The structural and dielectric properties of lanthanum substituted strontium based spinel ferrites nano-materials for high frequency device applications

Lanthanum substitution on spinel Nano particles has a significant repercussion on the several features of materials. Sol–gel (auto-ignition) allowed for the substitution of the rare earth lanthanum ion (La3+) for spinel ferrite with composition SrLaxFe2-xO4 (where 0.00 x 0.10). As a result of the base’s presence, the systems structural, magnetic, electrical, and magnetic properties are all modified. X-ray diffraction (XRD) assessed the structural characteristics. Using X-ray diffraction, we could determine that each sample had cubic structures. Based on electrical investigation, it can be concluded that the synthesized ferrites are semiconductors, as their direct current (dc) resistivity reduces with increasing temperature. The Curie temperature (Tc ) was lowered from 532 K to 462 K with the substitution of lanthanum ions. The values for activation energy increased from 0.24 eV to 0.38 eV when lanthanum is used as a substitute. The values of magnetization decreased from 50 eV to 27 eV as lanthanum content rose. We could see the gentleness of the material in M-H loops. All these parameters show that the prepared sample is a potential material for use in high-frequency applications.


Introduction
Ferrite nanoparticles are extremely small fragments of the metal ferrite.When iron oxides (Fe 2 O 3 or Fe 3 O 4 ) combine with a number of different metal oxides, such as Zn, Ni, Mn, or Co, ferrites are formed [1].Ferrites Nano particles were synthesized for lengthy period due to the numerous uses they seek in the technical and industrial domains.Their applicability span from spintronics to drug-shifting to microwave-absorbing substances to magnetically detecting disorders [2].Among some of the technologies developed in the twenty-first century, nanotechnology has surfaced as the most compelling.Numerous fields in science, industry, biodiversity, and technology are currently utilizing major advancements in the science and engineering of nanoscale materials.The main fields those benefitted from the great scientific advance made in nano-science are biology, agriculture, medicine, pharmacy, and food industry [3].Due to their magnetic and optical properties, they can be used for administration of medication, diagnosing, MRI, therapy, unobtrusive tumour imaging, and premature ailment detection [4].Spinel cubic ferrites (SCF) are suitable substances for electronic applications [5] and magnetic fluids [6] because of their distinct and unusual characteristics.Spinel-shaped ferrite Spintronics [7], optoelectronics [8], magneto electronics [9], electrolytic science and technology [10], and biological science are only a few disciplines that already have rendered AB 2 O 4 the focus of extensive study over a long time period.Various modulation processes and microwave applications took usage of the spinel ferrites [11].
Among the most significant MNPs are ferrites, which seem to be the metal oxides along spinel ferrites [12].Spinel cubic ferrites exhibit very compelling magnetic properties, including excellent electrical resistivity, mechanical hardness, and chemical stability.Here, oxygen uses the FCC closed packing mode, and the tetrahedral (site-A) pore spaces and octahedral (site-B) crystals filled with the metal ions M 2+ and Fe 3+ , respectively [13][14][15][16][17]. Cubic spinel holds 32 oxygen atoms in its unit cell, 8 of which reside in tetrahedral (A) positions and 16 in octahedral (B) locations [18].The basic tetrahedral unit cell observed in spinel is composed of two AB 2 O 4 molecular units.These cells combine the four essential components of a spinel cubic unit cell [19,20].In recent times, intensive research has been carried out on rare earth-substituted ferrites to improve these materials' magnetic properties, permittivity, and permeability [21].
In the crystal lattice structure of strontium-based spinel ferrites, which are ferrite materials, strontium (Sr) ions are present.Strontium spinel ferrites have various qualities that make them well-suited for diverse of applications.Because the iron ions' magnetic spins are not oriented, they behave in a manner called "ferromagnetic", which makes a net magnetic moment.These ferrites are therefore compatible with magnets and other magnetic objects [21].SrFe 2 O 4 can be classified as a mixed spinel ferrite.The distribution of strontium and iron cations in the spinel lattice of SrFe 2 O 4 is not consistent with either a normal or inverse spinel structure.SrFe 2 O 4 is a mixed spinel ferrite, with supporting data from studies using X-ray diffraction, Mössbauer spectroscopy, and other characterization techniques.The mixed spinel structure of SrFe 2 O 4 has unique magnetic and electrical properties, which have been investigated in several studies [22,23].
Sr 1−x La x Fe 2−y Zn y O 4 nanoferrites containing spinel and zinc co-doping have been effectively manufactured.Studies of these materials' frequency-dependent properties demonstrate a steady reduction in their respective values at lower frequencies, followed by a plateau-like behaviour at higher frequencies.The incorporation of La-Zn dopants leads in increased DC resistivity ranging from 1.08 × 10 9 Ω-cm to 1.67 × 10 9 Ω-cm, as well as lower dielectric loss values.These findings suggest that these doped strontium nanoferrites may be suitable for high-frequency device applications in material science [24].
The sol-gel approach was used to prepare CoSrFe 2 O 4 , and X-ray diffraction (XRD) analysis showed that all of the samples had cubic spinel phases.Furthermore, the dielectric characteristics within the frequency range of 3 MHz to 1 GHz were investigated.The obtained results exhibited reduced dielectric loss and improved direct current resistance, indicating the potential suitability of these samples for high-frequency applications in materials science [25].The comparison of different materials is reported in 1.
Strontium ferrites (SrFe 2 O 4 ) stand out from other ferrites due to their mesoporous architecture, huge surface area, and possibility for oxygen vacancies due to the size of the Sr 2+ ions [13,14].The synthesis of SrFe 2 O 4 was selected as the best option because of its dependability, economics, output consistency, and smallest crystalline size.Due to their higher DC resistivity and lower dissipation factor, these nanocrystals are well-suited for possibly high-powered devices.Investigations are underway to improve the permittivity, penetrability, and saturation magnetization of these rare earth-substituted ferrites (SrFe 2 O 4 ).Using strontium ferrites enhances the material's already impressively high dc resistivity and low dielectric loss, enabling its use in high frequency applications [15].The rare earth ion (RE +3 ) substitution in spinel nanoparticles is widely used to improve their electric and magnetic properties needed for microwave absorption applications.Rare earth metal ions, such as lanthanum (La 3+ ), transform the structure and texture [26].The substitution of La 3+ , whose larger ionic radii effect a change in structural features of ferrites via the size of the spinel lattice, is the most fundamental need for adjusting the optimum physical and structural properties of ferrites [27].

