2 Magneto-crystalline anisotropy energy dependence on the interface and cappings.

It is well known that the magneto-crystalline anisotropy energy could be affected by the reduction of the symmetry of the system such as the existence of interfaces, relaxation of the lattice or the presence of the capping in the system. We have tried to analyze how these effects contribute the MCA energy of our systems.

Table 6.5: In this table we present the values of the band energies difference $\Delta E_{B}$ (in meV) for different systems of Co(bulk), Co(100), Co(111), $Co(100)\setminus Ag_{n}$ and $Co(111)\setminus Ag_{n}$.

System	Interface	$\mathbf{\Delta E_{B}}$ (meV)
Cobalt bulk(fcc)	$(100)$	8.63e-4
Cobalt (fcc)	$(100)$	-0.09
Cobalt (fcc)	$(111)$	0.12
Ag monolayer on Co(fcc)	$(100)$	0.03
Ag monolayer on Co(fcc)	$(111)$	0.22
2 Ag ML on Co(fcc)	$(100)$	-0.01
2 Ag ML on Co(fcc)	$(111)$	0.06

In Tab. 6.5 we present the results of the total MCA energy calculated from the difference of the band energies. We have evaluated the MCA energy in Co fcc (bulk), semi-infinite Co fcc with (100) or (111) surface, and semi-infinite Co fcc with a capping of one or two Ag atomic monolayers (MLs) and (100) or (111) interfaces.

The first thing that we observe is that the bulk value of the MCA is lower than $1 \mu eV$, this value is in the limit of the accuracy of our method. For this reason we should take this value just as the order of magnitude of the MCA for cobalt bulk. Then, from the results of MCA energy shown in Tab. 6.5 we can assert that the MCA energy of the surface of semi-infinite Co systems suffers a huge increment from the bulk value.

From equation (6.35) we can conclude that $\Delta E>0$ means that the energy of the system is lower when the moments of the system are oriented in the direction perpendicular to the surface, therefore the magnetization prefers to align perpendicular to the surface. In the opposite situation, when $\Delta E<0$ the magnetization of the system prefers to align parallel to the surface of the sample. In the case of a $Co(100)$ system, $\Delta E<0$, this implies that the easy axis of the system is contained in the plane of the surface of the sample. At the same time, the magnitude of the MCA energy has increased two order of magnitude with respect to the bulk case. Now if we cover this system with an atomic monolayer of silver, an abrupt change of MCA energy occurs and the easy axis becomes perpendicular to the surface of the sample.

On the other hand, in the case of a $Co(111)$ system, $\Delta E>0$ then the easy axis of the system is perpendicular to the surface of the sample, and the value of the MCA is higher than that of the $Co(100)$. Now we analyze what happens with the MCA energy if we coat the system with a ML of Ag, we find that the easy axis of the system does not change, being also in this case perpendicular to the surface.

Our calculations of the MCA energy of the Co (fcc) system with one atomic monolayer of Ag capping suggest us that the existence of this capping induces an increase in the MCA energy and favors a perpendicular orientation of the magnetization. Nevertheless, if the width of the Ag capping is increased up to $2MLs$ the MCA energy is reduced with respect to the case of a capping by one ML of Ag and is close to the results of the corresponding pure surface.

In brief, the MCA energy depends closely on the proximity and characteristics of the surface or interface. The MCA energy is larger in the case of (111) surface or interface than in the (100) case.

Understanding the origin of the MCA is a complex task, due to the difficulty that involve this kind of studies. Nevertheless several authors suggest an interpretation of the MCA as a function of the symmetry breaking or asymmetry of the bonding at magnetic surface or interface [179]. The concept of Bruno suggests that under certain assumptions the magnetic anisotropy energy is related with an anisotropy of the orbital moment and the anisotropy bonding [45], see section 1.4.2. In the case of fcc lattice with surface/interface (111) the number of NN in the surface/interface is 6 in contrast to the 4 NN in surface/interface (100). Then the effect of the ligand field is stronger in the case of (111) than in the case of (100), this suggests that the orbital moment of the surface/interface (111) is more quenched than of the surface/interface (100), in agreement with our calculated $\mu _{L}$ for this surface. Therefore the variation of the magnetic moment in-plane and out-of-plane is stronger in the (111) case. Basing on the Bruno's model this fact can explain the larger value of the MCA energy on (111) surface/ interface.