Thermal Treatment-Induced Coercivity Modulation in Magnetodielectric LaFe_0.7Ni_0.3O₃

Abstract

This study investigates the modulation of coercivity and magnetodielectric coupling in heat-treated, nickel-substituted lanthanum ferrite. LaFe_0.7Ni_0.3O₃ samples were synthesized by high-energy ball milling and sintered at temperatures between 1073 and 1473 K. Chemical composition, crystalline structural evolution, surface morphology, magnetic, dielectric, and electrical properties, as well as magnetodielectric coupling, were analyzed. The XPS spectra revealed the presence of adsorbed oxygen, associated with the high oxygen affinity of the material. This behavior is interpreted as a charge-compensation mechanism, related both to the formation of oxygen vacancies and to the partial oxidation of Fe³⁺ to Fe⁴⁺. XRD and Rietveld refinement confirmed a single-phase orthorhombic Pnma structure, and structural simulations revealed progressive octahedral distortions with increasing temperature, affecting the octahedral tilting and electronic bandwidth. Magnetic characterization revealed that thermal processing modifies the magnetic behavior, inducing weak ferromagnetism and a significant increase in coercivity, correlating with progressive densification, greater domain stability, and reduced microstrain. Impedance measurements revealed magnetodielectric coupling, the Maxwell–Wagner interfacial polarization mechanism, and reduced dielectric losses. These findings demonstrate that the coercivity and magnetodielectric response in cationic nickel-substituted lanthanum ferrite can be tuned through thermal processing. A semi-empirical magnetocrystalline anisotropy model is proposed to explain the coercivity evolution and associated multiferroic behaviors, thus contributing to the study of functional ferrites as sustainable alternatives to rare-earth magnetic materials with potential in sensors and memory devices.

1. Introduction

Ferrites are ceramic compounds based on iron oxides combined with transition metals or rare earths, which mainly crystallize in perovskite or spinel-type structures. They are distinguished by their magnetic behavior, exhibiting ferromagnetic, ferrimagnetic, or antiferromagnetic properties depending on the composition and arrangement of the metallic ions that constitute them, as well as by their electrical resistivity, which makes them useful in transformer cores, magnetic memories, or microwave devices. Furthermore, some ferrites exhibit more complex behaviors, including magnetoresistance and multiferroicity, where two or more ferroic orders coexist, such as ferromagnetism, antiferromagnetism, ferroelectricity, or ferroelasticity [1]; this expands their applications in the design of functional materials for spintronics and nanoelectronics [2]. Among multiferroics, magnetoelectrics stand out as particularly attractive materials because they not only integrate multiple ferroic orders but also exhibit a direct coupling between them, which allows one property to be modulated through another. The versatility of these materials lies in the feature that relatively small modifications in their chemical composition and crystalline structure, through doping or cationic substitutions, make it possible to modulate their physical properties.

Among ferrites exhibiting multiferroicity, lanthanum ferrite (LFO) stands out, it is an improper multiferroic material in which ferroelectricity is driven by magnetism [3]. At room temperature, it is a single-phase compound that crystallizes in a perovskite-type structure with internal octahedral coordination of Fe³⁺ cations and O^2- anions. This structural arrangement is responsible for the G-type antiferromagnetic (AFM) order it exhibits, attributed to superexchange interactions [4]. Such ordering indirectly induces electric polarization through the inverse Dzyaloshinskii–Moriya (DM) mechanism [5], which gives the LFO its multiferroic behavior [6].

Due to the aforementioned versatility of this material, specific studies have been conducted to alter the magnetic order of LFO by distorting its structure and thus modifying the orientation of magnetic spins responsible for AFM order. This has been achieved through various methods, such as cationic doping with ions like Sr²⁺ [7], where the change in magnetic order is attributed to lattice distortions and increased coercivity due to spin reordering from the presence of Fe⁴⁺, along with reports of magnetodielectric coupling. Doping with Co²⁺ [6] which induces changes in magnetic order through double exchange interactions and spin canting (DM mechanism) due to structural distortion, with increased coercivity linked to the loss of orthorhombic symmetry and the emergence of a rhombohedral phase. Doping with Al³⁺ [8] has shown induced ferromagnetism due to structural distortion, FeO₆ octahedral tilting, spin canting, and substitution of a diamagnetic cation that modifies superexchange, as well as particle size reduction to the nanoscale. On the other hand, Ref. [9] complemented these investigations by studying pure and transition metal-doped LFO nanoparticles, observing induced ferromagnetism and modified dielectric properties. Furthermore, magnetodielectric coupling has been reported in these structurally distorted materials [10], referring to the interaction between magnetic order and the dielectric response of a material, that is, how the magnetic field affects dielectric permittivity. This is correlated to spin-phonon coupling, magnetoelastic effects, and exchange striction [11]. Finally, within the study of distorted materials, efforts have been made to correlate the causes of these structural modifications with their magnetic implications. For example, Ref. [12] analyzed the variation in the sintering temperature of the LFO to observe the effects of their microstructural evolution on magnetic properties, finding that changes in particle size, surface structure, and spin canting resulting from microstructural evolution modify magnetic coercivity, highlighting that grain growth, densification, microstrain relaxation, and changes in spin canting strongly influence coercivity. Coercivity is a key parameter in applications where the magnetic state must remain stable against external perturbations. Therefore, understanding the mechanisms responsible for coercivity and thus modulating it is imperative in the pursuit of precise control over the stability, efficiency, and functionality of magnetic devices in various applications.

In this context, it has also been found that cationic substitution with Ni²⁺ induces weak ferromagnetism due to the partial oxidation of Fe³⁺ to Fe⁴⁺, confirmed by XPS, which favors double ferromagnetic (FM) exchange, and the octahedral tilting caused by the structural distortion resulting from the cationic substitution, which in turn favors ferroelectric behavior [13]. For this reason, the compound LaFe_0.7Ni_0.3O₃ has been selected. This compound shows no evidence of secondary phases or precursor materials, exhibits induced ferromagnetic order with improved specific magnetization, low coercivity around 246 Oe, the presence of Fe⁴⁺, moderate structural distortion with Fe–O–Fe of 167.74°, an octahedral tilt angle of 7.49°, and an electron bandwidth of 0.09477. Additionally, it presents a homogeneous structure with controlled porosity and high permittivities, thereby offering a balance between structural stability, magnetic response, and multiferroic behavior. Finally, considering these characteristics and the feature that LFO is an improper multiferroic, the induction of magnetodielectric coupling in the compound can be anticipated.

This paper studies the modulation of coercivity induced both by thermal treatment and by cationic substitution, as well as the magnetodielectric coupling as an emergent property in distorted systems, taking the compound LaFe_0.7Ni_0.3O₃ as a case study. The study correlates structural and microstructural evolution of LaFe_0.7Ni_0.3O₃ with the modification of its magnetic, dielectric, and electrical properties. A semi-empirical model of magnetocrystalline anisotropy is proposed to explain the observed behavior, with the purpose of contributing to the design of functional ferrites as a sustainable alternative to the reliance on rare earths in magnetic materials. This opens an opportunity to deepen the study and understanding of fundamental phenomena such as hysteresis, domain coupling, spin–orbit interaction, and the contribution of crystalline defects to the stability of magnetization. Thus, the study introduces concepts of microstructure, magnetic anisotropy, and energy transfer, exploring the most fundamental aspects of materials with magnetic properties.