Synthesis of materials
In a sol-gel auto-ignition process, nitrate-containing ferrites were used to prepare pure strontium nanoparticles, to which lanthanum was added.The sample formulation is SrLa x Fe 2−x O 4 (where; 0.00-0.10).This synthesis method is widely used to fabricate nanoparticles because it has several benefits, including homogenous mixing of the constituent parts, quick processing, and being eco-friendly.Sr•(NO 3 ) 2 , Fe(NO 3 ) 3 •9H 2 O, La(NO 3 ) 3 •6H 2 O, and Citric acid (C 6 H 8 O 7 ), all of which are incredibly pure chemicals (Sigma Aldrich, 99.9%), were used to generate the strontium nanoferrite system.Deionized water was diluted to its smallest possible volume in individual beakers to provide clear solutions for each sample.These solutions were mixed and then heated on a magnetic stirrer for two hours at 100°C to evaporate the water content.Only a few drops of a one-molar ammonia solution (NH 3 ) were used to get the pH level up to 7. After 8 h of agitation, the substance had gelled altogether.The "gel" spontaneously ignites with the blazing flints during continuous heating; the resulting creamy product was reduced to powder form and sintered in a hot oven at 700°C for 5 h to ensure complete crystallization (Table 1).
The following methods were used to figure out what kinds of nano-ferrites were prepare: Instrument for X-ray powder diffraction (XRD) with a 2θ ranges from 15°to 70°.Saturation M s , remanence M r , and coercivity H c were all monitored using a VSM at ambient temperature.The Wayer-Kerr 6500B impedance analyser was used to determine the dielectric properties of the samples at room temperature.Pellets of synthesized nanoferrites were made in a hydraulic press at 3 kPa for 7 min were used in dielectric tests.

