Neodymium-engineered relaxor bismuth ferrite nanoparticles: structural tuning and dielectric enhancement for efficient sensor applications

Abstract

This paper investigates the synthesis and properties of neodymium-doped bismuth ferrite (BiFeO₃) nanoparticles, highlighting their enhanced functionality for advanced applications. The nanoparticles were successfully synthesized with a sol–gel method, where neodymium (Nd) was substituted into the A-site in concentrations ranging from 6% to 10%. A comprehensive analysis of the structural, morphological, and dielectric properties was conducted. X-ray diffraction (XRD) and Rietveld refinement confirmed that all samples maintained a rhombohedral crystal structure with the R3c space group. The incorporation of Nd cations was found to significantly alter the intrinsic distortion of the FeO₆ octahedron within the lattice, which is identified as a primary mechanism for property enhancement. Morphological studies showed that the nanoparticles were uniform, with grain sizes between 160 nm and 195 nm. Furthermore, XPS confirmed the presence of Fe²⁺ ions, which are directly linked to the improved ferroelectric performance. An extensive study of the dielectric properties revealed a change in the electrical conduction mechanism with temperature and notable relaxor behavior. A reduction in the Néel temperature and increased thermal sensitivity were also detected. These remarkable findings demonstrate that Nd substitution is highly effective in tailoring the properties of bismuth ferrite, making these modified nanoparticles excellent candidates for next-generation devices.

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I. Brief bibliography

The innovation potential is presented by modified relaxor bismuth ferrite nanoparticles, which find diverse applications.¹ Exploration of their structural, morphological, and dielectric properties can lead us to new pathways toward technology. A variety of fields stand to benefit from these nanoparticles, which are becoming essential in areas like electronics and energy storage, among others.² A substance that displays two or more ferroelectric properties at once, such as ferroelectricity and (anti)ferromagnetism, is known as a multiferroic.³ One of the most interesting multiferroic materials is BiFeO₃ (BFO), which has a rhombohedral distortion along the [111] direction and crystallizes in the R3c space group.⁴ This material exhibits both a ferroelectric transition temperature (T_c = 1103 K) and an antiferromagnetic NEEL temperature (T_N = 643 K),⁵ both of which are higher than the room temperature. However, achieving both electric (d⁰, lone pair) and magnetic order (dⁿ, fⁿ) at the same time is difficult. This is why single-phase multiferroics are rare.⁶ Producing pure BFO ceramics or thin films is difficult because the temperature range for stabilizing the phase is narrow.^7,8 Researchers have shown that rare earth (RE) substitution at the A-site of the ABO₃ perovskite materials is a proven method to address the aforementioned difficulties, improve the material's properties, and reduce impurity phases.^9,10 Moreover, the integration of rare-earth elements has been demonstrated to enhance the dielectric characteristics of bismuth ferrite, thereby augmenting the dielectric constant and mitigating dielectric dissipation. Significant changes in impedance, modulus, and intrinsic conduction mechanisms are observed. These improvements are in line with, and have been documented well in, existing literature.¹¹ We specifically selected neodymium among the various rare-earth elements considered because it has a proven ability to stabilize the host lattice and markedly enhance the overall functional properties, as our previous investigations confirmed.¹² As perovskite materials are recognized for their relaxor behavior,^13,14 which is advantageous for a broad spectrum of applications, they are considered a valuable asset in various fields. The introduction of these materials has created new possibilities for modern acoustic transducers,¹⁵ as well as for applications such as medical ultrasound, non-destructive sensing, marine seismic exploration, and energy harvesting.¹⁶

The present study aims to demonstrate the suitability of sol–gel synthesized samples as high-performance candidates for thermistor applications, while providing a comprehensive understanding of how neodymium incorporation governs the relaxor behavior of the system.

