The Study of the Influence of Matrix, Size, Rotation Angle, and Magnetic Field on the Isothermal Entropy, and the Néel Phase Transition Temperature of Fe₂O₃ Nanocomposite Thin Films by the Monte-Carlo 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 Fe₂O₃ 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 Fe₂O₃ 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 Néel (T_Ntot). When B and D increase, T_Ntot increases very strongly. The results also show that in Fe₂O₃ thin films, T_Ntot is always smaller than with Fe₂O₃ nano and Fe₂O₃ 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.

1. 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 Fe₂O₃ nano synthetic thin film as the subject of this paper. The reason is that the synthetic nano-Fe₂O₃ 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 equipment [20,21], printing devices [22], photocatalyst materials [23,24], ion recovery [25], magnetic nanofilm [26], magnetic resonance [27,28], magnetic fluids [29], pigments [30], gas sensors [31,32], the biomedical field [33,34,35,36,37], and spintronics [38,39]. In addition, Fe₂O₃ is also a material with many structures, such as α-Fe₂O₃ (hematite), β-Fe₂O₃ and γ-Fe₂O₃ (maghemite), and ε-Fe₂O₃ [40]. Among them, the α-Fe₂O₃ hematite structure is the most commonly used today, while the ε-Fe₂O₃ structure is the most difficult structure to be manufactured. The ε-Fe₂O₃ structure has not been studied extensively. Applying the experimental method, Zuohui Cheng et al. [41] have successfully determined the influence of enthalpy and entropy on the size of Fe₂O₃ materials. They concluded that when the size (D) increases from D = 19.3 nm to D = 140.5 nm, the total magnetization (M_tot) increases, and the Néel phase transition temperature (T_Ntot) increases from T_Ntot = 742 K to T_Ntot = 897 K [41]. Liu et al. [42] proposed an approximate expression for the enthalpy and entropy of the nanoparticles without considering the change in temperature during the transition. Cui et al. [43] demonstrated that the crystallinity densities of the two transitions are equal. Zhang et al. [44] successfully determined the Néel phase transition temperature of Fe₂O₃ when considered system transitioned from the maghemite phase to the α hematite phase, namely T_Ntot = 623 K at D = 4 nm and T_Ntot = 723 K at D = 24 nm. Mendili et al. [45], Wenger et al. [46] have successfully determined that the magnetization (M) of γ-Fe₂O₃ nanoparticles increases when D increases (from D = 2 nm to D = 4 nm). Hou et al. [47] suggested that the coercivity of nano-Fe₂O₃ increases when the D size increases (from D = 12 nm to D = 40 nm). Jun Wang et al. [48] successfully determined the T_Ntot of α-Fe₂O₃ as T_Ntot = 930 K. In the study of José Luis García-Muñoz [49] for ε-Fe₂O_3, this temperature was T_Ntot = 850 K. The obtained results show that, for bulk or nanomaterials, an increase in D leads to an increase in T_Ntot. Additionally, it was found that, for Fe₂O₃ synthetic thin film at room temperature T = 300 K, there exists a very large magnetic force of approximately 20 kOe. Due to this, Fe₂O₃ materials are used frequently in high-density magnetic recording devices, high-frequency electromagnetic waves absorbers [50,51,52]. On the other hand, ε-Fe₂O₃ is a ferromagnetic material [53]. In some kind, it is an intermediate form between γ-Fe₂O₃ and α-Fe₂O₃, so its Neel phase transition temperature ranges from T_Ntot = 150 K to T_Ntot = 500 K. It has a lowest temperature of T_Ntot = 150 K [54,55], and it is characterized by a decrease in coercivity and saturation magnetization [54]. In addition, at the cooling condition of Fe₂O₃, the T_Ntot value is in the range of T_Ntot = 85–150 K, when there is a change in spins in the rhombohedral structure and a non-monotonic change in the Fe-O bond [49,56], even though the Curie temperature (T_Ntot) of ε-Fe₂O₃ has a value of T_Ntot ∼ 500 K [57,58,59,60]. Additionally, magnetic ε-Fe₂O₃ nanoparticles can also be obtained by vibrating magnetometer, which shows T_Ntot > 500 K and a maximum value of T_Ntot = 850 K. The simulation method has received great attention, because researchers can study materials in extreme conditions, such as at high temperature (T), T = 700 K, pressure (P), P = 360 GPa, and with atomic size below 2 nm, where the experimental methods cannot be applied [61,62,63]. Meanwhile, the magnetic refrigeration technology industry attracts great attention from researchers. Cooling technology is based on the magnetic effect of materials, which was first discovered by E. Warburg in 1881 [64]. When the material is placed in the magnetic field (B), under the action of the magnetic field, the spins rotate in the direction of the magnetic field, leading to the total magnetic (M_tot) and total entropy (S_tot) change (increase or decrease), depending on the nature of magnetic materials. In the field of refrigeration technology at room temperature T = 300 K, researchers often use soft magnetic materials which operate at room temperatures, such as Gadolinium with T_Ntot = 296 K. It is the first used material with very high cost, poor oxidation resistance, and low magnetic value. For that, researchers try to find materials with high oxidation resistance and low cost, which exhibit high magnetic properties that are essential for future applications. Presently, materials such as transition metals or rare earth materials are still considered materials with great potential for refrigeration applications at room temperature [65,66,67,68,69,70,71]. When materials with high or low magnetism are considered, researchers concentrate on the entropy change (ΔS). This quantity is determined through the Maxwell 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~150 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-Fe₂O₃ at room temperature. Recently, M.R. Dudek et al. have successfully determined the giant magnetic field of Fe₃O₄ 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 Fe₃O₄ 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 Fe₂O₃ 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 Fe₂O₃ nano synthetic thin films to smart devices and refrigeration equipment.

