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.
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