In this section we investigate the temperature scaling of the macroscopic anisotropy with respect to the magnetization
. As it has been reported in the literature for a pure magnetic material the scaling factor at low temperature follows the well known CallenCallen law [106,129]. However, when the magnetic system has a more complex structure than a pure ferromagnet the scaling is unknown a priori and is a coordination number, thin film thickness, configuration of easy axes and material dependent. Nevertheless, it is this scaling which would be measured experimentally. To illustrate this effect we have calculated the scaling exponent at lowtemperatures for various thin films with different easy axes configurations.
First we analyze the scaling exponent as a function of the thin film thickness when the bulk has a uniaxial magnetocrystalline anisotropy and the surface anisotropy is modeled by the Néel surface anisotropy with a
. We consider two cases, the Néel surface anisotropy is perpendicular to the thin film plane in both cases while the bulk easy axis is in the first case (Fig. 4.20) also perpendicular to the surface and in the second case (Fig 4.21) lays in the plane parallel to X axis.
Figure 4.20:
The scaling exponent as a function of the ratio for a set of thin films with sc lattice, uniaxial anisotropy in the bulk and Néel surface anisotropy
, the bulk and surface easy axis are outofplane.

The scaling exponent is shown in the Fig. 4.20. We can observe that when our system is a pure surface system it is well modeled by an uniaxial anisotropy and
, practically we recover the value expected from the Callen Callen theory. But when the size of the system is increased and we have both surface and bulk moments the system presents two different anisotropy constants from the surface and bulk magnetization which have different dependence on temperature, and this affects the value.
Next, we consider that the configuration of the easy axis of the bulk and surface corresponds to that shown in Fig. 4.16 (right).
Figure 4.21:
The scaling exponent as a function of system size of a thin film with sc lattice, uniaxial anisotropy in the bulk with the easy axis parallel to X axis, and Néel surface anisotropy with
and easy axis outofplane.

As we have analyzed in the previous section 4.3.3, the existence of several easy axes with different origins could lead to a reorientation transition with temperature. The reorientation transition has the consequence that the scaling exponent of the effective anisotropy constant with the average magnetization is not monotonic, as we can see in Fig. 4.21, where the scaling exponent of systems with
is plotted.
The scaling exponent increases from
(when the total anisotropy is dominated by the surface anisotropy), diverges at the reorientation transition and later recovers the value
(when the total anisotropy is dominated by the bulk one) with increased thin film thickness (see Fig. 4.21). The negative value of the scaling exponent is related to a nonmonotonic dependence of the effective anisotropy on temperature, related to the orientation transition, as seen in Fig. 4.19.
In general the magnetic behavior of a thin film depends on the character of it surface, if it is coated, if it is free, etc. It is natural to ask the question on how the scaling exponent is affected by the surface. With the aim to answer this question, we have simulated the effect of different cappings on our thin film, varying the value of the Néel surface anisotropy parameter. We have modeled a thin film with a sc lattice structure (), the bulk anisotropy is uniaxial and the Néel surface anisotropy constant is changed in the range from
to
. In Fig. 4.22 we present the calculated value of the scaling exponent. When the value of the surface anisotropy per spin is smaller than the bulk one, the scaling factor is smaller than that predicted by the CallenCallen theory (). If the effective surface anisotropy per spin is larger than the bulk one, the scaling factor calculated is larger than 3 and
at
.
Figure 4.22:
Scaling exponent of a thin film, with sc lattice structure, as a function of the Néel surface anisotropy normalized to the bulk anisotropy constant ().

To summarize this section we conclude that the scaling exponent for thin films with uniaxial anisotropy in the bulk and surface anisotropy at low temperatures is different than that predicted by the CallenCallen theory, and depends on the system size. The increment of has its origin in the fact that the moments on the surface have a reduction of the number of the nearest neighbors, therefore the surface magnetization is more sensible to the temperature than the total magnetization. Also we have to note that the value of the surface anisotropy constant itself is different from the bulk one, and this could induce a change in the value of .
Rocio Yanes