coatings The Study of the Inﬂuence of Matrix, Size, Rotation Angle, and Magnetic Field on the Isothermal Entropy, and the N é el Phase Transition Temperature of Fe 2 O 3 Nanocomposite Thin Films by the Monte-Carlo Simulation Method

The Study of the Inﬂuence of Matrix, Size, Rotation Angle, and Magnetic Field on the Isothermal Entropy, and the N é el Phase Transition Temperature of Fe 2 O 3 Nanocomposite Thin Films by the Simulation Method. Abstract: In this paper, the study of the inﬂuence of the matrix structure (mxm) of thin-ﬁlm, rotation angle ( α ), magnetic ﬁeld (B), and size (D) of Fe 2 O 3 nanoparticle on the magnetic characteristic quantities such as the magnetization oriented z-direction (M zE ), z -axis magnetization (M z ), total magnetization (M tot ), and total entropy (S tot ) of Fe 2 O 3 nanocomposites by Monte-Carlo (MC) simulation method are studied. The applied MC Metropolis code achieves stability very quickly, so that after 30 Monte Carlo steps (MCs), the change of obtained results is negligible, but for certainty, 84 MCs have been performed. The obtained results show that when the mxm and α increase, the magnetic phase transition appears with a very small increase in temperature N é el (T Ntot ). When B and D increase, T Ntot increases very strongly. The results also show that in Fe 2 O 3 thin ﬁlms, T Ntot is always smaller than with Fe 2 O 3 nano and Fe 2 O 3 bulk. When the nanoparticle size is increased to nearly 12 nm, then T Ntot = T = 300 K, and between TNtot and D, there is a linear relationship: T Ntot = − 440.6 + 83D. This is a very useful result that can be applied in magnetic devices and in biomedical applications.


Introduction
Thin films, two-dimensional physical systems with different structures and materials, have been studied for many decades, especially after the discovery of the so-called giant magnetoresistance effect [1,2] and topological phase transitions [3,4]. Research in this domain not only gives us theoretical results concerning the foundations of modern physics but also leads to the nanotechnology of thin layers, which gives different new functional materials with significant applications in practice. Recently, Prof. Mirosław Dudek et al. [5][6][7][8][9] intensively studied mechanical and magnetic characteristics of different two-dimensional nanocomposite materials. We will discuss their results obtained in detail below. We would like to emphasize that our paper follows this direction of study.
Nowadays, magnetic nanocomposites play an important role in science and technology [10,11]. It is important in practical applications to find materials that can be used in devices such as force sensors and refrigeration equipment. These devices always ensure requirements such as high load capacity, good corrosion resistance, and lightweight. To ensure these requirements, we choose Fe 2 O 3 nano synthetic thin film as the subject of this paper. The reason is that the synthetic nano-Fe 2 O 3 thin film is an antiferromagnetic material [12][13][14][15][16][17] with many advantages relative to the original material. Moreover, it is widely used in practical applications such as recording equipment [18,19], refrigeration relationship and the isothermal magnetization curve. However, up to now, the use of the Maxwell relationship to determine entropy change is still causing much controversy and discussions about it [72][73][74][75][76]. Giguère et al. [72] successfully determined the entropy change based on the Maxwell relationship. Balli et al. [75] and Liu et al. [76] also successfully demonstrated that the Maxwell relationship no longer exists when the material is near the phase transition (paramagnetic phase, ferromagnetic phase, antiferromagnetic). They suggested that the cause of the appearance of the giant magnetic field at room temperature is due to the morphological leading to changes in the magnetic transition temperature at the T~110 K. In addition, there is an arrangement of the magnets. Different cations lead to a structural change from tetrahedral to octahedral, driven by spin-oriented ions at the magnetic particles, and a magnetic transition occurs at T Ntot~1 50 K. T. Muto et al. [77] studied the entropy, whereas P. Fratzl [78] and Dieter [79] determined the phase transition diagram. In 2010, Jirı Tucek et al. [80] successfully determined the existence of the huge magnetic field of nano ε-Fe 2 O 3 at room