Cubic ferrite has the formula , where M is a divalent atom (e.g., Fe²⁺, Co²⁺, Zn²⁺, Ni²⁺, etc). The general chemical formula of inverse spinels family can be written in the form , where the parentheses represents the cations occupying tetrahedral A sites, whereas the brackets denote the cations occupying octahedral B sites in the spinel structure. The parameter x is the degree of inversion defined as the fraction of tetrahedral sites occupied by Fe³⁺ cations. Ideal bulk MFe₂O₄ is a fully inverse spinel ferrite (x = 1). The experimental determination of x is of vital importance because the physicochemical properties exhibited by the ferrites critically depend on this parameter.^[1] Besides technological applications in spin-filtering^[2] and multiferroic devices,^[3,4] several biomedical applications on targeting, diagnosis, and disease treatment have also been reported. In addition, the cations M and Fe in the lattice respectively occupy the tetrahedral (1/8, 1/8, 1/8) sites, Wyckoff position 8a, and octahedral (1/2, 1/2, 1/2) sites, Wyckoff position 16d. Therefore, the single positional parameter u plus the unit-cell parameter a are sufficient to determine the spinel structure. It is known that altering the occupancies of the M and Fe sites can give rise to slight variations of the spinel structure, which can affect the corresponding nature and magnitude of superexchange interactions,^[5] namely, A–A, B–B, and A–B, of which the A–B interactions tend to be highly significant.

Different types of ferrite-based composites have been studied, including magnetic ferrites composed of nonmagnetic oxides and polymers.^[6–8] For practical applications, however, Fe₃O₄ is the most promising candidate due to its high Curie temperature, T_C=858 K, as compared to other half metals. These characteristics of Fe₃O₄ can be utilized in the development of tunneling magnetoresistance based devices that would operate at room temperature with high efficiency.^[9,10] Experimentally, the properties of ferrites are dependent on the synthesis technique and synthesis conditions.^[11] NiFe₂O₄ has the inverse spinel structure in bulk form where divalent metal ions (Ni²⁺) occupy octahedral sites and mixed spinel structure in case of nanoparticles where Ni²⁺ ions can occupy both tetrahedral and octahedral sites.^[12] CuFe₂O₄ is ferromagnetically ordered for a weak magnetic field and it has wide applications due to its high thermal stability.^[13] Nickel ferrite NiFe₂O₄ is a good example of inverse spinel ferrite and exhibits ferromagnetic nature with a Curie temperature of 858 K where half of the Fe³⁺ ions occupy the tetrahedral sites and the remaining ions occupy the octahedral sites, it is used in repulsive suspension for levitated railway systems, microwave absorber, magnetic liquids, magnetic refrigeration, and high density magnetic recording media.^[14,15] In previous work,^[16] the CoFe₂O₄ nanowire arrays become superparamagnetic at 600 K. High magnetostriction and strain-field derivative with low saturation field make (0 0 1) fiber textured CoFe₂O₄ attractive for sonar detector and force sensor applications.^[17] Theoretically, the formation condition and energy of possible intrinsic point defects have been investigated by the first-principles calculations, and the effects of the intrinsic point defects on the electronic and magnetic properties of CoFe₂O₄ have been analyzed.^[18] The experimental and numerical results using Monte Carlo simulation of the reduced magnetization and coercivity cobalt ferrite/polyvinylpyrrolidone are in relatively good agreement with each other.^[19] The first-principles, mean-field, and series expansions calculations have been used to study the electronic and magnetic structures of Fe₃O₄.^[20] Green’s function theory has been used to study the magnetic properties of Cu_xNi_1−xFe₂O₄ spinels.^[21] The effect of Zn doping on magnetic properties of MnFe₂O₄ ferrite spinel has been studied by using high temperature series expansion combined with the Padé approximant and mean field theory.^[22] In the previous work,^[23] silicene is an intriguing two-dimensional (2D) topological material which is closely analogous to graphene but with stronger spin–orbit coupling effect and natural compatibility with current silicon-based electronic industry. It could realize a ferromagnetic phase, super-counter-fluidity phase, and antiferromagnetic phase of polarized light that are of interest for studying spin-dependent photon-photon interactions.^[24] The effect of spin transport on nonlinear excitations was investigated in a ferromagnetic nanowire with nonuniform magnetizations.^[25]

In the present work, we have used the Monte Carlo simulation to study the magnetic properties of inverse ferrite , , , and spinels. The critical and Curie Weiss temperatures are obtained for each system. The magnetic hysteresis cycles have been found for the four systems. The critical exponents associated with magnetization, magnetic susceptibility, and external magnetic field have been found.

