  We experimentally study the dynamical modes excited by current-induced
spin-orbit torque and its electrostatic gating effect in a 3-terminal planar
nano-gap spin Hall nano-oscillator (SHNO) with a moderate interfacial
perpendicular magnetic anisotropy (IPMA). Both quasilinear propagating
spin-wave and localized "bullet" modes are achieved and controlled by varying
the applied in-plane magnetic field and driving current. The minimum linewidth
shows a linear dependence on the actual temperature of the active area,
confirming single-mode dynamics based on the nonlinear theory of spin-torque
nano-oscillation with a single mode. The observed electrostatic gating tuning
oscillation frequency arises from voltage-controlled magnetic anisotropy and
threshold current of SHNO via modification of the nonlinear damping and/or the
interfacial spin-orbit coupling of the magnetic multilayer. In contrast to
previously observed two-mode coexistence degrading the spectral purity in
Py/Pt-based SHNOs with a negligible IPMA, a single coherent spin-wave mode with
a low driven current can be achieved by selecting the ferromagnet layer with a
suitable IPMA because the nonlinear mode coupling can be diminished by bringing
in the PMA field to compensate the easy-plane shape anisotropy. Moreover, the
simulations demonstrate that the experimentally observed current and
gate-voltage modulation of auto-oscillation modes are also closely associated
with the nonlinear damping and mode coupling, which are determined by the
ellipticity of magnetization precession. The demonstrated nonlinear mode
coupling mechanism and electrical control approach of spin-wave modes could
provide the clue to facilitate the implementation of the mutual synchronization
map for neuromorphic computing applications in SHNO array networks.