2. Materials and Methods

LaFe_0.7Ni_0.3O₃ was synthesized via high-energy ball milling using powdered oxides as precursors: La₂O₃ (Sigma-Aldrich, 99.9 wt.% purity, St. Louis, MO, USA), Fe₂O₃ (99.9 wt.% purity, Sigma, Aldrich, St. Louis, MO, USA), and NiO (99.8 wt.% purity, Sigma Aldrich, St. Louis, MO, USA), which were mixed according to the stoichiometric reaction

$${\mathrm{L}\mathrm{a}}_2{\mathrm{O}}_3+0.7{\mathrm{F}\mathrm{e}}_2{\mathrm{O}}_3+0.6\mathrm{N}\mathrm{i}\mathrm{O}+0.15{\mathrm{O}}_2\to 2\mathrm{L}\mathrm{a}{\mathrm{F}\mathrm{e}}_{0.7}{\mathrm{N}\mathrm{i}}_{0.3}{\mathrm{O}}_3.$$

(1)

The powders were loaded into steel vials with a ball-to-powder weight ratio of 10:1, starting with a total of 5 g of powder. The milling process was carried out at room temperature in a Mixer/Mill (SPEX model 8000D, Spex SamplePrep, Metuchen, NJ, USA) at 90 min intervals with 30 min pauses between intervals until 5 h of milling were reached. The mechanically activated powders were uniaxially pressed at 1000 MPa in a cylindrical steel matrix to obtain pellets of 10 mm diameter and approximately 1 mm thickness, which were subjected to 4 h of sintering process using a tubular muffle furnace (Lindberg/Blue M model STF54459C, Thermo Fisher Scientific Inc., Riverside, MI, USA) at different temperatures ranging from 1073 K to 1473 K in 100 K increments.

This processing was used since it has been demonstrated that after 3 h of milling, the crystalline phase of the LFO becomes predominant [14] and that after 5 h it is possible to obtain advanced ceramics with structural homogeneity [9]. Moreover, it was found that the precursor phases and structural defects due to milling are eliminated above 1073 K [15] and that the ferrite phase remains stable up to 1473 K [6], thus ensuring the successful synthesis of the crystalline material.

The chemical composition was determined by X-ray photoelectron spectroscopy (XPS) using a Thermo Scientific K-Alpha spectrometer (East Grinstead, West Sussex, UK) equipped with a monochromatic Al Kα X-ray source (1486.6 eV) operating at 150 W. Survey spectra were acquired with a pass energy of 200 eV, while high-resolution spectra were recorded with a pass energy of 20 eV. The X-ray spot size was set to 400 µm².

The pellets were characterized by X-ray diffraction (XRD) with a Bruker D-8 diffractometer (Bruker D8 Advance, Bruker, Karlsruhe, Germany) using Cu radiation (wave length λ = 1.5419 Å) over a 2θ (total angle between the incident and the scattered X-ray beams) range of 20 to 80°, from which diffraction patterns were obtained for each sintering temperature. Rietveld refinement was performed using the Material Analysis Using Diffraction (MAUD) [16] software v 2.93 to obtain the corresponding crystallographic data employing the Crystallographic Open Database (COD) [17,18,19,20,21,22,23,24,25]. For surface morphology and microstructure analysis, a scanning electron microscope (SEM, HITACHI model TM3030, Hitachi, Tokyo, Japan) with energy-dispersive X-ray spectroscopy QUANTAX 75 (EDS) was used to obtain micrographs at each sintering temperature. Dimensional measurements were performed to estimate the physical density of the sintered pellets. Magnetic properties were studied by vibrating sample magnetometry MicroSense EV7 (MicroSense LLC, Lowell, MA, USA) in a magnetic field range of ±18 kOe. Impedance spectroscopy was carried out using a LCR (Hioki 3532-50, Hioki, Nagano, Japan). over a frequency range of 50 Hz to 5 MHz. To study the magnetodielectric coupling, a sample holder coupled to the LCR meter placed inside the magnetometer was used. This system measures the change in permittivity relative to the magnetic field in the range of 18 kOe spreading, with a variation of 1 kOe.

It is worth to note that all measurements corresponding to the characterizations were carried out at room temperature. The only parameter that varied in the study was the sintering temperature applied during the material processing.

3. Results and Discussion

3.1. Chemical Composition

Surface composition analysis was performed using XPS to determine the valence state of each element in the compound LaFe_0.7Ni_0.3O₃, with the analysis shown for a sintering temperature of 1073 K. Figure 1 shows the XPS spectra obtained for lanthanum (Figure 1a), iron (Figure 1b), nickel (Figure 1c), and oxygen (Figure 1d), each one with their respective deconvolution. Figure 1a, corresponding to La3d, shows doublets associated with the 3d_5/2 and 3d_3/2 states, with main peaks at 833.5 and 850.3 eV, respectively, and satellite peaks at 827.4 and 854.5 eV for each region of the spectrum. The presence of c4f⁰ and c4f¹L indicates charge transfer between the metal and the oxygen, a characteristic behavior of perovskites with high oxygen mobility. The c4f⁰ state, where c refers to the presence of a hole in the nucleus, 4f⁰ to the absence of electrons in the 4f orbital indicating states in lanthanum without charge transfer, and c4f¹L, which marks the transfer of an electron from the ligand atom L to the 4f orbital denoting charge transfer; together with the aforementioned main and satellite peaks, shows evidence of trivalent lanthanum ions in the sample [26]. Figure 1b presents the Fe2p spectrum, whose deconvolutions show the main peaks characteristic of the 2p_3/2 and 2p_1/2 states located at 709.9 eV and 723.6 eV, respectively, associated with the presence of Fe³⁺. Likewise, peaks shifted towards higher bond energies corresponding to the same states are observed, which is evidence of oxidation to Fe⁴⁺. The formation of Fe⁴⁺ is related to the charge compensation mechanisms necessary to maintain electrical neutrality in the material [27].

The oxidation to Fe⁴⁺ does not occur arbitrarily, but as a direct consequence of the charge neutrality in the perovskite lattice. Structurally, each substitution of a divalent cation in a site occupied by a trivalent cation implies the loss of one positive charge per substitution. As compensation, a percentage of the 3+ cation formally transforms into a 4+ cation to ensure the system’s stability without the need for additional compensation processes. On the other hand, a cation in the 2+ state does not directly resolve the charge deficit but requires a greater energy cost to reach equilibrium. From a thermodynamic perspective, the system tends to minimize its Gibbs free energy, so the oxidation to Fe⁴⁺ constitutes a more favorable route to stabilize the crystal lattice, as confirmed by the XPS results. This mechanism is not exclusive to ferrite; it has been observed in lanthanum manganites with divalent substitutions, even at A sites, with a confirmed oxidation of a percentage of Mn³⁺ to Mn⁴⁺ [28].

Figure 1c shows the spectrum corresponding to Ni2p, where a main peak is observed at 854.1 eV, associated with the predominant position for Ni²⁺ (Ni2p_3/2) [29]. This cation is characterized by a multicomponent envelope spectrum that includes the Ni2p_1/2 state at 871.9 eV, as well as satellite peaks at 861.9 and 878.9 eV, which are visible in the deconvolutions. The presence of these satellites is considered unequivocal evidence of Ni²⁺ states, since in the case of Ni⁰ there are no satellite peaks and the main peak is typically located below 852 eV. whereas for Ni³⁺ the associated states occur at higher binding energies and lack characteristic satellites.

The presence of Ni²⁺ is related to charge transfer processes associated with satellite peaks, indicating quite a strong electron-ligand interaction. Therefore, although Ni is present in relatively low proportions, its presence modulates the electronic structure and favors oxidation to Fe⁴⁺.

It is worth to note that the XPS spectra provide direct evidence supporting the stabilization of Fe⁴⁺ instead of Ni³⁺ in the LaFe_0.7Ni_0.3O₃ samples. This occurs because Ni³⁺ is an uncommon and thermodynamically less stable cation in oxide structures such as LFO, due to several factors. First, its ionic radius is significantly smaller than that of Fe³⁺, which induces considerable distortions in the crystal lattice. In addition, the energy required to oxidize Ni²⁺ to Ni³⁺ is higher, pointing that this oxidation may only occur under special conditions such as high pressure or highly oxidizing atmospheres. Consequently, the presence of Ni³⁺ in perovskites is quite unlikely.

On the other hand, Fe³⁺ with a 3d⁵ configuration exhibits a stable half-filled state, since all d orbitals are singly occupied. When oxidized to Fe⁴⁺ with a 3d⁴ configuration, it loses this stability with minimal energy cost, owing to the ability of iron to accommodate different oxidation states in oxide environments. In perovskites, this oxidation is common due to the strong covalent interaction with oxygen.

