Mixed-spin systems composed of two magnetic A and B sublattices constitue a very vast area of research. Their magnetic properties are very important to study certain types of ferrimagnets, such as molecular-based magnetic materials,^[1–4] pure mixed MnNi(EDT)6H₂O systems,^[5] bimetallic compounds AMn(S₂C₂O₂)₂(H₂O)₃:4.5H₂O (A = Cu, Ni, Pd, Pt),^[6] pure and diluted complexes mixed A_k[B(CN)₆]_l:nH₂O systems (where A and B are transition metal ions with different spin values).^[7] Mixed systems with ferromagnetic and ferrimagnetic interactions showed several results, such as the presence of multiple-shifted hysteresis loops, compensation behaviors and multicritical phenomenon, including tricritical and reentrance behaviors. In fact, the hysteresis loops illustrating the applied magnetic field dependence of magnetization, is one of the most important features of magnetic systems.

Unlike ferromagnets, ferrimagnetic mixed systems show, under certain conditions, the existence of a compensation temperature T_k, below the curie temperature, , at which the magnetization vanishes.^[8] The existence of such compensation point is of great technological importance, since at this point, only a slight variation of the field is necessary to change the sign of the resultant magnetization. This property is very useful in thermomagnetic recording. At temperatures , the magnetization of the system is proportional to the applied magnetic field and vanishes with it. However, for , a spontaneous magnetization appears in the absence of any external sources and vanishes at T_k. The appearance of the zero magnetization before the critical temperature results from an exact compensation of the magnetic moments of the A- and B-sublattices at . This is due to competition between the exchange interactions in the system, when the temperature dependence of the sublattice magnetizations becomes complex.^[9] The unstable compensation phenomenon depends also on the values of spins S_A and S_B, the structure of the system, the applied magnetic field, as well as the value of the anisotropy D_z.^[10–17] Generally, the total magnetization can take different shape types, such as Q-, R-, S-, P-, N-, W-, M-, and X-types. The N-, W-, and X-types are the case for which a one, two, and three compensation temperatures exist, respectively.^[18–20]

Various theoretical methods were employed to study these structures. Within exact methods,^[21] magnetic properties in two-dimensional spin-1/2 systems decorated with S = 1 spins were investigated. By both three-dimensional (3D) and one-dimensional (1D) micromagnetic calculations, the effect of exchange coupling on magnetic property in Sm–Co/α–Fe layered system was examined,^[22] the generalized theoretical approach is applied to evaluate the negative magnetoresistivity of ErBi in external magnetic fields.^[23] Applying the mean-field approach, magnetic properties of rare-earth antiferro-nanoparticle,^[24] two-sublattice with alternate layers,^[25] TbNi₂B₂C with a two-sublattice model,^[26] and the bilayer system^[27] were discussed. By means of the effective-field theory, phase diagrams and magnetic properties in the mixed Ising system with two alternative layers,^[28] hexagonal and ferrimagnetic Ising nanowire,^[19,29] and ferrimagnetic Ising System with a random longitudinal field^[30] were studied. Finally, by the use of Monte Carlo calculations, magnetic behavior and thermodynamic properties of core–shell nanoribbon^[31] and graphyne core/shell nanoparticles^[32] were studied in detail. A number of characteristic behaviors were observed, such as the occurrence of several compensation temperatures, the existence of two new and non-classified types of compensation behavior. Experimmentally, two compensation points were obtained in the system [(NiMnFe)[Cr(Cn)₆]7.6H₂O].^[33] In addition, a single and triple hysteresis loops with different steps and shapes are observed. A triple hysteresis loop patterns may have potential applications in multi-state memory devices.^[4] Recently, interesting results on mixed Ising ferro/ferrimagnetic bilayer systems^[18] within the framework of effective-field theory (EFT) were reported. It was shown that hysteresis loops can be modified as function of the temperature and the sign of the exchange interaction between A and B atoms.