In inverse spinel ferrites , the B site is occupied by two types of ions, Mn²⁺ and Fe³⁺, while the A site is occupied by Fe³⁺ ions. The Hamiltonian of the Ising system of inverse spinel ferrites with formula Fe³⁺( )O₄ (with M=Co, Cu, Fe, and Ni) in an external field H_ext may be put in the form 1 with or or or , .

The values of exchange interactions J_AA, , , , , and used in this work for four inverse ferrite , , , and spinels systems are given in Ref. [26] as listed in Table 1.

Table 1.

Table 1.

Table 1. The values of exchange interactions used in this work for the spinels. . Inverse spinels JAA/K[26] /K[26] /K[26] /K[26] /K[26] /K[26] –14 –28 –22.7 –9 –10.0 +46.0 –14 –28 –27.0 –9 –10.0 +29.0 –14 –28 –20.2 –9 –11.0 +44.0 –14 –28 –27.4 –9 –10.0 +44.0

Table 1. The values of exchange interactions used in this work for the spinels. .

The Fe³⁺( Fe³⁺)O₄ is assumed to reside in the unit cells, and the system consists of total number of sites , where N_A and N_B are the numbers of ions in site A and site B, respectively. We have used the Monte Carlo simulation to simulate the Hamiltonian given by Eq. (1). Periodic boundary conditions on the inverse ferrite spinels have been applied. Configurations have been generated by sequentially traversing the lattice and making single-spin flip attempts. Meanwhile, Monte Carlo update was performed by choosing random spins and then flipped with Boltzmann based probability. This can be done using the conventional Metropolis algorithm (where is the energy difference between the before and the after flip, and is the Bolzmann’s constant, and T is the absolute temperature).^[27] To obtain the numerical results, the data were generated with Monte Carlo steps per spin where the first Monte Carlo simulations are discarded and each Monte Carlo simulation started from a different initial condition.

The total internal energy is given by 2 The magnetizations of spinels Fe³⁺( Fe³⁺)O₄ (with M=Co, Cu, Fe, and Ni) are 3 4 where S_i and are the magnetic spins of the i ^th magnetic ions. The total magnetization is given by 5 The corresponding magnetic susceptibilities are given by 6 7 8 The critical exponents α, β, and γ associated with magnetization, magnetic susceptibility, and external magnetic field are given by 9

In the present work, we have used the Monte Carlo simulation to study the magnetic properties of inverse ferrite , , , and spinels. The critical temperatures T_C of , , , and are deduced from Figs. 1(a), 2(a), 3(a), and 4(a), respectively. The obtained values are given in Table 2. The Curie Weiss temperatures are deduced from the inverse of magnetic susceptibilities for the four inverse ferrite spinels from Figs. 1(a), 2(a), 3(a), and 4(a), respectively (see Table 1). The obtained values of T_C and are near to those obtained by experiment results^[28,29] and the molecular-field approximation.^[30]

Fig. 1.

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	Fig. 1. (a) The thermal magnetization and magnetic susceptibility and (b) inverse magnetic susceptibility of spinels. The straight line in panel (b) is used to deduce the Curie Weiss temperature.

Fig. 2.

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	Fig. 2. (a) The thermal magnetization and magnetic susceptibility and (b) inverse magnetic susceptibility of spinels. The straight line in panel (b) is used to deduce the Curie Weiss temperature.

Fig. 3.

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	Fig. 3. (a) The thermal magnetization and magnetic susceptibility and (b) inverse magnetic susceptibility of spinels. The straight line in panel (b) is used to deduce the Curie Weiss temperature.

Fig. 4.

