  The perpendicular magnetic anisotropy (PMA) arising at the interface between
ferromagnetic transition metals and metallic oxides are investigated via
first-principles calculations. In this work very large values of PMA up to 3
erg/cm$^2$ at Fe$|$MgO interfaces are reported in agreement with recent
experiments. The origin of PMA is attributed to overlap between O-$p_z$ and
transition metal $d_{z^2}$ orbitals hybridized with $d_{xz(yz)}$ orbitals with
stronger spin-orbit coupling induced splitting around the Fermi level for
perpendicular magnetization orientation. Furthemore, it is shown that the PMA
value weakens in case of over- or underoxidation when oxygen $p_z$ and
transition metal $d_{z^2}$ orbitals overlap is strongly affected by disorder,
in agreement with experimental observations in magnetic tunnel junctions.
