Phase binarization in mutually synchronized bias field free spin-Hall nano-oscillators for reservoir computing

Mutually coupled spin-Hall nano-oscillators (SHNO) can exhibit a binarized phase state, offering pathways to realize Ising machines and efficient neuromorphic hardware. Conventionally, phase binarization is achieved in coupled identical SHNOs via injecting an external microwave at twice the oscillator frequency in the presence of a strong biasing magnetic field. However, this technology poses potential challenges of higher energy consumption and complex circuit design. Moreover, differences in the individual characteristic frequencies of SHNOs resulting from fabrication-induced mismatch in SHNO dimensions may hinder their mutual synchronization. Addressing these challenges, we demonstrate purely dc current-driven mutual synchronization and phase binarization of two nonidentical nanoconstriction SHNOs without biasing magnetic field and microwave injection. We thoroughly investigate these phenomena and underlying mechanisms using micromagnetic simulation. We show how the localized fundamental mode of the spin wave emerging from the magnetization auto-oscillation reinforces the mutual synchronization, while the second-harmonic spin wave induces the phase binarization in the coupled SHNO pair. We further demonstrate the bias field free synchronized SHNO pair efficiently performing a reservoir computing benchmark learning task: sin- and square-wave classification, with 100% accuracy, utilizing the current-tunable phase binarization phenomenon. Our results showcase promising magnetization dynamics of coupled bias field free SHNOs for future computing applications.