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	Fig. 4. (a) The thermal magnetization and magnetic susceptibility and (b) inverse magnetic susceptibility of spinels. The straight line in panel (b) is used to deduce the Curie Weiss temperature.

Table 2.

Table 2.

Table 2. The critical and Curie Weiss temperatures obtained by Monte Carlo simulations, experiments (Ref. [25] for TC and Ref. [29] for ), and molecular-field approximation[30] for the spinels. . Inverse spinels TC/K (Monte Carlo) TC/K (experiment[28]) TC/K (molecular-field approximation[30]) /K (Monte Carlo) /K (experiment[29]) 830 793 826 888 520 743 728 739 883 455 834 858 858 784 585 834 858 857 960 585

Table 2. The critical and Curie Weiss temperatures obtained by Monte Carlo simulations, experiments (Ref. [25] for TC and Ref. [29] for ), and molecular-field approximation[30] for the spinels. .

The magnetic loops of , , , and spinels are shown in Figs. 7–10, respectively, for several values of temperatures T = 600 K, 850 K, and 1000 K. From these curves, we deduce that the magnetic coercive field, remanent magnetization, and saturation magnetization decrease with increasing temperature (see Figs. 5(b), 6(b), 7(b), and 8(b)). This behavior is similar in magnetic properties of multilayer BaTiO₃/NiFe₂O₄ thin films prepared by solution deposition technique in Ref. [31]. The well saturated hysteresis loops are recorded in the range from 0.2 T to −0.2 T (Figs. 5–8) and their shapes resemble the ferrimagnetic behavior typical for cobalt (or nickel, copper-iron) ferrite based materials.

Fig. 5.

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	Fig. 5. (a) Magnetic loop of for different temperatures T = 600 K, 850 K, and 1000 K, and (b) the enlarged low field region.

Fig. 6.

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	Fig. 6. (a) Magnetic loop of for different temperatures T = 600 K, 850 K, and 1000 K, and (b) the enlarged low field region.

Fig. 7.

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	Fig. 7. (a) Magnetic loop of for temperatures T = 600 K, 850 K, and 1000 K, and (b) the enlarged low field region.

Fig. 8.

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	Fig. 8. (a) Magnetic loop of for different temperatures T = 600 K, 850 K, and 1000 K, and (b) the enlarged low field region.

Figure 9–11 present the magnetic loop of , , , and for temperatures T = 600 K, 850 K, and 1000 K, respectively. The saturation magnetization and the magnetic coercive field decrease as the number of electrons in the valence shell of the magnetic atom (3d) increase, such as given in the periodic table. For a weak external magnetic field, the four systems present the ferromagnetical order and the superparamagnetic order for a higher external magnetic field such as given in Ref. [13].

Fig. 9.

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	Fig. 9. Magnetic loops of , , , and at temperature T = 600 K.

Fig. 10.

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	Fig. 10. Magnetic loops of , , , and at temperature T = 850 K.

Fig. 11.

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	Fig. 11. Magnetic loops of , , , and at temperature T = 1000 K.

We have calculated the critical exponents β, γ, and δ associated with the magnetization, magnetic susceptibility, and external magnetic field using Eq. (7). The obtained values of β, γ, and δ are given in Table 3. The obtained values are comparable with those given by 3D Ising model (β=0.22, γ=1.24, and δ=4.78).^[31–37]

Table 3.

Table 3.

Table 3. The critical exponents associated with the magnetization, magnetic susceptibility, and external magnetic field of the spinels. . Inverse spinels β γ δ 0.202 1.234 4.765 0.257 1.267 4.782 0.212 1.233 4.764 0.222 1.243 4.773

Table 3. The critical exponents associated with the magnetization, magnetic susceptibility, and external magnetic field of the spinels. .

The magnetic properties of inverse ferrite , , , and spinels are investigated. The critical temperatures and the Curie Weiss temperatures are found using Monte Carlo simulation. The obtained values of T_C and are near to those obtained by experiments. The magnetic coercive filed, remanent magnetization, and saturation magnetization decrease with increasing temperature. The critical exponents β, γ, and δ are deduced. The obtained values are comparable with those given by 3D Ising model.^[31–37]