Magnetic properties of thin magnetic films with magnetic-layered structures have been the subject of intense experimental and theoretical researches due to their great potential for technological applications, such as electronic semiconductor devices, optical coatings, computer memory, protection of substrate materials against corrosion, oxidation and wear.^[1] Many works have been experimentally investigated on the magnetic properties of the bilayers and multilayers thin films, such as Co/Pt, Pt/Co/Pt, Pt/Co/AlO_x, Gd-Co/Ti, W/B₄C, CoSiB/CSiB, Cd/Pd, Ni-Zr, Ni/Ti, Fe/Ni, Fe/Co, Ni/Au, Co/Cu/Ni₈₀Fe₂₀, La_2/3Ca_1/3MnO₃/SrTiO₃/YBa₂Cu₃O_{7 − x}, FePt/Fe, and Fe/TbFe etc. (see Refs. [2–17] and references there in). Theoretically, the spin-1/2 (see Refs. [18–30] and references therein), spin-1 (see Refs. [31–35] and references there in) and also mixed Ising systems, namely spin (2, 5/2),^[36–37] spin (1, 3/2),^[38] spin (3/2, 2)^[39] and spin (3/2, 5/2),^[40] have been used to study magnetic behaviors of the bilayers and multilayers thin films within the methods of the equilibrium statistical physics. Moreover, the bilayer spin-1,^[41] spin-3/2,^[42] spin^[43] as well as mixed-spin^[44–45] systems on the Bethe lattices have been also investigated to observe some magnetic properties of thin films. Dynamic magnetic properties of thin films have been examined by using the spin-1/2,^[46–50] spin-1,^[51–53] mixed-spin (2, 5/2),^[54–55] mixed-spin (3/2, 2)^[56–58] bilayer systems. In recent years, magnetic phase diagrams of ferrimagnetic mixed-spin system have been studied comprehensively (see Refs. [59–61] and references therein).

In spite of these studies, the dynamic magnetic critical properties of bilayer systems have not been investigated in detail. In particular, the crystal-field interaction, in which one of the important interaction parameter, either taken as zero or taken as same (D) for A and B sublatices in all these studies, except Ref. [57]. In Ref. [57], the crystal-field interaction for sites on A lattice as D_A and for sites on B lattice as D_B were taken and only the dynamic phase diagrams (DPDs) in ferromagnetic spin (3/2, 2) bilayer system was investigated. In this paper, influences of crystal-fields (D_A and D_B) and interlayer coupling interactions (J₃) on dynamic magnetic behaviors of a mixed-spin (3/2, 2) bilayer system under a time-dependent magnetic field are investigated by the Glauber-type stochastic (GTS) dynamics based on the mean-field theory (MFT) in detail. For this aim, the DPDs are presented in the reduced temperature (T) and magnetic field amplitude (h) plane for the ferromagnetic/ferromagnetic (FM/FM), antiferromagnetic/ferromagnetic (AFM/FM) and AFM/AFM interactions. The effect of the D_A, D_B as well as J₃, which is related to the thickness, on the critical behaviors of the system are extensively studied and discussed.

We take into account a mixed-spin (3/2, 2) model with a two-layers, namely L₁ and L₂, on a square lattices, shown in Fig. 1. Each layer of the model contains two sublattices (A and B) with spin variables , occupy L₁ layer and , layer. Thus, the system can be portrayed with four sublattice magnetizations that are introduced as follows: , , , denotes the thermal expectation value. Each layer has N sites and interacts with its nearest-neighbor (NN) and the corresponding adjacent spins in the other layer whose sites are labeled by i, i′, j and j′, as illustrated in Fig. 1.

Fig. 1

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	Fig. 1 Schematic representation of a two-layer square lattice: L1 and L2 refer to the upper and lower layers containing the spins labeled as , , and , .

The Hamiltonian of the system can be given by

The set of mean-field dynamic equations for sublattice magnetizations are found by using the GTS dynamics; hence the system evolves according to the GTS process at a rate of 1/τ transitions per unit time. We identify P(σ₁,σ₂,..., σ_N,S₁,S₂,...,S_N;t) as the probability that the system has σ- and S-spin configurations in each layer σ₁,σ₂,...,σ_N,S₁,S₂,...,S_N at time t. Let W_i be the probability per unit time that the i-th spin changes from the values to , while the others, i.e., (S₁,S₂, …, S_N) and spins on sublattice B, remain momentarily fixed, then the master equation can be written in which describes the interaction between the spins and the heat reservoir as

The probabilities satisfy the detailed balance condition. Utilizing Eqs. (1)–(4), we calculate the mean-field dynamical equation for the average magnetization on the sublattice A, i.e., as