Among mixed Ising models that are extensively studied, we find mixed spin−1 and spin-3/2 Ising system. Using techniques, such as: effective-field theory,^[34–39] mean field theory (MFT),^[40] Monte Carlo simulations (MCS),^[41] pair approximation,^[42] real space renormalization group theory,^[43] phase diagrams and magnetic properties were investigated. The results obtained present a variety of phase diagrams with first and second-order phase transitions. The author found that the tricritical behavior exists and strongly depend on both the lattice coordination number and the crystal field. One and two compensation temperature are found in ferrimagnetic systems in the presence of a crystal field D.^[35] The effect of random longitudinal field was also discussed.^[44–46] By performing EFT with the probability distribution technique, it was shown that the tricritical behavior is very sensitive to the negatives values of and for specific exchange interaction couplings between A and B atoms.^[47] Part of the numerical results obtained are in good agreement with those reported in the framework of MFT.^[48] Lately, it was reported that three points of compensation could be observed in mixed spin-1 and spin-3/2 with equal random crystal field.^[26]

The purpose of this paper, is to go beyond the results of Ref. [47] by employing EFT with the probability distribution technique to study the hysteresis behavior and the compensation phenomenon of mixed spin−1 and spin−3/2 ferromagnetic and ferrimagnetic Ising model in the presence of random crystal field. The system consists of two interpenetrating inequivalent magnetic sublattices A and B with different spins σ = 1 and S = 3/2, respectively. To our knowledge, such a study has not been carried out before. Our results show that the compensation phenomenon and the hysteresis loops can be controlled by varying both random crystal field and/or exchange interaction between atoms of the A- and B-sublattices. The effect of the temperature on the hysteresis loops is also discussed and other interesting results are found.

The outline of this paper is as follows: in Section 2, we describe briefly the model and the formulation. In Section 3, we discuss the obtained numerical results for magnetizations and phase diagrams. Finally, in Section 4, a brief conclusion is given.

We study mixed Ising system, consisting of two interpenetrating inequivalent magnetic sublattices A and B with different spins σ = 1 and S = 3/2, as depicted in Fig. 1, in the presence of longitudinal random crystal fields and . The sites of sublattices A are occupied by with spin values 1, while those of sublattices B are occupied by spins taking spin values 3/2. The Hamiltonian of the system is given by

Fig. 1.

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	Fig. 1. (color online) Two-dimensional mixed spin system consisting of two kinds of magnetic atoms with spin values and on the interpenetring sublattices A and B. White and black cercles refer to A- and B-atoms, respectively in the square lattice.

In the single site cluster theory, attention is focussed on the cluster comprising just a single selected spin and its neighboring spins with which it directly interacts. The Hamiltonian of the system can be divided into two parts one denoted H₀ which includes all contributions associated with the site 0, the other (denoted by ) does not depend on the site 0. Then, the Hamiltonian at site 0 can be rewritten in the following form

Because H₀ and commute, the longitudinal ordering parameters of the system are approximatively given by^[16,17]

To perform thermal averaging on the right-hand sides of Eqs. (5), one can follow the general approach described in Ref. [49]. First, in the spirit of the EFT, the multi-spin correlation functions are approximated by products of single spin averages. In basing on the integral representation of the Dirac delta distribution, equations (5) take the new following form

To evaluate the averages in the right-hand side of Eqs. (7) and with the use of the probability distribution technique of spin variables and (for details, see Refs. [49] and [50])

These five coupled equations for ordering parameters m_Az, q_Az, m_Bz, q_Bz, and r_Bz can be solved directly by numerical iteration without any further algebric manipulations. In order to determine the transition temperature , we can linearize and easily solve Eqs. (10 and (11), since , , when the temperature approaches . This leads to the matrix equation

In the following, we examine the temperature dependence of the longitudinal magnetizations m_Az, m_Bz, and the total longitudinal magnetization M_z of the system defined by the following expression

We discuss the effect of random crystal field and exchange interaction on magnetic properties as well as compensation behavior and hysteresis loops of square lattice . For simplicity, we take J as the unit of energy and we introduce the reduced exchange interaction .

3.1. Random crystal field effect

Figures 2(a)–2(c) show thermal variations of total magnetization when the system is strongly influenced by the random crystal field for p = 0 and q = 0.01. At fixed values of and for selected values of , the total magnetization has Q-type standard shape^[18] that usually can be observed in a ferromagnet. For , we can see that exhibits S-type shape.^[18–20] All curves decrease monotonically from the saturation values to vanish at the critical temperature . The effect of field is clearly observed, in particular, the total magnetizations become smaller in negative regions of and vanish rapidly with decreasing the value of .

Fig. 2.

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	Fig. 2. (color online) The temperature dependence of the average magnetizations in the ferrimagnetic mixed spin-(1, 3/2) Ising system for selected values of at p = 0 and q = 0.01. The field takes the value , , .