temperature. Recently, M.R. Dudek et al. have successfully determined the giant magnetic field of Fe 3 O 4 nanomaterials at room temperature [6], the magnetic domain in auxetic materials [7,8], and successfully constructed a single-body Hamiltonian function [6,12,17]. Combining with the phase space average field in [9] to study the magnetism of Fe 3 O 4 nanoparticles [81], Dung et al. [82] determined the magnetic properties of Fe nanomaterials by the Monte-Carlo simulation method with the classical Heisenberg model. In addition, researchers also successfully studied the influence of factors such as temperature, number atomic, pressure, annealing time on structure, electronic structure, phase transition, and crystallization progress of material metals [83][84][85][86][87], alloys [88][89][90][91][92], oxide [61][62][63], and polymers [93,94]. Here, a question appears: how to determine the magnetic characteristic quantities of materials such as magnetization in all directions and entropy of the material when the size of the material is less than 10 nm. To answer this question, in this article, we focus on a study of the influence of factors such as material size (mxm), magnetic field (B), and nanoparticle size on magnetization in the direction priority z-axis (M zE ), z-axis magnetization (M z ), total composite magnetization (M tot ), and total entropy of Fe 2 O 3 nano synthetic thin films. For this purpose, we used the Monte-Carlo simulation method. The obtained results will serve as a basis for future experimental studies when we try to apply Fe 2 O 3 nano synthetic thin films to smart devices and refrigeration equipment.

Method of Calculation
Initially, the two-dimensional model for Fe 2 O 3 nano synthetic thin film is constructed by creating a 2D matrix square (mxm). These matrices are composed of nonmagnetic squares and linked together by hinges defined as the intersection points between the corners of the squares (red color in the figure), and these 2D square matrices may be deformed [95]. Then, spherical Fe 2 O 3 nanoparticles were put into the 2D matrix square. For simplicity, we treat each nanoparticle as a magnetic spin. When the thin film model is not deformed, the rotation angle of these 2D square matrices has the value α = 0 • (Figure 1a). When the 2D square matrices rotate, the corresponding rotation angle varies from α = 0 • to α = 90 • (Figure 1b).
In this model, the size of each 2D square is the diameter of the inserted Fe 2 O 3 nanoparticle, with D = 2R, where R is the radius of the nanoparticle. We studied the magnetic properties of Fe 2 O 3 nano synthetic thin films by applying a potential force field to the nano synthetic thin film, with the value of the Hamilton function of the form (Equation (1)) [6], and numerical simulation was performed by the Monte-Carlo method.
where K a = 1 k B T · µ 0 4πM 3 , V = 4 3 πa 2 0 , a 0 = 8.394 Å, and K ij = µ 0 M 2 4πd 3 , d = √ 2asin α+ π 4 , a = D + a 0 , D = 2R g .  In this model, the size of each 2D square is the diameter of the inserted Fe2O3 nanoparticle, with D = 2R, where R is the radius of the nanoparticle. We studied the magnetic properties of Fe2O3 nano synthetic thin films by applying a potential force field to the nano synthetic thin film, with the value of the Hamilton function of the form (Equation (1)) [6], and numerical simulation was performed by the Monte-Carlo method. In it, Ka is the uniaxial magnetic anisotropy energy; the magnetic coefficient is 7 0 μ 4 10 π − = × m/A; V is the volume of the nanoparticle; a0 is the lattice constant; Kij is the interaction energy between the i th nanoparticle and the nearest j th nanoparticle; and Boltzmann's constant kB = 1.38 × 10⁻ 23 J/K = 8.617 × 10⁻⁵ eV/K, where B is the magnetic field, M is magnetic moment, αi is the rotation angle of the i th 2D square matrix, mi and mj are the magnetic moments of the i th and j th atom, and d, a, Rg, and D are the distance between the centers of the two nearest nanoparticles, the size of the square edge, the displacement radius, and the size of the Fe2O3 nanoparticle, respectively. Conversely, the size of the model is determined by the following formula (Equation (2)): Here, m is the number of rows (columns) of the matrix. The interaction between the Fe2O3 nanoparticles was determined by the magnetic dipole interaction. Then, Fe2O3 nanoparticles are affected by magnetic moments in all directions.
