Structural, Dielectric, Electrical, and Magnetic Characteristics of Bi_0.8Ba_0.1Er_0.1Fe_0.96Cr_0.02Mn_0.02O₃ Nanoparticles

Abstract

Bi_0.8Ba_0.1Er_0.1Fe_0.96Cr_0.02Mn_0.02O₃ (BBEFCMO) multiferroic ceramic was synthesized through the sol-gel route. The impact of incorporating various dopants into both A and B sites of the BiFeO₃ was investigated, and structural, Raman, dielectric, electric, and magnetic properties were studied. X-ray diffraction analysis and Raman spectroscopy revealed a rhombohedral structure with the R3c space group for the doped material (BBEFCMO). Dielectric properties were examined across a frequency range of 10²–10⁶ Hz. The present multiferroic material exhibits a colossal dielectric constant and minimal dielectric loss tangent, making it suitable for applications in energy storage. Furthermore, the Cole-Cole type of relaxation was deduced from the imaginary part of the modulus for both grain and boundary-grain contributions. Overall, this study indicates that substituting ions in both A and B sites of BiFeO₃ significantly enhances its multiferroic properties, as evidenced by dielectric and magnetic measurements.

1. Introduction

The multiferroic compounds with an ABO₃ formula have drawn the attention of researchers thanks to their unique properties and their various technological applications over the past decades in fields such as sensors, optoelectronic devices, spintronics, and transducers [1,2,3,4]. As a material extensively researched for its multiferroic properties, BiFeO₃ exhibits a rhombohedral structure with the R3c space group, displaying ferroelectricity, ferro-elastic, and anti-ferromagnetism properties simultaneously at room temperature [5,6,7,8].

While the ferroelectricity behavior of this compound emanates from the distortion of 6s² of the Bi³⁺ ions, their magnetic characteristics are linked to the super-exchange Fe-O-Fe interaction [9]. However, the significant leakage current densities, the weak anti-ferromagnetic behavior, and the poor ferroelectric hysteresis block their application. Unfortunately, the reduced polarization value is ascribed to the existence of secondary phases for the BiFeO₃ compound [10].

To address these drawbacks and enhance the characteristics of undoped BiFeO₃, researchers have explored partial substitution of Bi³⁺ (A-site doping) or Fe³⁺ (B-site doping), potentially leading to a highly dielectric constant, improvement of ferroelectric and ferromagnetic (FM) characteristics, and the elimination of the small impurity phase of the BiFeO₃ [11,12,13].

G. Catalan et al. [14] showed that the magnetic Néel temperature increased in the Ca-doped BiFeO₃ sample. Furthermore, the substitution of the Bi³⁺ ions for Ba²⁺ ones led to a pronounced ferromagnetic hysteresis loop for the BiFeO₃ sample [15].

The replacement of Iron (3+) ions by other transition metal ones, such as Mn ones, increases not only the Iron–Oxygen–Iron angle, but also the super-exchange interaction between Iron ions, and, therefore, enhances the magnetic properties [16].

Moreover, the concurrent substitutions of A and B ions with Er, Ba, Cr, and Mn ions exhibit an enhancement impact on dielectric, electric, and magnetic properties, as has recently been proven by the studies on Ba/Cr [17,18], Ba/Mn [19,20], Er/Mn [21], and Ba/Er [22] co-doped BFO materials. Furthermore, V.R. Palkar et al. [23] observed an improvement in the magnetization of the La/Mn co-doped BiFeO₃. Conversely, some reports have demonstrated that the incorporation of Eu³⁺ and Co³⁺ ions into the A-site and B-site of the BiFeO₃ boosts the saturation magnetization (M_S) and coercivity (H_C) (20 times more than the pure BiFeO₃) [24].

Indeed, over the last decade, researchers have adopted a new strategy for further enhancing the ferromagnetic and ferroelectric properties in BFO multiferroic material. This involves the simultaneous doping of both A and B sites with more than two ions [25,26,27,28]. It is worth noting that only a few studies have focused on the co-doping of multiple elements into both A and B sites of BiFeO₃ materials. This approach fosters increased ferromagnetic interactions and, concurrently, regulates structural transition and oxygen vacancy concentrations, thus obtaining the perfect parameters of ferroelectric properties, such as the high remnant polarization, the small coercive field, and reduced leakage current, etc. [29].

