Magnetism in ultrathin film is now a fairly well established field of research, and has attracted much attention, both experimentally and theoretically.^[1–4] The magnetic properties of hexagonal compact superlattices of materials A and B: a helimagnetic and a ferromagnetic phase, at low temperature, were studied in Ref. [5]. The ferromagnetic/antiferromagnetic superlattices, which are composed of a spin-1/2 ferromagnetic monolayer and a spin-1 antiferromagnetic monolayer with a single-ion anisotropy, respectively, have been investigated in Ref. [6]. The magnetization in (Fe 3 nm/Dy 2 nm) multilayers have been studied experimentally and numerically in Ref. [7]. The influences of the exchange coupling and single-ion anisotropy parameters in the model Hamiltonian on the martensitic–austenitic transformations were studied and analyzed in comparison with the results for hexagonal nanoparticles in Ref. [8]. To analyze the dependence of the critical temperature T_C on size, some authors^[9–13] have considered the ferromagnetic nearest-neighbor Ising model of the simple cubic lattices for film of c layer. The magnetic coupling at the surface, J_S, may be identical with or different from that in the bulk, J_b. Transition temperature of a multilayer consisting of spin-1/2 and spin-3/2 ferromagnetic Ising layers has been computed by using the effective field theory^[14–16] in order to explain the experimental data of rare earth/transition metal multilayer films, respectively. Mean-field theory has also been used to study magnetic properties of rare earth/transition multilayers,^[17] but very few Monte Carlo studies dealt with the rare earth/transition metal multilayer systems. The phase transition and magnetic properties of a ferromagnet spin-S, a disordered diluted thin and semi-infinite film with a face-centered cubic lattice are investigated by using the high temperature series expansions technique extrapolated with Padé approximants method for Heisenberg, XY and Ising models in Ref. [18]. The magnetic properties of ferromagnetic Ni/Au core/shell have been studied by using Monte Carlo simulations within the Ising model in Ref. [19]. Hysteresis loops, micromagnetic structures, and hysteresis loop area curves, as well as dynamic correlation between the magnetization and the external field have been studied each as a function of the field and film parameters in Ref. [20]. Recently, we have used the Monte Carlo simulations to study the magnetic properties of a nanowire system based on a honeycomb lattice in the absence and presence of both an external magnetic field and crystal field.^[21] The purpose of this work is to explore the variations of magnetization and magnetic susceptibility with the reduced temperature, exchange interaction between bilayers and zero crystal field for a fixed value of exchange interaction in surface and in bulk with different sizes. The critical temperatures are obtained for different values of size l_a and for different values of exchange interactions between bilayers. The magnetic hysteresis cycle with reduced crystal field, exchange interaction between bilayers and reduced critical temperatures are established. From these curves we deduce the coercive field and magnetization remnant.

We consider superlattices of ferromagnetic materials a and b, with a simple cubic structure. The l_a and l_b are the corresponding numbers of atomic layers in materials a and b, respectively. l = l_a + l_b is the thickness of the unit cell (see Fig. 1). In all text, l_b = 7 unless otherwise stated.

Fig. 1.

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	Fig. 1. Ferromagnetic materials a and b superlattices with the thickness of the cell being la + lb.

The Hamiltonian for the a and b ferromagnetic superlattices includes the interactions between the nearest neighbors, external magnetic field, and the crystal field is given as

The ferromagnetic materials a and b superlattices are assumed to reside in the unit cells and the system consists of the total number of spins N = (l_a + l_b)l, with l = 8, l_b = 7 and l_a = 5, 7, 9, 11 where (l_a + l_b) is the system size as given in Fig. 1. We use a standard sampling method to simulate the Hamiltonian given by Eq. (1). Cyclic boundary conditions on the lattice are imposed and the configurations are generated by sequentially traversing the lattices and making single-spin flip. The flips are accepted or rejected according to a heat-bath algorithm under the Metropolis approximation. Our data are generated with 10⁵ Monte Carlo steps per spin, discarding the first 10⁴ Monte Carlo simulations. Starting from different initial conditions, we perform the average of each parameter and estimate the Monte Carlo simulations, averaging over many initial conditions. Our program calculates the following parameters:

the magnetization of superlattices

the internal energy E per site

the magnetic susceptibility

The superlattices of ferromagnetic materials a and b are investigated by Monte Carlo simulation. The transition temperatures each as a function of surface exchange constant are studied. Figure 2 shows the variations of magnetization and magnetic susceptibility with reduced temperature T/J_aa for R₂ = 0.2, respectively, with Δ/J_aa = 0.0, R₁ = J_Sa/J_aa = R₃ = J_Sb/J_aa = R₄ = J_bb/J_aa = 1 and l_a = 5, 7, 9, and 11. The temperature dependences of the susceptibility are calculated for different lattice sizes in order to estimate the reduced critical temperature, T_C/J_aa. The position of the susceptibility maximum is allowed to be used to estimate the range of values of the critical temperature. The critical temperatures are obtained for each layer from the divergence (or the situation of peak) of magnetic susceptibilities at the reduced critical temperature. The obtained values are all 9 for l_a = 7, 9, 11 with R₂ = 0.2 and h/J_aa = 0.4. This value is superior to those obtained by high temperature series expansions^[22] and those obtained by effective field theory.^[23,24]

Fig. 2.

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	Fig. 2. Variations of magnetization and magnetic susceptibilities with reduced temperatures T/Jaa for R2 = 0.2 with Δ/Jaa = 0.0 and h/Jaa = 0.4.

Figure 3 shows the variations of magnetization and magnetic susceptibilities with reduced temperature T/J_aa for R₂ = J_ab/J_aa = 0.2, 1.5, 3, 5 with Δ/J_aa = 0.0, h/J_aa = 0.4, and R₁ = R₃ = R₄ = 1.0. The critical temperature is not very sensible with exchange interaction increasing. This behavior is comparable to those given in Refs. [22] and [25]. This behavior is similar to those obtained in Ref. [26] by using the Monte Carlo simulation in the case of mixed spins (2-1) hexagonal Ising nanowire with core–shell structure.

Fig. 3.

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	Fig. 3. Variations of magnetization and magnetic susceptibility with reduced temperature T/Jaa for R2 = 0.2, 1.5, 3, 5 with Δ/Jaa = 0.0 and h/Jaa = 0.4.

Figure 4 shows the variations of magnetization and magnetic susceptibilities with reduced temperature T/J_aa for different values of crystal field Δ/J_aa = 0, −1, −1.5, −2 with R₂ = 1 with R₁ = R₃ = R₄ = 1.0. The critical temperature decreases with increasing the absolute value of a crystal field (see Table 1). A similar behavior is observed in the previous work^[27] in which Monte Carlo simulation was conducted to investigate the phase diagrams of a ferrimagnetic cubic nanoparticle (nanocube) with a spin-3/2 core surrounded by a spin-1 shell layer with antiferromagnetic interface coupling. The variations of magnetization with reduced exchange interaction R₂ are given in Fig. 5. The plots of magnetization versus the R₂ with different reduced temperatures T/J_aa = 1, 1.5, 2, 2.5 with L_a = 7 are presented in Fig. 5(a). Presented in Fig. 5(b) are the plots of magnetization versus exchange interaction R₂ with L_a = 5, 7, 9, 11 for T/J_aa = 1.5. The magnetization increases with increasing R₂ and size of system and decreases with increasing reduced temperatures until it reaches saturation value. The same qualitative results have been obtained in early work.^[28,29] The system may be ordered in the bulk earlier than at surfaces; and when the number of layers increases, the ordering temperature of the system tends to the bulk one. For R₂ > 1.0, the system may order in interface layers earlier than in the other layers, i.e., the interface magnetism dominates and when the number of layers is very large, the system can be considered practically as a two-constituents superlattice with a critical temperature depending on magnetic coupling at the interface R₂.

Fig. 4.

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	Fig. 4. Variations of magnetization and magnetic susceptibilities with reduced temperature T/Jaa for Δ/Jaa = 0.0, −1.0, −1.5, −2.0, R2 = 1, h/Jaa = 0.4, and la = 5.

Fig. 5.

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	Fig. 5. Variations of magnetization with reduced exchange interaction R2: (a) for Δ/Jaa = 1.0, 1.5, 2.0, 2.5, la = 7 and (b) for la = 5, 7, 9, 11, T/Jaa = 1.5, and h/Jaa = 0.4.

Table 1.

Table 1.

Table 1. Values of reduced critical temperatures for different values of crystal field Δ with la = 5 and R2 = 1. . TC/Jaa 7.85 6.23 5 3.52 Δ/Jaa 0 −1 −1.5 −2

Table 1. Values of reduced critical temperatures for different values of crystal field Δ with la = 5 and R2 = 1. .

