  In this study, we theoretically investigate neuron-like spiking dynamics in
an elliptic ferromagnet/heavy metal bilayer-based spin Hall nano oscillator
(SHNO) in bias field-free condition, much suitable for practical realization of
brain inspired computing schemes. We demonstrate regular periodic spiking with
tunable frequency as well as the leaky-integrate-and-fire (LIF) behavior in a
single SHNO by manipulating the pulse features of input current. The frequency
of regular periodic spiking is tunable in a range of 0.5 GHz to 0.96 GHz (460
MHz bandwidth) through adjusting the magnitude of constant input dc current
density. We further demonstrate the reconfigurability of spiking dynamics in
response to a time varying input accomplished by continuously increasing the
input current density as a linear function of time. Macrospin theory and
micromagnetic simulation provide insights into the origin of bias field-free
auto-oscillation and the spiking phenomena in our SHNO. In addition, we discuss
how the shape anisotropy of the elliptic ferromagnet influence the bias
field-free auto oscillation characteristics, including threshold current,
frequency and transition from in-plane to out-of-plane precession. The SHNO
operates below $10^{12} A/m^2$ input current density and exhibits a large
auto-oscillation amplitude, ensuring high output power. We show that the
threshold current density can be reduced by decreasing the ellipticity of the
ferromagnet layer as well as enhancing the perpendicular magnetic anisotropy.
These findings highlight the potential of bias field-free elliptic SHNO in
designing power-efficient spiking neuron-based neuromorphic hardware.