In contrast, Ni²⁺ with a 3d⁸ configuration is highly stable in octahedral coordination because its d orbitals are nearly filled. Oxidizing it to Ni³⁺ with a 3d⁷ configuration requires removing an electron at a higher energetic cost compared to iron. Furthermore, Fe–O bonds exhibit greater covalent character, which favors the stabilization of Fe⁴⁺; whereas nickel tends to form more ionic bonds in its Ni²⁺ state, so the transition to Ni³⁺ would demand a more oxidizing chemical environment.

This interpretation is consistent with the findings of Ref. [26], who demonstrated that oxygen non-stoichiometry in LaFeO₃ becomes more pronounced at elevated temperatures, and vacancy formation strongly influences both electrical transport and local structural distortions. In this case, it is expected that at higher sintering temperatures, oxygen vacancies will become thermodynamically favorable.

Finally, Figure 1d presents the spectrum corresponding to oxygen, where the deconvolutions show two distinct contributions. The peak at 528.6 eV corresponds to structural oxygen present in the ferrite lattice, associated with La–O and Fe–O bonds in the crystal lattice. Meanwhile, the peak at 531.1 eV is associated with oxygen adsorbed or bound to oxygen vacancies on the surface. The presence of these vacancies provides high surface reactivity, so the adsorbed oxygen occupies them or interacts with them, influencing material properties such as conductivity and even generating an apparent increase in Maxwell–Wagner (MW) polarization, an intrinsic effect related to the material’s surface response. In this context, the XPS spectra also confirm the coexistence of Fe³⁺ and Fe⁴⁺ states, supporting the stabilization of Fe⁴⁺ rather than Ni³⁺.

Table 1. Element content in powders of LaFe_0.7Ni_0.3O₃ treated at 1073 K.

Element Content
Name	Atomic %	Weight %
La3d	7.5	41.78
Fe2p	2.8	6.35
Ni2p	1.5	3.57
O1s	35.2	22.54
C1s	51.7	24.85

shows the relative proportion of ions present in the sample, expressed as atomic percentage and by weight. A significant percentage of C1s, representing adventitious carbon, is observed, related to the uncontrolled atmosphere during the material’s synthesis. O1s reflects the contribution of structural oxygen, with a significant percentage in the composition, also related to the oxidizing atmosphere during synthesis and processing. La3d and Ni3p ions appear in proportions consistent with their expected presence in the compound, with some slight variations attributed to surface sensitivity and interaction factors. Specifically, Fe2p and Ni2p show proportions consistent with doping, as both atomic percentage and weight are around 34%. The remaining composition is associated with relatively small amounts of F1s, N1s, and Na1s, corresponding to common surface contaminants and traces related to material handling.

3.2. Crystal Structure

X-ray diffractograms were obtained for sintered pellets at different temperatures, which were analyzed by Rietveld refinement. The results are presented in Figure 2 and in

Table 2. Results of Rietveld refinement of X-ray diffractograms of LaFe_0.7Ni_0.3O₃ pellets at 1073 K to 1473 K temperatures: cell parameters, phase percentage, crystal size (D_m), microstrain (µ_s), and goodness of fit (χ² and R_wp).

Sintering Temperature (K)
		1073	1173	1273	1373	1473
Phase		LaFeO3	LaFeO3	LaFeO3	LaFeO3	LaFeO3
Space group		Pnma	Pnma	Pnma	Pnma	Pnma
Phase wt.%		100	100	100	100	100
Lattice Parameters (Å)	a	5.5532 ± 0.0004	5.5036 ± 0.0004	5.5175 ± 0.0004	5.5167 ± 0.0002	5.5227 ± 0.0003
	b	7.8594 ± 0.0006	7.8471 ± 0.0005	7.8103 ± 0.0007	7.8130 ± 0.0003	7.8224 ± 0.0005
	c	5.4994 ± 0.0004	5.5457 ± 0.0003	5.5486 ± 0.0005	5.5470 ± 0.0002	5.5441 ± 0.0002
Dm (Å)		1057.5537 ± 26.4439	2624.7898 ± 121.9684	3467.6555 ± 123.0093	5079.4260 ± 217.3670	9142.9620 ± 684.7188
Microstrain µs (×10−4)		20.7605 ± 0.3703	29.0880 ± 0.2649	23.0385 ± 0.2601	11.9828 ± 0.1863	8.2344 ± 0.1083
χ2		0.7895	0.6879	0.9531	0.6744	0.8310
Rwp		12.5981	10.5220	13.3334	10.0148	13.8405

, showing the values of cell size, crystallite size, microstrain, and fitting parameters, where χ² statistics and weighted profile residuals R_wp indicate an adequate fit between the calculated and experimentally obtained data. An orthorhombic phase of LFO (Pnma, COD 1561807) was identified for all samples at different sintering temperatures without the presence of secondary phases, confirming that the stoichiometric reaction (1) was successfully completed. The introduction of 0.3 mol nickel does not change the orthorhombic phase, so it is considered that a solid solution was formed.

All the X-ray patterns exhibit a main diffraction peak around 32–32.5° in 2θ, characteristic of the orthorhombic phase of LFO. As the sintering temperature increases, this peak shifts toward higher angles, indicating a contraction of the unit cell. This behavior is consistent with two contributing factors: initial doping and subsequent thermal effects. The lattice contraction is expected, as oxidation from Fe³⁺ to Fe⁴⁺ has been reported as a charge compensation mechanism when lower-valence cations are introduced [13]. Since Fe⁴⁺ (0.585 Å) has a smaller ionic radius than Fe³⁺ (0.645 Å), the lattice contracts accordingly. Additionally, the ionic radius of Ni²⁺ (0.69 Å) is comparable to that of Fe³⁺, allowing the formation of a solid solution without significant structural distortion.

The reflections such as (101), (211), and (240) in diffraction patterns, which are not listed in the ideal cubic perovskite but characteristic of the orthorhombic Pnma symmetry. These reflections arise from antiferrodistortive rotations of the FeO₆ octahedra, where adjacent octahedra rotate in opposite directions around the crystallographic axes. This distortion doubles the unit cell and breaks inversion symmetry at the midpoints between magnetic cations. Although tilting modes and ferroelectric displacements are structurally competitive, the observed superlattice peaks confirm that antiferrodistortion is the dominant structural feature in our samples, and it plays a crucial role in enabling weak ferromagnetism.

In addition, Figure 2 includes a zoom-in on the main diffraction peak, where a progressive narrowing is observed as the sintering temperature increases. This narrowing indicates an increase in crystallite size, attributed to: (i) enhanced atomic diffusion in the solid state, facilitated by thermal energy that enables atoms to overcome activation barriers and migrate within the crystal lattice [30]; (ii) the reduction in defects such as grain boundaries, which are thermodynamically unstable regions that hinder crystallite growth and tend to diminish at higher temperatures; and (iii) the Ostwald ripening effect, whereby smaller crystallites dissolve and their constituent atoms contribute to the growth of larger ones [31]. These processes collectively promote material densification.

On the other hand, the values corresponding to the microstresses decrease, which is consistent with the narrowing, with the increase in crystallite size, and this decreases the energy of the crystalline structure, promotes relaxation of local stresses and is related to the increase in temperature. That is, since smaller crystallites have a larger relative surface area and more defects, they generate a greater amount of microstresses in the unit cell, so that as the crystallites grow, as a consequence of the atomic distribution due to diffusion and a lower presence of defects, there is less distortion and tension in the lattice. That is, larger crystallites reduce the system’s overall energy by relieving local stresses and minimizing lattice distortions. Smaller crystallites, due to their higher surface-to-volume ratio and greater defect density, introduce greater microstrain into the unit cell. As atomic diffusion progresses and defects decrease, the relaxation of the crystal lattice increases, resulting in lower stress and greater structural stability.

In order to visualize the deformation in the octahedron of the ferrite structure as a consequence of the increase in sintering temperature, a simulation was performed using the VESTA software v 3.5.5 (Visualization for Electronics and Structural Analysis) [32] from the refinement data to estimate the Fe–O–Fe bond angles and Fe–O distances, given that the octahedron formed by these cations is responsible for the magnetic behavior of the material, so it is essential to analyze the modification of the angles. The obtained values, as well as the simulated structure of the material, are shown in

Table 3. Bond angles, bond lengths obtained using VESTA simulation data from XRD patterns.