In our recent investigation, we explored the structural, morphological, ac-electrical, and dielectric characteristics of the (Bi_0.8Ba_0.1Er_0.1)(Fe_0.96Cr_0.02Co_0.02)O₃ nanoparticles synthesized through the sol-gel method. We have reported that the substitution of Lanthanum and Calcium enhances the magnetization (10 times higher than the BiFeO₃ sample) [30].

The present study accounts for the synthesis of Bi_0.8Ba_0.1Er_0.1Fe_0.96Cr_0.02Mn_0.02O₃ (BBEFCMO) nanoparticles via the sol-gel method, followed by calcination at 800 °C. It explores the impact of Cr³⁺ and Mn³⁺ ion incorporation into the B site on structural, dielectric, electric, and magnetic properties. Remarkably, BBEFCMO material demonstrates interesting dielectric and magnetic characteristics, making it a potential candidate for diverse technological applications, particularly in energy storage.

2. Materials and Methods

Bismuth, Barium, Erbium, Iron, Chromium, and Manganese nitrates (from Sigma-Aldrich) have been used to prepare Bi_0.8Ba_0.1Er_0.1Fe_0.96Cr_0.02Mn_0.02O₃ (BBEFCMO) nanoparticles with respect to the sol-gel Pechini method [31,32]. According to the respective stoichiometric proportion, sufficient quantities of each nitrate were mixed and dissolved in distilled water. Subsequently, citrate acid (C₆H₈O₇) was introduced as a chelating agent into the mixture of all nitrates, ensuring the specified molar ratio n(metal ions):n(citric) = 1:2. The solution was kept at 70 °C under magnetic stirring until the obtention of a brown viscose gel. The gel was heated at 170 °C to get a dark powder, which was ground-heated at 300 °C for 12 h in order to eliminate all organic entities. Later, the obtained powder was finely ground and then compressed into small cylindrical forms, measuring roughly 8 mm in diameter and about 3 mm in thickness. These pellets were subsequently heated at 800 °C for a duration of 4 h.

A Bruker 8D Advance X-ray powder diffractometer (Karlsruhe, Germany) was employed to study the phase and structural properties by adjusting the RXD pattern with FullProf software (version: September-20) [33].

Transmission electron microscopy (TEM) (Hitachi H-800, Tokyo, Japan) was conducted in a FEI Tecnai G2 microscope employing an accelerating voltage of 200 kilovolts. For TEM observations, the powder was dispersed in ethanol by sonication and dropped on carbon-coated copper grids.

Mössbauer spectroscopy was used to record spectra at room temperature using a WissEl (Germany) conventional transmission geometry spectrometer and a ⁵⁷Co/Rh source.

Raman spectroscopy (Horiba, Vénissieux, France) at room temperature (RT) was conducted with backscattering geometry, utilizing a Jobin Yvon HR 800 system and employing an excitation wavelength of 473 nm.

Dielectric measurements were conducted across a temperature range of 120 to 400 K, employing a cryostat setup with a nitrogen bath. During conducting measurements, the materials were maintained in a Helium atmosphere to reduce thermal gradients, and the sample temperature was regulated using an Oxford Research IT-C4 system in conjunction with a platinum sensor. We measured the impedance of the samples using an Agilent 4294 Network Analyzer, which operated in the range of 100 Hz to 1 MHz, with a configuration involving capacitance in parallel with resistance (C_p–R_p configuration) [34,35,36].

Hysteresis loops (M vs. µ₀H) were recorded at room temperature, subject to an applied field ranging from −10 to 10 Tesla. The measurements were conducted using a VSM system from Cryogenics, with the sample axis aligned parallel to the direction of the applied magnetic field.

