  We investigate the individual impacts of critical magnetodynamical
parameters-effective magnetization and magnetic damping-on the auto-oscillation
characteristics of nano-constriction-based Spin Hall Nano-Oscillators (SHNOs).
Our micromagnetic simulations unveil a distinctive non-monotonic relationship
between current and auto-oscillation frequency in out-of-plane magnetic fields.
The influence of effective magnetization on frequency tunability varies with
out-of-plane field strengths. At large out-of-plane fields, the frequency
tunability is predominantly governed by effective magnetization, achieving a
current tunability of 1 GHz/mA-four times larger than that observed at the
lowest effective magnetization. Conversely, at low out-of-plane fields,
although a remarkably high-frequency tunability of 4 GHz/mA is observed, the
effective magnetization alters the onset of the transition from a linear-like
mode to a spin-wave bullet mode. Magnetic damping primarily affects the
threshold current with negligible impact on auto-oscillation frequency
tunability. The threshold current scales linearly with increased magnetic
damping at a constant out-of-plane field but exhibits a parabolic behavior with
variations in out-of-plane fields. This behavior is attributed to the
qualitatively distinct evolution of the auto-oscillation mode across different
out-of-plane field values. Our study not only extends the versatility of SHNOs
for oscillator-based neuromorphic computing with controllable frequency
tunability but also unveils the intricate auto-oscillation dynamics in
out-of-plane fields.
