Spin wave modes of two magnetostatic coupled spin transfer torque nano-oscillators

Abstract

A detailed analytical and numerical study of the spin wave modes of two nanopillar spin torque nano-oscillators coupled by magnetostatic interactions is presented under the macrospin approximation. Results show that the normal modes of the system oscillate with the magnetizations in-phase or anti-phase in both disks. The frequencies and critical current densities necessary to induce auto-oscillations of the spin wave modes of the coupled system depend on the relative position of the nanopillars and the applied magnetic field. If the oscillators are identical, these modes are degenerate at a certain relative position of the nanopillars, while if the oscillators are non-identical, such degeneracy is removed. Then, we can conclude that the magnetostatic coupling between two spin transfer torque nano-oscillators is a powerful mechanism to control the spin wave modes of these systems.

Synchronization of two spin-transfer-driven nano-oscillators coupled via magnetostatic fields

“…25 . We note that for θ = 0, we have ω ap < ω ip , that is, the antiphase mode has lower frequency and consequently requires less energy to be excited 31 . Thus, it is natural to observe this mode at the onset of the Andronov-Hopf instability [see the left panels of the Figs.…”

“…In the case of single-valve devices, the self-sustained magnetization oscillations emerge when the applied current density injects enough energy to counterbalance the dissipation. For two nano-oscillators coupled via magnetostatic fields, there are two critical current densities which correspond to the ones of the in-phase and antiphase modes of oscillation 31 . If the applied current density exceeds the minimum critical current density of the system, the magnetizations of the free layers oscillate.…”

“…When the current density attains its critical value β J = β J,c , the linear modes of the system correspond to an in-phase oscillation δm ip = A 0 v ip e iωipt + c.c, or an antiphase one δm ap = B 0 v ap e iωapt + c.c, where (ω ip , v ip ) and (ω ap , v ap ) are the frequencies and eigenvectors of the in-phase and antiphase modes 31 , respectively. Explicit expressions for the vectors and frequencies are given in Appendix B.…”

“…Despite the advances made by these authors, it is important to note that in an array of oscillators, the relative position between them plays an essential role because it may favor a parallel or antiparallel magnetization alignment. In a previous work 31 , developing a linear study, we concluded that it is possible to control the normal modes of STNOs as well as the critical current densities necessary to induce oscillations of the magnetization by changing the relative position of the oscillators. Then, in order to understand the role that the position of the oscillators could play in the synchronization phenomenon, it is necessary to study the nonlinear dynamics of the magnetization.…”

“…Furthermore, the difference between the oscillation phases remains constant. This synchronized motion can have two modes, namely the in-phase and the antiphase ones 31 , where the phase difference is zero and π, respectively. To elucidate the behavior of the modes for larger currents, we derive simplified equations for the envelope of the in-phase (IP) and antiphase (AP) modes from the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation.…”