Equations (5)--(8) were solved by the Adams-Moulton predictor-corrector method to find phases in the system for the FM/FM (J₁ > 0 and J₂ > 0), AFM/FM (J₁ < 0 and J₂ > 0) and AFM/AFM (J₁ < 0 and J₂ < 0) interactions for the given system parameters and initial values. Since the solution of these kind of dynamic equations were presented^{[46,50,54–57]} in detail, the solutions are not discussed and figures are not given. From these investigations we find that the system comprises, depending on system parameters, the paramagnetic (p), , the ferromagnetic (f): , the compensated (c): and , fundamental phases, and the f + p, f + c mixed phases for the FM/FM interaction. The p, af, antiferromagnetic (af): , , the surface (sf): , phases and the f + p, af+p, sf + f, sf + af mixed phases for the AFM/FM interaction. The p, af, and mixed (m): phases and the af + p, m+p, m+f, sf + f, sf + af mixed phases for the AFM/AFM interaction. We should also mention that since the some similar typical results for the dynamic magnetization curves for different magnetic phase states were given by Refs. [50, 56, 62–64], we do not present the dynamic magnetizations curves in order to avoid some overlaps.

To study the effects of D_A, D_B, and J₃ on the DPDs, one has to investigate the temperature dependence of the dynamic magnetizations as a function of the reduced temperature (T) that are defined as

Fig. 2

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Fig. 2 Phase diagrams of the spin (3/2, 2) Ising system on two-layer square lattice in (T, h) plane for the FM/FM interactions (J1 > 0 and J2 > 0), J1 = 1.0 and J2 = 1.0. The solid and dashed lines are the second- and first-order lines, respectively. TR, B, TP and QP represent thedynamic tricritical, dynamic critical end, triple and quadruple points, respectively. (a) J3 = 5, DA = 0.0 and DB = 0.0, (b) J3 = 0.25, DA = − 0.5, DB = 0.0; (c) J3 = 5, DA = 0.0, DB = −0.55, (d) J3 = 0.25, DA = −0.5, DB = −0.25, (e) J3 = 5, DA = −0.25, and DB = −0.5, (f) J3 = 0.25, DA = −0.25, and DB = −0.5.

Fig. 3

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	Fig. 3 Same as Fig. 2, but J3 = −5 for (a), (c), (e) and J3 = −0.25 for (b), (d), (f). The special points are the dynamic multicritical (A) and dynamic tetracritical (M) points.

Fig. 4

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	Fig. 4 Same as Fig. 2, but for the AFM/FM interactions (J1 < 1.0 and J2 > 0), J1 = −1.0, J2 = 1.0. J3 = 5.0 for (a), (c), (e) and J3 = 0.25 for (b), (d), (f).

Fig. 5

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	Fig. 5 Same as Fig. 4, but for J3 = −5 for (a), (c), (e) and J3 = −0.25 for (b), (d), (f).

Fig. 6

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	Fig. 6 Same as Fig. 2, but for the AFM/AFM interactions (J1 < 0 and J2 < 0), J1 = − 1.0, J2 = −1.0, J3 = 5 for (a), (c), (e) and J3 = 0.25 for (b), (d), (f).

Fig. 7

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	Fig. 7 Same as Fig. 6, but J3 = −5 for (a), (c), (e) and J3 = −0.25 for (b), (d), (f).

Figures 2 and 3 show the DPDs of the FM/FM interactions for the repulsive (J₃ > 0) and attractive (J₃ < 0) interlayer coupling constant, respectively, and J₁ = 1.0, J₂ = 1.0, various values of D_A and D_B. From Fig. 2, we have seen following interesting phenomena: (i) For D_A = D_B = 0, the system contains the p, f and f+p phases as well as T_R and B special critical points. The dynamic phase boundaries between the f+p and f phases and between the f+p and p for high values of h are first-order lines. The dynamic boundaries between the f and p phase and between the f+p and p for low values of h are second-order lines, seen Figs. 2(a) and 2(b). (ii) For D_A ≠ 0 and D_B = 0 or wise versa, the DPD is like to Fig. 2(a), except the B disappears and the first-order line between the p+f and f terminates at T_R, illustrated Figs. 2(c) and 2(d). (iii) For D_A ≠ 0 and D_B ≠ 0, the phase diagrams are similar to Fig. 2(c), except the A special critical point and the f+p appear for the low values of h and high values of T. The phase boundaries among the f, p+f and p phases are second order-phase lines (Fig. 2(e)). Moreover, if D_A > D_B the A point and a new p+f mixed phase appear at lower values of h and higher values of T, compare Fig. 2(c) with Fig. 2(e). For small values of J₃, the A critical point and p+f phase do not appear (Fig. 2(f)). Figure 3 (J₃ < 0) illustrates the effect of J₃, if one compares with Fig. 2, one can see the basic and important difference is that the c phase occurs instead of the f phase. Moreover, the following influences have been also observed: (i) The B point is not observed in Fig. 3. (ii) The TP appeared in Fig. 3(d). We should also mention that the behavior of Figs. 3(a), 3(c), and 3(e) are akin to Figs. 2(a), 2(c), and 2(e). Figure 3(f) displays more richer critical behaviors in which for high values of T and low values of h, the A, M special critical points and two more the p+c phase are observed. The boundaries among these phases are second-order lines.