Structural measurements
Figure 1 depicts the crystallinity and phase identification of synthesized La-substituted strontium ferrite (SrLa x Fe 2−x O 4 ) by XRD method.The peaks obtained were matched JCPDS#49-0266 with space group (Fd3m).The patterns contain no extra diffraction lines, which is consistent with the cubic spinel shape and lends credence to the development of single-phase spinel.The existence of sharp, narrow peaks was conclusive evidence that appropriate crystallization had occurred in each and every one of the ferrites.When La 3+ ions are introduced into strontium ferrites in place of Fe 3+ ions, the resulting spinel has a more extensive crystal structure.Because iron (Fe 3+ ) ions and rare earth ions have different radii, substituting lanthanum (La 3+ ) in spinel ferrites can change how the crystals are built.The structural distortion caused by doping increases the lattice size [26].The very crystalline structure of the material is shown in the prominent diffraction peak at (311).The size of the individual crystallites in Scherrer's algorithm is used to calculate the composites.This finding suggests how varying the La-concentration affects the typical crystallite size of the as-prepared ferrite samples graphically in Figure 2. Consistent with earlier research, this diminution in crystallite size with escalating Lacontent is interesting.Bonding of La 3+ -O 2− requires more work than Fe 3+ -O 2− because the binding energy of La 3+ -O 2− is higher.As a result, La 3+ ions amass close to the grain boundaries, exerting pressure and preventing the grains from expanding.Additionally, the larger size of the La 3+ ions may produce internal tension, which further prevents grain formation and leads to a smaller grain size of La 3+ -substituted nanoparticles [27].
The ionic radius difference between the substituted rare earth ion and the Fe 3+ ion influences the unit cell size and lattice characteristics of this substitution.The La 3+ ion is bigger than the Fe 3+ ion in terms of radius, with a value of (1.061 Å) compared to (0.67 Å), respectively.La 3+ ions often occupy particular positions within the spinel lattice as a result of interactions between the crystal field, ionic size, and valence state.The symmetry and magnetic interactions of the crystal can be altered by the addition of La 3+ ions and the subsequent reorganization of cations in the spinel lattice.It should be highlighted that the structural stability of spinel ferrites can be compromised by the addition of La 3+ ions.It follows that, given adequate space in the B-sublattice, the La 3+ ion in ferrites may reveal activity analogous to that of other ions in the lanthanide elements, such as (Dy, Tb, Pr, Yb, Er, and Ce).If La 3+ ions were put in place of Fe 3+ ions at the octahedral (B-sites), the unit cell would get bigger, which would make the lattice constants bigger.The dielectric properties and other properties of SrLa x Fe 2−x O 4 can be influenced by changes in the lattice constant.This is because the lattice constant affects the crystal structure of the material, which in turn affects its electronic and magnetic properties.Research has shown that by changing the La content in SrLa x Fe 2−x O 4 , the lattice constant can be tuned, leading to changes in the dielectric properties of the material.For example, a study by N. Alzate-Cardona et al. [22] found that the dielectric constant of SrLa x Fe 2−x O 4 decreased with increasing La content, which was attributed to an increase in the lattice constant.Similarly, other studies have shown that changes in the lattice constant can affect the magnetic properties of SrLa x Fe 2−x O 4 , such as the magnetic moment and magnetic anisotropy.For example, a study by S. M. Lee et al. [28] found that the magnetic moment and magnetic anisotropy of SrLa x Fe 2−x O 4 decreased with increasing lattice constant due to spin-orbit coupling.Therefore, changes in the lattice constant can have significant effects on the dielectric and magnetic properties of SrLa x Fe 2−x O 4 , making it an important parameter to consider when tuning the properties of this material for high frequency applications.
In Figure 2, the relationship between the size of the crystal and lattice characteristics of the prepared Nano ferrites.Because of the larger size of the former, crystalline anisotropy can be formed by substituting La 3+ for Fe 3+ .As a result of the crystalline anisotropy, the addition of La 3+ causes an increase in strain in the crystal's volume.Because of the interaction involving crystal anisotropy and volume strain, the present structure maintains equilibrium.As a result, as S lowers, so does the tension as the La 3+ content increases.
Figure 3(a-c) depicts a scanning electron micrograph (SEM) of ferrite and ferrite/polymer nanocomposites.This demonstrates that ferrite particles in ferrite/polymer nanocomposites are not uniformly distributed, with some areas having more ferrite particles than others.The physical and mechanical characteristics of the nanocomposites as a whole may alter as a result of this lack of consistency.Polymer is characterized by the presence of small globules and holes, which may aid in polaron diffusion.It has been shown that the inclusion of the nanofiller has no effect on the particles' morphological qualities.Due to weak interparticle interactions, polymer's sphere form covers nano oxide particles, resulting in several agglomerated particles and some vacancies [3].Aggregation of particles may have been caused by particle interactions.In addition, homogenous ferrite nanoparticles enclosed in polymer were found.This demonstrates the polymers' capacity to enclose ferrite particles efficiently.According to the micrograph, the average size ranged between 2 and 6 μm.

