Conduction Mechanisms and Permittivity: Conductivity Correlation in the Gd-Based Perovskite Structure

1. Introduction

Advanced ceramics represent a fascinating material class that has developed various industries due to their unique properties and applications [1, 2, 3, 4]. These ceramics, often engineered at the molecular or nanostructure level, offer exceptional mechanical strength, thermal stability, chemical inertness, and electrical insulation properties. Advanced ceramics are typically based on oxides, carbides, nitrides, and borides of metals. The development of advanced ceramics has been driven by the need for materials that can withstand extreme conditions in the aerospace, automotive, electronics, biomedical, and energy sectors [5]. Manufacturing advanced ceramics often involves sophisticated techniques such as sintering, hot pressing, and chemical vapor deposition to achieve the desired microstructure and properties. Modern technology and industry rely heavily on advanced ceramics, which offer solutions to challenges that traditional materials cannot meet. The ongoing research and development in this field promise further innovations and applications in the years to come, contributing to advancements in areas ranging from electronics to healthcare and beyond [6].

Materials exhibiting colossal permittivity have garnered significant attention in recent years due to their exceptional electrical properties and potential applications in various technological fields. Colossal permittivity refers to a phenomenon where materials exhibit extremely high dielectric constants, often several orders of magnitude greater than those of conventional dielectrics. This remarkable characteristic allows them to store significant amounts of electrical charge under an applied electric field, making them ideal candidates for capacitive devices and energy storage applications [7]. Studying and exploring materials with colossal permittivity have opened new avenues for research in materials science, solid-state physics, and engineering. The exploration of materials with colossal permittivity has become a new frontier in materials science and engineering in recent decades. These materials, characterized by their extraordinary ability to store electrical charge under an applied electric field, hold immense promise for revolutionizing various technological applications. Among the notable examples of such materials are BaTiO₃ (ABO₃), Gd₂CuO₄(ABO₄), and CaCu₃Ti₄O₁₂ (AAʼB₄O₁₂) (Figure 1) [8, 9, 10]. Due to its high relative permittivity, barium titanate (BaTiO₃) is used in dielectric composites [3]. BaTiO₃’s reported results indicate that the εr′ is high but is affected by temperature. The presence of the transition temperature event is generally a problem when discussing these materials’ applications. Recently, calcium copper titanate (CCTO) has been receiving a lot of attention because of its high and stable dielectric permittivity [10].

The exceptional physical characteristics of CaCu₃Ti₄O₁₂ perovskite, such as its elevated dielectric properties, have made it a significant consideration for functional materials. The electrical and dielectric properties of ceramic compounds are influenced by dipolar species and localized charges, which are the parameters that determine strength, kinematics, and charge carrier interactions [10]. The process of coordinating the structural, dielectric, and electrical properties of ceramics to create multifunctional compounds and utilize them in future applications is challenging. The recent proposal of fabrication of ceramic capacitors using CaCu₃Ti₄O₁₂ complex ceramics for their high permittivity (ε′), excellent performance, low dielectric loss, and excellent temperature stability over a large temperature domain have been made due to its motivation [6, 7]. The evolution of materials with colossal permittivity has been marked by continuous advancements in synthesis techniques, characterization methods, and theoretical understanding.

This chapter would open up an opportunity to develop new high dielectric permittivity material (GdCa₂Cu₃O_δ). These efforts to optimize other critical properties such as conductivity, stability, and environmental sustainability [11, 12]. The crystal structure of GdCa₂Cu₃O_δ was discovered recently [11, 12]. GdCa₂Cu₃O_δ is a ternary oxide compound with exceptional dielectric permittivity values, which has been considered for capacitor applications.

2. Ceramics synthesis

The ceramic sample was synthesized using a conventional solid-state reaction (SSR) method. The compositions GdCa₂Cu₃O_δ dictated the weight of the reagent grades Gd₂O₃, CaCO₃, and CuO, which were determined by weighing them. The raw materials were weighed according to the composition GdCa₂Cu₃O_δ and mixed thoroughly in an agate mortar. The powders were pressed into pellets and calcined in the air in a box furnace for 24 h at 800°C. After the pellets were sintered at 950°C in the air for 24 h (Figure 2).

