  We study the stabilization of an isolated magnetic skyrmion in a magnetic
monolayer on a nonmagnetic conducting substrate via the
Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction. Two different types
of the substrate are considered, usual normal metal and single-layer graphene.
While the full stability analysis for skyrmions in the presence of the RKKY
coupling requires a separate effort that is outside the scope of this work, we
are able to study the radial stability (stability of a skyrmion against
collapse) using variational energy estimates obtained within first-order
perturbation theory, with the unperturbed Hamiltonian describing the isotropic
Heisenberg magnet, and the two perturbations being the RKKY exchange and the
easy-axis anisotropy. We show that a proper treatment of the long-range nature
of the RKKY interaction leads to a qualitatively different stabilization
scenario compared to previous studies, where solitons were stabilized by the
frustrated exchange coupling (leading to terms with the fourth power of the
magnetization gradients) or by the Dzyaloshinskii-Moriya interaction (described
by terms linear in the magnetization gradients). In the case of a metallic
substrate, the skyrmion stabilization is possible under restrictive conditions
on the Fermi surface parameters, while in the case of a graphene substrate the
stabilization is naturally achieved in several geometries with a
lattice-matching of graphene and magnetic layer.
