Experimental and Theoretical Investigation of the Synthesis, Electronic and Magnetic Properties of MnFe₂O₄ Spinel Ferrite

Abstract

MnFe₂O₄ ferrite nanoparticle was synthesized via the sol–gel method, and structural, morphology and magnetic characteristics were investigated. X-ray diffraction analysis showed that the synthesized sample was in a single phase with a spinel-ferrite-like structure (space group Fd-3m). The scanning electron microscopy displayed homogenous spherical grains with an agglomeration of the particles. The chemical composition determined by energy-dispersive spectroscopy shows the absence of any impurities. To understand the role of magnetic interaction in MnFe₂O₄ spinel ferrites, the structural and magnetic properties of MnFe₂O₄ have been explored theoretically. Based on the first-principles methods via density functional theory and Monte Carlo simulations, the magnetic hysteresis cycle has been plotted. Using the generalized gradient and GGA-PBE approximation in the full-potential linearized augmented plane wave (FP-LAPW) method, the exchange coupling interactions between magnetic elements and local magnetic moment were evaluated. Furthermore, the theoretical magnetic properties of MnFe₂O₄ were found to match the experimental ones. They both revealed that MnFe₂O₄ is a soft ferromagnetic material. The theoretical curve of magnetization versus temperature indicates that the transition occurred at T_c = 580.0 K. This was also in good agreement with the experimental Curie temperature.

1. Introduction

Nowadays, the study of new materials becomes very interesting and continues to develop rapidly, especially those which can be included in industrial, ecological and technological applications in optoelectronic and piezoelectric fields for instance, where several criteria are to be considered. The field of condensed matter is a very large domain of research. In seeking to satisfy technological needs, several categories of materials have surfaced from three dimensions to zero dimension, such as diluted magnetic semiconductors [1,2,3], pyrites [4,5], Heusler or half Heusler [6,7], perovskite [8,9,10] for 3D and phosphorene [11,12], graphene [13,14,15,16], MoS₂ [17,18,19], composite materials [20,21,22,23] and silicon-based materials [24] for 2D. Recently, magnetic nanoparticles have been implicated in a wide application domain such as in experimental cancer treatments called magnetic hyperthermia [25], Magnetic immunoassay [26], electrochemical sensing [27], catalyst or catalyst supports [28,29] and high-density storage [30]. Among the magnetic nanoparticles, there is the class of spinel ferrite oxides (SF) where the main structural formula may be written as MFe₂O₄ (where M = Mn, Fe, Co, Ni, Cu and Zn) [31,32]. Due to their very important characteristics at nanometric sizes, they have emerged as a forwarded category of nanostructured material in the fields of nanoscience and technology. These compounds have soft physical and chemical properties, such as excellent chemical stability, high electromagnetic performance, high cubic magnetocrystalline anisotropy and mechanical hardness [33].

The principal elements that define the magnetic performance of spinel ferrite nanoparticles (NPs) are often acknowledged on several levels: whether on the atomic level, or at the single-particle level as a balance between the magnetically and crystallographically ordered fractions of the NP, as well as at the mesoscopic level by means of mutual interparticle interactions and size distribution phenomena. Generally, the SFs compounds crystallize in a face-centered cubic structure (f.c.c) and belong to the space group Fd3m [34]. The classification of these systems is realized on the basis of the degree of inversion x, and there are three classes, which are based on the spreading of cations at both the tetrahedral (A) and octahedral (B) sites. This degree is the parameter used to distinguish the cation disorder and it is defined as the fraction of M²⁺ ions located in octahedral sites, according to ^T[M_1−xFe_x]^O[M_xFe_2−x]O₄ [35]. As Soufi et al. [36] have reported: when the SF is termed normal spinel, then x = 0; while when it is called an inverse spinel, then x = 1; when 0 < x < 1, it is called mixed spinel. In order to obtain a suitable nanomaterials ferrite, various routes have been used, such as sono-chemical, hydrothermal, sol–gel, co-precipitation and micelle microemulsion [37,38]. These approaches led to different sizes and shapes and hence, to tailoring their properties. It has been signalized that the MnFe₂O₄ exhibits a magnetic shift from ferrimagnetic to paramagnetic order around 559.26 K [39]. Using the Monte Carlo simulation, the magnetic characteristics of CoFe₂O₄ nanoparticles have been studied [40]. It is found that at room temperature for a size around 21 nm, in the absence of an external magnetic field, the total magnetic moment of the compound is about 1.60 μ_B [40]. On the other hand, while realizing the three-sublattice model it has been displayed that this last one fits well with the super-exchange and the Néel theory of ferrimagnetic material that provide the results of the magnetic properties of ferrites [41]. Bercoff et al. [42] have revealed while studying the compounds Fe₃O₄, CoFe₂O₄, NiFe₂O₄ and CuFe₂O₄ that the condition that allows the validity of the Néels two-sublattice collinear model and the super-exchange theory, is that the interaction between ions in tetrahedral sites should be very weak. Nasri et al. have established the second-order ferromagnetic–paramagnetic (FM-PM) phase transition exhibited by the CoFe₂O₄ material at a Curie temperature of 688 K while it decreased when substituting iron by copper [43].