In the spherical coordinate system: mx = sinαcosφ, my = sinαsinφ, mz = cosα; 0 < α < 180°, 0 < φ < 360°. The total magnetic moment (Mtot) is expressed by the following expression (Equation (3)) [6]: where Si = +1 with spin up, and Si = −1 with spin down of Fe2O3 nanoparticles. In it, K a is the uniaxial magnetic anisotropy energy; the magnetic coefficient is µ 0 = 4π × 10 −7 m/A; V is the volume of the nanoparticle; a 0 is the lattice constant; K ij is the interaction energy between the i th nanoparticle and the nearest j th nanoparticle; and Boltzmann's constant k B = 1.38 × 10 −23 J/K = 8.617 × 10 −5 eV/K, where B is the magnetic field, M is magnetic moment, α i is the rotation angle of the i th 2D square matrix, m i and m j are the magnetic moments of the i th and j th atom, and d, a, R g , and D are the distance between the centers of the two nearest nanoparticles, the size of the square edge, the displacement radius, and the size of the Fe 2 O 3 nanoparticle, respectively. Conversely, the size of the model is determined by the following formula (Equation (2)): Here, m is the number of rows (columns) of the matrix. The interaction between the Fe 2 O 3 nanoparticles was determined by the magnetic dipole interaction. Then, Fe 2 O 3 nanoparticles are affected by magnetic moments in all directions.
In the nano synthetic thin film with the size L, V s = b(Ld) 2 is the thin film body and b, d, α, and ϕ are, respectively, the nanoparticle thickness, the distance between the nanoparticles, the polarization angle (pointing the direction of the magnetic moment in the x-y plane), and the azimuth (pointing the direction of the magnetic moment for the z-axis).
To study the magnetic properties of Fe 2 O 3 nano synthesized thin films, various authors have applied the Monte Carlo method in numerical simulations [96][97][98]. To increase the accuracy of the results, we used periodic boundary conditions to eliminate surface effects. The obtained results are also compared with the results of the density functional theory method to increase the accuracy of the obtained results.
Ising's 2D model is placed in the magnetic field B = 0.1 T while the influencing factors, such as model size (mxm), rotation angle (α), and the external magnetic field (B), are changed. To simulate numerically, we used the Metropolis algorithm in the framework of the Monte-Carlo method and surveyed magnetic characteristic quantities in temperatures from T = 0 to 600 K, with a total number of MC simulation steps 5 × 10 4 corresponding to 84 MC steps for each temperature T = 1 K. It has been emphasized in [6] that the Monte-Carlo Metropolis code becomes stable very quickly. It follows from Figure 3c of this paper that, from the vicinity of room temperature to the larger temperatures, the results obtained after 20 Monte Carlo steps (MCS) are practically the same as those after 200 MCS. Our calculations show that, after 30 MCS, the change in obtained results is negligible, but for certainty, 84 MCS have been performed.
The simulation method is based on a random generation of energy variation of the system.
