  The magnetic anisotropy energy defines the energy barrier that stabilizes a
magnetic moment. Utilizing density functional theory based simulations and
analytical formulations, we establish that this barrier is strongly modified by
long-range contributions very similar to Frieden oscillations and
Rudermann-Kittel-Kasuya-Yosida interactions. Thus, oscillations are expected
and observed, with different decaying factors and highly anisotropic in
realistic materials, which can switch non-trivially the sign of the magnetic
anisotropy energy. This behavior is general and for illustration we address
transition metals adatoms, Cr, Mn, Fe and Co deposited on Pt(111) surface. We
explain in particular the mechanisms leading to the strong site-dependence of
the magnetic anisotropy energy observed for Fe adatoms on Pt(111) surface as
revealed previously via first-principles based simulations and inelastic
scanning tunneling spectroscopy (A. A. Khajetoorians et al. Phys. Rev. Lett.
111, 157204 (2013)). The same mechanisms are probably active for the
site-dependence of the magnetic anisotropy energy obtained for Fe adatoms on Pd
or Rh(111) surfaces and for Co adatoms on Rh(111) surface (P. Blonski et al.
Phys. Rev. B 81, 104426 (2010)).