II. Methods for preparing and characterizing the samples

(1) Sol–gel process

High-purity precursors were used to ensure the production of high quality Nd_xBi_1−xFeO₃ nanoparticles (x = 0.06, x = 0.08 and x = 0.1). The precursors contain bismuth nitrate pentahydrate [Bi(NO₃)₃·5H₂O], ferric nitrate nonahydrate [Fe(NO₃)₂·9H₂O], neodymium nitrate hexahydrate [Nd (NO₃)₃·6H₂O] and citric acid (C₆H₈O₇). The other two reagents used included ethylene glycol (C₂H₆O₂) and nitric acid (HNO₃). These precursors were mixed in the right amounts. The resulting solution was heated to form a dark red mixture. Heating and stirring continuously took 2 hours. A clear brownish gel formed. The product was ground into powder and heated to 500 °C for 5 hours. The obtained nanoparticles were pressed into pellets. This was done using a hydraulic press. Then they were sintered at 600 °C for 4 hours. Fig. 1 illustrates the entire preparation process in detail.

Fig. 1 Preparation process of NdBFO NPs using the sol–gel synthesis process.

(2) Characterization

To determine the crystal structure and phase analysis of the generated compounds, XRD measurements were taken at room temperature using a Bruker D8 instrument (source = Cu-Kα radiation, wavelength = 1.5418 Å). The analysis of the collected data was performed using the Rietveld refinement technique in the FullProf program (version January 2021). The morphology of the synthesized materials was examined using a JEOL JSM-1644 HR scanning electron microscope at the GREMAN laboratory in Tours, France. The surface chemical composition and oxidation states of the samples were investigated by X-ray photoelectron spectroscopy (XPS), employing a standard Al Kα radiation source (Omicron DAR 400, 1486.6 eV). The spectra were collected in constant analyzer energy (CAE) mode, and survey scans were recorded across a binding energy range of 0 to 1400 eV. Finally, the dielectric measurements were carried out over a frequency range of 100 Hz to 1 MHz at varying temperatures from room temperature up to 673 K.

III. Structural and morphological results

(1) Analysis of the sample's structure

As shown in Fig. 2(a), the XRD patterns of all the synthesized samples are depicted. The main indexed peaks indicate the presence of planes with Miller indices (hkl) belonging to the rhombohedral structure with the R3c space group. The experiment diffraction peaks showed good agreement with the standard crystal data [JCPDS No. 01-075-6667]. The FullProf software was utilized to model the peak profile of the NdBFO (x = 0.06, 0.08, and 0.10) samples. The Rietveld analysis was conducted, and the results are displayed in Fig. 2(b). During the refinement process, the pseudo-Voigt function was used to determine the peak shape, and linear interpolation was used to calculate the background. The observed and calculated patterns demonstrate excellent agreement, as confirmed by the satisfactory reliability factors.¹⁷Table 1 shows the extracted parameters and indicating that an increase in Nd content slightly influences the lattice parameters and the volume of the structure. These effects could be attributed to the difference between the ionic radii of Nd and Bismuth. In the same way, the electronic density of NdBFO nanostructured materials was obtained from the Rietveld refined XRD patterns. Electron density plots (x, y, z) are essential for understanding atomic-level interactions.^18,19 These plots are derived from the reverse Fourier transform^20,21 of the structure factors obtained from the Rietveld refinement of XRD information. As the neodymium concentration increased, an up-and-down shift of high-density contours around the neodymium sites along the c axis was observed in the 2D map of electronic density (Fig. 3(a)). The same trend was observed in the 3D map of density variation (Fig. 3(b)). These results reinforce the crystallographic findings and demonstrate a consistent trend across all studied lattice parameters. The influence of neodymium on the intrinsic distortion of the samples, particularly on the FeO₆ octahedron in the structures, was investigated by studying parameter bonds using VESTA software. The results obtained are shown in Fig. 2(c) and summarized in Table 2. The Fe–O–Fe angle in the FeO₆ octahedron plays a crucial role in the electrical, magnetic, and electronic properties of materials, particularly at the nanoscale. In a perfect structure, this angle would be 180°.²² However, if the angle is less than 145°, it can affect the materials’ electrical properties and if the angle is greater than 160°, the materials can become less magnetic.^23,24 Our earlier tests showed that adding more neodymium could disrupt the structure of the FeO₆ octahedron and reduce the Fe–O–Fe angle from 158.06° to around 156.29° (Fig. 4). In this case, a good balance between ferroelectricity and magnetism was achieved. The aforementioned properties make the samples Nd_xBi_1−xFeO₃ NPs (x = 0.06, x = 0.08, x = 0.1) suitable candidates for ferroelectric capacitors and ferroelectric random-access memory (FeRAM).²⁵

Fig. 2 (a) X-ray diffraction patterns for Nd_xBi_1−xFeO₃ (x = 0.06, 0.08, 0.1) nanoparticles. (b) NdBFO's FullProf Rietveld refinement simulation. (c) The visualization structure of the NdBFO NPs using Vesta software.

