Synchronization and Bistability of Two Uniaxial Spin-Transfer Oscillators with Field Coupling

Abstract

A spin-transfer oscillator is a nanoscale device demonstrating self-sustained precession of its magnetization vector whose length is preserved. Thus, the phase space of this dynamical system is limited by a three-dimensional sphere. A generic oscillator is described by the Landau – Lifshitz – Gilbert – Slonczewski equation, and we consider a particular case of uniaxial symmetry when the equation yet experimentally relevant is reduced to a dramatically simple form. The established regime of a single oscillator is a purely sinusoidal limit cycle coinciding with a circle of sphere latitude (assuming that points where the symmetry axis passes through the sphere are the poles). On the limit cycle the governing equations become linear in two oscillating magnetization vector components orthogonal to the axis, while the third one along the axis remains constant. In this paper we analyze how this effective linearity manifests itself when two such oscillators are mutually coupled via their magnetic fields. Using the phase approximation approach, we reveal that the system can exhibit bistability between synchronized and nonsynchronized oscillations. For the synchronized one the Adler equation is derived, and the estimates for the boundaries of the bistability area are obtained. The two-dimensional slices of the basins of attraction of the two coexisting solutions are considered. They are found to be embedded in each other, forming a series of parallel stripes. Charts of regimes and charts of Lyapunov exponents are computed numerically. Due to the effective linearity the overall structure of the charts is very simple; no higher-order synchronization tongues except the main one are observed.