  In previous experiments, thin films of L1$\mathrm{_0}$ FeNi with different
surfaces, including (001), (110) and (111), were produced and studied. Each
surface defines a different alignment of the crystallographic tetragonal axis
with respect to the film's plane, resulting in different magnetic anisotropies.
In this study, we use density functional theory calculations to examine three
series of L1$\mathrm{_0}$ FeNi films with surfaces (001), (010), and (111), and
with thicknesses ranging from 0.5 to 3 nm (from 4 to 16 atomic monolayers). Our
results show that films (001) have perpendicular magnetic anisotropy, while
(010) favor in-plane magnetization, with a clear preference for the tetragonal
axis [001]. We propose calling this type of in-plane anisotropy fixed in plane.
A film with surface (111) and a thickness of four atomic monolayers has the
magnetization easy axis almost perpendicular to the plane of the film. As the
thickness of the (111) film increases, the direction of magnetization rotates
towards a tetragonal axis [001], positioned at an angle of about 45$^{o}$ to
the plane of the film. Furthermore, the magnetic moment of ultrathin films
increases by a maximum of 5%, and the most significant changes in spin and
orbital magnetic moments occur at a depth of about three near-surface atomic
monolayers. The presented results could be useful for experimental efforts to
synthesize ultrathin L1$\mathrm{_0}$ FeNi films with different surfaces.
Ultrathin L1$\mathrm{_0}$ FeNi films with varying magnetic anisotropies may
find applications in spintronic devices.
