1 Curie temperature

The Curie temperature $Tc$ is the temperature beyond which a ferromagnetic material becomes paramagnetic and perhaps the loss of the ferromagnetic character with the temperature is the most important temperature effect in magnetic systems. For this reason, to determine the Curie temperature through the constrained Monte Carlo method is a good test. The Curie temperature is related to the value of the exchange constant and in the Mean Field Approximation (MFA) it could be expressed as follows:

$$Tc=\frac{z\cdot J}{3k_{B}}$$ (72)

where $J$ is the exchange constant, $\textit{z}$ is the number of the first neighbors and $\textit{k}_{B}$ is the Boltzmann constant.

Figure 4.2: Temperature dependence of the magnetization in a ferromagnetic material with $8000$ spins with periodic boundary conditions in $3D$ and with a sc and fcc lattice structures.

On the other hand, it is well known that MFA overestimates the value of the Curie temperature, this overestimation is due to the inaccuracy in the consideration of the correlations between spins within this approximation. In the literature we find other approaches, such as the classical spectral density method (CSDM) [127] or the hight temperatures expansion method (HTS) [128] which is used to extract the well known expression for the Curie temperatures for simple cubic lattices.

$$Tc=1.44\frac{J}{k_{B}}$$ (73)

We have considered two systems: a bulk system with periodic boundary conditions in $3D$ with a simple cubic or fcc lattice and evaluated the magnetization as a function of the temperature. We consider that the Curie temperature is defined when the magnetization becomes zero (see Fig. 4.2), excluding the finite-size effects. The exchange constants in both systems are different. In the case of a simple cubic lattice, each spin has 6 first neighbors, the exchange constant is $J=10\times10^{-14} erg$ and the Curie temperature obtained by the constrained Monte-Carlo simulation is close to $Tc=1100 K$. For a thin film with fcc lattice with a $J=5.6\times10^{-14} erg$, the number of first neighbors is $12$ and the Curie temperature determined by (CMC) simulations is close to $Tc=1320 K$.

The Curie temperature obtained by the constrained Monte Carlo simulation for sc and fcc thin films presents an important discrepancy with the theoretical Mean Field prediction. In contrast, our values of $Tc$ are in a good agreement with the values predicted by the classical spectral density method [127] (see Table 4.2).

Table 4.2: Curie Temperature ($Tc$) values for thin films with sc ( $J=10\times10^{-14}\quad erg$) and fcc ( $J=5.6\times10^{-14}\quad erg$) lattices, predicted by the Mean Field Approximation (MFA), Classical Spectral Density Method (CSDM) and extracted from the constrained Monte Carlo (CMC) simulation.

Lattice	(MFA)	(CSDM)	(CMC)
sc	$1448.58$	$1064.71$	$\sim 1080$
fcc	$1622.41$	$1309.29$	$\sim 1320$