Table 1 Crystallographic parameters of NdBFO NPs

Crystallographic parameters	NdxBi1−xFeO3
	x = 0.06	x = 0.08	x = 0.10
a = b (Å) (±0.0001)	5.5748	5.5753	5.5748
c (Å) (±0.0002)	13.8397	13.8479	13.8397
c/a	2.4825	2.4837	2.4825
D sch size (nm)	51.45	52.35	47.29
Volume (Å^3) (±0.01)	372.498	372.787	372.496
R p	9.46	9.47	9.58
R wp	9.67	9.75	9.75
χ ^2	2.38	2.5	2.43

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Fig. 3 (a) 2D contours for electron density of the NdBFO samples. (b) 3D Fourier map for electron density of NdBFO samples.

Table 2 NdBFO sample's parameters bond angle and bond length

Molecular formula	Space group	Bond name	Bond length (Å)	Bond type	Bond angle (deg)
Nd0.06Bi0.94FeO3	R3c	Bi/Nd–O	2.518, 2.367	Bi–O–Bi	108.252
		Fe–O	2.133, 1.899	Fe–O–Fe	158.064
Nd0.08Bi0.92FeO3	R3c	Bi/Nd–O	2.499, 2.383	Bi–O–Bi	108.381
		Fe–O	2.172, 1.869	Fe–O–Fe	156.993
Nd0.1Bi0.9FeO3	R3c	Bi/Nd–O	2.525, 2.358	Bi–O–Bi	108.307
		Fe–O	2.132, 1.902	Fe–O–Fe	156.290

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Fig. 4 Representation of the bond angle Fe–O–Fe of NdBFO NPs.

(2) Morphological analysis of synthetic samples

The samples’ microstructures, as revealed by SEM, show distinct spherical shapes and irregular agglomeration, forming stronger interconnections among them (Fig. 5). Thorough investigation of the size distribution histograms, employing ImageJ software, reveals that the synthesized samples, namely Nd_0.06Bi_0.94FeO₃, Nd_0.92Bi_0.08FeO₃, and Nd_0.1Bi_0.9FeO₃, possessed particle sizes of 195 nm, 192 nm, and 160 nm, respectively.

Fig. 5 SEM images of the Sol–gel synthesized samples and histograms of particle size distribution.

(3) XPS characterization of the nanoparticles

To verify the presence of the expected elements in each synthesized sample and to confirm the successful synthesis of high-quality samples free of residues or impurities, measurements using X-ray photoelectron spectroscopy (XPS) were performed. Fig. 6 illustrates the survey spectrum of the synthesized samples, confirming the presence of Nd, Bi, Fe and O, as indicated by their characteristic peaks. As shown in Fig. 7, two broad peaks correspond to Fe 2p_3/2 and Fe 2p_1/2. These peaks are located in the binding energy range of 700 eV to 730 eV. This indicates the presence of both Fe³⁺ and Fe²⁺ oxidation states. The XPS peak fitting was executed using the Thermo Avantage software, employing a mixed Gaussian–Lorentzian (GL) peak shape.²⁶ The background subtraction and peak fitting parameters were carefully optimized within the software to ensure that accurate and reliable deconvolution results were achieved. The C 1s peak, at 284.7 eV, was used as a reference for the binding energy calibration. The deconvolution of the data reveals a decrease in the atomic percentage of Fe²⁺ ions compared to Fe³⁺ ions with increasing neodymium content (Table 3), confirming the significant role of neodymium in the structural configuration. This observation is consistent with previous research results.²⁷

Fig. 6 XPS spectra of the NdBFO NPs.

Fig. 7 Fe 2p XPS spectral fitting for NdBFO samples.

