  The reduction of phase noise in electronic systems is of utmost importance in
modern communication and signal processing applications and requires an
understanding of the underlying physical processes. Here, we systematically
study the phase noise in mutually synchronized chains of nano-constriction spin
Hall nano-oscillators (SHNOs). We find that longer chains have improved phase
noise figures at low offset frequencies (1/f noise), where chains of two and
ten mutually synchronized SHNOs have 2.8 and 6.2 dB lower phase noise than
single SHNOs. This is close to the theoretical values of 3 and 10 dB, and the
deviation is ascribed to process variations between nano-constrictions.
However, at higher offset frequencies (thermal noise), the phase noise
unexpectedly increases with chain length, which we ascribe to process
variations, a higher operating temperature in the long chains at the same drive
current and phase delays in the coupling between nano-constrictions.
