1 The studied system

We have calculated the magnetic properties for a system of Co fcc (bulk), as well as semi-infinite system with surface terminations Co(111), Co(100), $Co(100)\setminus Ag_{n}$ and $Co(111)\setminus Ag_{n}$, where n is the number of the atomic layers of Ag ($n=1$, or 2).

We would like to indicate that the coordinate system has been chosen in such form that the Z axis is perpendicular to (100) or (111) plane depending the case under study.

Figure 6.2: Sketch of the systems studied in the calculations (a) Cobalt bulk system; (b) Films systems. The systems are divided in three regions, one corresponding to a semi-infinite substrate of Co, a intermediate region which correspond to: 3 atomic layer of Co in the case of a Co bulk system or, in the case of film systems, several (up to six) atomic layers of Co with or without a Ag capping and 3 Vacumm layers. The last region corresponds to a semi-infinite Co capping in Co bulk, or a Vacuum region in film systems.

The system under study is divided in three regions, see Fig. 6.2, corresponding to a semi-infinite substrate of Co, an intermediate region which corresponds to: 3 atomic layers of Co in the case of a Co bulk system or, in the case of semi-infinite systems, several (up to six) atomic layers of Co with or without Ag capping and 3 Vacumm layers. The last region corresponds to a semi-infinite Co capping in Co bulk, or a Vacumm region in film systems. We have to note that only interactions between spins belonging to the intermediate region are evaluated.

In our calculations when capping layers of Ag on Co exists, it is assumed that the Ag layer(s) adjust to the in-plane lattice parameter of Co fcc. However the lattice parameter of fcc Ag is bigger than that of Co. Then with the aim to deal with this mismatch between the lattice constants of the Co fcc and the Ag, we supposed a "rigid relaxation" of the lattice. In our "rigid relaxation", we kept the in-plane lattice constant of Co fixed, for the whole system since we are modeling epitaxial growth of Ag on Co fcc, and relax the perpendicular lattice constant. For normal-to-plane distance relaxation of the Ag layer(s), we consider an expansion of the interlayer distances, $d_{Co-Ag}$ and $d_{Ag-Ag}$ of r% with respect to the bulk Co lattice constant. The relaxation ($r$) was considered to range from 6 to 12 $\%$, depending on the system.

Figure 6.3: Calculated relaxation of the lattice in the systems:(a) $Co(100)\setminus Ag_{1}$; (b) $Co(100)\setminus Ag_{2}$; (c) $Co(111)\setminus Ag_{1}$ and (d) $Co(111)\setminus Ag_{2}$. This calculations have been performed with the code SIESTA [156].

With the aim to have an estimation of the actual relaxation of the system we have compared it with the relaxation obtained for these system with SIESTA code [156], see Fig. 6.3. The first thing that we can conclude from these results is that the relaxation of the lattice in these layered systems is complex. In the SIESTA results we can observe not only a relaxation of the lattice in the surface layer but also there is a contraction between the subsurface layers of Co. In the cases of one ML of Ag capping, the relaxation of the lattice range from $9.25\%$ to 11.56 %. Therefore, it is always in the range assumed in the study with the Budapest-Vienna code.

In what follows the value of the relaxation which is selected in the multiscale study is $r=10\%$. This relaxation of the lattice has been chosen because this value coincides with the one which provides the minimum of the total energy and it is close to the SIESTA results. It should be noted that the value of the relaxation has little effect on the spin and orbital magnetic moments, and the influence on the work function and anisotropy will be discussed separately.