Table 3 Oxidation states of iron in the synthesized nanoparticles

Samples	Fe^3+ (%)	Fe^2+ (%)
Nd0.06Bi0.94FeO3	73.51	26.49
Nd0.08Bi0.92FeO3	79.41	20.59
Nd0.1Bi0.9FeO3	80.46	19.54

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Based on the structural and morphological analysis, it can be concluded that well-structured and impurity-free samples were successfully synthesized using the simple Sol–Gel method. The dielectric properties of the samples will now be explored.

IV. Examination of the nanoparticles’ dielectric qualities

(1) Frequency-dependent dielectric relaxation response of samples at a given temperature

The analysis of dielectric properties is a pivotal aspect of the study of ceramic materials. This investigation elucidates various phenomena, including the relaxation process, the degree of polarization, the impact of defects, and the underlying mechanisms contributing to dielectric loss. The electrodes were prepared by coating both surfaces of the pellets with a thin and uniform layer of silver lacquer to ensure good electrical contact during the measurements. The pellets were oval in shape, with an average diameter of approximately 8 mm and a thickness of about 1 mm. The measurements of dielectric parameters were performed over a frequency range from 100 Hz to 1 MHz, and within a specified temperature range between 537 K and 575 K. As depicted in Fig. 8, the dielectric parameters (ε′, ε′′, and tan δ) of all samples demonstrate significantly elevated values in the low-frequency domain, a typical behavior for dielectrics, attributed to the contribution of various types of polarization, including dipolar, space-charge, electronic, and ionic polarizations. At elevated frequencies, the phenomena mainly involve atomic and electronic polarizations, which yield a decrease in the losses. Furthermore, the Maxwell–Wagner model, in alignment with Koops’ phenomenological theory, provides insight into the variations in dielectric parameters with frequency.^28,29 This model suggests that in low-frequency regimes, grain boundaries have a significant impact, whereas grains become more pronounced at higher frequencies. In the low-frequency domain, the substantial presence of highly resistive grain boundaries facilitates significant space charge polarization by electrons. Conversely, at higher frequencies, the dielectric constant decreases as the electrons move between grains and grain boundaries. The curve depicting the tangent loss follows a similar trend to that of the dielectric constant, with high values at low frequencies due to the increased resistance of grain boundaries, which necessitates more energy for electron mobilization. As the frequency increases, the resistance decreases, resulting in a corresponding reduction in the tangent loss values.³⁰ This low frequency dispersion effect is commonly seen in ferroelectric and dielectric materials.^31,32 As the temperature increased, the dielectric constant of the two samples (Nd_0.06Bi_0.94FeO₃ NPs and Nd_0.08Bi_0.92FeO₃ NPs) exhibited an upward trend at low frequencies. However, a decline in the dielectric constant was observed with temperature at the high neodymium concentration (Nd_0.1Bi_0.9FeO₃ NPs). The presence of neodymium disorder cations leads to the relaxor characteristics of this sample.³³ Yadav et al. suggest another possible explanation for this: the electrons may be jumping between Fe³⁺ and Fe²⁺ ions.³⁴ Similar types of behavior have been observed in other eco-friendly ABO₃-type relaxor-ferroelectric ceramics.^35–37

Fig. 8 Frequency dielectric parameters in a specific temperature range of Nd_xBi_1xFeO₃ (x = 0.06, x = 0.08 and x = 0.1) NPs.