Next, we rotated their magnetic moment from → m = m x , m y , m z to → m = m x , m y , m z and calculated the energy values H m , H m ' with the corresponding probability distribution (Equation (4)) [6]: where P(E) is the probability value of finding spin min (1, exp(−β∆E)) in a state; β = 1/k B T; T is the temperature, Z is the partition function, ∆E is energy variation of the system generated randomly. To analyze the model, we calculate the magnetic characteristic quantities of the considered system, such as the total entropy (Equation (5)) [6] of the Fe 2 O 3 nano synthetic thin film with the following expression: To determine the Néel phase transition temperature (T Ntot ) of Fe 2 O 3 nano synthetic thin films, the intersection between the magnetization curve with the entropy curve is fixed. The entire numerical simulation was carried out based on the Python programming code provided by Prof. M.R. Dudek [6]. This code was properly modified for our purpose and was applied on the computational server system of the Institute of Physics, Department of Physics and Astronomy, Zielona Gora University, Poland.

Magnetic Characteristic Quantities
We determine the magnetic characteristic quantities of Fe 2 O 3 nano synthesized thin films, such as the preferred magnetization in the z-axis (M zE ), the magnetization in the z-axis (M z ), the total magnetization (M tot ), total entropy (S tot ), and Néel phase transition temperature (T Ntot ).
The Néel phase transition temperature is the phase transition temperature of a material from an antiferromagnetic state to a superparamagnetic state. To determine the characteristic quantities, as has been emphasized above, we treat each spherical nanoparticle as a spin (with D = 6 nm and the magnetic moment is determined by Equation (3)). The results are shown in Figure 2.
The results show that when the Fe 2 O 3 nano synthetic thin film is placed in the magnetic field (B), B = 0.1 Tesla (T), and the spin of the nanoparticles is rotated by an angle α = 90 • , the shape of the synthesized thin film nano Fe 2 O 3 with mxm = 5 × 5, nano size (D), D = 6 nm corresponding to the size L = 27 nm is given in Figure 2a. The relationship between the magnetization oriented in the direction of the z-axis (M zE ) is shown by the black line in Figure 2b; magnetization in the z-axis (M z ) is shown by the green line in Figure 2c. Synthetic magnetization of Fe 2 O 3 materials is given by dark blue line in Figure 2d and synthetic entropy (S tot ) is drawn in red color when the temperature increases. The Néel phase transition temperature is the phase transition temperature of a material from an antiferromagnetic state to a superparamagnetic state. To determine the characteristic quantities, as has been emphasized above, we treat each spherical nanoparticle as a spin (with D = 6 nm and the magnetic moment is determined by Equation (3)). The results are shown in Figure 2. The results show that when the Fe2O3 nano synthetic thin film is placed in the magnetic field (B), B = 0.1 Tesla (T), and the spin of the nanoparticles is rotated by an angle α = 90°, the shape of the synthesized thin film nano Fe2O3 with mxm = 5 × 5, nano size (D), D = 6 nm corresponding to the size L = 27 nm is given in Figure 2a. The relationship between the magnetization oriented in the direction of the z-axis (MzE) is shown by the black line in Figure 2b; magnetization in the z-axis (Mz) is shown by the green line in Figure 2c. Synthetic magnetization of Fe2O3 materials is given by dark blue line in Figure  2d and synthetic entropy (Stot) is drawn in red color when the temperature increases.
Dudek et al. [6] successfully determined the Néel phase transition temperature of the Fe3O4 nano synthetic thin film and showed that the cause of this phenomenon is due to the magnetic effect of the spins. For this reason, we omit the determination of magnetic characteristic quantities such as magnetization (M), specific heat (Cv), magnetic suscep- Dudek et al. [6] successfully determined the Néel phase transition temperature of the Fe 3 O 4 nano synthetic thin film and showed that the cause of this phenomenon is due to the magnetic effect of the spins. For this reason, we omit the determination of magnetic characteristic quantities such as magnetization (M), specific heat (C v ), magnetic susceptibility (χ), and energy (E) of the thin film synthesized Fe 2 O 3 nano and only focus our attention on studying the relationship between the characteristics of the magnetization M (M zE , M z , M tot ) with the total entropy (S tot ) when the temperature (T) increases (what it has been demonstrated above in Figure 2). The total entropy is determined according to the formula between M z and S tot is T Nz = 68 K; and between M tot and S tot is T Ntot = 68 K. This is the Néel phase transition temperature from the antiferromagnetic state to the superparamagnetic state. This result is completely consistent with the magnetic effect results previously obtained with Fe 3 O 4 nano synthetic thin films at room temperature [6]. To confirm that, we study the factors affecting the isotherm entropy and Néel temperature of Fe 2 O 3 nano synthesized thin films with D = 6 nm.