2. Method of Calculation

Initially, the two-dimensional model for Fe₂O₃ 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 Fe₂O₃ 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 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.

$${\mathrm{H}}_{\mathrm{mfa}}^{\left(\mathrm{i}\right)}=-{\mathrm{K}}_{\mathrm{a}}{\mathrm{Vcos}}^2{\mathsf{\alpha }}_{\mathrm{i}}-{\mathrm{BMcos}\mathsf{\alpha }}_{\mathrm{i}}-{\displaystyle \sum _{\mathrm{j}}{\mathrm{K}}_{\mathrm{ij}}{\overrightarrow{\mathrm{m}}}_{\mathrm{i}}.\langle {\overrightarrow{\mathrm{m}}}_{\mathrm{j}}\rangle }$$

(1)

where ${\mathrm{K}}_{\mathrm{a}}=\frac{1}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}·\frac{{\mathsf{\mu }}_0}{{4\mathsf{\pi }\mathrm{M}}^3}$, $\mathrm{V}=\frac{4}{3}{\mathsf{\pi }\mathrm{a}}_0^2$, a₀ = 8.394 Å, and ${\mathrm{K}}_{\mathrm{ij}}=\frac{{\mathsf{\mu }}_0{\mathrm{M}}^2}{{4\mathsf{\pi }\mathrm{d}}^3}$, $\mathrm{d}=\sqrt{2}\mathrm{asin}\left(\mathsf{\alpha }+\frac{\mathsf{\pi }}{4}\right)$, ${\mathrm{a}=\mathrm{D}+\mathrm{a}}_0$, ${\mathrm{D}=2\mathrm{R}}_{\mathrm{g}}$.

In it, K_a is the uniaxial magnetic anisotropy energy; the magnetic coefficient is ${\mathsf{\mu }}_0=4\pi \times {10}^{-7}$ m/A; V is the volume of the nanoparticle; a₀ 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 Fe₂O₃ nanoparticle, respectively. Conversely, the size of the model is determined by the following formula (Equation (2)):

L = (m − 1) × d

(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: m_x = sinαcosφ, m_y = sinαsinφ, m_z = cosα; 0 < α < 180°, 0 < φ < 360°. The total magnetic moment (M_tot) is expressed by the following expression (Equation (3)) [6]:

$${\mathrm{M}}_{\mathrm{tot}}=\sqrt{{\mathrm{M}}_{\mathrm{x}}^2{+\mathrm{M}}_{\mathrm{y}}^2{+\mathrm{M}}_{\mathrm{z}}^2}=\pm 1,{\mathrm{M}}_{\mathrm{tot}}=\frac{1}{\mathrm{N}}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{S}}_{\mathrm{i}}},{\mathrm{S}}_{\mathrm{i}}=\pm 1,$$

(3)

where S_i = +1 with spin up, and S_i = −1 with spin down of Fe₂O₃ nanoparticles.

