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.
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