Analytical theory of modulated magnetic solitons

Abstract

Droplet solitons are coherently precessing solitary waves that have been recently realized in thin ferromagnets with perpendicular anisotropy.In the strongly nonlinear regime, droplets can be well approximated by a slowly precessing, circular domain wall with a hyperbolic tangent form. Utilizing this representation, this work develops a general droplet modulation theory and applies it to study the long range effects of the magnetostatic field and a nanocontact spin torque oscillator (NC-STO) where spin polarized currents act as a gain source to counteract magnetic damping. An analysis of the dynamical equations for the droplet's center, frequency and phase demonstrates a negative processional frequency shift due to long range dipolar interactions, dependent on film thickness. Further analysis also demonstrates the onset of a saddle-node bifurcation at the minimum sustaining current for the NC-STO. The basin of attraction associated with the stable node demonstrates that spin torque enacts a restoring force to excursions of the droplet from the nanocontact center, observed previously in numerical simulations. Large excursions lead to the droplet's eventual decay into spin waves

Deterministic drift instability and stochastic thermal perturbations of magnetic dissipative droplet solitons

“…The phase φ and position ξ dynamics have a leading order linear coupling to the frequency ω and velocity v equations, respectively. The second, additional terms in the phase and position equations proportional to current σ correspond to higher order corrections from soliton perturbation theory, which prove to be essential for describing perturbed droplet dynamics 8,22 , in particular, the finite temperature effects explored here. The terms W i , with i = φ, ξ, ω, and v, are scaled Weiner processes, with nontrivial covariance structure.…”

“…In order to achieve sufficient current density to oppose magnetic damping, a nanocontact (NC) of radius R * is placed on top of the free layer, confining the current to flow in an approximately cylindrical path 20 and therefore defining a region of effectively zero damping in the free layer. An external, perpendicular applied field H 0 is generally used in NC-STOs both to tilt the polarizer (useful for increasing STT and magnetoresistance), to provide an external source for the Larmor frequency, and to stabilize the droplet 8 .…”

Chiral excitations of magnetic droplet solitons driven by their own inertia

“…Droplets are unique as they are inherently dynamical solitons. Due to their dynamical features and internal degrees of freedom, droplets can be considered dynamical particles carrying an effective mass [16,23], which they gain from the applied STT. This means that, in the absence of the STT, they lose their effective mass and dissipate.…”

Stochastic ejection of nanocontact droplet solitons via drift instability

“…Dependence of the diffusion tensor on the state variables precludes a definite statement that this is true, but we have observed it in all cases we have computed. Since this exit mechanism occurs strictly via changes in ω, thereby mimicking the impact of damping on solitons [13], we refer to it as a decay mode of exit.…”

“…The existence and dynamics of droplets in nanocontacts have been successfully described using soliton perturbation theory [13,14], that yields a system of ordinary differential equations (ODEs) for the droplet's parameters that we call the modulation equations. The ODEs' fixed points correspond to sustained droplets, which, for a certain parameter regime, are unstable and manifest an increase in the droplet's speed, i.e., the drift instability [15].…”