  By means of systematic first-principles calculations based on density
functional theory we search for suitable materials that can host
antiferromagnetically coupled skyrmions. We concentrate on fcc-stacked
(111)-oriented metallic $Z$/Co/Pt ($Z=4d$ series: Y$-$Pd, the noble metals: Cu,
Ag, Au, post noble metals: Zn and Cd) magnetic multilayers of films of
monatomic thickness. We present quantitative trends of magnetic properties:
Magnetic moments, interlayer exchange coupling, spin-stiffness,
Dzyaloshinskii-Moriya interaction, magnetic anisotropy, and the critical
temperature. We show that some of the $Z$ elements (Zn, Y, Zr, Nb, Tc, Ru, Rh,
and Cd) can induce antiferromagnetic interlayer coupling between the magnetic
Co layers, and that they influence the easy magnetization axis. Employing a
multiscale approach, we transfer the micromagnetic parameters determined from
$ab$ $initio$ to a micromagnetic energy functional and search for
one-dimensional spin-spiral solutions and two-dimensional skyrmions. We
determine the skyrmion radius by numerically solving the equation of the
skyrmion profile. We found an analytical expression for the skyrmion radius
that covers our numerical results and is valid for a large regime of
micromagnetic parameters. Based on this expression we have proposed a model
that allows to extrapolate from the $ab$ $initio$ results of monatomic films to
multilayers with Co films consisting of several atomic layers containing
$10\,$nm skyrmions. We found thickness regimes where tiny changes of the film
thickness may alter the skyrmion radius by orders of magnitude. We estimated
the skyrmion size as function of temperature and found that the size can easily
double going from cryogenic to room temperature. We suggest promising material
systems for ferromagnetically and antiferromagnetically coupled spin-spiral and
skyrmion systems.
