  Fourth order perturbation was applied to study a small variation of the
azimuthal angle of spin of fcc structured ferromagnetic thin films with two
spin layers. The variation of magnetic energy and the orientation of magnetic
easy axis with the fourth order magnetic anisotropy constant in two spin layers
was investigated. When the second order magnetic anisotropy constant at the top
spin layer is higher than that of bottom spin layer, the total magnetic energy
is slightly higher. Some spikes appeared in the 2D plot of of magnetic energy
versus azimuthal angle of spin. According to the 3D plots, the peak value of
the magnetic energy gradually decreases with the increase of the stress induced
anisotropy constant, and thereafter the peak value of the magnetic energy
gradually increases with the increase of the stress induced anisotropy
constant. The angle between magnetic easy and hard directions was not 90
degrees. The magnetic easy and hard directions of the film with a higher second
order magnetic anisotropy constant of top layer are different from the magnetic
easy and hard directions of the film with a lower second order magnetic
anisotropy constant of top layer. When the second order magnetic anisotropy
constant of the bottom layer is increased, the total magnetic energy does not
change.