Electrical analysis
At cellar temperature, a Two-Probe approach was deployed to investigate the DC-resistivity of (SrLaxFe2-xO4) nanoparticles.The values of the DC-resistivity for different compositions of La 3+ are depicted in Figure 4, which is drawn at cellar temperature.The sample (SrLa x Fe 2−x O 4 ) has a higher resistivity value after adding La 3+ ions, which causes this effect.The value of the DC resistivity grows in a linear fashion with growing lanthanum concentrations.This is due to the fact that La 3+ ions are concentrated at the grain boundaries, which causes greater resistivity.The conduction mechanism among spinel ferrite nanoparticles is referred to as "electron hopping", which refers to the process in which electrons hop among ferrous and ferric ions.A rise in La 3+ content rises resistivity due to ionic growth affecting electron and hole mobility across octahedral and tetrahedral sites [28].
Arrhenius plots illustrate activation energies in Figure 5.Both activating force resistance and La-concentration acted in a same fashion.The activation energy of spinel ferrites rises when La 3+ is substituted for Fe 3+ , which can introduce disorder and distortion in the lanthanum-doped Sr-ferrites.This disorder and distortion hinder the path of the charged particles and increase activation energy (E a ) for their mobility.The presence of the La atoms disturbs the arrangement of atoms in the lattice, hindering the charge carriers from  leaping between the lattice sites and increasing the activation energy.This changes the local environment and electronic structure of lanthanum compared to Fe 3+ because La 3+ is bigger in size.This also changes the activation energy for different processes, such as oxygen diffusion or charge carrier hopping, by changing the electronic interactions and landscape energy within the crystal structure.The magnetic order and interactions between the particles affect activation energies, including the movement of a magnetic domain boundary or the rebounding of a spin-dependent carrier.La 3+ increases the resistivity of spinel, which may prevent the production of spinel because they are larger than the host cations and by increasing the number of carriers, the material becomes stronger.
A correlation between sample resistivity and activation energy was discovered, with higher resistivity corresponding to greater activation energy.Since the resistivity increased with increasing La-content, it was also observed that activation energies increased.The stronger blockage of the conduction process across ferrous and ferric ions was confirmed by higher activation energy values with increased La-concentration at B-sites.Activation energies were also greater because electron shifting among Fe 2+ and Fe 3+ ions was reduced [29].
DC resistivity values for the completely built system are shown in Figure 5 as a variation of temperature.
The Arrhenius equation makes claims about each sample's semiconducting characteristics.The plots demonstrate that the resistivity reduces with increasing temperature, demonstrating that the samples act as semiconductors.All SrLa x Fe 2−x O 4 samples had a sharper steepness of (log vs 1/T), which can be ascribed to the charge carriers mobility being thermally activated rather than their formation.All of the specimens show signs of being degenerate type semi-conductors.A paramagnetic state is one that is above the Curie temperature, whereas a ferromagnetic state is one that is below it.The computed activation energies for the ferromagnetic and paramagnetic areas for each composition are shown in Figure 5. Figure 6 depicts the modified Arrhenius plot that is used to determine the Curie temperature (T c ).For spinel ferrites, these values are rather usual.The opposite is also true: a high La 3+ content is associated with a low T c .A shift in the (Fe 3+ -o-Fe 3+ ) and (Fe 3+ -Fe 2+ ) angles underlies the linear decrease in T c with La-concentration.Thus, the mutual attraction between magnetic moments is less.The R-E substituted samples were found to have a lower T c than the control samples.La-Fe interactions at octahedral sites are weaker than Fe-Fe interactions, which likely accounts for the lower T c .During this process, the valence of the iron atom will change from one with a high spin to one with a low spin.The change from a collinear configuration to a non-collinear configuration is the causes of a downturn in T c that is a consequence of these adjustments.Aside from this, it is likely that the reduce in T c is due to weaker La-Fe contacts at octahedral sites compared to interactions among Fe-Fe.These interactions could be responsible for the decrease [30][31][32].