In Figs. 3(a)–3(c), the thermal variations of total magnetization is depicted for selected values of parameter q and crystal field when the values of and p are fixed at and p = 0 (all spins for the sublattice A are affected by the random crystal fields . The corresponding results show a low magnetizations S-type,^[18], for q = 0.01, in particular for values of the random crystal field (Fig. 3(a)). Afterwards, at , where much of spins of sublattices B are not effected by , the different curves tend to merge with increasing or decreasing the value of (Fig. 3(b)). For the limit value q = 1, where all spins of the sublattice B are not affected by the field , the magnetization presents one curve with the Q-type shape for several values of , as shown in (Fig. 3(c)). This implies that when increasing the value of q, the effect of decreases and disappears in the pure case. The obtained results are in good agreement with those reported in Refs. [47] and [48].

Fig. 3.

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	Fig. 3. (color online) The temperature dependence of the average magnetizations in the ferrimagnetic mixed spin-(1, 3/2) Ising system for p = 0 and q = 0.01 (a), p = 0 and q = 0.81 (b), p = 0 and q = 1.0 (c), and for the selected values of . The value of is fixed at .

3.2. Compensation behavior

3.2.1. Random crystal field effect

To shed light on the existence of compensation points in mixed spin-(1, 3/2) ferrimagnetic system, figures 4(a) and 4(b) display thermal variation of magnetization for selected values of p and q (where lattice is strongly affected by the random crystal field), and for several values of at which tricritical points of phase transition temperature of the system are found.^[47] In Fig. 4(a), the magnetization of the system exhibits three standard shape types Q–R–S^[18–20] and vanishes at the critical temperature . Further, it becomes smaller for negative values of , in particular, when . In these cases, no compensation points are found as shown in the figure.

Fig. 4.

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	Fig. 4. (color online) Average magnetization as a function of temperature for the ferrimagnetic mixed spin-(1, 3/2) Ising system in the tricritical values of the , when the probability parameters p and q are fixed at p = 0 and q = 0.01 (a), and p = 0.2185 and q = 0 (b).

In Fig. 4(b), when increasing the value of parameter p to p = 0.2185, the random crystal field effect on the A-sublattice decreases and the ferrimagnetic system exhibits one or even two points of compensation in the regions and . Our numerical results show four shape types N–P–R–W of the total magnetization. The N- and W-types are the case for which, a one and two compensation temperature exist, respectively. Moreover, the P-type behavior appears and the total magnetization is smallest without presenting compensation points, particularly, in the region . It follows that the random crystal field and have a strong effect on compensation temperature. Note that p = 0.2185 corresponds to the critical value of the probability parameter p at which the tricritical behavior of the transition temperature disappears for q = 0 and .^[47]

Figures 5(a) and 5(b) plot the temperature dependence of the total magnetization for some typical values of and . The results show that the compensation behavior appears approximately when and for fixed values of and , respectively. This compensation behavior disappears with increasing the values of p and q as shown in Figs. 6(a) and 6(b).

Fig. 5.

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	Fig. 5. (color online) The temperature dependence of the magnetizations for tricritical values of and (a), and and (b) with q = 0 and for several values of p.

Fig. 6.

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	Fig. 6. (color online) Magnetizations for tricritical values of and with a typical value of p = 0.22 (a), and of and with p = 0.2188 (b) and for several values of q.

3.2.2. Exchange interaction effect

Figures 7(a)–7(c) show the influence of the strength of the exchange interaction parameter R on the compensation behavior for the selected values of the random crystal field (, when the parameters p and q are fixed at values p = 0.2185 and q = 0. The average magnetization exhibits a compensation point only when the (as shown in Figs. 7(a) and 7(b)). This compensation behavior disappears with increasing () or decreasing () the values of the parameter R. Finally, whatever the value of R, the system does not exhibit any compensation point if the values of the and do not belong to the intervals [−0.8588,−1.1863] and [−0.3278,−0.5], respectively as displayed in Fig. 7(c).

Fig. 7.

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	Fig. 7. (color online) The temperature dependence of the magnetizations for tricritical values of and (a), and and (b), and and (c) with p = 0.2185 and q = 0 and for several values of R.

3.3. Hysteresis behaviors

3.3.1. Random crystal field effect

This subsection evaluates hysteresis loops at fixed values of temperature T = 0.2 and parameter . Figures 8(a) and 8(b) show M_z curves versus the applied magnetic field at fixed values of p = 0.25 and q = 0.02. The (M_z–h) ferrimagnetic and ferromagnetic hysteresis loops are obtained by changing cyclically the value of the magnetic field for selected values of the random crystal field . From these figures we can see that, in the absence of the random crystal field , the magnetization curves are symmetric for both positive and negative values of the longitudinal magnetic field. Moreover, the hysteresis loops become narrower with decreasing in the region of the negative values and disappear completely for .

