2 Magneto-crystalline anisotropy: Bruno model

A relatively simple model that takes into account the bonding and magnetic moment is the Bruno model of the magnetic anisotropy of transition metals [45]. Following his concept, MCA for transition metal elements can be related with the anisotropy of the orbital magnetic moment under certain conditions in such way that:

$$\Delta E_{SO}=\zeta[\langle\mathbf{L}\cdot \mathbf{S}\rangle_{har... ...athbf{S}\rangle_{easy}]=\frac{\zeta}{4\mu_{B}}(\mu_{L}^{easy}-\mu_{L}^{hard})>0$$ (10)

where $\mu_{L}^{hard,easy}$ is the orbital magnetic moment in the hard or easy axis. From that model we can conclude that the easy axis of the magnetization coincides with the direction which has the maximum orbital magnetic moment.

Figure 1.5: Illustration of the directional quenching of the orbital moment of an atom by the ligand field effects in a thin film (from [48]).

The layered thin films present an inherent in-plane/out-of-plane asymmetry in contrast to the bulk. The magneto-crystalline anisotropy values up to two orders of magnitude larger than that for the typical magnetic elements Fe, Co, Ni have been reported for layered thin films. Therefore these systems are perfect examples to study the origin of the surface anisotropy. According to the Bruno's model, the MCA is related with the anisotropy bonding and its relation with the ligand field. To illustrate this concept we consider a d electron in a free atom and in an atom contained in a planar geometry with other four atoms which can have negative or positive charge, see Fig. 1.5. In the planar geometry the d electron suffers the effects of the Coulomb repulsion or attraction depending on the charge of its neighbors, altering its orbital with respect to the free one. This way we can see the effects on the magnetic moment due to the ligand field. We can observe a partial break of the degeneracy of the d orbital, one can group the d orbitals into in-plane and out of plane, and we may quantitatively relate the anisotropy of the orbital moment with the anisotropy of the bonding environment. The corresponding orbital moment along the normal of the bonding plane is quenched, however in the case of the moment in-plane with respect to the bond plane this is not so. Due to the loss of neighbors at the surface the orbital motion perpendicular to the bonding plane is less disturbed and the in-plane orbital momentum is unquenched. This leads to an anisotropic orbital moment and to the surface anisotropy, according to the Bruno's model.

Therefore the exchange interaction is responsible for the creation of the spin magnetic moment and the ligand field creates anisotropic orbitals. The spin and orbital moments are linked by the spin-orbit coupling and the orbital moment is locked in a particular direction. The interplay of all these factors creates the surface magnetrocrystalline anisotropy.

We should have in mind that the Bruno model is illustrative and may be too simple in real situations. As it has been indicated by Andersson [49], the Bruno model could be not adequate in the analysis of magnetic systems with high spin-orbit coupling.