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 Fe₂O₃ 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 × 10⁴ 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 $\overrightarrow{\mathrm{m}}=\left({\mathrm{m}}_{\mathrm{x}}{,\mathrm{m}}_{\mathrm{y}}{,\mathrm{m}}_{\mathrm{z}}\right)$ to ${\overrightarrow{\mathrm{m}}}^{\prime }=\left({\mathrm{m}}_{\mathrm{x}}^{\prime }{,\mathrm{m}}_{\mathrm{y}}^{\prime }{,\mathrm{m}}_{\mathrm{z}}^{\prime }\right)$ and calculated the energy values H_m, H_m^’ with the corresponding probability distribution (Equation (4)) [6]:

$$\begin{gathered} \mathrm{P}\left(\mathrm{E}\right)=\frac{\exp (-\mathsf{\beta }\Delta \mathrm{E})}{\mathrm{Z}},\mathrm{Z}=\sum _{\mathrm{i}}^{\mathrm{N}}\exp (-\mathsf{\beta }\Delta \mathrm{E}) \\ \mathrm{and}\Delta \mathrm{E}={\mathrm{H}}_{\mathrm{m}}-{{\mathrm{H}}_{\mathrm{m}}}^{\prime }, \end{gathered}$$

(4)

where P(E) is the probability value of finding spin min (1, exp(−βΔE)) in a state; β = 1/k_BT; T is the temperature, Z is the partition function, ΔE 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 Fe₂O₃ nano synthetic thin film with the following expression:

$$\frac{{\mathrm{S}}_{\mathrm{tot}}}{{\mathrm{Nk}}_{\mathrm{B}}}=\frac{1}{{\mathrm{Nk}}_{\mathrm{B}}}{\displaystyle \sum _{\mathrm{i}}{\mathrm{S}}_{\mathrm{i}}}$$

(5)

To determine the Néel phase transition temperature (T_Ntot) of Fe₂O₃ 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.

3. Results and Discussion

3.1. Magnetic Characteristic Quantities

We determine the magnetic characteristic quantities of Fe₂O₃ 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 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 (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 Fe₂O₃ materials is given by dark blue line in Figure 2d and synthetic entropy (S_tot) 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 (C_v), magnetic susceptibility (χ), and energy (E) of the thin film synthesized Fe₂O₃ 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 (5). At T = 10 K, M_zE = 0.886, M_z = 0.941, M_tot = 0.941, S_tot = −0.069, when T increases from T = 10 K to T = 600 K with the temperature shift dT = 5 K, all values of magnetization M (M_zE, M_z, M_tot) decrease as M_zE decreases from M_zE = 0.886 to M_zE = 0.156, M_z decreases from M_z = 0.941 to M_z = 0.390, and M_tot decreases from M_tot = 0.941 to M_tot = 0.394, which leads to an increase in S_tot from S_tot = −0.069 to S_tot = 2.354. The displacement of the spins corresponds to the probability of finding the existence of spins in a given state at a certain temperature. The lines of the magnetization M_zE, M_z, M_tot, and the total entropy S_tot intersect at a point, which is called the magnetic phase transition point or the Néel phase transition temperature (T_Ntot). The intersection between M_zE and S_tot is T_NzE = 60 K; 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 Fe₃O₄ nano synthetic thin films at room temperature [6]. To confirm that, we study the factors affecting the isotherm entropy and Néel temperature of Fe₂O₃ nano synthesized thin films with D = 6 nm.

3.2. The Influence of Different Factors

3.2.1. Effect of the Synthetic Thin-Films

Effect of the Synthetic Thin Film Size

To study the effect (mxm) of Fe₂O₃ nano synthetic thin films, the size model increases from mxm = 5 × 5 (L = 27 nm) to mxm = 10 × 10 (L = 62 nm), 15 × 15 (L = 96 nm), 20 × 20 (L = 130 nm), 30 × 30 (L = 198 nm), 40 × 40 (L = 267 nm) with D = 6 nm. The considered system is placed in a magnetic field (B), B = 0.1 T with a rotation angle of α = 90°. The result obtained is shown in Figure 3.

The obtained results show that the Fe₂O₃ 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 Fe₂O₃ 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 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 results show that when the Fe₂O₃ nano synthetic thin film of size L = 130 nm is placed in a magnetic field (B) with B = 0.1 T and the rotation angle (α) increases from α = 0° to α = 30°, 45°, 60°, 90°, the shape of the thin film is shown as in Figure 4a. The M_tot increases slightly from M_tot = 0.939 to M_tot = 0.940, 0.940, 0.941, 0.941, because an increase in the size L leads to an increase in the density of spins, whereas S_tot increases slightly from S_tot = −0.249 to S_tot = −0.133, −0.107, −0.088, −0.076. This leads to the decrease in magnetic phase transition temperature (T_Ntot) from T_Ntot = 93 K to T_Ntot = 86, 75, 70, 68 K (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 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 T_Ntot? To study the influencing factors of Fe₂O₃ 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.