VSM analysis
Figure 7 depicts the hysteresis cycles of SrLa x Fe 2−x O 4 materials evaluated at room temperature in order to ascertain magnetic properties, involving saturation (M s ), remanence (M r ), and coercivity (H c ).All of the material' magnetization values were recorded while still being exposed to an applied field between −2 KOe to +2 KOe.The samples in each plot have a noticeable S-shaped distribution.The M-H curves for un-substituted SrFe 2 O 4 show the origin of the material's ferrimagnetism, which arises from the alignment of magnetic moments of the iron (Fe) ions occupying the octahedral sites within the spinel structure.Specifically, the Fe ions have a net magnetic moment arises from the partially filled d-orbitals, and this moment aligns with an applied magnetic field to produce a ferrimagnetic response.When La 3+ is added as a dopant to SrFe 2 O 4 , it can impact the magnetic properties of the material.One possible explanation for the lower magnetic properties observed with the addition of La 3+ is that the dopant ions may disrupt the ordered arrangement of Fe ions in the spinel structure, leading to a reduction in the net magnetic moment of the material.Additionally, the larger size of the La 3+ ions compared to the Sr ions may introduce lattice strain and disorder, further reducing the magnetic properties.Another possible explanation is that the La 3+ ions may preferentially occupy the tetrahedral sites in the spinel structure, which are typically occupied by non-magnetic ions in un-substituted SrFe 2 O 4 .This could lead to a decrease in the quantity of Fe ions occupying tetrahedral sites, which would reduce the total magnetic moment of the material.Overall, the exact mechanism by which La 3+ addition lowers the magnetic properties of SrFe 2 O 4 may rely on the specific experimental conditions and details of the doping process.Further research would be needed to fully understand the influence of La 3+ on the magnetic attributes of SrFe 2 O 4 .Several magnetic properties values could be determined by employing the M-H loop as a reference.It was found that the M-H loops are not linear, exhibiting reversible behaviour, coercive indicators, and low retentivity.Lanthanum atoms changed the magnetic properties of barium ferrite.As the lanthanum content increased, the magnetic properties of the output diminished [33].When injected, a magnetic field causes magnetization to rise and eventually saturate.All of the samples had low H c values (a few hundred oersted), showing the relative softness of these spinel ferrites.
Lanthanum substitution occurs when lanthanum ions replace some metal ions in spinel ferrite.This substitution can significantly impact the coercive field and remanent magnetization of the material.Lanthanum plays a vital role in influencing the magnetic characteristics of spinel ferrites, such as magnetic anisotropy and exchange interactions.Adding lanthanum can change the coercive field by causing strain and breaking up magnetic connections in the crystal structure, which is more than the metal ions could do on their own.As a result of these disturbances, the coercive field and magnetic anisotropy increase, making demagnetization more challenging.Interactions within the crystal structure and variations in the material's magnetic anisotropy are the causes of these changes [34,35].
Furthermore, as La 3+ concentration increases, the samples' M s , M r , and H c values appear to decrease, as seen in Figure 8.As a result, the steady decline in all magnetic parameters can be attributed to diminishing AB-exchange interactions.Lower magnetic ordering temperatures distinguish rare earth ions, and magnetic moments are often derived from localized 4f-electrons.Because the magnetic dipole orientation of La 3+ ions is disturbed at room temperature, they may have no effect on the magnetization of substituted ferrite [36].
As the concentration of lanthanum increases, the exchange connections weaken, and the saturation magnetization values decreases.Because there are more surface spins on smaller particles, the net magnetic moment reduced.Because of a decline in the magnetic moments of unit cells and a weakening of the sub-lattice interaction, saturation magnetization (M s ) and coercivity (H c ) decreases with increasing lanthanum concentration [37].One likely interpretation for the drop in saturation magnetization is the movement of ferric ions (B-to A-sites).
Lanthanum substitution in spinel ferrites introduces several factors that can impact the saturation magnetization of the material.In this process, lanthanum ions replace some metal ions in the ferrite structure.The quantity of lanthanum introduced, the location of lanthanum ions within the crystal structure, and the presence of other components in the material all impact saturation magnetization.These factors collectively determine the resulting saturation magnetization of the lanthanum-substituted spinel ferrites [38].It has also been demonstrated that the saturation magnetization of formed spinel ferrites tends to decrease as the grain size of the material decreases, possibly as a result of an increase in the surface disorder of magnetic moments.Consequently, the resulting samples have less net magnetization [39].

