Propagation and control of nanoscale magnetic-droplet solitons

The propagation and controlled manipulation of strongly nonlinear, two-dimensional solitonic states in a thin, anisotropic ferromagnet are theoretically demonstrated. It has been recently proposed that spin-polarized currents in a nanocontact device could be used to nucleate a stationary dissipative droplet soliton. Here, an external magnetic field is introduced to accelerate and control the propagation of the soliton in a lossy medium. Soliton perturbation theory corroborated by two-dimensional micromagnetic simulations predicts several intriguing physical effects, including the acceleration of a stationary soliton by a magnetic field gradient, the stabilization of a stationary droplet by a uniform control field in the absence of spin torque, and the ability to control the soliton's speed by use of a time-varying, spatially uniform external field. Soliton propagation distances approach 10 $\mu$m in low-loss media, suggesting that droplet solitons could be viable information carriers in future spintronic applications, analogous to optical solitons in fiber optic communications.

Figure 1

(a), (b), (c) Droplet trajectories in the $V$-$\omega$ plane from numerical integration of the modulation equations (3) (solid curves) and micromagnetics (dashed curves). The dotted curves correspond to the edges of the precomputed droplet library and the thick, solid parabolic curve is the spin wave band. (a) Low-loss $\alpha =0.001$. (b) $\alpha =0.01$, positive bias field. (c) $\alpha =0.01$, negative bias field. See text for further details. (d) Droplet velocity time dependence for the solid/dashed trajectories in panels (b) and (c). (e), (f) Droplet profiles from micromagnetics corresponding to the circles in panels (b) and (c), respectively. The gray (color) scale represents the out-of-plane magnetization $m_z$ and the arrows represent the in-plane components.

Figure 2

Droplet acceleration by current flowing through two nanowires.

Figure 3

Droplet trajectories with uniform, static magnetic field and $\alpha =0.01$. (a) Modulation solution showing acceleration or deceleration of a propagating droplet due to damping and $h_0=-0.4$ (solid curves), $h_0=0$ (dashed curves), and $h_0=0.9$ (dash-dotted curves). (b) $h_0=1$. (c) $h_0=-0.7$. In panels (b) and (c), the dashed curves are micromagnetic simulations. In panel (c), the dash-dotted curve is the separatrix between the switched state $(V,\omega )=(0,0)$ and spin wave states.

Figure 4

Stabilization of a stationary droplet with ${\omega }_{*}=0.5$ by the linear feedback control law (8) with $G=2$. (a) Stationary droplet relaxation. (b) Trajectories in the $V$-$\omega$ plane.

Figure 5

Micromagnetic simulation giving the spatially averaged magnetization frequency due to the control field (8), with ${\omega }_{*}=0.4$, $G=1.5$, $\alpha =0.01$, and an update period of 74. The droplet is locked when $\Omega =0$.

Figure 6

Open-loop control of droplet speed. (a) Two trajectories (solid and dash-dotted curves) in the $V$-$\omega$ plane from micromagnetic simulations with (b) the corresponding control field (solid and dash-dotted curves, respectively) determined from the modulation equations (3).