Sintering Temperature (K)	Fe–O–Fe Bond Angle (Degree)		Fe–O Bond Length (Å)
	Fe–O1–Fe	Fe–O2–Fe	Fe–O1	Fe–O2
1073	176.0	153.9	1.966	2.160
1173	156.4	163.4	2.004	2.050
1273	164.0	159.6	1.973	2.030
1373	156.7	160.7	1.994	1.972
1473	134.4	168.2	2.121	1.970

and Figure 3.

The proposed structural model has already considered the coexistence of the cations present in the system (Fe³⁺/Fe⁴⁺ and Ni²⁺), whose presence has been confirmed by XPS analysis. Moreover, the difference in ionic radii is compensated within the structural model by fulfilling the relation (Ni²⁺-Fe⁴⁺) - (Ni²⁺-Fe³⁺) = (Fe³⁺-Fe⁴⁺), which ensures that these variations do not significantly modify the structural framework or the crystallographic symmetry.

As observed, the simulations reveal different values for the Fe–O1–Fe, Fe–O2–Fe angles, as well as for the Fe–O1 and Fe–O2 distances. At 1073 K a tendency towards structural linearity can be noticed, that is, when the central atom is bonded to the other two at an angle of 180°, presenting values close to 176° for Fe–O1–Fe and 153.9° for Fe–O2–Fe, which suggests a reduction in angular distortion due to processing compared to literature-reported values around 167 degrees for Fe–O1–Fe, and that for the bond with O2 the distances are so large that they are not reported [13]. Considering that LFO has a distorted orthorhombic perovskite structure due to the mismatch in ionic radii between La³⁺, Fe³⁺, and O²⁻, these values indicate a compensated angular symmetry and greater internal deformation within the local symmetry of the present octahedron [33].

At 1173 K, both angles show similar values, of 156.4 and 163.4°, reflecting a slight structural distortion with possible redistribution of octahedral tilts. At the intermediate temperature of 1273 K, both angles stabilize around 160°, indicating a balanced configuration between symmetry and distortion.

The non-monotonic behavior observed between 1173 and 1273 K can be explained by the competition of microscopic processes occurring in this temperature range. On the one hand, enhanced atomic diffusion promotes crystallite growth and partial relaxation of microstrain, leading to a more symmetrical configuration of the Fe–O–Fe octahedra. On the other hand, the redistribution of oxygen vacancies and local tilts introduces transient distortions that temporarily stabilize bond angles around 160°, as shown in Table 3. This balance between structural relaxation and defect reorganization produces a metastable state, reflected in the non-linear evolution of structural properties. At higher temperatures, vacancy concentration and anisotropic distortions increase again, driving the subsequent reorganization of the lattice and the changes in hysteresis behavior.

At 1373 K, the Fe–O1–Fe angle decreases while Fe–O2–Fe increases, suggesting a compensated symmetry within the structure. However, at the highest temperature of 1473 K, an inversion in the trend occurs, Fe–O1–Fe decreases considerably up to 134.4°, while Fe–O2–Fe reaches its maximum at 168.2°, indicating a notable structural reorganization. At such high temperatures, diffusion mechanisms may cause atoms to move across grain boundaries, for example, dopants such as Ni²⁺, to migrating towards grain boundaries or other surfaces, even causing some oxygen vacancies or the generation of structural defects.

These vacancies alter the local coordination environment, producing distortions in bond lengths and angles, as observed in Table 3. Such structural modifications are consistent with the stabilization of Fe⁴⁺ cations, since oxygen deficiency promotes the oxidation of Fe³⁺ to maintain charge neutrality. Therefore, the structural changes tabulated are rationalized by the combined effect of oxygen vacancy formation and Fe oxidation, rather than by the unlikely stabilization of Ni³⁺. This interpretation is consistent with the observations of Cao [26], who reported with vacancy formation strongly influencing electrical transport and local structural distortions.

This latter phenomenon promotes the formation of crystalline anisotropies, which impact the magnetic and dielectric behavior of the material.

Regarding the changes in bond distances, they reflect a definite thermal evolution. At low temperature (1073 K), axial elongation is observed, as shown in Figure 3, with Fe–O2 measuring 2.120 Å and Fe–O1 at 1.966 Å. At 1173 and 1273 K, the distances progressively equilibrate around 2 Å, which, together with the bond angles at those temperatures, suggests structural relaxation and a possible tendency towards optimization of the octahedral lattice, that is, to achieve a more stable and symmetrical configuration.

At 1373 K, a partial inversion of these bond distances is observed, with Fe–O1 elongating to 1.994 Å and Fe–O2 shortening to 1.972 Å. This distortion becomes more pronounced at 1473 K, where Fe–O1 reaches a maximum of 2.121 Å and Fe–O2 a minimum of 1.970 Å. These modifications reflect structural distortions mainly associated with octahedral tilting within the same orthorhombic symmetry, which may favor the ferroelectric response of the material [34].

It is hypothesized that, under high-temperature processing conditions and in the presence of oxygen vacancies, local regions with partial loss of centrosymmetry could emerge due to structural distortions, thereby promoting extrinsic polarization phenomena. Furthermore, variations in bond angles and distances affect the bandwidth (W), i.e., the extent of electronic itinerancy that mediates magnetic interactions through diamagnetic bridges between magnetic ions (Fe, Ni), favoring ferromagnetic-type electronic superexchange interactions.

This distance is calculated from the previously simulated bond angles and distances as follows [35]:

$$W\approx {\displaystyle \frac{\cos \left(\frac{\pi -\omega }{2}\right)}{d_{\mathrm{F}\mathrm{e}–\mathrm{O}}^{3.5}}} ,$$

(2)

where W is the bandwidth, ω is the Fe–O–Fe bond angle, and d_Fe–O is the angle distance. On the other hand, to correlate the electronic connectivity and the structural distortion of the FeO₆ lattice with the change in the heat treatment, the angular deviation with respect to the ideal geometry mentioned above (180°) is quantified through the estimation of the tilt angle, which is calculated as follows for the orthorhombic perovskite structures [36]:

$${\phi }_1={\cos }^{-1}\sqrt{\left|{\displaystyle \frac{2-2\cos {\omega }_1}{5+\cos {\omega }_1}}\right|}$$

(3)

and

$${\phi }_2={\cos }^{-1}\sqrt{\left|{\displaystyle \frac{3\cos {\omega }_2-1}{-4}}\right| },$$

(4)

where ω₁ refers to the Fe–O1–Fe angle along the Z-axis (axial angle), and ω₂ to the Fe–O2–Fe angle along the X-axis (perpendicular tilt), such that φ₁ represents the structural distortion along the Z-axis and φ₂ the structural distortion along the X-axis.

The values obtained from these calculations are given in

Table 4. Bandwidth in Z-axis (W_Fe–O1–Fe), bandwidth in X-axis (W_Fe–O2–Fe), structural distortion in Z-axis (φ_Fe–O1–Fe) and structural distortion in X-axis (φ_Fe–O2–Fe).

Sintering Temperature (K)	WFe–O1–Fe (adim)	φ1 (degree)	WFe–O2–Fe (adim)	φ2 (degree)
1073	0.0937	2.4490	0.0658	16.0543
1173	0.0859	14.3519	0.0802	10.1834
1273	0.0918	9.7664	0.0826	12.5261
1373	0.0875	14.1719	0.0915	11.8472
1473	0.0663	27.2410	0.0927	7.2324

, where a decrease in W_Fe–O1–Fe is observed at high temperatures, associated with a deoptimization of the FeO₆ lattice, that is, a loss of symmetry, reflecting a structural reorientation. Similarly, the tilting values at 1073 K indicate an almost ideal symmetry along the Fe–O1–Fe axis, as previously mentioned, while at 1473 K a notable increase confirms severe distortion of the octahedral environment. In contrast, the tilt for Fe–O2–Fe decreases with temperature, from 16.05° to 7.23°, suggesting partial relaxation along the perpendicular axis, possibly as a compensatory mechanism for the excessive octahedral rotation in φ₁ through redistribution of internal stresses. Considering all the parameters analyzed above, the sintering range between 1173 and 1273 K appears to be structurally optimal due to the balanced values of W and φ; that is, W is not as low as at 1473 K, where symmetry and electron connectivity are lost, nor as high as at 1073 K, where symmetry is almost linear with limited electronic connectivity. Therefore, presenting intermediate values in the working temperature range implies that the itinerant electrons can still move with sufficient connectivity to favor FM super-exchange interactions over AFM interactions. For its part, the octahedral lattice is neither too rigid nor excessively distorted but rather compensated, which suggests a favorable structural configuration.

