) skyrmion for spin structure with $c=1,q=1$, where the arrows indicate the in-plane components ($m_{x,y}$) and the color code gives the perpendicular component of the magnetization ($m_z$).