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Spin-orbit torque can drive auto-oscillations of propagating spin-wave modes in nano-constriction spin Hall nano-oscillators. These modes facilitate both long-range coupling and the possibility of controlling their phase, which is a crucial aspect for device application. Here, we demonstrate variable-phase coupling between two nano-constriction spin Hall nano-oscillators and their mutual synchronization driven by propagating spin waves. Using electrical measurements and phase-resolved micro-focused Brillouin light scattering microscopy, we show that the phase of the mutual synchronization can be tuned by modulating the drive current or the applied field. Our micromagnetic simulations explore the phase tunability using voltage gating. Our results advance the capabilities of mutually synchronized spin Hall nano-oscillators and open the possibilities for applications in spin-wave logic-based devices.
Kumar et al. (Wed,) studied this question.