Spin-orbit torque can drive auto-oscillations of propagating spin wave (PSW) modes in nano-constriction spin Hall nano-oscillators (SHNOs). These modes allow both long-range coupling and the potential of controlling its phase -- critical aspect for nano-magnonics, spin wave logic, and Ising machines. Here, we demonstrate PSW-driven variable-phase coupling between two nano-constriction SHNOs and study how their separation and the PSW wave vector impact their mutual synchronization. In addition to ordinary in-phase mutual synchronization, we observe, using both electrical measurements and phase-resolved μ−Brillouin Light Scattering microscopy, mutual synchronization with a phase that can be tuned from 0 to π using the drive current or the applied field. Micromagnetic simulations corroborate the experiments and visualize how the PSW patterns in the bridge connecting the two nano-constrictions govern the coupling. These results advance the capabilities of mutually synchronized SHNOs and open up new possibilities for applications in spin wave logic, unconventional computing, and Ising Machines.
@article{arxiv.2402.00586,
title = {Spin wave-driven variable-phase mutual synchronization in spin Hall nano-oscillators},
author = {Akash Kumar and Avinash kumar Chaurasiya and Victor H. González and Nilamani Behera and Roman Khymyn and Ahmad A. Awad and Johan Åkerman},
journal= {arXiv preprint arXiv:2402.00586},
year = {2025}
}