  Within Density Functional Theory, we have studied the interplay between
vortex arrays and capillary waves in spinning prolate He-4 droplets made of
several thousands of helium atoms. Surface capillary waves are ubiquitous in
prolate superfluid He-4 droplets and, depending on the size and angular
momentum of the droplet, they may coexist with vortex arrays. We have found
that the equilibrium configuration of small prolate droplets is vortex-free,
evolving towards vortex-hosting as the droplet size increases. This result is
in agreement with a recent experiment [S.M. O'Connell et al., Phys. Rev. Lett.
124, 215301 (2020)], where it has been disclosed that vortex arrays and
capillary waves coexist in the equilibrium configuration of very large drops.
Contrarily to viscous droplets executing rigid body rotation, the stability
phase diagram of spinning He-4 droplets cannot be universally described in
terms of dimensionless angular momentum and angular velocity variables:
instead, the rotational properties of superfluid helium droplets display a
clear dependence on the droplet size and the number of vortices they host.