This work reports on the MnFe₂O₄ nanoparticles synthesis and structural, morphological and magnetic properties. The obtained nanoparticles by the sol–gel method were analyzed via X-ray diffraction (XRD), scanning electron microscopy (SEM) and the DC extraction method for magnetization measurements. The magnetic properties were explored using DFT and Monte Carlo simulations.

This paper is presented in the following organization: In the experimental section, we present the synthesis process and characterization techniques. After that, we introduce in the simulation section the computational details used in this study. Then, we discuss and compare the synthesis, the calculation magnetization and hysteresis cycles. We finally summarized our results and comparison.

2. Experimental Methods

2.1. Synthesis of MnFe₂O₄ Nanoparticles

Based and inspired by the work of C. Brinker et al. [44], we have modified some experimental conditions for the experiments, such as the nature of precursors and the reaction medium, with the aim of developing a synthesis protocol further to obtain MnFe₂O₄ nanoparticles with all the optimized structural and magnetic parameters.

The MnFe₂O₄ nanoparticles synthesis was obtained according to the following chemical reaction:

MnCl₂ + 2FeCl₃ + 8NaOH → MnFe₂O₄ + 4H₂O + 8NaCl

The MnFe₂O₄ Nanoparticles were prepared by combining the manganese chloride (MnCl₂) (3.7 mmol) and iron chloride III(FeCl₃) (7.4 mmol), which were liquified separately in 10 mL of purified water and they were mixed in a flask. The obtained solution was kept at 60 °C for 30 min under magnetic stirring. The metal ions were chelated by adding citric acid using the 1:1 ratio. A few drops of a basic solution (NaOH) (11.1 mmol, 10 mL) were used to correct the pH value. The pure solution was treated for 3 h at 80 °C. The mix was positioned in a beaker, then heated softly for several hours at the same temperature until the making of a viscous gel. The gel was also heated overnight at 100 °C to obtain the powder. The obtained sample was grounded and calcined at a temperature of 600 °C for 6 h.

2.2. Characterization Techniques

In this work, the phase, particle size and lattice parameters of the samples were performed by X-ray diffraction (XRD) using a diffractometer with Cu K α radiation (λ = 1.5406 Å). The scanned angular range (2θ) covered from 10° to 70° at a 0.02° step. The applied voltage and current were 45 kV and 100 mA, respectively. The Scanning Electron Microscope (FEI Quanta 450 FEG; Zurich, Switzerland) was utilized to analyze the shape and microstructure of the samples. The magnetic properties were determined using a superconducting quantum interference device (SQUID, Quantum Design, Inc., Darmstadt, Germany).