The first step in sintering is the rapid formation of bridges between the particles in contact. Several mechanisms described in Figure 2 explain bridge formation. There is a transport of material from the surface of the grain or the center of the grain joint to the bridges. The diffusion of material can occur through surface, volume, vapor phase, or grain boundaries [13]. Obtaining thermodynamically stable phases at high temperatures through solid-state diffusion is achievable using the solid-state reaction method, also known as the ceramic method [14]. Ceramics are a diverse class of compounds such as monolithic or composite ceramics [15] (Figure 3).

3. Dielectric properties of ceramics

Insulating materials are known for not conducting current in an electric field. However, a microscopic approach to these phenomena allows us to distinguish more subtly between the dielectric aspect.

The nineteenth-century English Michael Faraday studied the ability of a system to store an electrical charge, he is honored with the unit ‟Farad” (symbol F) [16]. The capacitance of a system of two charged parallel plates with electrodes of area s separated by a distance e and filled with a material of relative permittivity ε′_r is given by:

With ε₀ being the dielectric constant of vacuum.

3.1 Different types of polarization mechanism in dielectrics materials

The dielectric constant (ε_r) is the property of a material that can polarize in an electric field with an alternating current. Its expression is as follows (Eq. 2):

ε∗f=ε′f−iε′′fE2

The polarization response of the dielectric to an applied electric field can be represented by the real part εr′of the dielectric constant, which is related to the capacitance of the material. Energy dissipation such as heat or other forms of energy loss is represented by the imaginary part of the dielectric constant εr′′. The imaginary part is a result of the material’s inability to produce real-time polarization when changing the oscillation frequency [17].

The total polarization of a dielectric is the sum of four sources of charge displacement: space charge (or interfacial) displacement (P→i), the orientation of permanent dipoles (P→o), ionic (atomic) displacement (P→a), and electronic displacement P→e [18]:

P→=Pi→+P→o+P→a+P→eE3

Figure 4 shows the contribution of space charge, dipole orientation, ions, and electrons to the overall polarization mechanism in dielectric materials.

The frequency dependence of polarization means that the dielectric constant must also depend on the frequency, as shown in Figure 4. Each polarization mechanism has a characteristic frequency that is limited. The frequency range for space-charge (interface-charge) polarization is generally between 0 and 10² Hz. The accumulation of charge carriers is the name for interfacial polarization, which is a term used to describe Maxwell-Wagner Sillars interfacial polarization [19]. Dipolar polarization occurs when a material initially possesses dipoles that are randomly oriented with an overall dipole moment of zero. Under the effect of an applied electric field, the dipoles orient according to the field. This results in the appearance of a non-zero dipole moment. This polarization is strongly influenced by temperature and frequency [20]. Ionic polarization is the physical transport of ions (positive and negative) in materials. The cations and anions become attracted to opposite directions when an electric field is applied. A relatively large ionic displacement is created, which can result in high dielectric constants in ceramics commonly used in capacitors [21]. The electronic polarization occurs in all atoms when an electric field is applied. The occurrence of electronic polarizations is caused by the distorted negative electrons and positive nuclei in an atom, which are in opposition to the external electric field, resulting in electric dipole moments, which occur at infrared frequencies. Electrons can follow high-frequency fields up through the optical range because of their small mass [22].

3.2 Polarization and relaxation mechanisms in dielectric materials GdCa₂Cu₃O_δ

By measuring the real and imaginary parts of the impedance, we can calculate the frequency-dependent permittivity:

ε′f=e2πfε0s−Z′′fZ∗2E4

ε′′f=e2πfε0sZ′fZ∗2E5

The relative permittivity and capacitance of GdCa₂Cu₃O_δ were analyzed by measuring the frequency (100–1.3 MHz) and the results are shown in Figure 5(a, b).

Figure 5(a) shows that the low-frequency range in the dielectric GdCa₂Cu₃O_δ ceramic has an elevated permittivity plateau that is a result of the grain boundary (GB) contribution. The intrinsic bulk (grain) response of the studied compound is represented by the high-frequency plateau of the permittivity. At low frequencies, the data indicate that the permittivity from the grain boundary remains constant (εr′=104) over a wide frequency range, but drops in the MHz region. Below 270 K, it is seen to have a signature of a second high-frequency plateau that εr′= 10²) is visible in the high-frequency range. In the framework, the grain boundary and grain capacitance plateaus shown in the capacitance plot versus frequency (Figure 5(b)) are both only an approximation for C_gb (C_gb = 10⁻¹¹ F) and C_g (C_g = 10⁻⁹ F) respectively. The dielectric behavior of the investigated ceramic is comparable to those reported in the CCTO compound [23, 24].