Dielectric properties
How frequency influences the dielectric constant see Figure 9 for a breakdown of temperature bands.It has been noted that the dielectric constant is highly significant for both low and high frequencies; hence, for each set of frequency values, the dielectric constant decreased with the frequency.According to Koop's theory, ferrites produce an inhomogeneous dielectric structure formed by high conductivity grains, and low conductivity grain borders which would explain the observed dielectric behaviour.Charges in this configuration take some time to spin in order to face the direction of a variable electric field is being applied.Charge carriers in space will begin to move as the field's frequency rises, and they will continue doing so until the field reverses and stops severely polarizing space charge.As a result, as the frequency increases, the dielectric constant gradually drops until it almost becomes constant [40,41].The localized states can be introduced within spinel ferrites by incorporating La +3 ions, and these localized states act as a trap for the charge carriers.The carrier travels less as a result and experiences greater resistance; their mobility decreases, and La 3+ ions may affect the formation of oxygen gaps in spinel structures.The oxygen vacancies act as charge carriers for themselves, and the material's dielectric constant and dielectric properties may be affected.The presence of La +3 enhances the ionic polarization in the spinel, and this polarization occurs due to the movement of charged species, and the La +3 ions creates dipoles in the material.The ionic polarization improves the dielectric characteristics by increasing the potential of the material to store and unleash energy, and lanthanum ions can influence the dielectric relaxation processing of the spinel [42].
The local electric field and the rate of material softening might both alter simply by having La 3+ ions present and by increasing the dielectric constant and lowering the dielectric loss, which improves the material's dielectric properties.With the addition of La 3+ , defects are created, charge carriers are dispersed, oxygen vacancy formation occurs, and ionic polarization increases.Spinels that have been treated with La 3+ are suitable for electronics, sensors, and capacitors due to their enhanced conductivity and insulating properties.The shape, size, and overall microscopic structure at the nanolevel refer to the morphology of the ferrites, and when La is incorporated into the material, it changes the sample's morphology due to different factors and can influence the nucleation process and growth of the crystals.La 3+ ions may rapidly alter the environment, which may affect the rate of crystal growth and the number of new nuclei that form, and these ions can induce phase transformation in the material, which results in the new phase formation between host materials and La 3+ ions.The phase formation and chemical reactions change the morphology of the material, such as the formation of a new crystal or the rearrangement of the atoms [43].
The interface properties of the material and surface energy are influenced by incorporating La atoms, which can affect the stability and growth of the surface of the sample.The facets, grain boundaries, and formation of the different surface textures can lead to changes in the morphology of the material.By changing the lattice structure, the size disparity between La ions and the host material may result in internal stress.The formation and arrangement of crystal structures may change under various pressures and stresses, which can have an impact on the material's configuration.The chemical composition and development of a substance are influenced by the distribution of La 3+ ions.This can lead to the segregation of dopant ions at different locations within the material.The changes in a material's morphology are often caused by La doping's effects on crystal growth, nucleation, chemical reactions, phase formation, surface energy, contact effects, strain and stress, and dopant segregation.Form, size, and microor nanoscale dimensions all these factors collectively contribute to the structure of the material and morphological changes at the nanoscopic level [44].
Increasing lanthanum concentration decreases the dielectric constant.Electronic interaction between Fe 2+ and Fe 3+ at octahedral sites governs ferrites' electric polarization [45].Furthermore, La 3+ ions located the octahedral sites, which decrease the content of iron ions because of their large ionic radius.Therefore, polarization is diminished, and the dielectric constant is lowered, because no electrons are being exchanged ions (Fe 2+ and Fe 3+ ) ions.The drop in dielectric constant (ε ) may be attributed to impurity phase and a shift in cationic distribution [46].The distortion of the lattice is due to the substitution dopant having a substantially larger size than the iron.Low lattice distortion due to cubic symmetry allows for fine tuning of the dielectric properties very near to the point.
Figure 10 shows how the change in dielectric loss depends on the frequency.In Maxwell-Wagner effect, the values of dielectric loss get closer to infinity as the frequency gets closer to zero.In a Debye system, the ε values get closer to zero as the frequency gets closer to zero.Plots in Figure 10 showed that there are Maxwell-Wagner-style relaxations.Dielectric loss significantly declines with frequency, with the decline being more pronounced at low frequencies.As the frequency rises, the dielectric loss remains essentially constant across all temperatures.It is frequency independent.Low frequencies experience a slight reduction in dielectric loss as temperature rises [47,48].