3. Results and Discussion

3.1. Structural Properties

3.1.1. Powder X-ray Diffraction (XRD)

Figure 1a presents the XRD pattern of the (Bi_0.8Ba_0.1Er_0.1)(Fe_0.96Cr_0.02Mn_0.02)O₃ compound. As expected, the highest reflection peaks in the pattern precisely correspond to those documented and recognized for the rhombohedral crystal structure of the BiFeO₃ (JCPDS file no. 71-2494) [37]. The pronounced intensities of these diffraction peaks indicate the good crystallization of the compound under investigation. Nevertheless, supplementary peaks with low intensities were observed and analyzed with Xpert-height score software. The presence of the triclinic Bi₄O₇ (P-1 space group) and Ba₂FeO₄ (P12/c1 space group) secondary phases were identified. These impurities emanate from the volatility of bismuth ions that were identified in doped BiFeO₃ materials [38,39]. To further analyze the substitution effect on structural properties, we executed a Rietveld refinement using FullProf software. We began by refining the profile, adjusting the lattice parameters for all phases. Subsequently, atomic positions, the B_iso, and the corresponding occupancy of all atoms were individually adjusted for accuracy and enhancement of the structure. As presented in Figure 1a, the observed (Y_obs) data aligned closely with the refined XRD data (Y_cal), giving almost a linear behavior of their differences (Y_obs–Y_cal: blue line). The various resulting parameters of all phases are listed in

Table 1. Refined structural parameters obtained for the BBEFCMO sample using rhombohedral crystal structure with R3c space group.

Lattice Parameters
a = b (Å)			5.561 (3)
c (Å)			13.769 (3)
V (Å3)			368.818 (12)
Interatomic distances
Fe/Cr/Mn–O (Å)			2.2861/1.8025
Bi/Er/Ba–O (Å)			2.6444/2.1023
Fe–O–Fe (°)			149.431
R-factors (%)
Rwp			4.873
Rexp			3.132
GOF			1.555
χ2			2.420
Atomic position parameters
Atom	x	y		z
Bi/Er/Ba	0	0		0.2206
Fe/Cr/Mn	0	0		0
O	0.8812	0.6558		0.4533

.

As regards the quantitative phase compositions of the R3c, P-1, and P12/c1 phases in the studied compound, they were found to be around 95.30%, 2.63%, and 2.07%, respectively. It is worth mentioning that the c lattice parameter and cell volume V were observed to be slightly lower than values reported in an earlier study about the Bi_0.8Ba_0.1Er_0.1Fe_0.96Cr_0.02Co_0.02O₃ nanoparticles [30], which can be attributed to the smaller ionic radius of Mn³⁺ ions compared to Co²⁺ ones. Figure 1b presents the 3D schematic representation of the crystal structure of the studied sample using the Vista software, according to the structural refinement results. It is evident that the substitution in both the A site and B site of the BiFeO₃ oxide induces a tilt and a compaction of the Octahedron FeO₆.

On the other side, we analyzed the average crystallite size of the studied compound according to the Williamson–Hall formalism (Equation (1)) [40].

$${\beta }_{hkl}cos(\theta )=\frac{0.9\times \lambda }{D_{W-H}}+4\times \epsilon \times sin(\theta )$$

(1)

ε represents the strain attributed to the nanoparticles, θ denotes the peak position, D_W-H indicates the crystallite size (measured in nanometers), λ stands for the wavelength of the X-ray radiation source (which is 1.5406 angstroms), and β refers to the integral width. The average calculated crystallite size was found to be equal to 52 nm.

3.1.2. Raman Spectroscopy

Research findings indicate that in pure BiFeO₃ exhibiting rhombohedral distortion and the R3c space group, there are eighteen optical modes that can be represented as follows [41,42]:

Γ= 4A₁+ 5A₂ + 9E

(2)

where the A₁ mode and E mode are Raman active and IR. Conversely, A2 modes exhibit Raman inactivity.

