### 2 Comparison of the constrained Monte Carlo method at low temperature with the Lagrange multiplier method

At this stage of our study we decided to check and compare the results of the effective anisotropy constants obtained with the constrained Monte Carlo method with those obtained with the Lagrange multiplier method for . For this purpose we have simulated a set of thin films. This set is made up of thin layers of different thicknesses from with total spins to with total spins. Here is the number of atomic layers of the system, consequently the films have different ratios of surface to volume number of spins . All thin films have a sc structure with periodic boundary conditions in 2D, uniaxial anisotropy in the bulk with the easy axis of the system parallel to the Z axis and a Néel surface anisotropy constant . The bulk anisotropy constant, exchange constant, and the saturation magnetization value are presented in Tab. 4.1.

First we obtain the effective anisotropy constant at by the Lagrange multiplier method for each of the thin films that conform the set (for more details about this method see section 2.4). After that, we extract the effective anisotropy constants of the same systems but with the constrained Monte Carlo method at (note that due to the limitation of the method it is impossible to perform simulations of the CMC at ). And finally we compare the effective uniaxial anisotropy constants obtained by both methods. In the case of the Lagrange multiplier method the effective uniaxial anisotropy constant ( ) is calculated from the expression of the energy barrier of the system, in the CMC method from the restoring torque curve.

The results of this test are plotted in Fig. 4.3, the squared dots represent the data obtained by the Lagrange multiplier method and the circular ones are obtained by the CMC method. The value of the effective uniaxial anisotropy constant is normalized by the value of the macroscopic volume anisotropy constant at (). As we can see, the data show total agreement between both methods at low temperature. The results show a linear behavior of the effective anisotropy as a function of ratio as predicted by the formula:

 (74)

at the bulk anisotropy is recovered , as we can see in Fig. 4.3.
Rocio Yanes