The reported results in Figure 6 indicate that the dielectric characteristics of ceramic GdCa₂Cu₃O_δ are highly significant, with a low dielectric loss tanδ=0.2 at 1 KHz. Furthermore, we discovered the existence of tanδ relaxation moves to high-frequency values but is suppressed after 415 K. The density of the material and the grain boundary resistance are two factors that affect the dielectric loss value [25]. The ceramic’s low dielectric loss, elevated dielectric constant (10⁴), and stability over a wide temperature range make it a suitable candidate for various capacitor types [26]. Various parameters, like annealing temperature, substitution, and nanoparticle addition, can be used to enhance the dielectric performance of ceramic materials [27, 28].

At room temperature, both samples showed high permittivity (10⁴) with low losses (tanδ = 0.2). This behavior is also present in a renowned high permittivity material, CaCuTiO [29]. The activation of dielectric relaxation is indicated by the dielectric loss peak which accompanies the decrease in relative permittivity. The frequency decreases result in the appearance of dielectric loss peaks (Figure 7(a)). A heterogeneous microstructural system is responsible for these behaviors, which can be represented by a sequence of semiconductor grains and insulating grain boundaries. Impedance spectroscopy can define and characterize each of these contributions. According to the literature, the ceramic has an electrically heterogeneous microstructure that is caused by the significant variance in grain and boundary resistances (Figure 7(b)) [30]. The GdCa₂Cu₃O_δ ceramic’s dielectric constant stability, order, and losses are controlled by the formation of internal barrier layer capacitance, which is the main responsible factor. It has been established that impedance spectroscopy detects different dielectric relaxation processes, including those found at the GB and bulk regions. In the simplest case, the brickwork layer model describes each grain’s boundary or grain type contribution as a single RC circuit element that consists of a resistor and capacitor in parallel. The RC model works particularly well for insulators like dielectrics, where the capacitor describes the material’s charge-storage capability and the parallel resistor describes the leakage current due to some un-trapped carriers bypassing the ideal charge-storage element [31]. In reality, the dielectric relaxation processes in electrical ceramics are usually not ideal, and, therefore, experimental data cannot usually be modeled by ideal RC elements. To fit the data to an adequate equivalent circuit model, the non-ideality in a particular dielectric relaxation can be accounted for in the element by connecting a constant phase element (CPE) by replacing the ideal capacitor (Figure 7(c)) [32].

Generally, the variation of Z* as a function of frequency is given by the following relationship:

Z∗CPE=1QjωαE6

The complex impedance depends strongly on the pseudo-capacitance Q and the exponent α (0 < α <1) [33]. For a pure capacitance α = 1 and a pure resistance α = 0 (Figure 8). It is possible to determine the phase angle (angle of depression) and capacitance (C) by analyzing the value of the exponent α.

β=1−απ2E7

C=Q.R1−α1αE8

The high relative permittivity of this material is associated with an effective internal insulating barrier or IBLC model, resulting in strong interfacial polarization [34]. Therefore, the high permittivity levels in the tested compound are related to the average area of the GB layer, the thickness of the GB layer, and the charge carriers that have accumulated at the GBs. Based on the IBLC dielectric model, conducting crystalline regions, insulating barriers, and grains/sub-grains effects are all factors that affect the dielectric loss of the ceramic being studied. This phenomenon is primarily attributed to various mechanisms of polarization and hopping (Figure 7(d, e)). The ceramic has a complex microstructure where the grain boundaries play a crucial role. The interfaces between grains can form capacitor-like structures due to the accumulation of charges, contributing to the overall dielectric constant. Based on the IBLC model, total relative permittivity (ε_r) is related to grain boundary permittivity εjg, grain thickness (l_g), and grain boundary thickness (l_gb) by the following relationship [35]:

εr=εgblglgbE9

According to the above relationship, the relative permittivity of a ceramic can be improved by reducing the thickness of grain boundaries and increasing grain size. In this case, Maxwell-Wagner polarization can significantly enhance the dielectric response (the high dielectric constant and low loss tanδ).