The Raman spectra of the Bi_0.8Ba_0.1Er_0.1Fe_0.96Cr_0.02Mn_0.02O₃ (BBEFCMO) sample with an excitation wavelength of 532 nm at different temperatures (25 °C, 100 °C, and 400 °C) are shown in Figure 2a. The experimental Raman spectra of the prepared sample were fitted using the Lorentzian approach. The adjustment results are shown in Figure 2b–d, and the peak positions are collected in

Table 2. The Raman modes of the prepared BBEFCMO sample.

Raman Mode [21]	25 °C/(cm−1)	100 °C/(cm−1)	400 °C/(cm−1)
A1-1	127.90	128.57	123.67
A1-2	138.22	164.27	158.15
A1-3	166.28	184.96	192.74
A1-4	238.88	248.22	248.15
E	258.74	273.73	279.74
E	283.14	307.18	310.92
E	374.10	376.39	397.65
E	463.25	449.73	458.66
E	490.03	475.36	484.28
E	512.37	500.71	507.51
E	597.08	595.86	589.96
E	617.02	617.50	612.65
E	640.04	643.85	653.32

.

It can be clearly seen that the Raman spectra obtained for the BBEFCMO ceramic are consistent with previously reported findings in the literature for undoped BiFeO₃ [43]. Moreover, the observed spectra accord well with the Raman active modes identified for the Er, Ba, Cr, and Co-doped BiFeO₃ reported in our previous research [30].

In the low Raman range, A₁ modes are related to the Bi-O vibration [44]. Compared to BiFeO₃, we can plainly observe a shift in these modes. The change in this Bi-O vibration can be explained by the effect of the Ba and Er substitution in the A site. Moreover, the Er-O bond (606 KJ/mol) and Ba-O bond (562 ± 13.4 KJ/mol) are stronger than the Bi-O bond (337.2 ± 12.6 KJ/mol), which induces more displacement in the center of the BiFeO₃ sample [45].

Furthermore, the high-frequency E modes are attributed to the Fe-O vibrations [46]. The shift of the modes in this range is due to the introduction of the Mn and Cr atoms in the Fe site which form more stable Cr-O and Mn-O bonds, thus causing significant distortion of FeO₆ octahedra [47].

We will discuss the evolution of the Raman peaks of our sample around the anti-ferromagnetic phase transition T_N (≈643 K) of the non-doped BiFeO₃. In this regard, Figure 2d shows the mode frequencies of the Raman modes as a function of temperature for the Bi_0.8Ba_0.1Er_0.1Fe_0.96Cr_0.02Mn_0.02O₃ ceramic, obtained from Lorentzian fittings.

We can confirm that the overall spectral signature remains consistent within the temperature range of 25 °C and 400 °C, suggesting that our sample maintains its room temperature structure up to 400 °C. However, it is worth noting that, upon heating, there is a noticeable alteration in both spectra shape and intensity. Furthermore, a distinct step-like evolution of the wave number for several bands is clearly observed.

3.1.3. Transmission Electron Microscopy (TEM)

Figure 3a displays a selected micrograph obtained from transmission electron microscopy (TEM) of the synthesized BBEFCMO compound. Nanoparticles with a combination of spherical and rectangle shapes are observed in Figure 3a. The distribution profile of the particles size was obtained using Image-J software and is presented in Figure 3b. The average size of the BBEFCMO particles was found to be around 76 nm, which accords well with the crystallites size found from the Williamson–Hall formalism.

3.2. Magnetic Properties

Figure 4a presents the magnetic hysteresis loops of the BBEFCMO nanoparticles in an applied field in the range of −10 T to 10 T, measured at 7 K and 300 K. The magnetization data displayed a significant hysteresis loop, indicating finite values for both coercive and remanent magnetization (M_S), and importantly an unsaturated magnetization under high fields. The unsaturated behavior indicates an antiferromagnetic (AFM) behavior [48,49,50]. Moreover, the important remanent magnetization value (nonzero), below which the magnetization sharply increased, serves as evidence for the presence of a ferromagnetic (FM) contribution presence within the compound [51]. As reported by D. Lebeugle et al., it is worth mentioning that the M(H) plots of the pure BiFeO₃ (BFO) depict an almost unsaturated linear behavior, indicating a G-type antiferromagnetic (AFM) configuration with the pitching of Fe³⁺ ions [52]. In the present work, the magnetization of the BBEFCMO compound was found to be around 27 times higher than what was found for the undoped BFO system [53,54], and 1.5 times higher than what was found for the (Bi_0.8Er_0.1Ba_0.1)(Fe_0.96Cr_0.02Co_0.02)O₃ compound [55], which is essentially due to the presence of Manganese ions, as was previously reported [56,57,58]. Moreover, it was confirmed that the introduction of small quantities of Manganese ions in the BiFeO₃ system induces a John–Teller distortion which would enhance the magnetization [58].

