Phase-Binarized Spin Hall Nano-Oscillator Arrays: Towards Spin Hall Ising Machines

Ising machines (IMs) are physical systems designed to find solutions to combinatorial optimization (CO) problems mapped onto the IM via the coupling strengths between its binary spins. Using its intrinsic dynamics and different annealing schemes, the IM relaxes over time to its lowest-energy state, which is the solution to the CO problem. IMs have been implemented on different platforms, and interacting nonlinear oscillators are particularly promising candidates. Here we demonstrate a pathway towards an oscillator-based IM using arrays of nanoconstriction spin Hall nano-oscillators (SHNOs). We show how SHNOs can be readily phase binarized and how their resulting microwave power corresponds to well-defined global phase states. To distinguish between degenerate states, we use phase-resolved Brillouin-light-scattering microscopy and directly observe the individual phase of each nanoconstriction. Micromagnetic simulations corroborate our experiments and confirm that our proposed IM platform can solve CO problems, showcased by how the phase states of a $2\ifmmode\times\else\texttimes\fi{}2$ SHNO array are solutions to a modified max-cut problem. Compared with the commercially available D-Wave ${\mathrm{Advantage}}^{\mathrm{TM}}$, our architecture holds significant promise for faster sampling, substantially reduced power consumption, and a dramatically smaller footprint.