Real and imaginary impedance
The real component (Z ) and imaginary component (Z ) of impedance vary with alterations in frequency.At low frequencies, where resistance is most important, the Z of impedance is bigger than the Z .The research was conducted at room temperature, with frequencies ranging from 1 MHz to 3 GHz.The Z and Z are calculated as; Evidence from the past implies that the grain and grain boundary influences, which is induced by external factors such as temperature and frequency, is the primary source of conduction in ceramic materials.However, the bouncing of Fe 2+ and Fe 3+ ions is unproductive in ferrites nanoparticles, and grain boundaries that are poor electrical conductors perform better at low frequencies.If you increase the frequency, the Fe 2+ and Fe 3+ ions in the electrically active grains will travel more quickly as they leap from one location to another.The main goal is to determine if conduction happens inside the grain or at its edges.Using impedance, you can find places on a ferrite that are primarily reactive or resistant.The component that resists is real, but the part that reacts is imaginary [49].
As the frequency went up, the results showed that both the Z and Z of the went down (as Figure 11(a,b)).The fact that the Z value went down shows that the sample was conducting.It was found that the impedance of the pure sample was the highest.The highest impedance was found in a pure sample.Conduction can come from either the grains themselves or the spaces between them.This relationship is visually represented on a complex impedance graph, depicting the interplay between the Z and Z .Either a Cole-Cole plot or a Nyquist plot (Figure 11(c)) can be used to show how the impedance of a system changes with frequency.On the X and Y-axes, it shows the Z and Z of the impedance.From this graph, you can figure out how to make a half circle.Since ionic polarization doesn't happen in ferrite materials, even simple shapes like single semicircles can exist.The resistance of grains is depicted by the semicircle in the high-frequency zone.The height of the circular arcs was used to figure out the capacitance, and the points where the arcs met the X-axis were used to figure out the resistance.Lower in frequency, the cole-cole plot only shows a single semicircular arc for all samples.A curve like this shows that conduction is happening at the edges of the grains.It showed that grain boundaries are a big reason why RE-substituted La-ferrites conduct electricity.This shows that in modern RE-substituted strontium spinel ferrites, conduction is most likely to happen at the grain boundary [50].
The grains and grain boundaries considerably impact the characteristics of SrLa x Fe 2−x O 4 spinel ferrites.These materials' conductivity, magnetism, and dielectric characteristics are all directly influenced by the grains that make up their crystalline structure.The grain boundaries, which are the interfaces where these grains converge, also significantly impact these properties.Grain boundaries can change a material's conductivity and other electrical properties by restricting the movement of electrons and other charge carriers.Additionally, they can affect the material's thermal and mechanical properties by increasing stress concentration or acting as locations where faults or phase transitions can occur.Understanding the complex principles behind the behaviour of grains and grain boundaries is essential for producing and improving SrLa x Fe 2−x O 4 spinel ferrites for various applications.
The size, shape, and arrangement of the A and B ions inside the lattice, as well as the grain size and structure of the grain borders, all significantly impact properties of spinel ferrites.Each discrete crystalline grain that makes up the substance substantially affects the material's characteristics.These grains' size and shape greatly influence the material's mechanical, thermal, magnetic, and electrical properties.In contrast to materials with smaller grains, materials with larger grains typically show improved saturation magnetization and fewer magnetic losses.Smaller grains, on the other hand, usually exhibit increased magnetic anisotropy and coercivity [51,52].The shape of granules can impact their magnetic properties, as longer or flatter granules often exhibit asymmetrical magnetic behaviour.Grain boundaries are present at the points where spinel ferrite crystals intersect.These grain boundaries possess considerable potential to modify the characteristics of the material [53].When electrons and other charge carriers encounter obstacles at grain boundaries, the resistance increases, and conductivity decreases [50].Apart from affecting the material's thermal and mechanical properties, grain boundaries can amplify magnetostriction and magnetic anisotropy.The composition, crystallographic orientation, and structure of grain boundaries can alter the properties of a material [52].