Effect of the Spins Rotation Angle
Similarly, as in the case of analyzing the effect of synthetic thin film size, we consider the influence of spin angle using Fe2O3 nano synthetic thin film with L = 130 nm (D = 6 nm), B = 0.1 T, with an angle α that changes from α = 0° to α = 90°. The obtained results are shown in Figure 4. The obtained results show that the Fe 2 O 3 nano synthetic thin film of size L = 27 nm has the shape given in Figure 3a. The total magnetization (M tot ) is shown by the blue line in Figure 3b, and the total entropy (S tot ) is shown by the red line in Figure 3c. When temperature (T) increases from T = 10 K to T = 600 K, the magnetization (M tot ) decreases from M tot = 0.941 to M tot = 0.394, S tot increases from S tot = −0.069 to S tot = 2.354, and Neel's magnetic phase transition temperature (T Ntot ) increases slightly from T Ntot = 68 K to T Ntot = 68, 68, 68, 71, 73 K (Figure 3d). The reason for these changes is that increasing temperature T leads to a shift of the domain walls. When the thin film size increases from L = 27 nm to L = 62, 96, 130, 198, 267 nm, the M tot increases slightly from M tot = 0.941 to M tot = 0.944, 0.947, 0.952, 0.972, 0.983, respectively, because the increase in the thin film size of L leads to an increase in the density of spins. The obtained results are completely consistent with the simulation results of amorphous Fe nanoparticles [82]. The cause of this phenomenon is due to the size effect (when increasing the lattice size L leads to an increase in T Ntot ) with a negligible increase in results (about 9%). We chose Fe 2 O 3 nano synthetic thin film with the nano size D = 6 nm, mxm = 20 × 20 corresponding to the size L = 130 nm as standard to study other influencing factors. Further, we investigated the influence of the spin angle of spin on the magnetic characteristic quantities.

Effect of the Spins Rotation Angle
Similarly, as in the case of analyzing the effect of synthetic thin film size, we consider the influence of spin angle using Fe 2 O 3 nano synthetic thin film with L = 130 nm (D = 6 nm), B = 0.1 T, with an angle α that changes from α = 0 • to α = 90 • . The obtained results are shown in Figure 4.   (Figure 4b). The cause of the change in TNtot is the fact that an increase in the rotation angle leads to an increase in spin spacing (d) and to a decrease in the magnetization Mtot. The distance between the nanoparticle centers increases d > a, and in consequence, total entropy Stot decreases. It follows from obtained results that when L = 27 nm increases to L = 62, 96, 130, 198, 267 nm, the TNtot increases from 68 K to 75 K, and when the rotation angle increases from α = 0° to α = 90°, TNtot decreases from TNtot = 93 K to TNtot = 68 K with L = 130 nm. The increase in size leads only to an insignificant change of TNtot, and the rotation angle of the matrix will be a very convenient parameter for experimental studies with different types of materials used to manufacture thin films. Through the research results on the influence of thin films on the magnetic properties of Fe2O3 nano synthetic thin films, we conclude that the influence factor of the thin film is very small, almost negligible. So, a question arises: what causes the increase or decrease in TNtot? To study  (Figure 4b). The cause of the change in T Ntot is the fact that an increase in the rotation angle leads to an increase in spin spacing (d) and to a decrease in the magnetization M tot . The distance between the nanoparticle centers increases d > a, and in consequence, total entropy S tot decreases. It follows from obtained results that when L = 27 nm increases to L = 62, 96,130,198, 267 nm, the T Ntot increases from 68 K to 75 K, and when the rotation angle increases from α = 0 • to α = 90 • , T Ntot decreases from T Ntot = 93 K to T Ntot = 68 K with L = 130 nm. The increase in size leads only to an insignificant change of T Ntot , and the rotation angle of the matrix will be a very convenient parameter for experimental studies with different types of materials used to manufacture thin films. Through the research results on the influence of thin films on the magnetic properties of Fe 2 O 3 nano synthetic thin films, we conclude that the influence factor of the thin film is very small, almost negligible. So, a question arises: what causes the increase or decrease in T Ntot ? To study the influencing factors of Fe 2 O 3 nanoparticles, we chose a thin film with a size L = 130 nm with a rotation angle of α = 90 • . To answer this question, we continued the study of the influence of nanoparticles and the impact factors of the external magnetic field on experiments.

