## 5 Magneto-crystalline anisotropy energy

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 :

 (109)

In this general case we have a triaxial anisotropy. This sort of anisotropy could be present in a system with a reduced symmetry such as surfaces, edges, etc. The contribution due to the on-site anisotropy to the total energy has the following form:

 (110)

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:

 (111)

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.

Subsections
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