Other research groups have attributed the colossal dielectric response to the effect of hopping polarization. In this case, free electrons cause a distribution of space charges (oxygen vacancies V0∘∘ and electrons) at the grain boundaries [36]. In our case, such conduction may be governed by electron transport based on mixed valence Cu^2+/Cu³⁺ (p-type) [37]. The following relationships can express the process of Cu³⁺ reduction to Cu²⁺ and the appearance of oxygen vacancies in the compound GdCa₂Cu₃O_δ [38, 39].

O0x→V0∘∘+e′+1/2O2↑E10

Cu3++1e′→Cu2+E11

The dielectric constant of ceramic can fluctuate due to changes in Cu density, different oxidation states because of the Cu element, and particle size variations.

3.3 Analytical models of uniform dielectrics

To characterize the properties of dielectric materials, the Debye equation is one of the most classical models, which explains relaxation phenomena in ceramics well. In the ideal case, the dipoles have the same relaxation time τ. These dipoles do not interact with each other.

The complex permittivity (ε*) of the Debye function can be expressed as follows [40].

ε∗=ε∞+εs−ε∞1+iωτE12

where ε_∞ is the dielectric constant at high frequencies, ε_s is the static dielectric constant at low frequencies, ω is the angular frequency, and τ is the relaxation time.

For the Debye model case, the Nyquist diagram corresponds to a semicircle with a maximum loss factor ε′_max at the top of the circle obtained for ωτ=1. In this case, ε′_max is related to ε_s and ε_∞ by the expression as follows:

ε′max′=εs−ε∞2E13

ε_s and ε_∞ represent the intersections of the circle with the x-axis (Figure 9(a)). The frequency spectrum obtains the dielectric constant results according to the Debye equation (Table 1), as demonstrated in Figure 9(b). A Debye-type dielectric relaxation is characterized by a steady ε′_r value that then drops rapidly, while ε′′_r shows a broad peak that passes through a maximum at relaxation frequency f_r (Figure 9(b)).

Modèle	ε′	ε″
Modèle de Debye	ε′=ε∞+εs−ε∞1+ω2τ2	ε″=εs−ε∞ωτ1+ω2τ2
Modèle Cole-Cole	ε′=ε∞+εs−ε∞1+ωτ1−αsinπα21+2ωτ1−αsinπα2+ωτ2−2α	ε″=εs−ε∞1+ωτ1−αcosπα21+2ωτ1−αsinπα2+ωτ2−2α
Modèle de Havriliak Negami	ε′=ε∞+εs−ε∞cosβφ1+2ωταcosπα2+ωτ2αβ2	ε″=εs−ε∞sin−βφ1+2ωταcosπα2+ωτ2αβ2

Table 1.

Variation of ε′ and ε′′ as a function of temperature using different models.

Generally, in a Debye system, while in Maxwell- Wagner system, at low-frequency region. The origin of heterogeneity-generated Maxwell-Wagner relaxations is likely due to the surface’s roughness in a Schottky-diode mechanism or from the distribution of grain and grain boundary sizes in ceramic compounds [41]. Furthermore, the omnipresent hopping conductivity commonly leads to the typical power-law frequency dependences of the UDR. The Debye equation describes the phenomenon of relaxation for a pure dielectric. To reflect this mixed behavior, the contribution of the static conductivity σ_dc can then be inserted into the Debye equation, which becomes the modified Debye equation:

ε∗=ε∞+εs−ε∞1+iωτ−iσdcωε0E14

The σ_dc/(ε₀ω) term in the loss expression corresponds to the contribution of DC conductivity (σ_dc).

The imaginary part of the permittivity versus frequency at room temperature as shown in Figure 10. In this system, the dielectric loss εr′′ασdcf decreases linearly with increasing frequency. The movement of charge carriers within the sample becomes significant and results in an increase in dielectric losses.

