Parametric autoexcitation of magnetic droplet soliton perimeter modes

Recent experiments performed in current-driven nanocontacts with strong perpendicular anisotropy have shown that spin-transfer torque can drive self-localized spin waves [W. H. Rippard, A. M. Deac, M. R. Pufall, J. M. Shaw, M. W. Keller, S. E. Russek, G. E. W. Bauer, and C. Serpico, Phys. Rev. B 81, 014426 (2010); S. M. Mohseni, S. R. Sani, J. Persson, T. N. A. Nguyen, S. Chung, Y. Pogoryelov, and J. Åkerman, Phys. Status Solidi RRL 5, 432 (2011)], that above a certain intensity threshold can condense into a nanosized and highly nonlinear dynamic state known as a magnetic droplet soliton [S. M. Mohseni, S. R. Sani, J. Persson, T. N. A. Nguyen, S. Chung, Y. Pogoryelov, P. K. Muduli, E. Iacocca, A. Eklund, R. K. Dumas, S. Bonetti, A. Deac, M. A. Hoefer, and J. Åkerman, Science 339, 1295 (2013)]. Here we demonstrate analytically, numerically, and experimentally that at sufficiently large driving currents and for a spin polarization direction tilted away from the normal to a nanocontact plane, the circular droplet soliton can become unstable against the excitations in the form of periodic deformations of its perimeter. We also show that these perimeter excitation modes (PEMs) can be excited parametrically when the fundamental droplet soliton precession frequency is close to the double frequency of one of the PEMs. As a consequence, with increasing magnitude of a bias magnetic field the PEMs with progressively higher indices and frequencies can be excited. Full qualitative and partly quantitative agreement with experiment confirm the presented theoretical picture.

Figure 1

Perimeter excitation modes. (a)–(e) Two-dimensional snapshots of the magnon drop PEM mode $n=3$ during one half-period of oscillations. The color code shows the magnitude of the out-of-plane $m_z$ component of the magnetization, with the arrows indicating the in-plane component. (f) Overlayed snapshots from the time evolution of the shape of the drop. (g) The numerically simulated PEM amplitude $a_n\left(t\right)$ for $n=3$. Dots in the frames (h) and (i) show the numerically calculated magnitudes of the PEM frequencies ${\mathrm{\Omega }}_n$ and damping rates ${\mathrm{\Gamma }}_n$ as functions of the PEM index $n$, respectively. Solid lines in these frames are showing the results of analytical calculations of the same quantities using Eqs. (12) and (13), respectively, for $\mathrm{\Omega }/2\pi =1.5\mathrm{GHz}$ [see Eq. (10)] and ${\alpha }_p=0.031$.

Figure 2

Spin transfer torque driven magnetic droplet solitons and the perimeter excitation modes. (a)–(c) Snapshots of the free layer magnetization in a micromagnetically simulated NC spin torque oscillator at a low current density ($j=0.2\times {10}^{12}\mathrm{A}/{\mathrm{m}}^2$). (d)–(f) Same simulation as in panels (a) to (c), however, at a much higher current density ($j=2.5\times {10}^{12}\mathrm{A}/{\mathrm{m}}^2$). High amplitude PEM excitations are observed at progressively higher mode numbers when the applied field is increased. The color indicates the perpendicular component $m_z$ with red (positive $m_z$) corresponding to spins aligned with the applied field, and blue (negative $m_z$) to spins pointing away from the applied field. The black arrows provide the corresponding in-plane direction and magnitude of the magnetization. The white circles outline the NCs.

Figure 3

Phase diagrams of the droplet PEMs parametrically excited by the uniform droplet self-oscillation mode: (a) on the plane “spin polarization angle ${\theta }_p$ vs bias magnetic field $H$”; (b) on the plane “droplet frequency ${\omega }_0$ vs bias magnetic field $H$.” The driving DC current density is $j=2.5\times {10}^{12}\mathrm{A}/{\mathrm{m}}^2$. High-amplitude PEMs are observed at progressively higher mode indices $n$ when the bias magnetic field $H$ (and, therefore, the droplet self-oscillation frequency ${\omega }_0$) is increased.

Figure 4

Comparison of micromagnetic simulations with experimental data on droplet PEM. Experimentally measured phase diagram of NC-STO on Co/Cu/Co-${\left[\text{Ni/Co}\right]}_{\times 4}$ orthogonal spin valve with an applied magnetic field of (a) 0.8 and (b) 0.9 T with a tilt angle of $7.5^{\circ }$. Simulated phase diagram of the same device with slightly different field strengths and tilt angle as compared to the experimental data: (c) $H=0.7\mathrm{T}$, (d) $H=0.8\mathrm{T}$ at a tilt angle of $3^{\circ }$. Other material parameters used in the simulation include $R_c=60\mathrm{nm}, A_{\mathrm{ex}}=30\mathrm{pJ}/\mathrm{m}, K=447\mathrm{kJ}/{\mathrm{m}}^3$, and $M_s=716\mathrm{kA}/\mathrm{m}$.