The variations of magnetization with reduced crystal field Δ/J_aa for (T/J_aa = 1, 2, 2.5, 3, R₂ = 1 and l_a = 5), for (R₂ = 1, 2, 3, 5, T/J_aa = 1 and l_a = 5) and for (l_a = 5, 7, 9, 11, T/J_aa = 1 and R₂ = 1), are given in Figs. 6(a)–6(c), respectively with R₁ = R₃ = R₄ = 1.0 and h/J_aa = 0.4. The magnetization increases with increasing the crystal field and deceases with increasing the reduced temperatures; magnetization increases with increasing the exchange interaction R₂ as shown in Figs. 5(a) and 5(b). The magnetization increases with increasing the size of system for Δ/J_aa > −2.47 and decreases with reducing the size of system for Δ/J_aa −2.47.

Fig. 6.

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	Fig. 6. Variations of magnetization with reduced crystal field Δ/Jaa (a) for T/Jaa = 1.0, 2.0, 2.5, 3.0, R2 = 1 and la = 5, (b) for R2 = 1.0, 2.0, 3.0, 5.0, T/Jaa = 1 and la = 5, and (c) for la = 5, 7, 9, 11, T/Jaa = 1 and R2 = 1 with h/Jaa = 0.4.

The magnetic hysteresis cycles with the reduced exchange interactions R₂ = 0.4, 3.0, for a zero reduced crystal field and T/J_aa = 1 are shown in Figs. 7(a) and 7(b), respectively. The coercive field and remnant magnetization decrease with increasing the reduced exchange interaction R₂ between the bi-layers for fixed value of reduced temperature and crystal field. This behavior is observed in previous work.^[30–32] The saturation magnetization and coercive field are obtained for given specific unit-cell sizes. Our results are also in good agreement not only with those obtained experimentally,^[33–37] but also with those estimated by the ab initio calculations in Ref. [38]. The magnetic hysteresis cycles with reduced crystal field Δ/J_aa = 0 and −1 for R₂ = 1 and T/J_aa = 3 are shown in Figs. 8(a) and 8(b), respectively. The coercive field and remnant magnetizations decrease with increasing the reduced crystal field in the bi-layers for fixed values of reduced temperature and reduced exchange interactions R₂. A similar compositional dependence of the coercivity was also observed in CoPt and CoO nanoparticles synthesized using the reverse micelle method.^[30,39] The increase of the reduced exchange interactions R₂ has no significant effect on coercive field nor on the remnant magnetization. The coercive field and remnant magnetizations decrease with increasing the reduced temperature as noted in Ref. [7]. The obtained results are similar to those obtained in Ref. [40]. The loop magnetic hysteresis cycle is observed at R₂ = 1, Δ/J_aa = −3 and T/J_aa = 3 with l_a = 5. Moreover, the increase of the temperature softens the jumps of the hysteresis curve (Figs. 7 and 8), which disappears for higher temperatures, as has been experimentally observed in the above-mentioned Ce system.^[41] Although the general trends of the experimental results are well described by the present results (symmetric hysteresis loops and evanescence of the jumps with increasing temperature), at this stage it is difficult to carry out a direct comparison with the experimental results. Finally, we show in Figs. 9(a) and 9(b) the magnetic hysteresis cycles with l_a = 5 and l_a = 11, respectively for a zero reduced crystal field, R₂ = 0.4 and T/J_aa = 1. The magnetic coercive field h_C increases with increasing the size. The observed increase in h_C is similar to previously reported results showing that h_C increases with film thickness decreasing.^[42,43]

Fig. 7.

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	Fig. 7. Magnetic hysteresis cycles with the reduced exchange interaction R2 = 0.4 (a), 3.0 (b) for a zero reduced crystal field, T/Jaa = 1 and la = 5.

Fig. 8.

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	Fig. 8. Magnetic hysteresis cycles with the reduced crystal field Δ/Jaa = 0.0 (a), −1.0 (b) for R2 = 0.4, T/Jaa = 3, and la = 5.

Fig. 9.

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	Fig. 9. Magnetic hysteresis cycles with la = 5 (a) and 11 (b) for a zero reduced crystal field, R2 = 0.4 and T/Jaa = 1.

The Monte Carlo simulations are used to investigate the magnetic properties of the Ising model of infinite a and b ferromagnetic superlattices having simple cubic crystal structure. The effects of the reduced interlayer exchange coupling R₂, crystal field and the thickness of the unit cell on a and b ferromagnetic superlattices are given. The transition temperatures as a function of exchange interaction constants and crystal field are calculated. The obtained values are comparable to those given by the other theoretical results. The coercive field and the remnant magnetization have no influence when increasing the reduced exchange interaction R₂. The coercive field decreases with increasing the reduced crystal field and increases with increasing the size of the system.