  We use He$^+$ irradiation to tune the nonlinearity, $\mathcal{N}$, of
all-perpendicular spin-torque nano-oscillators (STNOs) using the He$^+$
fluence-dependent perpendicular magnetic anisotropy (PMA) of the [Co/Ni] free
layer. Employing fluences from 6 to 20$\times10^{14}$~He$^{+}$/cm$^{2}$, we are
able to tune $\mathcal{N}$ in an in-plane field from strongly positive to
moderately negative. As the STNO microwave signal properties are mainly
governed by $\mathcal{N}$, we can in this way directly control the threshold
current, the current tunability of the frequency, and the STNO linewidth. In
particular, we can dramatically improve the latter by more than two orders of
magnitude. Our results are in good agreement with the theory for nonlinear
auto-oscillators, confirm theoretical predictions of the role of nonlinearity,
and demonstrate a straightforward path towards improving the microwave
properties of STNOs.