2.3. Simulation and Computational Analysis

To compare experimental results with theoretical ones, we used both the Density Functional Theory (DFT) calculations and Monte Carlo simulations. Our calculations were based on the full-potential linearized augmented plane wave (FP-LAPW) through the Wien2k package [45]. Based on the Generalized Gradient Approximation with the Coulomb and exchange interaction effects (GGA + U), which is usually used to explore 3d electrons [46], the exchange coupling interactions have been calculated. The self-consistency is achieved by 650 K-points in the irreducible symmetry wedge of the first Brillouin zone. The crystal field of Mn is about 0.51 × 10⁻³ J/m³ and for Fe atoms is almost equal to 1.69 × 10⁻³ J/m³ and the muffin tin-radii (R_MT) of Mn, Fe and O ions are 1.87, 2.13 and 1.5, respectively [39]. We carried out, from our experimental study, the space group and the initial lattice parameter of the MnFe₂O₄ compound. It belongs to the Fd-3m space group and the lattice parameter is (a = 8.404 Å).

To study the magnetic behavior of MnFe₂O₄ ferrite, Equation (1) shows the Hamiltonian describing this system. As input parameters for Monte Carlo simulation, we used the spin values (±5/2) for both Mn²⁺ and Fe³⁺. These values have been used as input parameters to simulate the magnetic properties, for a predefined system size L = 30 nm, with temperature variation by the Monte Carlo method under the Metropolis algorithm. Resulted data is performed with 10⁶ Monte Carlo steps per spin, neglecting the first 10⁵ configurations.

$$\mathrm{H}=-{\mathrm{J}}_{\mathrm{Mn}-\mathrm{Mn}}{\sum }_{\langle \mathrm{i},\mathrm{j}\rangle }{\mathrm{s}}_{\mathrm{i}}{\mathrm{s}}_{\mathrm{j}}-{\mathrm{J}}_{\mathrm{Fe}-\mathrm{Fe}}{\sum }_{\langle \mathrm{k},\mathrm{l}\rangle }{\mathsf{\sigma }}_{\mathrm{k}}{\mathsf{\sigma }}_{\mathrm{l}}-{\mathrm{J}}_{\mathrm{Mn}-\mathrm{Fe}}{\sum }_{\langle \mathrm{m},\mathrm{n}\rangle }{\mathrm{S}}_{\mathrm{m}}{\mathsf{\sigma }}_{\mathrm{n}}-{\mathsf{\Delta }}_{\mathrm{Mn}}{\sum }_{\mathrm{i}}{\mathrm{s}}_{\mathrm{i}}{}^2-{\mathsf{\Delta }}_{\mathrm{Fe}}{\sum }_{\mathrm{i}}{\mathsf{\sigma }}_{\mathrm{i}}{}^2-\mathrm{h}({\sum }_{\mathrm{i}}\left({\mathrm{s}}_{\mathrm{i}}+{\mathsf{\sigma }}_{\mathrm{i}}\right))$$

(1)

where S and σ are the spins of Mn²⁺ and Fe³⁺, respectively, J_Mn-Mn, J_Mn-Fe and J_Fe-Fe represent, respectively, the exchange couplings among the first nearest neighbors of Mn-Mn, Mn-Fe and Fe-Fe, Δ_Mn and Δ_Fe are the octahedral crystal field due to oxygen atoms for Mn and Fe, and h is the external magnetic field.

After the system equilibrium, the following parameters have been calculated for every iteration:

The internal energy:

$$\mathrm{E}=\frac{\langle \mathrm{H}\rangle }{{\mathrm{L}}^3}$$

(2)

The partial magnetizations:

$${\mathrm{M}}_{\mathrm{s}}=\langle \frac{2}{{\mathrm{L}}^3}{\sum }_{\mathrm{i}}{\mathrm{S}}_{\mathrm{i}}\rangle$$

(3)

$${\mathrm{M}}_{\mathsf{\sigma }}=\langle \frac{2}{{\mathrm{L}}^3}{\sum }_{\mathrm{i}}{\mathsf{\sigma }}_{\mathrm{i}}\rangle$$

(4)

The total magnetizations:

$${\mathrm{M}}_{\mathrm{T}}=\frac{{\mathrm{M}}_{\mathrm{s}}+{\mathrm{M}}_{\mathsf{\sigma }}}{2}$$

(5)

3. Results and Discussion

3.1. Experimental Results: X-ray Diffraction and SEM Analysis

To determine the average particle size and phase of the synthesized MnFe₂O₄ nanoparticles, the X-ray diffraction patterns are depicted in Figure 1. It is clear that the elaborated sample fits well with the spinel ferrite model with a Fd-3 m space group belonging to the reference (JCPDS NO 00-022-1086) and with a lattice constant close to the previously obtained values of bulk MnFe₂O₄ [39].