3.3. Morphological Characterization

The surface micrographs of the pellets sintered at different temperatures (Figure 4) show a definite evolution of the microstructure as the thermal treatment increases. At 1073 K, the material still exhibits a highly porous surface, with what appear to be primary particles loosely compacted and without full coalescence, typical of an early sintering stage with limited densification. At 1173 K, porosity begins to decrease, although grain-grain connections are still incipient and sintering necks are not yet well developed. By 1273 K, the microstructure becomes noticeably denser and better interconnected. At 1373 K, the grains start to display more rounded edges, suggesting the onset of coalescence driven by surface diffusion. Finally, at 1473 K, quite large and uniformly rounded grains dominate the surface, consistent with advanced mass-transport processes that smooth out high-energy grain boundaries and lead to the highest degree of densification.

As the crystallite size increases, structural disorder decreases, which enhances the easy axis of magnetization and increases anisotropy energy due to the alignment of magnetic moments along preferential crystallographic axes, particularly in the more densified samples as shown in Figure 4f (1273 and 1373 K).

An analysis of the surface porosity at each sintering temperature shown in Figure 5 was performed to examine the progressive evolution of porosity and its relationship with grain connectivity and the consolidation of the crystalline network. In the micrographs, the red regions correspond to pores, while the gray areas represent the solid matrix. This color coding was applied to facilitate the identification and quantification of porosity. In Figure 5a, a high level of porosity is confirmed, corresponding to incomplete sintering characterized by low densification, as previously observed under limited diffusion conditions. In Figure 5b, the reduction in porosity indicates the onset of a transition toward a more consolidated microstructure, while in Figure 5c exhibits higher densification and a noticeable decrease in porosity compared to samples sintered at lower temperatures. In Figure 5d, a slight increase in porosity is observed, which is attributed to the release of internal stresses during the densification process. Finally, in Figure 5e, consolidated grains of a larger size are evident, resulting in a markedly reduced porosity compared to the other sintering temperatures.

Energy dispersive X-ray spectroscopy (EDS) characterization in Figure 6 confirms the presence of the elements that make up the LFO with Ni substitution in the LaFe_0.7Ni_0.3O₃ sample sintered at 1473 K. The SEM micrograph shows a granular morphology with contrasts that suggest surface heterogeneity. Element distribution maps use specific color coding to identify each element: red indicates lanthanum (La–LA), blue represents iron (Fe–KA), yellow corresponds to oxygen (O–KA), and green highlights nickel (Ni–KA). These maps show homogeneous dispersion of La, Fe, Ni, and O, with good enough integration of nickel into the perovskite matrix. The atomic percentage table indicates a dominant proportion of oxygen (55.62%), followed by lanthanum (22.23%), iron (16.87%), and nickel (5.28%), which supports the partial substitution of Fe by Ni at the B site. These results validate the nominal composition and suggest effective incorporation of the cations into the crystal structure.

3.4. Magnetic Properties

It is known that lanthanum ferrite exhibits a G-type antiferromagnetic order in its pure state, resulting from the cancellation of magnetic moments due to the opposite orientation of the Fe spins. However, as mentioned, through doping, the induction of its magnetic order to weak ferromagnetic has been achieved, as is the case for the LaFe_0.7Ni_0.3O₃ compound.

The observed weak ferromagnetism can be explained by the spin tilt induced by antiferrodistortion rotations of the FeO₆ octahedra. Antiferrodistortion rotations are alternating tilts of the octahedra in opposite directions, consequently destroying the inversion centers between Fe sites. These rotations eliminate the inversion centers between neighboring Fe sites, thus activating the DM interaction. As a result, the antiferromagnetic sublattices exhibit a slight spin tilt, producing a net magnetization. This mechanism is consistent with structural evidence obtained by X-ray diffraction and with the observations of Ederer [37], where weak ferromagnetism in perovskites is directly linked to antiferrodistortion. Therefore, the magnitude of the magnetization is sensitive to the degree of octahedral rotation, which can vary with the sintering temperature and processing.

This change in order has been attributed to the structural distortion caused during synthesis and processing, as well as the substitution of Fe³⁺ by Ni²⁺, which modifies the superexchange interactions, favors double exchange mechanisms upon oxidation of Fe³⁺ to Fe⁴⁺ due to a charge compensation mechanism, and shows the appearance of coercivity [13]. To analyze the magnetic behavior of the material, the magnetic hysteresis loops of the LaFe_0.7Ni_0.3O₃ compound at different sintering temperatures are shown in Figure 7.

The magnetic hysteresis loop of the sample at 1073 K is shown with a weak antiferromagnetic order with a saturation magnetization (M_s) of 0.3533 emu/g and an coercivity (H_c) of 236.0527 Oe (

Table 5. Magnetic parameters of LaFe_0.7Ni_0.3O₃ sintered pellets. See text for details.

Sintering Temperature (K)	Ms at 18 kOe (emu/g)	Mr (emu/g)	Hc (Oe)	Mr/Ms (adim)
1073	0.3533	0.0615	236.0527	0.1741
1173	0.1597	0.0083	373.794	0.0520
1273	0.1992	0.0386	2781.947	0.1938
1373	0.2489	0.0898	6046.8357	0.3608
1473	0.2842	0.1327	7565.1201	0.4669

), which coincides with previous studies [13]. In the magnetic hysteresis loops, a change in magnetic behavior can be observed, where starting from the 1173 K sample, H_c increases and M_s decreases, suggesting structural distortions attributed to the increase in sintering temperature [12]. Subsequently, it is observed that the increase in temperature leads to a behavior typically exhibited by hard magnetic materials; however, it is worth to mention that since the material is weak ferromagnetic, it is not considered magnetically hard.

In Table 5, the values of M_s, remanent magnetization (M_r), H_c, as well as the squareness ratio (M_r/M_s) are presented. Generally, an abrupt decrease can be observed between the temperatures of 1073 and 1173 K, which subsequently recovers progressively. The minimum at 1173 K is associated with the possible inhibition of magnetic alignment by structural defects or local disorder, as well as the subsequent increase suggesting grain growth that favors a better magnetic order, meaning the alignment of domains, which is compatible with a multi-domain behavior. The grain growth is supported by the refinement values in Table 2. This evolution in grain size is key, since larger grains allow for self-subdivision into regions with distinct magnetic orientation to minimize internal energy [6]. On the other hand, the radically decreased microstrain values (from 20.7605 to 8.2344) indicate a relaxation of internal stresses, modifying the mobility capacity of the domain walls (pinning/depinning) and forming more stable domain structures, which impacts coercivity and results in higher remanence, as shown by the results. Regarding the M_r/M_s ratio, which indicates the material’s magnetic retention capacity, a value of 0.1741 is observed for 1073 K, an expected value in soft magnetic materials [10]; however, at 1173 K it decreases to 0.0520 before increasing up to 0.47 at 1473 K with the sintering temperature, which also aligns with more stable and defined magnetic domains as mentioned, which resist demagnetization in coherence with the Rietveld refinement values. The LFO phase and the Pnma space group remaining constant suggest that the magnetic changes are not due to phase transformations as is usual, but to microstructural evolution (grain growth and defects).