Effect of Fe 2 O 3 Nanoparticles
To study the influence of Fe 2 O 3 nanoparticles, we used again a matrix of size L = 130 nm, D = 6 nm with a rotation angle of the matrix α = 90 • .

Effect of the External Magnetic Field
Let us consider the Fe 2 O 3 nano synthetic thin film with nanoparticle size D = 6 nm, L = 130 nm into the external magnetic field B with different intensities. The obtained results are shown in Figure 5.  The results show that when Fe2O3 nano synthetic thin film with matrix size L = 130 nm, nanoparticle size D = 6 nm is placed in a magnetic field (B) with B = 0.1 T, the shape of Fe2O3 thin film is as in Figure 5a (Figure 5c), TNtot = 228 K at B = 0.5 T (Figure 5d), TNtot = 300 K at B = 0.7 T (Figure 5e), TNtot = 376K at B = 0.9 T (Figure 5f). The cause of the change in TNtot is that an increase in B leads to the stronger orientation of the spins in the preferred direction of the The results show that when Fe 2 O 3 nano synthetic thin film with matrix size L = 130 nm, nanoparticle size D = 6 nm is placed in a magnetic field (B) with B = 0.1 T, the shape of Fe 2 O 3 thin film is as in Figure 5a. The M tot composite magnetization decreases from M tot = 0.941 to M tot = 0.404, the S tot composite entropy increases from S tot = −0.076 to S tot = 2.353, and the magnetic phase transition temperature is T Ntot = 68 K (Figure 5b). When the external magnetic field increases from B = 0.1 T to B = 0.3, 0.5, 0.7, 0.9 T, the point above of total magnetization M tot increases slightly from M tot = 0.941 to M tot = 0.943, the total entropy always increases from S tot = −0.076 to S tot = −0.932, the lower point of M tot increases again from M tot = 0.404 to M tot = 0.754, 0.844, 0.881, 0.889, and S tot again decreases from S tot = 2. 353, 1.789, 1.479, 1.295, 1.165. This leads to a decrease in the magnitude of M, while the magnitude of S tot increases. It implies that S tot tends to shift towards the negative axis, which leads to a corresponding increase in T Ntot : T Ntot = 68 K at B = 0.1 T (Figure 5b), T Ntot = 148 K at B = 0.3 T (Figure 5c), T Ntot = 228 K at B = 0.5 T (Figure 5d), T Ntot = 300 K at B = 0.7 T (Figure 5e), T Ntot = 376K at B = 0.9 T (Figure 5f). The cause of the change in T Ntot is that an increase in B leads to the stronger orientation of the spins in the preferred direction of the magnetic field and they rotate very strongly with a large magnetic field. So, there is another problem: how to increase T Ntot with a small external magnetic field?

