Title: Robust mutual synchronization in long spin Hall nano-oscillator chains

Abstract: Mutual synchronization of N serially connected spintronic nano-oscillators increases their coherence by a factor $N$ and their output power by $N^2$. Increasing the number of mutually synchronized nano-oscillators in chains is hence of great importance for better signal quality and also for emerging applications such as oscillator-based neuromorphic computing and Ising machines where larger N can tackle larger problems. Here we fabricate spin Hall nano-oscillator chains of up to 50 serially connected nano-constrictions in W/NiFe, W/CoFeB/MgO, and NiFe/Pt stacks and demonstrate robust and complete mutual synchronization of up to 21 nano-constrictions, reaching linewidths of below 200 kHz and quality factors beyond 79,000, while operating at 10 GHz. We also find a square increase in the peak power with the increasing number of mutually synchronized oscillators, resulting in a factor of 400 higher peak power in long chains compared to individual nano-constrictions. Although chains longer than 21 nano-constrictions also show complete mutual synchronization, it is not as robust and their signal quality does not improve as much as they prefer to break up into partially synchronized states. The low current and low field operation of these oscillators along with their wide frequency tunability (2-28 GHz) with both current and magnetic fields, make them ideal candidates for on-chip GHz-range applications and neuromorphic computing.