To confirm and quantify the ferromagnetic contribution in the Bi_0.8Er_0.1Ba_0.1Fe_0.96Cr_0.02Mn_0.02O₃ compound, the M-H magnetic loop was adjusted according to the following equation [59,60]:

$$M\left(H\right)=\left\{2\frac{M_{FM}^S}{\pi }{\tan }^{-1}\left(\left(\frac{H\pm H_{ci}}{H_{ci}}\right)\tan \left(\frac{\pi \times M_{FM}^R}{2\times M_{FM}^S}\right)\right)\right\}+\left\{\chi H\right\}$$

(3)

where the first term is related to the ferromagnetic contribution, while the second one represents the linear behavior from the antiferromagnetic contribution (AFM). $M_{FM}^S$, $M_{FM}^R$, and H_ci are the ferromagnetic saturation magnetization, remnant magnetization, and intrinsic coercive field, respectively. The resulting adjusted curves are presented in Figure 4b together with the experimental data showing a good adjustment quality. From the adjusted magnetic parameters (

Table 3. Obtained adjusted parameters of the room temperature magnetic hysteresis loop of BBEFCMO compound.

Observed Magnetization (emu/g)	AFM Contribution (×105 emu*Oe/g)	FM Contribution
		HCI (kOe)	MFMS (emu/g)	MFMR (emu/g)
8.051	0.325	4.208	4.801	1.947

), the ferromagnetic contribution was found to be equal to 40.37%. With the replacement of Cobalt ions by Manganese ones, the magnetization increased from 5.068 to 8.05 emu/g, which is related to a 2.05% increase in the ferromagnetic contribution.

On the other hand, Figure 5 illustrates the room temperature Mössbauer spectrum of the BBEFCMO compound. The spectrum was fitted with three Fe³⁺ sextets and a Fe³⁺ doublet, as in the case of the replacement of Mn by Co [55]. The fitted parameters are tabulated in

Table 4. Hyperfine parameters resulted from the fitting procedure of the Mössbauer spectrum of the BBEFCMO compound. Isomer shifts are given relatively to α-Fe.

Site	δ (mm/s)	Δ (mm/s)	B (T)	Γ (mm/s)	%	Coordination
Doublet	0.23 (1)	0.550 (1)	-	0.47 (1)	5.3	Fe3+
Sext 1	0.39 (1)	0.034 (1)	50.7 (1)	0.45 (1)	42.3	Fe3+
Sext 2	0.38 (1)	0.032 (1)	49.3 (1)	0.47 (1)	47.2	Fe3+
Sext 3	0.25 (1)	0.049 (1)	41.5 (1)	0.67 (1)	5.2	Fe3+

. The presence of a doublet is likely to emanate from the Bi₂FeO₉ impurity phase (5.3%). Indeed, despite being non-magnetic, it could be associated with the pure BFO compound, as well as the first sextet. The second sextet with a hyperfine field of 49.3 T, accounting for 47%, is attributed to Fe³⁺ ions in the tetrahedral coordination and AFM ordering in the pure BFO compound. The first and third sextets are ascribed to Fe³⁺ at octahedral environments and are related to FM. Together, their relative amounts total 47.5%, representing 45% of the magnetic part of the sample.

Compared to the Mössbauer results of the (Bi_0.8Er_0.1Ba_0.1)(Fe_0.96Cr_0.02Co_0.02)O₃ compound, the introduction of Mn ions induces a reduction in the third sextet and a slight change in its isomer shift. This sextet has the lowest hyperfine field (41.5 T), and its relative area decrease is related to a higher saturation magnetization of the sample, compared with the one with Co²⁺ in the B site.