Conclusion
Sol-gel (auto-ignition) approach was utilized successfully prepared lanthanum substituted Strontium spinel ferrites with formulations SrLa x Fe 2−x O 4 (where; 0.00 ≤ x ≤ 0.10).X-ray diffraction analysis was used to investigate the formation of the spinel structure in nanoferrites.As the lanthanum ion (La 3+ ) concentration increased, the lattice parameters (8.37 Å to 8.50 Å) increased and crystallite size (61.5 nm to 28 nm) decreased.Resistance falls with temperature, confirming the finding from electrical testing, while the activation energy (0.24 eV to 0.38 eV) and resistivity both increase as the substitution of La 3+ , suggesting that the produced ferrites are semiconductors.Samples are ferromagnetic at low temperatures, but they turn paramagnetic at high temperatures (Curie temperature, T c ). Due to the element's saturation, M s (51 emu/g to 24 emu/g), remanence magnetization, M r (13 emu/g to 2 emu/g), and coercivity, H c (322 Oe to 100 Oe) all decreased as lanthanum concentration increased.The resulting loops are all distinguished by a soft ferrite structure.Higher frequencies have been found to have smaller real and imagined dielectric properties.As the concentration of La 3+ ions rises, dielectric constant falls.The link between tangent loss and frequency indicated that electrons switched between ions.The use of synthetic spinel ferrites in ultrahigh-frequency electromagnetic systems is possible.

Figure 1 .
Figure 1.X-ray diffraction pattern of all SrLa x Fe 2−x O 4 the samples.

Figure 2 .
Figure 2. Lattice constant "a" (A°) and Crystallite size (D) of SrLa x Fe 2−x O 4 ferrites as a function of La-concentration.

Figure 3 .
Figure 3. (a-c) SEM micrograph of all the samples.

Figure 6 .
Figure 6.Critical temperature vs La concentration for SrLa x Fe 2−x O 4 ferrites.

Figure 8 .
Figure 8. Magnetic properties of all the samples.

Figure 9 .
Figure 9. Dielectric constant of all the samples.

Figure 10 .
Figure 10.Dielectric loss of all the samples SrLa x Fe 2−x O 4 .

Figure 11 .
Figure 11.(a) Real part impedance, (b) imaginary part and (c) cole-cole plot of the real and imaginary part of all the samples.

Table 1 .
Comparison of SrLa x Fe 2−x O 4 spinel ferrites with well reputed research data.