Theory for a dissipative droplet soliton excited by a spin torque nanocontact

A distinct type of solitary wave is predicted to form in spin torque oscillators when the free layer has a sufficiently large perpendicular anisotropy. In this structure, which is a dissipative version of the conservative droplet soliton originally studied in 1977 by Ivanov and Kosevich, spin torque counteracts the damping that would otherwise destroy the mode. Asymptotic methods are used to derive conditions on perpendicular anisotropy strength and applied current under which a dissipative droplet can be nucleated and sustained. Numerical methods are used to confirm the stability of the droplet against various perturbations that are likely in experiments, including tilting of the applied field, nonzero spin torque asymmetry, and nontrivial Oersted fields. Under certain conditions, the droplet experiences a drift instability in which it propagates away from the nanocontact and is then destroyed by damping.

Figure 1

(Color online) Schematic of nanocontact (yellow central disk of radius ${\rho }_{\ast }$), the domain of $\mathbf{m}$ (${\mathbb{R}}^2$ in polar coordinates with radius $\rho$, azimuthal angle $\phi$), the range of $\mathbf{m}$ (unit sphere ${\mathbb{S}}^2$ with polar angle $\Theta$, azimuthal angle $\Phi$), orientations of the applied field ${\mathbf{h}}_0$ and fixed layer ${\mathbf{m}}_{\text{f}}$.

Figure 2

(Color online) Conservative droplet profiles with $m_z=\text{cos}{\Theta }_0$.

Figure 3

(Color online) Dissipative droplet soliton for the high-symmetry case with ${\theta }_{\text{f}}=0$. The color scale corresponds to $m_z$ while the vector field corresponds to the in-plane components $\left(m_x,m_y\right)$. The circle here and in future plots represents the boundary of the nanocontact. Parameters are ${\sigma }_{\text{sus}}/\alpha =0.94$, $\omega =0.17$, ${\rho }_{\ast }^{\prime }=5.24$, and $\nu =0$.

Figure 4

(Color online) Dissipative droplet frequency (dashed and solid curves) as a function of sustaining current from Eq. (18). Parameters are ${\rho }_{\ast }^{\prime }=5.24$ and $\nu =0$. The dashed-dotted line is the FMR frequency.

Figure 5

(Color online) Dissipative droplet properties for varying contact radius ${\rho }_{\ast }^{\prime }$ and spin torque asymmetry $\nu$: (a) minimum sustaining current, (b) maximum frequency, (c) maximum $m_z$ at origin, and (d) droplet radius (dotted line is ${\rho }_{\text{drop}}^{\prime }={\rho }_{\ast }^{\prime }$, plotted for comparison).

Figure 6

(Color online) Dissipative droplet in the presence of the Oersted field and with the applied field and fixed layer nearly perpendicular to the plane. Parameter values are $h_k=1.25$, $\alpha =0.03$, ${\rho }_{\ast }=5.24$, $h_0=1.8$, ${\theta }_0=0.00001°$, ${\theta }_{\text{f}}=0.40°$, $\nu =0.257$, and $\sigma =0.196$.

Figure 7

(Color online) Dots: perturbed droplet frequency as a function of current with the Oersted field computed from micromagnetic simulations. Crosses: perturbed droplets that undergo a drift instability; the frequency is calculated before the instability manifests. Solid curve: droplet frequency as a function of the sustaining current from Eq. (20). Dashed-dotted: the Zeeman $\left(h_0\right)$ and FMR $\left(h_0+h_k-1\right)$ frequencies. Triangle: the Slonczewski critical current and onset frequency for high-symmetry (Ref. 8) (see Appendix ). Solid vertical line: numerically computed threshold current in the presence of the Oersted field and the canted fixed layer. Parameter values are the same as in Fig. 6.

Figure 8

(Color online) Time sequence of a strongly perturbed droplet over one period of precession in the presence of a canted applied field, canted fixed layer, and Oersted field. Parameter values are $h_k=1.25$, $\alpha =0.03$, ${\rho }_{\ast }=5.24$, $h_0=1.8$, ${\theta }_0=5°$, ${\theta }_{\text{f}}=31.4°$, $\nu =0.257$, and $\sigma =0.189$. The time of the initial panel here and in Fig. 9 is set to 0 for comparison. The simulation actually began earlier.

Figure 9

(Color online) Time sequence showing the droplet drift instability for the same parameter values as in Fig. 8 but with smaller current $\sigma =0.121$. To facilitate visualization, the length of the in-plane magnetization vectors is normalized to the largest value in each frame.

Figure 10

(Color online) Birth of a dissipative droplet soliton for the current $\sigma =0.186$, above the Slonczewski critical current ${\sigma }_{\text{s}}=0.160$. Parameter values are the same as those in Fig. 8.

Figure 11

Modulation parameter $\text{Im}\left(\xi \right)$ as a function of $h_k$. When $\text{Im}\left(\xi \right)>0$, the weakly nonlinear Slonczewski mode is modulationally unstable. Parameter values are $\alpha =0.03$, ${\rho }_{\ast }=5.24$, $h_0=1.8$, and $\nu =0.26$.

Figure 12

(Color online) The critical anisotropy field $h_k^{\text{cr}}$ as a function of contact radius ${\rho }_{\ast }$ for various spin torque asymmetries $\nu$. Other parameters are $\alpha =0.03$ and $h_0=1.8$.