According to the densification measurements in Figure 4, the samples sintered at the lowest temperatures exhibit a highly porous and disordered microstructure, consistent with the low coercivity observed. At the intermediate sintering temperature of 1273 K, the microstructure becomes noticeably denser, a change that aligns with the increase in coercivity. In contrast, the sample treated at 1473 K shows the highest densification and signs of grain overgrowth associated with over-sintering. This microstructural state correlates with the maximum coercivity, believed due to reduced domain disorder and enhanced magnetic stability.

Regarding the local geometry provided by the angle data in Table 3, the ferromagnetic superexchange and double exchange interactions are evidenced as favored. At the highest temperature, a distorted angle of up to 134.4° is observed, breaking the 180° linearity found in AFM materials, which can induce weak ferromagnetic components due to spin canting. Similarly, the value at this same temperature of 2.121 Å suggests a significant lengthening of the Fe–O1 distance, indicating structural relaxation and a reduction in the antiferromagnetic superexchange interaction, pointing to the presence of more stable domains. Smaller Fe–O–Fe angles reduce AFM superexchange, which favors coercivity by preventing the cancellation of magnetic moments; a greater Fe–O length points to lower structural rigidity and microstrain as discussed in the Rietveld refinement; finally, the angular distortion and bond relaxation leads to the formation of more defined magnetic domains with more stable walls in a multi-domain behavior with high coercivity.

Table 4 includes the W values, as an indirect measure of the orbital overlap between the Fe and O atoms, which influences the intensity of magnetic superexchange. Higher W values indicate greater electronic coupling, which points to more efficient overlap, signifying a stronger transfer of magnetic interaction is allowed between the metallic ions through the oxygen. An efficient Fe–O–Fe coupling favors the alignment between the d orbitals of Fe and the p orbitals of O, generating greater energy dispersion. For 1073 K, presenting the highest W value (0.0937), a more linear structure is suggested, promoting opposite spin directions, consistent with the bond angle values in Table 3. This favors the superexchange interaction, which coexists with the double exchange due to the presence of Fe⁴⁺, resulting in a ferromagnetic hysteresis curve [13]. Conversely, for 1473 K, the lowest value (0.0663) is presented, suggesting a reduction in coupling, resulting in lower direct AFM interaction but fostering spin canting and coercivity due to domain stabilization, given that the angle distortion encourages the decrease in electronic coupling by breaking the structural symmetry (tilting), leading to spin canting, which is consistent with all the results above. In the particular case of the 1173 K temperature, a lower W is observed compared to 1073 K, unlike the higher tilt angles. This indicates a radical decrease in orbital coupling, resulting in an increase in canting, which is a possible local magnetic disorder. This explains the abrupt drop in M_s and M_r, as well as the moderate increase in H_c. These values progressively recover with the increase in temperature, attributing this to the moderate distortion that seeks a favorable structural reorganization, which in turn has an effect on an increase in coercivity and remanence, indicating the consolidation of stable magnetic domains, continuing the distinct multi-domain behavior.

To deepen the understanding of the coercivity trend, a magnetic anisotropy analysis was performed, particularly of magnetocrystalline anisotropy, which is the one that arises from the interaction of the magnetic moments and the crystal field generated by the atomic lattice of the material. Therefore, any non-uniform process, locally, implies a breaking of one or more magnetic moments with respect to the others (Figure 8).

This model relates the magnetization angle of anisotropy to effective anisotropy energy, a model that arises from the interaction between magnetic moments and is studied as a coercivity mechanism present in materials with significant coercivity that do not have phase change interactions, i.e., the coercivity or magnetization mechanisms are not influenced by the occurrence of a phase change; therefore, the model can be applied in materials with multi-domain behavior. Having an orthorhombic-type structure with the presence of structural strains as a consequence of sintering and bulk morphology, a second-order magnetocrystalline anisotropy model is considered. This model is commonly used in both uniaxial and cubic or distorted configurations, so the symmetry of the orthorhombic structure allows for second-order terms in the anisotropy energy, which is why the following expression is proposed [38]:

$$E_{\mathrm{m}\mathrm{a}}\left(\gamma \right)=K_{1 }{\sin }^2\gamma +K_2{\sin }^4\gamma ,$$

(5)

where E_ma is the magnetic anisotropy energy, K₁ and K₂ are anisotropy constants, and γ is the angle between the magnetization direction and the easy axis, the crystallographic orientation along which the material is most readily magnetized. The magnetocrystalline anisotropy arises primarily from the spin–orbit interaction, where the orbital motion of the electrons couples to the crystal’s internal electric field. This interaction defines the first-order contribution to the anisotropy. The second-order contribution, often treated as a correction, is associated with dipole–dipole interactions and subtle changes in the electron density driven by Coulomb forces.

The estimation of the easy axis of magnetization (Figure 9) has been proposed as a theoretical approximation based on the crystallographic symmetry of the orthorhombic system and on previous reports of similar materials. The XRD pattern exhibits a preferred orientation toward the (200) plane, whose normal is [200]. This crystallographic texture suggests that the crystallites tend to align with this plane parallel to the surface of the pellet. Nevertheless, it is worth to emphasize that the preferred orientation identified by X-ray diffraction does not directly determine the easy axis of magnetization.

By symmetry, the [100] direction, normal to the dominant plane, has been considered as a candidate for the easy axis, in agreement with what has been reported in related orthorhombic systems. To fit the anisotropy parameters (K₁ and K₂), the magnetic energy E_ma(θ) was simulated by taking θ = 0° as the minimum, aligned with [100].

The second-order magnetocrystalline anisotropy model, which describes how the magnetic energy varies with the orientation of the magnetization relative to the crystallographic axes, can be modeled by estimating the anisotropy field from the hysteresis loops. Taking H_k as the field necessary for magnetic saturation, where the subscript k refers to the crystallographic anisotropy constant, the following expression can be used [39]:

$$E_{\mathrm{e}\mathrm{f}\mathrm{f}}={\mu }_0{H}_k{M}_s,$$

(6)

where µ₀ is the vacuum permeability, and effective anisotropy energy (E_eff) can be used as an average estimation to adjust the second-order model. Therefore, K₁ and K₂ can be adjusted so that the minimum energy coincides with the expected direction of the easy axis and the maximum value approximates the effective anisotropy energy estimation, which is the material’s internal resistance to modifying the magnetization orientation. The adjustment values are shown in

Table 6. Effective anisotropy energy (E_eff) and anisotropy constants (K₁ and K₂) fitted for the sintered LaFe_0.7Ni_0.3O₃ pellets.

Sintering Temperature (K)	Eeff (erg/g)	K1 (erg/g)	K2 (erg/g)
1073	1.08 × 104	*	*
1173	2.87 × 106	1.92 × 106	9.58 × 105
1273	3.59 × 106	2.39 × 106	1.20 × 106
1373	4.48 × 106	2.99 × 106	1.49 × 106
1473	5.12 × 106	3.41 × 106	1.70 × 106

* for the SW model, the parameter α is reported instead of K₁ and K₂, which at 1073 K is 2885.65 Oe.

. For the 1073 K sample, the Stoner-Wohlfarth (SW) model is used because the hysteresis loop presents low coercivity compared to the samples at other temperatures, which is evidence of smooth transitions in the magnetization direction, characteristic of soft magnets. E_eff values of 1.08 × 10⁴ erg/g and alpha of 2885.65 Oe are reported, showing moderate anisotropy and a coherent response to the external field, pointing it does not require great energy to reorient the magnetization. The energy value suggests that the system maintains a stable ferroic structure but with sufficient sensitivity to respond to external fields.

As can be seen in Table 6, the lowest E_eff value appears at 1073 K, which is consistent with a poorly consolidated microstructure, where the combination of internal stresses and limited densification leads to reduced anisotropy. When the temperature increases to 1173 K, E_eff rises abruptly, indicating that the material begins to recover structural coherence as porosity decreases and the microstrain becomes more stable. From 1273 K, the progressive increase in E_eff reflects a most distinct tendency toward densification (Figure 4f), improved structural alignment, and stabilization of the anisotropy, in agreement with the microstrain evolution. This behavior also explains the increment in coercivity with temperature. According to the Givord model [40], higher anisotropy increases the domain-wall energy and reduces the activation volume, which naturally results in higher H_c. Finally, the presence of Fe⁴⁺, introduced as a charge-compensation mechanism, may also contribute to the observed coercivity. Fe⁴⁺ in octahedral coordination can act as a Jahn–Teller center, producing local lattice distortions (Fe–O bond compression), modifying the magnetic interactions through partial double-exchange pathways, and hindering charge transport, all of which are consistent with the experimental observations.