The Debye relaxation model accurately defines the dielectric behavior of various gaseous and liquid materials that contain dipolar molecules [42]. However, most solids have a loss peak much broader and cannot be expressed by a single relaxation time, but instead by a distribution of relaxation times. Since the Debye model cannot correctly predict the dielectric response of numerous materials, several relaxation functions have been developed from the Debye model. For this reason, Cole-Cole, Davidson-Cole, and Havriliak Negami have proposed models to account for this relaxation time distribution [43]. Furthermore, empirical exponents have been used to increase the number of relaxation times. To fit the equivalent circuit to the ideal Debye behavior, a constant phase element (CPE) is suggested.

According to the fitting result in Figure 11, the dielectric response of our sample cannot be explained by Debye’s model. The classical Debye equation does not fully explain the relaxation process in materials with multiple dispersions or possibly multiple defects. According to the Cole-Cole and Havriliak Negami models, the complex dielectric permittivity is given by the following relationship [44]:

ε∗=ε∞+εs−ε∞1+iωταβE15

The factors α (0 ≤ α ≤1) and β (0 ≤ β ≤ 1) control the distribution width of the relaxation time and the asymmetry of the loss peak, respectively. The case where β = 1 corresponds to the Cole-Cole model. The expression of ε′ and ε′′ as a function of temperature using different models is summarized in Table 1.

The experimental and fitting (Debye, Cole-Cole, and Havriliak Negami) variation of ε′′ as a function of ε′, ε′, and ε′′ versus frequency is shown in Figure 11(a). From Figure 11(b, c), the fitting of the Cole-Cole model and the Havriliak Negami model is significantly better than the Debye model.

The fitting value of the characteristic parameters ε_s, ε_∞, τ, σ_dc, α, and β of the three models are indicated in Table 2. Therefore, the best model to characterize the dielectric relaxation of GdCaCuO is the Havriliak Negami model with characteristic parameters α = 0.9 and β = 0.85.

Model	εs	ε∞	τ (s)	α	β	σdc (S.cm−1)
Debye	5171	41	3.10−5	—	—	—
Cole-Cole	5165	41	3.10−5	0.86	1	9.10−6
Havriliak Negami	5162	41	4.10−5	0.9	0.85	9.10−6

Table 2.

Parameters ε_s, ε_∞, τ, σ_DC, α, and β of dielectric models.

The ε_∞ generated by the dipole polarization is due to the fast oscillation of the alternating electric field, which results in the dipole following and the dipole remaining essentially in motion. The β factor is caused by one or more types of polarization.

4. Conduction mechanisms in dielectric materials

The microstructure of oxides significantly impacts their electrical transport properties and relaxation phenomena. The microstructure of perovskites frequently includes grains separated by grain boundaries. This morphology greatly influences the conductivity of perovskites and the activation of most conduction mechanisms. Ceramic structures have temperature-dependent conductivity variations influenced by various factors, including percolation thresholds, inter-grain or inter-particle distance, and the insulating layer or energy barrier. Conductive grains are arranged in particle chains separated by an insulating layer or barrier to ensure the transfer of charge carriers between conduction sites. Both factors that are characteristic of the insulating barrier have a profound effect on the electrical transport mechanisms. For perovskites, transport phenomena are also explained in terms of the resistance of grains (Rg) and grain boundary (Rgb).

Figure 12 depicts the variation of resistance (R) with temperature for grain and grain boundary. The graph shows that the resistance (R) for the sample decreases with temperature indicating the thermal activation of electrical conduction. The semiconductor nature of the synthesized ceramic was bolstered by the thermal decrease in electrical resistance.

Figure 12 indicates that the Rg resistance is usually lower than the R_gb resistance, which means that the conduction phenomena in the studied compound are mostly dominated by the intergranular domains [45].

Electrical conductivity is highly influenced by frequency and temperature variation in many materials. The double Jonscher law describes each conductivity spectrum in the present work [46].

The term σ_dc explains the variation of DC electrical conductivity, A and B two constants. Two frequency exponents, s₁ and s₂ (in the range 0–1), describe the degree of interaction between mobile charges and provide information about the origin of electrical conduction.

The conductivity (DC) and temperature dependences of the frequency exponents (s₁ and s₂) for the sample at the selected temperatures are depicted in Figure 13.