Fig. 8.

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	Fig. 8. (color online) The applied magnetic field h dependence of in the ferrimagnetic (a) and ferromagnetic (b) mixed spin-(1, 3/2) Ising system for selected values of the random crystal field with and . The values of p and q are fixed at p = 0.25 and q = 0.02.

With changing the value of the probability parameter q, figures 9(a)–9(e) present hysteresis behaviors of ferrimagnetic system for fixed p = 0 and selected values of and . One can notice that the coercive field in the hysteresis loops decreases with decreasing q, , and (Figs. 9(a)–9(c)). This reduction on the coercive field is due to the collective response of the magnetic moments that lead to reduction of the energy barrier for the magnetization reversal. As shown in Figs. 9(d)–9(e), the hysteresis loop completely disappears when q = 0, in particular for and , and for and .

Fig. 9.

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	Fig. 9. (color online) Average magnetization as a function of applied magnetic field in the ferrimagnetic mixed spin-(1, 3/2) Ising system for selected values of q with p = 0 and R = −0.3. The values of and are fixed at and (a), and and (b), and and (c), and and (d), and and (e).

3.3.2. Temperature and exchange interactions effect

To examine the effect of ferromagnetic () and ferrimagnetic () exchange interactions on the hysteresis properties, figures 10(a) and 10(b) plot longitudinal magnetizations M_z dependence of the applied magnetic field h, by changing the value of exchange interactions parameter , 0.5, 1.0 at fixed p = 0.25 and q = 0.02. The results show that with increasing the value of the interaction R, the coercive field increases and the single central hysteresis loop gets larger. In (M_z–h) ferrimagnetic Ising model, the hysteresis curve exhibits a single loop whose width increases with decreasing R. The single hysteresis loop disappear at low values of R for both ferrimagnetic () and ferromagnetic () Ising systems.

Fig. 10.

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	Fig. 10. (color online) The applied magnetic field h dependence of the magnetization for selected values of the random crystal field with , R = −0.3 (a), and R = 0.3 (b). The values of p and q are fixed at p = 0.25 and q = 0.02.

Finally, the longitudinal magnetizations m_Az, m_Bz, and M_z dependences of the longitudinal magnetic field are presented in Figs. 11(a)–11(c) for different values of temperature and at fixed values of , , R = −0.3, p = 0.25, and q = 0.02. It is found that the average magnetization exhibits a triple hysteresis loop at low temperatures. The unusual behavior observed at is due to the competition between the exchange interactions, the magnetic field applied, and the thermal fluctuations. Indeed, when the value of the applied magnetic field is sufficient to align all magnetizations in its direction, the antiferrimagnetic interactions between A and B sublattices cause an opposite alignment of magnetization m_Az. Next, the applied field returns to overcome the competition again; which leads to align all magnetizations of the B sublattice in its direction as well. Therefore, the result is the appearance of the triple loops in the total magnetization. The outer loops become narrower and disappear with rising temperatures, as seen for .

Fig. 11.

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	Fig. 11. (color online) The applied magnetic field h dependence of the magnetizations mAz (a), mBz (b), and (c) in the ferrimagnetic and ferromagnetic mixed spin-(1, 3/2) Ising system for selected values of the random crystal field with and . The values of p and q are fixed at p = 0.25 and q = 0.02.

In conclusion, hysteresis and compensation behaviors of ferromagnetic and ferrimagnetic mixed spin-(1, 3/2) Ising model on square lattice in the presence of random crystal field were investigated by using EFT with probability distribution technique. Our system consists of two interpenetrating magnetic sublattices and where at each sublattice site there is an extra random crystal field ( and ) acting on the corresponding sublattice. It is shown that the model presents one and two compensation points depending on the probability parameters p and q, the values of the exchange interactions R between A and B atoms and the values of random crystal fields and . Afterwards, we have also demonstrated that hysteresis loops and coercive field are highly influenced by the random crystal fields, in particular for negative values. Another important finding is the observation of the triple hysteresis loops at low temperatures. The hysteresis loops disappear as the ferrimagnetic exchange interaction R decreases. This behavior has been investigated carefully and the physical explanation has also been given briefly. Some of our results are in good agreement with some theoretical works.