(2) Thermal dielectric characteristics at a particular frequency

The temperature dependence of the dielectric permittivity and tangent loss at different frequencies (1 kHz, 10 kHz, 103 kHz, and 1 MHz) is presented in Fig. 9 for x = 0.06, 0.08, and 0.1, respectively. As the temperature increases, a notable rise in the dielectric parameters is observed, which can likely be attributed to the involvement of the thermally activated dipoles.³⁸ Within the temperature range up to 373 K, the dielectric constant demonstrates consistent behavior across all frequencies. It exhibits a gradual increase, with a prominent peak observed around the NEEL temperature (T_N). A distinctive feature of relaxing ferroelectrics is that the broad peak around the T_N shifts toward higher temperatures and diminishes in intensity as the frequency increases from 1 kHz to 1 MHz for all samples. According to the literature,³⁹ magnetoelectric coupling may directly cause this type of abnormal dielectric behavior. The observed peak in the temperature, ranging from 573 K to 650 K in Nd_0.1Bi_0.9FeO₃ samples, confirms the progressive relaxor behavior induced by the addition of neodymium. This unusual behavior in magnetically ordered systems is predicted by Landau–Devonshire theory of phase transitions as an influence of the disappearance of the magnetic order on the electric order.⁴⁰ As the temperature increases, tanδ shows a marked increase in the higher temperature region for different frequencies (across all compositions), which is due to the dominance of the conductance.⁴¹ Finally, studying the material's electric properties (see Fig. 10) showed that adding neodymium causes a drop in the NEEL temperature from T_N = 619 K to T_N = 520 K making these samples suitable for many applications.

Fig. 9 Thermal dielectric parameters in a specific frequency range of Nd_xBi_1xFeO₃(x = 0.06, x = 0.08 and x = 0.1) NPs.

Fig. 10 Variation of the NEEL temperature of the NdBFO NPs.

(3) Modulus spectrum study

Electrical modulus spectroscopy is an advanced technique used to measure the properties of materials. These include the effects of grain boundaries, electrical conductivity, relaxation time, electrode polarization, conduction, and bulk properties. It can also be used to analyze the different electrical processes that occur within a material. To determine M′ and M′′, we used the formula given below:⁴²

As shown in Fig. 11, the value of M′ increases as the frequency increases for all the samples. These results could be explained by the lack of a restoring force. This is the force that moves charges when an electric field is formed. These characteristics show that the material is not significant impacted by electrode polarization.⁴³ The plot of the imaginary part M′′ of the modulus is characterized by the presence of distinct asymmetric peaks in the higher frequency range. As the temperature increased from 537 K to 575 K, the peak value of M′′ declines. This observation indicates that, as the temperature rises, the relaxation time decreases. The shift in the peak position further demonstrates that the samples undergo thermally driven relaxation processes, resulting in charge carrier hopping.^44–46 The Cole–Cole diagram of the modulus was plotted, where a change in its shape was observed. Initially, the diagram showed an S-shaped curve for 6% neodymium, however it gradually transformed into an arc at 10% neodymium. The presence of a single arc indicates that the grain boundary is more important for conductivity than the individual grains.⁴⁷ The plot shape changes when neodymium is substituted, and an incomplete arc appears at higher temperatures. This indicates that the material's relaxation behavior broadens, confirming the presence of non-Debye relaxation. These findings are consistent with those reported in several previous studies.^48–50

Fig. 11 Modulus spectra of Nd_xBi_1−xFeO₃ NPs.

