PHASE TRANSITION IN MAGNETIC ULTRA-THIN FILMS

In this paper, we study the influence of surface anisotropy on the phase transition in antiferromagnetic and ferromagnetic ultra-thin films using the functional integral method. Besides, spin fluctuations are also given to illustrate these phase transitions. We find that the phase transition temperature of the ultra-thin films may be higher or lower than that of the corresponding bulk systems, which depends on the surface anisotropy. Moreover, we also determine crossover points at which the phase transition temperature is not influenced by the thickness of the thin film.


Introduction
Two-dimensional (2D) systems have been extensively studied during the past decades due to the rich physical properties that they exhibit, especially the variety of their interesting magnetic phase transitions.
A large number of recent experimental and theoretical studies have shown that the order-disorder phase transition in magnetic ultra-thin films may differ significantly from that in the corresponding bulk systems [3,7,10]. In the general case, the phase transition temperature (Curie temperature for the ferromagnetic (FM) thin film and Néel temperature for the antiferromagnetic (AFM) thin film) of the ultrathin films is lower than that in the bulk and decreases when the thickness of the film reduces. However, in some special cases, such as Gd [1], Tb [5], and NbSe2 [2], the phase transition temperature of the ultra-thin films is higher than that of the bulk. In these works, the authors also suggested that the presence of very large surface anisotropies causes the magnetic order at the surfaces above the bulk Curie temperature. Hence, we can see that one of the most important contributions for the unusual properties in thin films is their anisotropy at the surface. In general, it can be said that atoms at the surface state create a new phase with special properties such as low symmetric order and a decrease of the number of nearest neighbors (NN), which may cause several interesting physical properties [8].
In this paper, we investigate the phase transition in the magnetic ultra-thin film on the basis of the Heisenberg model via spin fluctuations using the functional integral method [3] where n and n' are layer indices; We choose the Oz direction to be the average alignment of the spins, so the spin fluctuations are defined as follows: where N is the number of the spins in every monolayer, Hamiltonian (1) of the system is rewritten as In (6), where a is the distance between the two NN spins in a monolayer of the thin film and b(y) is the Brillouin function , , , The free energy of the thin film is calculated as follows: Using the functional integral method given in details in [3], we achieve the last expression of the free energy for the thin film The dependence of the phase transition temperature on the thickness of the thin film can be derived from the logarithmic singularity of the free energy in the zero field, y = 0, and in the long wavelength limit

Numerical results and discussion
The numerical results of the dependence of the reduced phase transition temperature with the experimental results given in [7] and [10].
In this case, the exchange interaction between spins in the bulk systems is more than that in the thin film due to a decrease in the number of NN spins, which results in a reduction in the magnetic order, and thus a decreased C . Case 2:   In [4], the authors also showed that a positive uniaxial anisotropic parameter (Ks/J > 0) favors large values of the spin's z-projection, and the thin film has an easyaxis in the Oz direction, which is an energetically favorable direction of spontaneous magnetization; with a negative uniaxial anisotropic parameter (Ks/J < 0), the spin tends to minimize the z-component of its magnetic moment so that the system has an easy-plane orthogonal to the Oz axis. These theoretical points can be explained from the spin fluctuations given in Fig. 3 and Fig. 4. From Fig. 3, we can see that the x/y-components

Conclusions
In this paper, using the functional integral method, we investigate the