3.3. Dielectric Study

3.3.1. Dielectric Constant and Dielectric Loss Tangent tg(δ)

In Figure 6a,b, we plot the frequency and temperature dependence of the dielectric constant ε′ of the Bi_0.8Ba_0.1Er_0.1Fe_0.96Cr_0.02Mn_0.02O₃ (BBEFCMO) compound. At low frequencies, ε′ presents extraordinarily high values (≈8 × 10⁵), indicating a colossal dielectric constant nature of the studied compound. This can be explained by a potential barrier generated by space charge polarization at the grain boundaries inducing a charge accumulation. Such materials with a high value of ε′ are candidates for energy storage devices [61].

Furthermore, it was observed that the dielectric constant rises as temperature increases, primarily ascribed to the increase in various polarization contributions, resulting from thermally activated charge carriers [62]. The presented curves of the dielectric constant show 3 plateaus (with a slight dependence on frequency) at around 400, 10⁴, and 3 × 10⁵, between which the curves decrease rapidly with increasing frequency. It is also important to mention that the step-like decrease shifted to high frequencies with the rise in temperature, indicating a thermally activated relaxation process [63]. These observations confirm that the studied compound is composed of three different contributions.

To further confirm the presence of different relaxation processes from dielectric constant curves, we calculated the logarithmic derivative of the dielectric constant component (known as the Kramers–Kronig transformation [64,65,66]) and we plotted the results at different temperatures in Figure 6c. The appearance of three different relaxation peaks aligns well with the presence of three different contributions.

On the other hand, typically, the step-like decrease in the dielectric constant is accompanied by a relaxation peak in the dielectric loss tangent tg(δ). We plotted in Figure 7a the frequency dependence of the dielectric loss tangent for the temperature range between 120 K and 400 K. In fact, it was observed that the dielectric loss tangent exhibits two distinct and well-defined relaxation peaks at low and high frequencies for each temperature. Moreover, a third relaxation peak is almost hidden in the intermediate frequency region, which is clear in the Kramers–Kronig transformation of the dielectric loss tangent at 170 K, as presented in Figure 6c. The dielectric relaxations at low and high frequencies are associated with the electrode and grain contributions, respectively, while the dielectric relaxation at intermediate frequencies is due to the grain boundary contribution. Figure 7b shows the temperature dependence of the relaxation frequency behavior of both relaxations at low and high frequencies, which obeys the linear Arrhenius law between Ln(f_max) and 1/T. The calculated activation energies obtained for electrode and grain contributions are equal to 0.553 eV and 0.319 eV, respectively.

3.3.2. Imaginary Part of Modulus

The complex modulus study has been adapted to investigate the dielectric relaxation behavior and to differentiate the grain, the grain boundaries, and the electrode contributions in our sample.

Figure 8a shows a 3D presentation of the imaginary part M″ of the complex modulus of the prepared compound. A close inspection of the M″ plot shows the appearance of two peaks, the first of which is located at a low-frequency region and the second one appears at a high-frequency range, which implies the existence of a relaxation process in this material [67]. The low frequency side of the peaks presents the range in which the ions are mobile over long distances and perform successful hopping from one site to the neighboring one. Furthermore, at the high frequency side, the ions are confined to their wells and can execute only localized motion [68].

We can clearly confirm that the peaks gradually shift to high frequencies and temperature, as shown in Figure 8b. This behavior suggests that the relaxation is thermally activated [69].

To better study the dielectric relaxation processes, a model function of Havriliak–Negami (HN) given by (Equation (3)) [70] is adopted to fit the experimental data of the imaginary part of Modulus M″.

$$M^{″}\left(\omega \right)=M_{\infty }+\frac{M_0-M_{\infty }}{{\left[1+{(j\omega {\tau }_{HN})}^{\alpha }\right]}^{\beta }}$$

(4)

where τ_HN is the average relaxation time, the parameters α (0 < α < 1) and β (0 < α < 1) are the shape factors related to the width of the relaxation and the asymmetry of curves, and M_∞ and M₀ are the dielectric modulus at low and high frequency sides of the relaxation.