3.2.2. Effect of Fe₂O₃ Nanoparticles

To study the influence of Fe₂O₃ 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 Fe₂O₃ 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. 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.

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. The M_tot composite magnetization decreases from M_tot = 0.941 to M_tot = 0.404, S_tot composite entropy increases from S_tot = −0.076 to S_tot = 2.353. The T_Ntot magnetic phase transition temperature is T_Ntot = 68 K (Figure 6b). When the size increases from D = 6 nm (L = 130 nm) to D = 8, 10, 12, 14 nm (L = 168, 206, 244, 282 nm), then upper point of total magnetization M_tot increases slightly from M_tot = 0.941 to M_tot = 0.983, the total entropy always increases from Stot = −0.076 to Stot = −0.083, the lower point of M_tot increases again from M_tot = 0.404 to Mtot = 0.876, 0.929, 0.940, 0.935, and S_tot again decreases from S_tot = 2.353, 1.281, 0.705, 0.310, 0.003, which leads to the decrease in magnitude of M_tot, whereas the magnitude of S_tot increases. S_tot tends to shift towards the negative axis, leading to an increase in T_Ntot: T_Ntot = 68 K at D = 6 nm, (L = 130 nm) (Figure 6b), T_Ntot = 164 K at D = 8 nm, (L = 168 nm) (Figure 6c), T_Ntot = 320 K at D = 10 nm, (L = 206 nm) (Figure 6d), T_Ntot = 560 K at D = 12 nm, (L = 244 nm) (Figure 6e), T_Ntot ≥ 600 K at D = 14 nm, (L = 282 nm) (Figure 6f). The M_tot curve and S_tot curve intersect at a point called the point of magnetic phase transition (T_Ntot). Thereby, it shows that there is a relationship between D and T_Ntot: for D = 6 nm (L = 130 nm), T_Ntot = 68 K; for D = 8 nm (L = 168 nm), T_Ntot = 164 K; for D = 10 nm (L = 206 nm), T_Ntot = 320 K; when D = 12 nm (L = 244 nm), T_Ntot = 560 K; and for D = 14 nm (L = 282 nm), T_Ntot > 600 K. This result is completely consistent with the results obtained in [99] for the magnetic phase transition temperature (T_Ntot) of Fe₃O₄ 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 Fe₂O₃ nano synthetic thin film is placed in the external magnetic field B = 0.025 T with the rotation angle α = 90° for the increase in the size of the nanoparticle D from D = 10 nm (L = 206 nm) to D = 12, 14 nm (L = 244, 282 nm), M_tot decreases from M_tot = 0.943 to M_tot = 0.740, and S_tot increases from S_tot = −0.597 to S_tot = 1.803; whereas for D = 10 nm (L = 206 nm), M_tot decreases from M_tot = 0.941 to M_tot = 0.871, and S_tot increases from S_tot = −0.875 to S_tot = 1.304; with D = 12 nm (L = 244 nm), M_tot decreases from M_tot = 0.936 to M_tot = 0.909, and S_tot increases from S_tot = −0.908 to S_tot = 1.002. For D = 14 nm (L = 282 nm) we also have a decrease in the magnitude of M_tot and an increase in S_tot. Therefore, as B and D increase, the magnetic phase transition temperature increases from T_Ntot = 185 K (Figure 7a1) to T_Ntot = 324 K (Figure 7a2), 515 K (Figure 7a3). For B = 0.045 T, when D increases from D = 10 nm (L = 206 nm) to D = 12, 14 nm (L = 244, 282 nm), the magnetic phase transition temperature increases from T_Ntot = 220 K (Figure 7b1) to T_Ntot = 383 K (Figure 7b2), 596 K (Figure 7b3). With B = 0.065 T, when D increases from D = 10 nm (L = 206 nm) to D = 12, 14 nm (L = 244, 282 nm), the magnetic phase transition temperature increases from T_Ntot = 255 K (Figure 7c1) to T_Ntot = 445 K (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 Fe₂O₃ 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.

4. 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 Fe₂O₃ 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 Fe₂O₃ 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.