(4) Nanoparticles complex impedance analysis

The study of how grains, grain boundaries, and electrode polarization affect the dielectric properties of a material is commonly performed using complex impedance spectroscopy (CIS). The real part, Z′, and the imaginary part, Z′′, of the complex impedance⁵¹ were measured at different temperatures (537–575 K) and frequencies (1 Hz–1 MHz). Fig. 12 shows the changes in the real Z′ and imaginary Z′′ parts of the complex impedance Z* and the Nyquist plots concerning frequency for the Nd_0.06Bi_0.94FeO₃, Nd_0.08Bi_0.92FeO₃, and Nd_0.1Bi_0.9FeO₃ NPs. As can be seen in this figure, Z′ and Z′′ attain their maximum values at lower frequencies, indicating heightened polarization within the nanoparticles. The observed decline in Z′ and Z′′ values at higher frequencies and temperatures indicates of a decrease in resistance. This abrupt decline is influenced by three primary factors: grain–grain contact, grain boundaries, and electrode interfaces. Collectively, these factors result in a reduction in resistance. This phenomenon could be the underlying cause of the material's enhanced conductivity with temperature at higher frequencies, exhibiting a negative temperature coefficient of resistance (NTCR).^52,53 A single arc is visible in samples containing 6% and 8% Nd, suggesting that grain resistance is the dominant factor. On the other hand, the formation of a double arc in the 537–575 K temperature range for 10% neodymium reveals simultaneous grain and grain boundary contributions to the conduction process. The change from a single to a double arc in the Nyquist diagram emphasizes non-Debye relaxation and is a relaxor effect confirmed to occur at high neodymium concentrations. According to existing literature,⁵⁴ the phenomenon under investigation may be attributed to numerous factors, including, but not limited to, grain orientation and distribution, grain boundary presence, atomic defect distribution, and stress–strain phenomena. This behavior has been observed in rare earth-doped ferroelectric materials. To investigate the potential impact of temperature on the distribution of relaxation times in our samples, Z′′ data is plotted using scaled coordinates, specifically vs. f/f_max, where and f_max represent the peak value and it corresponding frequency of the peak in Z curves. As can be seen in Fig. 13, all peaks converge into a single master curve at different temperatures, suggesting that charges undergo dynamic processes with a consistent activation energy across various timescales. Additionally, plots of the relaxation peak versus frequency show a gradual shift towards higher frequencies as the temperature increases. This further highlights the existence of multiple equilibrium states and the distribution of relaxation times in the prepared samples.⁵⁵ Within the studied temperature range (537–575 K), neodymium does not alter the fundamental nature of the relaxation process, but it does affect the relaxation time value. This reflects the high homogeneity of the material in terms of the dynamics of the charge carriers or dipoles responsible for dielectric relaxation. The corresponding relaxation time τ is obtained through the following equation:where f_max is the frequency at which Z′′ reaches its maximum value. To calculate the activation energy of the relaxation process, τ could be efficiently fitted using the Arrhenius equation:⁵⁶As shown in Fig. 14, the activation energies for the relaxation process of Nd_0.06Bi_0.94FeO₃, Nd_0.08Bi_0.92FeO₃, and Nd_0.1Bi_0.9FeO₃ NPs are 1.18 eV, 0.8 eV, and 1.15 eV, respectively. The fluctuation in these values highlights the impact of neodymium on the relaxation process.

Fig. 12 Impedance spectra of NdxBi_1−xFeO₃ NPs.

Fig. 13 Frequency-dependent scaling behavior of Z′′ at varying temperatures.

Fig. 14 Arrhenius plots of ln (τ) versus 1000/T for the nanoparticle samples.

(5) Transport characteristics of the synthetized NPs

A detailed analysis of the frequency-dependent electrical conductivity of the samples at various temperatures was conducted to investigate their conduction behavior and identify the parameters that influence the conduction process. Fig. 15 shows a plateau region at low frequencies and a dispersion region at higher frequencies. The plateau section indicates long-range DC conductivity, while the dispersion section indicates localized AC conductivity. The increase in conductivity (σ) is due to the presence of thermally activated carriers. The frequency dispersion of electrical conductivity is typically well described by the simple Jonscher equation. However, due to the modification of the relaxation process induced by neodymium, a modified form of the Jonscher equation was employed to accurately represent this behavior, as presented below:⁵⁷

Fig. 15 Variation in electrical conductivity with frequency at selected temperatures for the NPs.

The following is a list of the variables in this equation:

• σ_dc is the conductivity at low frequency

• σ_∞ is the conductivity at high frequency

• τ is a characteristic relaxation time

• A is a temperature-dependent parameter

• n is the power law exponent factor (0 ≤ n ≤ 1)

Fig. 16(a) shows how the parameter n changes with temperature. This helps us to understand the nature of the materials’ electric conductivity. For the Nd 6% and Nd 8% composites, the behavior follows OLPT (overlapping large polaron tunneling). This means that n decreases at low temperatures and then increases at higher temperatures. By contrast, the third composite (Nd 10%) exhibits a conduction mechanism called NSPT (non-overlapping small polaron tunneling), as evidenced by a continuous increase in the n parameter with temperature. Increasing the neodymium (Nd) content leads to a beneficial structural distortion, stabilizing the crystal lattice and reducing defect concentration. This improved structural order decreases oxygen vacancies and defects, modifying the conduction mechanism. Consequently, the charge carriers become more localized due to the distortion, shifting the conduction from OLPT to NSPT. This change reflects a transition to a more thermally activated, hopping-type transport consistent with the stabilized and less defective Nd-doped structure. In addition, the activation energy was calculated using the following Arrhenius relation:

where K_B and σ₀ stand for the Boltzmann constant and pre-exponential factor, respectively. According to Fig. 16(b), the decrease in the activation energy, observed with increasing neodymium, can be attributed to structural changes induced by the insertion of Nd³⁺ into the lattice. The angles and distances of the Fe–O–Fe bonds observed in the previous section are affected by these changes. The hopping barriers for charge carriers are reduced by these changes, which facilitates their mobility and results in a gradual reduction in activation energy. This variation has been reported by several researchers.⁵⁸

