5 Magneto-crystalline anisotropy energy of a $Co(111)\setminus Ag_{1}$ thin films

Here we present a detailed study of the MCA energy for a particular system of $Co(111)\setminus Ag_{1}$. The first step is to obtain the angular dependence of the energy. In Fig. 6.13, we show the band energy of the system $Co(100)\setminus Ag_{1}$ as a function of the angle ($\theta$) from the Z axis (axis perpendicular to the surface of the thin film). The value obtained from SKKR calculation has been compared with the analytical expression of the angular dependence of the uniaxial anisotropy energy, showing that both adjust perfectly.

$$E(\theta)= E_{0}- \Delta E \cos^{2}(\theta)$$ (114)

Figure 6.13: Band energy as a function of the angle of the magnetization axis from the Z axis of the system.

From the analysis of this result we can suppose that the $Co(111)\setminus Ag_{1}$ system presents an uniaxial anisotropy whose easy axis is perpendicular to the surface of the system, and the on-site anisotropy has the following form.

$$\sum_{i}\mathbf{S_{i}}\mathrm{d}_{i}\mathbf{S_{i}}=-\sum_{i}d_{i}^{zz}(\mathbf{S_{i}}\cdot e_{z})^{2}$$ (115)

Henceforth, we omit the super-index (zz) of the on-site anisotropy constant.

As we presented in previous section (6.3.5), the effective anisotropy constant extracted from the "ab-initio" calculations is related to the difference of the band energy between the magnetic moments configuration oriented at different directions, $\{\vec{S}_{i}\} \vert\vert \vec{x}$ and $\{\vec{S}_{i}\} \vert\vert \vec{z}$. The problem appears when we try to figure out which are different contributions to this effective anisotropy. In the first place, we have an on-site contribution ($d_{i}$). As we mentioned in section (6.3.4) the anisotropic part of the exchange tensor interaction also contributes to the anisotropy, with a two-site anisotropy contribution, in a such way that:

$$\Delta E = Eb_{x}-Eb_{z}\approx -\frac{1}{2}\sum_{i,j} J_{ij}^{S,xx}+\frac{1}{2}\sum_{i,j} J_{ij}^{S,zz}+\sum_{i}d_{i}$$ (116)

In the way reported in the appendix we can get the layer-resolved on-site anisotropy constant $d_{i}$ for $Co(111)\setminus Ag_{1}$, which we plotted in Fig. 6.14 together with the values of $\Delta E_{b}$. We observe a huge increment of the macroscopic magneto-crystalline anisotropy on the surface. However the discrepancy between the values of $\Delta E_{b}$ and the on-site anisotropy contribution suggests us that a large contribution of the two-site anisotropy to the total MCA energy should exist.

Figure 6.14: The layer-resolved MCA energy and the on-site anisotropy energy for $Co(111)\setminus Ag_{1}$ films.