Another relevant finding is the increase in the anisotropy constant K₁. According to the Kronmüller model [41], the coercivity of a ferromagnetic material depends on its microstructure and is proportional to K₁. This model therefore predicts a significant increase in coercivity, as observed in the results. Furthermore, the increase in densification and the reduction in microstrain, evidenced in the structural analysis, contribute to increasing the microstructural parameter that reflects the efficiency of the domain wall pinning mechanisms. On the other hand, octahedral distortions directly modify the magnetocrystalline anisotropy, which translates into the variation of H_c described by the Kronmüller equation. Taken together, these experimental results confirm that the coercivity in doped ferrites depends not only on the intrinsic anisotropy but also on the microstructure induced by the heat treatment. The fitted anisotropy parameters are consistent with the structural distortions observed in the Fe–O–Fe bond angles obtained from Rietveld refinement, providing independent validation of the model.

Since it has been verified that the modification of structural stresses has increased coercivity, pointing it has modified the magnetic behavior, the consideration of a magnetoelastic anisotropy, a stress-induced anisotropy model, can be explored. This internal stress can be modeled by Hooke’s Law

$$\sigma =E·{\mu }_s,$$

(7)

where σ is the internal stress, E is the Young’s modulus, and µ_s is the microstrain, so the magnetoelastic energy is then given by

$$E_{\mathrm{m}\mathrm{e}}={\displaystyle \frac{3}{2}}\lambda \sigma {\cos }^2\vartheta ,$$

(8)

with λ as the magnetostriction constant, which can be estimated using typical values for Fe, Ni, or their oxides, and ϑ as the angle between the magnetization and the stress axis.

From the literature, a Young’s modulus (E) value for material under use is obtained, which is approximately 213 ± 14 GPa. The magnetostriction constant (λ) for this material has not yet been measured; however, it has been reported that doping with Ni or Co tends to increase magnetostriction [42]. For example, for NiFe (70% Ni), it has a λ ₁₀₀ value of 22.7, exceeding that of Fe, which is approximately 20. The knowledge or estimation of these values opens the way to the application of a semi-empirical model that describes the total anisotropy energy for this type of system using the expression

$$E_{\mathrm{a}}=E_{\mathrm{m}\mathrm{a}}+E_{\mathrm{m}\mathrm{e}}.$$

(9)

3.5. Dielectric Properties

Figure 10 shows the effect of sintering temperature on the relative permittivity (ε_r) and dielectric losses (tanδ) of the material as a function of frequency. ε_r decreases with increasing frequency at all temperatures. This behavior is attributed to the space charge polarization mechanism, specifically the MW effect, due to the presence of charge accumulators such as grain defects, vacancies, or material porosity resulting from processing and charge compensation mechanisms. From Figure 10a, it is observed that permittivity values decrease with increasing sintering temperature, suggesting a reduction in interfacial polarization and defects, as well as greater material densification. Therefore, the density of each sample was calculated, and this assertion is supported by Figure 10, which aligns with the SEM data in Figure 4, allowing us to affirm that higher sintering temperatures lead to greater densification, reduced porosity, and consequently, lower MW polarization contribution, resulting in lower ε_r and tanδ values.

At relatively high frequencies, the ε_r and tanδ values reflect the intrinsic behavior of the material, falling within the reported range for bulk Ni²⁺ doped LFO [13]. The best condition is at 1473 K, along with the lowest dielectric losses. This condition is considered optimal because the relative permittivity remains within a stable and useful range without abrupt variations, while the dielectric losses reach their minimum. The combination of a sufficiently high εᵣ with the lowest tanδ value indicates that the material can efficiently store electrical energy with minimal dissipation. In other words, the dielectric response at 1473 K represents the best balance between energy storage capacity and energy efficiency, making this temperature the most favorable for practical applications. On the other hand, Ni²⁺ substitution has modified the crystal structure, generating distortion confirmed by structural analysis in Section 1, attributed to the mismatch in ionic radii, processing conditions, and the presence of Fe⁴⁺ cations as a result of charge compensation mechanisms [13]. Therefore, greater distortion can be associated with improved dielectric response, linked to the previously mentioned hypothesis of partial loss of centrosymmetry [34].

As shown in Figure 4, increasing sintering temperature leads to grain growth, which reduces grain boundaries and hinders charge movement, also contributing to the reduction of tanδ. In Figure 10b, as already mentioned, the decrease in dielectric losses is visible; however, the separation between curves is more pronounced than in ε_r, which may indicate that the loss factor is more sensitive to microstructural distortion induced by sintering. Regarding the trend, the sample sintered at 1173 K shows a decrease in both permittivity and tanδ compared to the 1073 K sample. These values increase at 1273 K and then decrease again with further temperature increase, a trend also observed in the XRD results, indicating that the behavior is largely dependent on changes in the crystal structure.

To understand the relaxation processes and the complex impedance response, the imaginary electric modulus (M″) and a Nyquist plot are used, shown in Figure 11a and Figure 11b, respectively. In M″, which indicates the energy dissipated by relaxation processes, no significant slope change is observed at 1073 K, indicating the absence of significant internal relaxation. For the other temperatures, a slope change is observed, associated with a characteristic relaxation frequency directly related to the relaxation time and dipole mobility dynamics in the material [10]. In the low-frequency region (below 10³ Hz), electrode effects tend to dominate, so no marked internal relaxation is observed, which is associated with the high interfacial permittivity in this range due to the MW effect.

In the regions where a change in behavior (slope) is observed, the change in relaxation processes can be seen as a consequence of the sintering temperature. This suggests that the temperature increase accelerates these processes due to a lower defect density, which is consistent with the structural analysis. Finally, at high frequencies, the dipoles can no longer follow the change in the field, so the response of the dense grains dominates, with less contribution from the interfaces. The behavior of M″ and the imaginary axis Z′ versus the real axis Z″ may also be influenced by the so-called “electron hopping”, a charge transport mechanism in materials lacking free electron movement and they jump between localized sites [43], due to the presence of mixed valence states between Fe³⁺/Fe⁴⁺ and Ni²⁺, favored by super and double exchange.

In Figure 11b, no characteristic semicircle is formed at 1073 K, suggesting the absence of relaxation processes, consistent with the M″ analysis. In contrast, for higher temperatures, well-defined semicircles are observed, indicating transport or relaxation processes associated with grain or grain boundary responses. The semicircles shift to the left, implying lower Z′, which is associated with reduced effective resistance to current flow, enriching the analysis of permittivity and losses. The reduction in semicircle size indicates lower Z″, which, being the component associated with energy storage, results in lower charge accumulation.

These behaviors are also associated with improved structural and magnetic alignment, which, being an improper multiferroic material with favored double exchange, directly impacts its dielectric properties. In conclusion, at low sintering temperatures, the behavior corresponds to a porous material with incomplete densification. As the sintering temperature increases, the microstructure becomes progressively consolidated, and the observed changes are due to bond-angle and bond-length distortions within the crystal lattice rather than a phase transition. This microstructural evolution improves dielectric behavior, as evidenced by reduced interfacial polarization, controlled dielectric relaxations, and changes in internal resistance, indicating a denser, less defective material, favorable for applications in capacitors, sensors, or dielectric devices.

3.6. Electrical Characterization

Figure 12 shows electrical conductivity as a function of frequency, with the highest value observed at 1073 K, reaching 44.45 S/m, a value in accordance with the literature [44], which decreases with increasing sintering temperature. The curves exhibit a smooth slope without abrupt transitions, suggesting a gradual charge transport mechanism associated with electronic hopping. The decrease in conductivity with temperature may seem counterintuitive; however, the induced structural transitions that alter bond angles reduce orbital overlap, hindering the hopping phenomenon. Additionally, the literature reports that at lower sintering temperatures, oxygen vacancies may be present, which, combined with mixed valence states, favor conductivity [12]. In contrast, higher temperatures reduce these defects, thereby decreasing conductivity.