Figures 4 and 5 display the DPDs of the AFM/FM interactions for the repulsive (J₃ > 0) and attractive (J₃ <0) interlayer coupling constant, respectively, J₁ = −1.0, J₂ = 1.0 and various values of D_A and D_B. From Fig. 4, we have seen following interesting phenomena: (i) For D_A = D_B = 0, the system contains the p, f and p+f phases as well as the T_R tricritical point. The dynamic phase boundaries among the p+f, f and p phases are first-order lines but the dynamic boundary between the f and p phase is a second-order, seen in Fig. 4(a). For small values of J₃, the TP point is also observed (Fig. 4(b)). (ii) For D_A ≠ 0 and D_A = 0 or wise versa, the DPD is akin to Fig. 2(a), except followings; A critical point and the p+f mixed phase appear for the low values of h and high values of T, seen in Fig. 4(c) and the new f+sf phase occurs at low values of T and h. (iii) For D_A ≠ 0 and D_A ≠ 0 (Figs. 4(e) and 4(f)), the phase diagrams are very similar to Fig. 4(c), except the A special critical point and one more the p+f phase occur at high values of T and low values of h, seen in Fig. 4(f). (iv) For small values of the J₃ the sf+f mixed phase occurs at low values of T and h, and the boundary between the sf+f mixed and f phases is a second-order phase line (Fig. 4(f). The behavior of DPDs in Figs. 5(a), 5(c) and 5(e) are similar to Figs. 4(a), 4(c) and 4(e), but the difference is that the f phase transforms into the c phase. Fig. 5(b) displays the same behavior as Fig. 5(a), except the c phase turns to the af and the TP point and the c+af mixed phase at low T appear. The DPD in Fig. 5(d) is similar to Fig. 5(b), except the reentrant behavior is observed. Fig. 5(f) illustrates the same behavior as Fig. 5(b), but differences are following. (i) The c+af at low T disappears and the new p+af phase is seen. (ii) The boundaries among the af, p+af and p phases are second-order lines.

Figures 6 and 7 display the DPDs of the AFM/AFM interactions for the repulsive (J₃ > 0) and attractive (J₃ < 0) interlayer coupling constant, respectively, J₁ = −1.0, J₂ = −1.0 and various values of D_A and D_B. First of all, only Figs. 6(a), 6(c), 6(e) display a reentrant behavior. For D_A = D_B = 0 (Figs. 6(a) and 6(b)), the system contains the p, m and p+m as well as the T_R tricritical point. The f+m mixed phase occurs for high values of J₃ (Fig. 6(a)) and the TP point is seen for low values of J₃ (Fig. 6(b)). Moreover, the dynamic phase boundaries among the p+m, m and p phases and the boundary between the f+m and m are first-order phase lines but the dynamic boundary between the m and p phase is a second-order line. Figure 6(c) is like Fig. 6(a), but the DPD in Fig. 6(c) contains the new p+m phase and the TP, A points at high values of T. The boundary between the m and f+m, and the boundaries among m, p+m and p are second-order lines. The DPD in Fig. 6(d) displays same behavior as Fig. 6(c), except the TP, A points and the p+m phase as well as the reentrant behavior do not occur. The topology of Fig. 6(e) is very akin to Fig. 6(c), except the p+m phase does not occur. The behavior in Fig. 6(f) is very similar to Fig. 6(d), except the TP point also appears. The behavior of Fig. 7 is similar to Fig. 6, except following two main differences: (i) The m phase turns into the c phase in Figs. 7(a), 7(c), 7(e) and the af phase in Figs. 7(b), 7(d), 7(f). (ii) The T_R appears for lower values of h.

In conclusion, we study influences of crystal-fields (D_A and D_B) and interlayer coupling (J₃) interactions on the DPDs of the mixed-spin (3/2, 2) bilayer system for the FM/FM, AFM/FM, and AFM/AFM cases in the presence of a time-varying magnetic field within the GTS dynamics based on the MFT. We investigate the influence of the D_A, D_B as well as J₃ on the DPDs and find that their effects on the dynamic magnetic critical properties of the DPDs more for the FM/FM, AFM/FM interactions, but less for the AFM/AFM. In particular, the sign and magnitudes of J₃ are mostly effect the phases in the system and the D_A and D_B mostly affect the critical behavior of the DPDs. Finally, it is worthwhile to mention that the mixed-spin (3/2, 2) bilayer system also shows the compensation behavior as the other mixed-spin bilayer systems^[59–63,65] that the comprehensively study should be worked out.