Effect of Nanoparticle Size
When nanoparticle size (D) increases, we obtain the results shown in Figure 6. Effect of Nanoparticle Size When nanoparticle size (D) increases, we obtain the results shown in Figure 6. The results show that when Fe2O3 nano synthetic thin film with nanoparticle size D = 6 nm (L = 130 nm) is placed in a magnetic field (B), with B = 0.1 T, rotation angle α = 90°, the shape of the film is as in Figure 6a  This result is completely consistent with the results obtained in [99] for the magnetic phase transition temperature (T Ntot ) of Fe 3 O 4 nanoparticles (as D increases, the T Ntot increases and reaches a maximum value T Ntot = 860 K). The cause of this phenomenon is due to the size effect. The results show that, as D increases, L increases. T Ntot increases nearly in a linear manner with D according to the approximated formula T Ntot = − 440.6 + 83D (Figure 6g). Through this formula, researchers can adjust the nanoparticle size and subtract the B field to be suitable for specific applications. For example, one can fabricate a nano synthetic thin-film operating at T Ntot = 300 K with the Earth's magnetic field. For this purpose, we study below the influence of magnetic field B, nanoparticle size D on T Ntot at room temperature T = 300 K. The results of the influence of B and D on the Néel phase transition temperature (T Ntot ) show that, when increasing both B and D, we have a decrease in Mtot and an increase in Stot. This leads to the conclusion that the magnetization of the material always decreases, and the entropy of materials increases. This is very interesting for future applications of magnetic nanomaterials.
Relationship between B and D at Room Temperature 300 K Considering the above research results, we investigate the influence at room temperature. We investigate the influence of nanoparticle size at the values of B = 0.025, 0.045, and 0.065 T with dimensions D = 10, 12, and 14 nm (corresponding to L = 114, 174, and 234 nm). The results are shown in Figure 7.
The results show that, when the  (Figure 7c2), and T Ntot > 600 K (Figure 7c3). The obtained results show that, at room temperature, because the magnetic field of the earth, B, is very small, one can increase the nanoparticle size to nearly 12 nm, then T Ntot = T = 300 K. In addition, between T Ntot and D there is a relationship that satisfies the equation T Ntot = −440.6 + 83D. This is a very useful result. In practice, researchers can manufacture Fe 2 O 3 thin films right at ambient conditions (at room temperature). Achieving the size D = 12 nm (L = 244 nm), these thin films can be used not only in magnetic devices.

Conclusions
In this study the following results were obtained:

•
We successfully studied the influence of the matrix structure (mxm) of thin-film, rotation angle (α), magnetic field (B), and size (D) of Fe 2 O 3 nanoparticle on the magnetic characteristic quantities such as the magnetization-oriented z-direction (M zE ), z-axis magnetization (M z ), total magnetization (M tot ), and total entropy (S tot ) of Fe 2 O 3 nanocomposites by Monte-Carlo simulation method.

•
We successfully determined the magnetic phase transition temperature Néel (T Ntot ). The obtained results show that when the mxm increases from mxm = 5 × 5 (L = 27 nm) to mxm = 10 × 10, 15 × 15, 20 × 20, 30 × 30, 40 × 40 (L = 62, 96, 130, 198, 267 nm), the T Ntot increases slightly from T Ntot = 68 K to T Ntot = 73 K. When the matrix rotation angle α increases from α = 0 • to α = 90 • , the T Ntot decreases slightly from T Ntot = 93 K to T Ntot = 68 K. The increase in B (from B = 0.1 T to B = 0.9 T) determines an increase in T Ntot (from T Ntot = 68 K to T Ntot = 148, 228, 300, 376 K). The increase in D (from D = 6 nm (L = 130 nm) to D = 8, 10, 12, 14 nm (L = 168, 206, 244, 282 nm)) determines an increase in T Ntot (from T Ntot = 68 K to T Ntot = 164, 320, 560, and higher 600 K). The results show that when B and D increase, T Ntot increases also. • In addition, between T Ntot and D, there is a linear relationship that satisfies the equation T Ntot = −440.6 + 83D. This is a very interesting result that can be used in practical applications from cooling technology. Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author upon reasonable request.