Fig. 16 (a) The mechanism of conductivity of NPs. (b) Activation energy from DC conductivity as a function of 1000/T.

In summary, this rise in conductivity dependent on temperature suggests that our samples show temperature-dependent hopping behavior. This indicates that the compounds have a negative temperature coefficient of resistance (NTCR).

(6) NTC thermistor behavior of the nanoparticles

In this section, we will demonstrate the reliability of our synthesized Nd_xBi_1−xFeO₃ NPs (x = 0.06, x = 0.08, x = 0.1) in NTC thermistor applications. Our previous dielectric results indicate that our samples are excellent candidates for thermal sensors, such as thermistors. In the following, we will assess their performance as NTC thermistor candidates by comparing them to traditional ones. Fig. 17 shows the resistance behavior at temperatures between 530 K and 580 K. We observed an exponential decrease in resistance with increasing temperature for each NPs. The development of a highly temperature-sensitive thermistor requires a high thermistor constant (β), a pronounced non-linearity in the temperature coefficient of resistance (TCR), and an extensive sensitivity range. These characteristics can be determined using the following formula:⁵⁹The obtained β values, for Nd_xBi_1−xFeO₃ (x = 0.06, x = 0.08, and x = 0.1) NPs, were around 574 K. This is higher than that published elsewhere⁶⁰ (see Table 4 for comparison). This underscores their potential for use in advanced applications. In addition, the sensitivity parameter, calculated using the Steinhart–Hart equation (eqn (10)), revealed that all synthesized samples showed a promising value, including in the range between −1% and 12%. This wider range is more advantageous and surpasses that of conventional thermistors.

Fig. 17 Thermal sensors parameters of samples ranging in temperature (530–580 K).

Table 4 Comparison of sensitivity values with other materials at T = 574 K

Compounds	β (K)	Ref.
NdxBi1−xFeO3	39 578	This work
Ba3Bi2WO9	8000	61
ZnFe2O4	5820	62
CaTiO3	10 795	63
Dimond thermistor	8012	64

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V. Conclusion

A cost-effective and straightforward sol–gel preparation technique was used to successfully synthesize environmentally friendly multiferroic materials based on neodymium-modified bismuth ferrite nanoparticles. Both experimental and simulation studies, including Rietveld refinement, confirmed the rhombohedral structure with the R3c space group and neodymium-induced intrinsic distortion modification. Our findings revealed that the relaxation behaviour was attributed to Maxwell–Wagner polarization in the dielectric properties. Analysis of the complex electric modulus indicated a conduction process, and impedance studies confirmed the negative temperature coefficient (NTC) thermistor behavior. The transport characteristics of the synthesized materials were observed with a transition from the OLPT mechanism to the NSPT. In summary, structural analysis revealed the enhancement and promotion of intrinsic distortion of the FeO₆ octahedron by the substitution of neodymium, resulting in improved structural stability of the synthesized materials. These structural enhancements make the materials ideal candidates for use in ferroelectric capacitors and ferroelectric random-access memory (FeRAM). Additionally, dielectric measurements revealed a noticeable decline in the NEEL temperature and a modification in the conduction mechanism that renders these materials suitable for spintronic devices, magnetic refrigeration, and magnetic drug targeting. Finally, analysis of thermistor properties showed that the synthesized structures are well-suited for use in NTC thermistors and exhibit higher sensitivity than conventional thermistors. These unique characteristics make the synthesized samples highly promising for many different applications.

Conflicts of interest

There are no conflicts to declare.

Data availability

Detailed data for current research findings and studies is available from the corresponding author upon request.

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