Pulse-coupled synchronization in an array of spin torque nano oscillators

In this work, we propose a novel synchronization scheme utilizing pulse current interactions to stabilize the mutual synchronization in a parallel array architecture. Specifically, we numerically investigate the effects of transmission delays on synchronization properties of an STNO array. Importantly, we demonstrate that complete synchronization can occur under the presence of delay. Here we consider the magnetization dynamics of an ensemble of STNOs at macroscopic level. For each STNO, we assumed the most conventional structure that consists of a fixed ferromagnetic layer, a free ferromagnetic layer, and a nonmagnetic spacer layer. The magnetization dynamics of the STNO can be well described by the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation with a macro-spin approximation. For simplicity, we consider the dynamical properties of the LLGS equation in spherical coordinates (θ, φ). By considering practical situations, we assumed that a peripheral CMOS circuit for each STNO generates pulse current instantaneously when the resistance R becomes close to a certain value near the maximum of R. Simultaneously, the pulse current was assumed to be pulled from all STNOs via CMOS current mirrors that act as a low-pass filter. The filtered pulse current can be approximately modeled using the alpha function. We numerically investigated the synchronization properties of an array of fully pulse-coupled STNOs with transmission delay. The system of LLGS equations with a fixed delay were simulated with XPPAUT, which is a solver for delay differential equations. Through the following simulations, the common parameters were fixed so that each STNO exhibits the out-of-plane precession mode and the number of STNOs as N = 5. We set the bias current as a control parameter. The time constant of the alpha function was tuned so that the pulse duration time became enough narrower than a period of the STNO in a free-running state. The time evolution of the pulse-coupled- STNO array is shown, in which we observed x_i = sinθ cosφ as the state of i-th STNO in the array. In the case of no transmission delay, the STNO array exhibits an asynchronous state. In contrast, in the presence of the delay within a certain range, the STNO array became a synchronous state, i.e, a complete synchronization state. The range of the delay for achieving the synchronous state were measured. The results indicate that synchronization conditions are determined by relative timing of pulse current interactions. Based on the results, we will design a practical synchronization scheme with CMOS peripheral pulse generator circuitries in the near future. For further understanding, we are going to systematically study the synchronization mechanism based on the phase reduction theory.