The strong reflection (hkl) is found to be the (311) plane, which is in good agreement with the same intensified reflection observed in previous works [39,47,48,49]. The lattice parameters were calculated using the Bragg formula, the cell parameter was estimated to be 8.404 Å, which gives an approximate cell volume of 593.614 ų.

The Debye-Scherrer formula [50] has been used to calculate the particles size ${\langle \mathrm{D}\rangle }_{\mathrm{XRD}}$ from the width of the XRD patterns using the following equation:

$${\langle \mathrm{D}\rangle }_{\mathrm{XRD}}=\frac{\mathrm{K}\mathsf{\lambda }}{\mathsf{\beta }\cos \mathsf{\theta }}$$

(6)

where θ is the Bragg angle, β is the full-width at half-maximum (FWHM) of the reflection peaks, λ is the wavelength of the radiation and the Scherrer constant K is a dimensionless factor with a value close to unity (∼0.9). The lattice parameter, crystallite size values with the same method, sol–gel, and other methods are summarized in

Table 1. XRD patterns of MnFe₂O₄ nanoparticles synthetized by several methods.

Samples	Crystallite Size dDRX (nm)	Lattice Parameter a (Å)
Our work	20.00	8.4043
Sample(sol-gel-wheat flour /MnFe2O4) [51]	16.87	8.4960
Sample (sol-gel-potato flour/MnFe2O4) [52]	23.12	8.4920
Sample(co-precipitation-calcin in air/MnFe2O4) [53]	5.0–15.00	8.3359
Sample(co-precipitation-calcin under argon/MnFe2O4) [54]	20.10–80.00	8.2399

. In this table, the calculations of the crystallite sizes are d_DRX = 10.91 nm for the samples calcined in air and d_DRX = 20.10–80 nm for the nanoparticles calcined under argon. The lattice parameters by Bragg’s law are a = 8.3359 Å and a = 8.2399 Å for nanoparticles calcined in air and under argon, respectively. The lattice parameter of the oxidized sample (calcined in air) corresponds to the value of the lattice parameter of γ-Fe₂O₃ in literature (a = 8.3300 Å), which supports the diffraction pattern indicating the oxidation of the sample. Regarding the sample calcined under argon, the lattice parameter value is different and lower than that of MnFe₂O₄ and γ-Fe₂O₃ (also, other cubic elemental lattice oxides such as Fe₃O₄ and Mn₃O₄), which confirms its low crystallinity.

Figure 2 displays SEM pictures and the energy-dispersive X-ray spectroscopy (EDX) spectra, in which the microstructure of the manganese ferrite was investigated, as well as the distribution of its chemical composition. The MnFe₂O₄ particles are homogeneous in size and morphology, but they are agglomerated to some degree, which can be justified by the interaction among magnetic nanoparticles. The EDX analysis proves the presence of the Mn, Fe and O elements, which shows the lack of impurities in our obtained compound.

To explore the magnetic behavior of the synthesized MnFe₂O₄ ferrite nanoparticles, the temperature and the applied magnetic field dependence of magnetization M (H) were also evaluated. The magnetic data of the synthesized MnFe₂O₄ ferrite powder were collected for the temperature from 200 K to 800 K, under 500 Oe. The obtained magnetization against temperature M(T) curve is given in Figure 3 at the Curie temperature T_C. This temperature was obtained by the variation of dM(T)/dT as shown in Figure 3. The transition occurred at Tc = 588.0 K, which is in good agreement with the literature [46].

The hysteresis cycle for MnFe₂O₄ nanoparticles at room temperature T = 300 K is displayed in Figure 4. The magnetization of the MnFe₂O₄ nanoparticles was accomplished under a magnetic field from 0 to 60 kOe. From the hysteresis loops, the saturation magnetization (Ms) is about 67.17 emu/g, meaning that this material is heavily magnetic, while the remnant magnetization is Mr = 50 emu/g and the coercive field obtained is about H_C = 113.31 Oe. The magnetization saturation is around M = 67.17 emu/g. The obtained hysteresis behavior of such a ferromagnetic system is interesting for several applications such as anti-lock brake system (ABS) sensor rings [55].

