The magneto-crystalline anisotropy (MCA) energy has its origin in the spin-orbit coupling under the ordering imposed by the crystalline lattice of the material. In the most general case, the magnetocrystalline anisotropy matrix can be written as a function of its eigenvectors (corresponding to the easy axes) and eigenvalues :
Epitaxially grown systems present usually uniaxial anisotropy. If we suppose that in this case the easy axis is parallel to the Z axis, then the on-site anisotropy has the form:
We have only described the on-site magneto-crystalline anisotropy. However, there could exist other contributions to the magnetic anisotropy of the system, as e.g. the two-site magnetic anisotropy, mentioned already above and discussed later in this chapter.