L.C. Brazzano et al. [71] have confirmed that the HN function is obtained by a simple generalization of a single time relaxation, which illustrates the response of a Debye process when α = β = 1. Nevertheless, if β = 1 (α ≠ 1) or α = 1 (β ≠ 1), the relaxation process is according to Cole-Cole [72] and Cole-Davidson [73] models, respectively. The result of the adjustment is plotted in Figure 8b.

The obtained parameters are used to identify the model of the dielectric relaxation. Figure 9a presents the temperature dependence of α and β parameters. We can deduce that the Cole-Cole model fits well with both the low and high-frequency ranges, which are accredited to boundary grains and grains, respectively. These results are in good agreement with those obtained in our previous work for the Bi_0.8(La_0.8Ca_0.2)_0.2FeO₃ compound [32].

Furthermore, the obtained HN relaxation time parameters were used to calculate the activation energies. The logarithmic variation in the relaxation time Ln(τ_HN) as a function of the 1000/T in terms of the Arrhenius law is depicted in Figure 9b, from which it can be inferred that the activation energy of the boundary grains exceeds those of the grains. On the other hand, the obtained values are very close to those previously calculated from the permittivity plots.

3.4. Electrical Investigation

3.4.1. DC Conductivity

Figure 10a presents the variation in the Ln(ρ/T) versus 1/T of the BBEFCMO sample. It is found that the resistivity decreases rapidly with the increase in temperature until reaching a slow variation above T = 363 K. This behavior implies a rise in material conductivity, reaching a saturation value at this temperature, indicating a semiconductor behavior. Beyond 370 K, the temperature-independence conductivity may be explained by the stability of the parameters governing the microstructure of the material (inter-site or inter-gain distances, the resistance of the insulating barrier). Additionally, the saturation of σ_dc can be attributed to the stability of the factors affecting the transfer of a small polaron between two sites (the jump and disorder energies, the radius of the polaron, the resistance of the grains, and the mobility of the charge carriers and their concentrations).

However, below 363 K, the conductivity steadily increases with temperature increases. This suggests that the DC conduction is described by the variable range-hopping (VRH) process at lower temperatures and transitions to a thermal activation by the small polaron-hopping (SPH) process at higher temperatures [74].

To verify the validity of SPH conduction model, we plotted the variation in Ln (σ_dc.T) versus the inverse of temperature according to the Arrhenius law. This plot exhibits three distinct slopes corresponding to three different activation energies. According to Mott-theory, in the high-temperature region above θ_D/2, where the obtained activation energy is around 338 meV, the conduction process is typically characterized by thermally activated SPH conduction. This activation energy, E_a, arises from polaron formation via a banding energy (E_P = E_H/2; E_H: polaron-hopping energy) and the disorder energy (E_D) due to the variations in the local arrangements of ions [75,76,77]. Conversely, at the low-temperature region, the Mott revealed that the conduction is purely governed by the disorder energy, which is approximately 33 meV, with negligible hopping and banding energies.

In light of these findings, the temperature dependence of the DC conductivity of the present sample can be described by the following expression [78,79]:

$${\sigma }_{dc}={\sigma }_{0}T.e^{\left(-\frac{Ea}{k_{B}T}\right)}$$

(5)

where σ₀ is a pre-exponential factor, T is the absolute temperature, and k_B is the Boltzmann constant.

