2 Separation of the surface torque

Figure 4.14: Total, surface and bulk torques in a thin film with $N=16000$, $Lz=10$, uniaxial anisotropy in the bulk and Néel Surface anisotropy $K_{s}=10\cdot K_{c}$. The bulk easy axis is parallel to Z axis.

The second method is based on the possibility to separate the contributions of the surface to the restoring torque.

Fig. 4.14 represents the total, bulk and surface torque curves, evaluated for a thin film with $Lz=10$, sc structure and the relative orientation between the bulk and surface easy axes, parallel to each other. Each curve is fitted separately to expression (4.19). The comparison between the values of effective surface anisotropy obtained by the two methods shows a good agreement, see Fig. 4.15.

In practice, the surface torque is a noisy quantity due to the relatively small number of atoms. In order to obtain a good thermodynamic average it is generally necessary to use a large number of steps in the CMC algorithm ( $N_{sweeps}=70000$). However, the total torque converges much more rapidly and requires relatively few steps.

Figure 4.15: The effective surface anisotropy constant $K_{s}^{eff}$ obtained at several temperatures by the separation of the surface torque (triangular dots) and the scaling method (circular dots).

The experimental results for Gd, Fe, Ni show a largely linear temperature dependence of surface anisotropy [119,133,134]. Note that low temperature behavior of the surface anisotropy presented in Fig. 4.15 resemble a linear behavior. However, when comparing with our results, various factors should be taken into account, including structural changes with increased temperatures or a lattice mismatch which could influence the bulk-type anisotropy. One other possibility is that of an enhanced exchange interaction at the surface of the material [135] (see also section 6.3.4 of the next chapter). An increased exchange interaction would lead to a reduction in the criticality of the surface layer and similarly the temperature scaling of the surface anisotropy will be affected.