3.2. Simulation Results: DFT and Monte Carlo Simulations

We have calculated the magnetic properties and the exchange coupling interactions of the compound MnFe₂O₄ spinel ferrite with both the GGA and the GGA + U (U = 3 eV) calculations. The ferromagnetic exchange mechanism has been found based on the calculated spin density and magnetic moment. This compound is ferromagnetic with a half-metallic nature [39]. This study obviously explains the important role of the Fe atom in MnFe₂O₄ [46,49].

Table 2. Exchange coupling values by GGA + U approximation.

Nearest Neighbors	JMn-Mn	JMn-Fe	JFe-Fe
Exchange couplings (meV)	1.18	−1.87	−1.87

shows the values of exchange coupling interactions between the first nearest neighbors.

We plot in Figure 5 the calculated magnetization versus temperature curve for the MnFe₂O₄ ferrite in the absence of an external magnetic field (h = 0). It is found that, for low-temperature values, the magnetization takes its maximum value, i.e., decreasing the temperature causes a decrease in thermal agitation, which leads to rises in magnetization. Furthermore, we notice a second-order phase transition that involves inversions of the Ising system from the ferromagnetic to the paramagnetic phase. This transition is important for a broad spectrum of technological applications [56,57,58]. Moreover, the obtained curve of magnetization shows a transition from the ferromagnetic to paramagnetic phase at Curie temperature, around Tc = 580.0 K. This is comparable to the experimental Curie temperature displayed in Figure 3. Furthermore, the experimental and theoretical curves exhibit the same pace of magnetization but not the same values of magnetizations. This may be caused by the absence of an external magnetic field in the theoretical part or by a diverse cationic distribution between the two sites A and B of the spinel structure. The calculated magnetization is given in μ_B while the experimental total magnetic moment is in emu/g, and the conversion is drawn from the total magnetic moment to the magnetization using the following formula:

$$\mathrm{M}\left(\mathrm{emu}/\mathrm{g}\right)=\frac{5585}{\mathrm{MW}} \times \mathrm{m} \left({\mathsf{\mu }}_{\mathrm{B}}\mathrm{s}\right)$$

(7)

where m is the total magnetic moment in μ_B, and MW is the molecular weight of MnFe₂O₄ (MW = 230.63 g/mol).

From the hysteresis loops, we explore the coercive field (H_c), the remanent magnetization (M_r) and the saturation of magnetization (Ms). The obtained coercive field for these nanoparticles is 116.15 Oe at room temperature, which is comparable to the experimental coercive field. The magnetization value at saturation Ms of MnFe₂O₄ is 52 emu/g at room temperature, which is in good agreement with the experimental one.

The difference between the saturation magnetization for the experimental and the calculated hysteresis loops is that for the calculated one we consider that the system is perfect which is not the case for the real system. From the hysteresis loops plotted in Figure 4 and Figure 6, MnFe₂O₄ is a soft ferromagnetic material, which can be used for transformers, electric machines and/or electromagnets. The cause of the difference in these results is due to the magnetic properties of the nanoparticles that depend on the particle size and the preparation method [56]. This variety is most probably linked also to a different cationic distribution between the two atoms Fe and Mn of the spinel structure.

4. Conclusions

MnFe₂O₄ ferrite nanoparticles were obtained via sol–gel method, and the structural, morphology and magnetic characteristics were investigated. The system crystalizes in a single phase with a spinel ferrite-like structure (space group Fd-3m). The structural properties and magnetic properties of MnFe₂O₄ have been examined using the first-principles methods via density functional theory and Monte Carlo simulations. The experimental and the calculated Curie temperature of the studied system are obtained at Tc ≈ 588.0 K. From the magnetic hysteresis cycles, MnFe₂O₄ is found to be a soft ferromagnetic material, which can be used for transformers, electric machines and/or electromagnets.