The activation energy is given by

$$E_a\approx \{\begin{array}{c}{E}_a=E_H+\frac{E_D}{2} for T>\frac{{\theta }_D}{2}\\ E_a=E_D for T<\frac{{\theta }_D}{4}\end{array}$$

(6)

where E_H and E_D are the polaron-hopping energy and the disorder energy, respectively. As can be seen from Figure 10b, describing the dependence of Ln (σ_dc) on T^−1/4. This figure clearly proves a linear slope at lower temperatures (below θ_D/4), confirming the validity of the Mott-VRH model to describe the transport process of the studied material at the low-temperature region, by the following expression of DC conductivity [80]:

$${\sigma }_{dc}=A{e}^{\left(-\frac{B}{T^{1/4}}\right)}$$

(7)

where A and B are constants and B is given by

$$B=2.1{\left(\frac{{\alpha }^3}{k_{B}N\left(E_F\right)}\right)}^{1/4}$$

(8)

where N(E_F) is the density of the state at the Fermi level.

Furthermore, in the intermediate temperature range, the DC conductivity is not considered a Mott-VRH Model. As shown in Figure 10c, Greaves revealed that the DC conductivity can be investigated based on the following expression [79]:

$${\sigma }_{dc}\frac{T}{2}=A{e}^{\left(-\frac{B}{T^{1/4}}\right)}$$

(9)

3.4.2. AC Conductivity

Figure 11a reveals the frequency dependence of the AC conductivity in the temperature range from 120 K to 400 K. It is clearly seen that σ_ac is affected only at the high-frequency region, attesting a dispersion behavior due to the ionic relaxation after the particles’ movement, following the universal power law of Aω^S [80].

According to the temperature dependence of the already-discussed dielectric features, it is clear that only one dispersion behavior is observed in the low-temperature region and a second one is detected in the high-temperature region. These findings reveal that the conductivity enhancement is likely to emanate from that of the hopping probability of charge carriers by the frequency rise.

On the other hand, the Ln(σ_ac × T) vs. 1000/T curves at different frequencies are plotted in Figure 11b. As it can be clearly seen, the curves show two different regions in which the electric conductivity behaves differently. While at high temperatures, the conductivity seems to depend only on temperature, at the low-temperature region, it becomes frequency and temperature dependent. The activation energy in the high-temperature region was calculated to be equal to 0.431 eV, which is close to that found for the boundary grains contribution (M″ vs. frequency curves).

Meanwhile, the AC conductivity curves have been fitted (red lines in Figure 11a) by the following relation [81]:

$${\sigma }_{ac}\left(\omega \right)=\left[\frac{{\sigma }_s}{1+{\tau }^2{\omega }^2}+\frac{{\sigma }_{\infty }{\tau }^2{\omega }^2}{1+{\tau }^2{\omega }^2}\right]+A.{\omega }^s$$

(10)

Here, σ_S represents the conductivity at low frequencies, while σ_∞ denotes the conductivity at high frequencies. ω stands for the angular frequency, τ signifies the relaxation time, A denotes a constant influenced by temperature, and s represents an exponent characterizing the interaction degree between mobile ions and their surrounding environment.

Figure 11c proves the evolution of the exponent s in the studied temperature range. The behavior of this exponent under temperature variation presents a great indicator of the origin of the conduction process. As previously observed, the decrease in this exponent between 120 K and 230 K indicates the correlated barrier-hopping (CBH) conduction mechanism while in the region above 240 K, the power law exponent s changes without any obvious trend. This is due to the fact that the AC conductivity at high temperatures is frequency independent, as deduced previously.

4. Conclusions

The synthesis of a multi-doped Bi_0.8Ba_0.1Er_0.1Fe_0.96Cr_0.02Mn_0.02O₃ nanomaterial was carried out using the sol-gel Pechini method. The characterization revealed that these nanoparticles (D_TEM = 76 nm) crystalized in the cubic structure with the R3c space group as confirmed by X-ray diffraction and Raman spectroscopy. Notably, the compound exhibited significant magnetization, primarily attributed to the 40.37% of ferromagnetic contribution from the total magnetization. Furthermore, the BBEFCMO compound displayed a colossal dielectric constant (>5.105), coupled with minimal dielectric loss tangent values, thus confirming its promising potential for energy storage applications. Three different relaxation processes were identified for the compound, related to electrode, boundary grains, and grain contributions.

Moreover, the conduction processes were studied through analysis of DC and AC conductivity values, proving that the conduction is due essentially to the polaron hopping between Iron states.