It has also been noted that greater structural order is achieved, which, although it enhances densification and thus physical conductivity, may reduce electronic disorder, limiting defect-assisted transport pathways. Supporting this argument is the Nyquist diagram (Figure 11b), where smaller semicircles at higher temperatures reflect lower charge accumulation and consequently higher conductivity, consistent with the behavior at 1273 K.

The frequency-independent values indicate mixed conductivity in the material, that is, the simultaneous presence of electronic and ionic conductivity. The contributions from hopping and diffused O²⁻ ions via vacancies can be related to Jonscher’s universal power law, which connects DC conductivity with frequency-dependent conductivity in dielectric materials, semiconductors, and functional oxides like this system. This law describes the variation in conductivity with frequency when charge transport is governed by mechanisms such as electronic hopping. This independent behavior is also associated with intrinsic charge carriers such as polarons.

3.7. Magnetodielectric Coupling

Lanthanum ferrite is a multiferroic material at room temperature that has undergone a change in magnetic order due to cationic substitution with Ni²⁺ [13], which has opened the possibility of coupling between magnetic and dielectric properties. Ferromagnetic oxides typically possess centrosymmetry, a property that hinders electric polarization. The compound LaFe_0.7Ni_0.3O₃, sintered at different temperatures, has enabled exploration of this coupling due to evidence of structural distortion that modifies its centrosymmetry without breaking the inversion symmetry. On the other hand, ferroelectric materials often contain transition metal ions that hinder magnetism, an issue resolved in this material. Magnetodielectric (MD) coupling is explored through the variation in relative permittivity in response to a static magnetic field applied within a range of ±18 kOe, allowing quantification of magnetocapacitance (MC) using the following expression [45]:

$$\mathrm{M}\mathrm{C}\left(\%\right)={\displaystyle \frac{C\left(B\right)-C\left(0\right)}{C\left(0\right)}}\times 100,$$

(10)

where C(B) is the capacitance under magnetic field B and C(0) is the capacitance in the absence of a magnetic field. The magnetodielectric coupling analysis is shown in Figure 13 (50 Hz to 10,000 Hz).

In general, MD coupling is observed in all samples, with a decrease in permittivity as frequency increases, consistent with Figure 10. This behavior is indicative of a direct MD effect, where electric order is modified by the magnetic field [46].

At 1073 K, where the microstructure is porous, a high dielectric response (high ε_r) and dispersive behavior are observed. MD coupling is favored by structural disorder, vacancies, and mixed valence states. This sample, dominated by interfacial effects, likely exhibits spin–charge coupling. At 1173 K, the temperature at which the first microstructural modification occurs, lower interfacial polarization and more controlled losses are observed. Thus, MD coupling may be related to defects and hopping mechanisms associated with densification while maintaining magnetic sensitivity. At 1273 K, with a more optimized microstructure, reduced losses, faster relaxation, and higher conductivity, the lowest MD coupling values are recorded. For the 1373 K system, with greater dielectric stability and fewer charge transport defects, the coupling appears more stable but less sensitive to the magnetic field.

Finally, at 1473 K, discussed above as potentially over-sintered, the result is a system that is more stable both magnetically and dielectrically, though not necessarily with enhanced magnetodielectric functionality.

In Figure 13f, magnetocapacitance values are shown to be negative across all temperatures, indicating that the magnetic field reduces the permittivity of the studied system. These negative values may be associated with the suppression of interfacial polarization, which is present in all samples. The magnetic field may reduce charge accumulation at grain boundaries, for example. If the magnetic field aligns spins, it may restrict hopping between mixed valence states, thereby reducing polarization. Additionally, in densified systems, the magnetic field may stabilize the structure, decreasing the dielectric response associated with defects. It can be said that in a more ordered system, the magnetic field suppresses polarization mechanisms. Negative MC values may be related to coercivity, as they follow a similar trend with sintering temperature. Materials with high coercivity have more stable magnetic domains that do not easily respond to external fields, influencing how the magnetic field affects electric polarization.

To rule out that the observed magnetodielectric coupling is extrinsic, magnetoresistance (MR) data were analyzed in parallel with magnetocapacitance (MC). Figure 14 shows the percentage of MR, calculated using the following expression:

$$\mathrm{M}\mathrm{R}\left(\%\right)={\displaystyle \frac{R\left(B\right)-R\left(0\right)}{R \left(0\right)}}\times 100$$

(11)

where R is the resistance, as a function of the applied magnetic field. The MR measurements exhibit only minor variations (below 5%) compared to the significant changes observed in MC (about −90%).

The absence of a direct correlation between both curves suggests that the observed MC is not dominated by resistive artefacts.

This analysis is consistent with the report by Gustau Catalan [47], who noted that MC may originate from MR via the MW mechanism, making it necessary to evaluate MC together with the resistive response of the system. The behavior of MC measured at 10,000 Hz further supports that the observed effect is not due to the MW mechanism, but rather reflects a magnetodielectric coupling of the material.

According to investigation [48], at a frequency of 10,000 Hz, the contribution of MW–Sillars (MWS) polarization is already strongly reduced. This is supported by Figure 10, where the pronounced drop in the frequency range of 10²–10³ Hz stabilizes at 10⁴ Hz with much lower values. A similar behavior is observed in the loss tangent (tanδ), whose characteristic peak appears in the intermediate region and decreases rapidly, so that at 10⁴ Hz the MWS contribution is essentially suppressed. At the frequency investigated for MR, the MWeffect is attenuated and suppressed, leaving the spectra dominated by dipolar and electronic polarization mechanisms associated with the intrinsic behavior of the system.

4. Conclusions

The effect of processing temperature on the LaFe_0.7Ni_0.3O₃ compound reveals a complex structural transition, where the bandwidth and tilting of Fe–O–Fe bonds modulate the intensity of magnetic interactions. At lower temperatures, the almost linear structure favors antiferromagnetic coupling, while at higher temperatures, angular distortion induces spin canting and stabilizes more defined magnetic domains. This structural reorganization, along with grain growth and microstrain relaxation, helps explain the progressive increase in coercivity and the strengthening of multidomain behavior. Structural relaxation, reflected in longer Fe–O bond lengths, promotes the formation of magnetic domains, improves crystal quality, enhances magnetic response, and increases resistance to demagnetization, supporting the concept of magnetoelasticity associated with these internal stresses. Greater electronic coupling indicates orbital overlap, resulting in a coherent magnetic response in less distorted structures within the optimal sintering range of 1173 to 1273 K. A decrease in dielectric losses was observed with increasing sintering temperature, indicating improved material efficiency. Although ε_r decreases, stability improves due to lower energy dissipation. The electric modulus suggests the presence of controlled dielectric relaxation, and the Nyquist diagram semicircles reflect lower internal resistance and better charge transport. Furthermore, the simultaneous evolution of magnetic and dielectric properties with sintering temperature provides evidence of magnetodielectric coupling in LaFe_0.7Ni_0.3O₃. The negative magnetocapacitance values observed across all samples indicate that the applied magnetic field suppresses interfacial polarization and charge accumulation at grain boundaries, consistent with spin–charge interactions and structural stabilization. The lack of correlation between magnetoresistance and magnetocapacitance further supports the intrinsic nature of this coupling. These findings highlight the multifunctional character of the LaFe_0.7Ni_0.3O₃ compound, where structural distortions, magnetic anisotropy, and dielectric relaxation converge to enable magnetodielectric functionality. The progressive densification of the material, accompanied by reduced defect density and enhanced coercivity, reinforces its coherent magnetic and dielectric response, favoring potential applications in advanced capacitive and sensing devices, as well as high-frequency dielectric components. A model based on magnetocrystalline anisotropy is proposed to understand the evolution of coercivity mechanisms, paving the way for the study of ferrites as a sustainable alternative to rare-earth-based hard magnetic materials.