  Rare-earth (RE) atoms on top of 2D materials represent an interesting
platform with the prospect of tailoring the magnetic anisotropy for practical
applications. Here, we investigate the ground state and magnetic properties of
selected $4f$-atoms deposited on a graphene substrate in the framework of the
DFT+$U$ approach. The inherent strong spin-orbit interaction in conjunction
with crystal field effects acting on the localized $4f$-shells results in a
substantial magnetic anisotropy energy (tens of meVs), whose angular dependence
is dictated by the $C_{6v}$ symmetry of the graphene substrate. We obtain the
crystal field parameters and investigate spin-flip events via quantum tunneling
of magnetization in the view of achieving a protected quantum-spin behavior.
Remarkably, the large spin and orbital moments of the open $4f$-shells (Dy, Ho
and Tm) generate a strong magneto-elastic coupling which provides more
flexibility to control the magnetic state via the application of external
strain.
