  Spin-Hall nano-oscillators (SHNO) are nanoscale spintronic devices that
generate high-frequency (GHz) microwave signals useful for various applications
such as neuromorphic computing and creating Ising systems. Recent research
demonstrated that hybrid SHNOs consisting of a ferromagnetic metal (permalloy)
and lithium aluminum ferrite (LAFO), a ferrimagnetic insulator, thin films have
advantages in having lower auto-oscillation threshold currents
($I_{\text{th}}$) and generating larger microwave output power, making this
hybrid structure an attractive candidate for spintronic applications. It is
essential to understand how the tunable material properties of LAFO, e.g., its
thickness, perpendicular magnetic anisotropy ($K_u$), and saturation
magnetization ($M_s$), affect magnetic dynamics in hybrid SHNOs. We investigate
the change in $I_{\text{th}}$ and the output power of the device as the LAFO
parameters vary. We find the $I_{\text{th}}$ does not depend strongly on these
parameters, but the output power has a highly nonlinear dependence on $M_s$ and
$K_u$. We further investigate the nature of the excited spin-wave modes as a
function of $K_u$ and determine a critical value of $K_u$ above which
propagating spin-waves are excited. Our simulation results provide a roadmap
for designing hybrid SHNOs to achieve targeted spin excitation characteristics.
