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Description
The ability for a Spin Torque Nano Oscillator (STNO) to perform as a nano-scaled microwave voltage oscillator continues to be the focus of extensive research. Due to their small size (on the order of 10 nm), low power consumption, and ultrawide frequency range STNOs demonstrate significant potential for applications in microwave generation. To date, the ability for a STNO to produce microwave signals is achievable, however, the low power output produced by a single STNO currently renders them inoperable for applications. In response, various groups have proposed the synchronization of a network of STNOs such that the coherent signal produces a strong enough microwave signal at the nanoscale. Achieving synchronization, however, has proven to be a challenging task and raises complex problems related to the field of Nonlinear Dynamical Systems. In this work we analyze the problem of synchronization for networks of STNOs connected in parallel. Bifurcation diagrams for small networks of STNOs are computed which depicts bistability between in-phase and out-of-phase limit cycle oscillations for much of the phase space. Numerical approximations of the relative sizes of the basins of attraction are generated and further suggest the inherent difficulty in achieving synchronization. To extend the analysis to large networks of STNOs, we exploit the SN symmetry exhibited by the system all-to-all coupled STNOs. We develop implicit analytic expressions for Hopf bifurcations which yield synchronized limit cycle oscillations, allowing for the computation of the Hopf loci for an arbitrarily large network of oscillators. Through stability analysis we determine the parameter space for which the Hopf bifurcation is supercritical and exhibits a stable center-manifold. This analysis is completed for large arrays and used to numerically demonstrate synchronization in up to N = 1000 STNOs. These results should help guide future experiments and, eventually, lead to the design and fabrication of a nanoscale microwave signal generator.