In terms of DC conductivity, Mott considered that the transport properties in dielectric materials are achieved by thermal activation of the self-caught small polarons into intermediate states [47]. The Small Polaron Hopping (SPH) mechanism is the appropriate model for describing DC conductance at high temperatures. In Figure 14 (bottom and left), the evolution of ln (σ_dc × T) is shown in relation to the inverse of the temperature (1000/T) for the material. The linear curve is able to approve the thermal activation of the SPH mechanism at elevated temperatures. The temperature dependence of DC conductivity can be described by the following relation at this temperature range.

The parameter σ₀ is a pre-exponential factor that is independent of temperature. The activation energy E_a = 0.24 eV is required to move the charge carrier between two conducting sites. Dielectric relaxation occurs because oxygen vacancies have a lower activation energy than dipoles due to the same reorientation process [48].

At lower temperatures, the conduction mechanisms of perovskite are typically impacted. The linear variation shows that the VRH conduction mechanism dominates in the lower temperature range, as confirmed by the evolution of ln (σ_dc) as a function of T^−1/4 (Figure 14 (top and right)). The VRH model defines the temperature dependence of electrical conductivity based on the Mott-VRH law [49].

The parameters σ0 = be 1.07x10⁷ S.cm⁻¹ and T₀ = 1.3x10⁸ K have been determined.

Jonscher’s power law can be used to describe the frequency-dependent dielectric response of localized carriers. The conductivity spectra for the studied compound were deduced from the dielectric plots using the universal dielectric response UDR model [50]:

Therefore, plotting f.εr′ as a function of frequency at a given temperature should result in a straight line with a slope of s₁ and s₂. In Figure 15, the lines show the fitting curves of experiments based on Eq. 19.

The thermal activation of the electrical transport mechanism is confirmed by the observed variation in the’s1 and s2 values in Figure 16, which shows that the temperature activates it. It also proves that hopping and tunneling are the potential mechanisms that govern transport properties [51]. According to the linear fit, the values of exponents s₁ and s₂ are depicted in Figure 16.

The value of s₁ confirms that the Quantum Mechanical Tunneling (QMT) conduction process governs electrical conductivity [52]. In the QMT model, s₁ is independent of temperature, but rather dependent on frequency. The QMT model is founded on particle transport through a network of interconnected chains of particles or sites. The particles are separated by a thin layer that prevents them from having direct contact with each other (inset Figure 16(a)). The insulating layer’s thinness allows for electron transfer from one conducting particle to another through tunneling.

The second frequency exponent (s₂) temperature dependence validates the influence of the correlated barrier hopping mechanism on the transport properties of the investigated compound. When the temperature increases, the frequency exponent decreases in the CBH model. This transport mechanism is influenced by frequency and temperature. The conduction is also supported by chains of particles or grains (inset Figure 16(b)). Specifically, electrons move between two conductive sites that are separated by distances, called inter-site or inter-particle distances, which are much larger than the tunnel barrier (tunnel characteristic distance).

5. Advanced ceramics for energy applications

Advanced ceramics play a crucial role in various energy applications due to their unique properties that can withstand extreme conditions such as high temperatures, corrosion, and wear. Advanced ceramics with additional functionality can greatly increase their potential impact on energy and environmental technologies. They are utilized in energy-related applications, including energy conversion, photovoltaics, energy storage, supercapacitors, lithium-ion, and capacitors (Figure 17) [53].

The discovery of a new ceramic (GdCa₂Cu₃O_δ) has recently drawn much attention due to its extremely high dielectric constant (ε′ = 10⁴) and low loss (tanδ = 0.2). This material can potentially be utilized in various microelectronics device applications, including multilayer capacitors (MLCC) [54]. Furthermore, for each frequency value examined, the permittivity of the studied ceramic reveals a temperature-independent region with elevated values of ε_r′ and excellent temperature stability. In addition to the high relative permittivity value (ε_r′) with good temperature stability (Figure 18(a)) and a low dielectric loss factor (tanδ), the temperature coefficient of the dielectric constant (Δε′) is a crucial factor to consider when analyzing the dielectric performance of commercial electronic devices. The relationship below was used to calculate the variation of Δε′(T) for the GdCa₂Cu₃O_δ ceramic at select frequency values.

where ε_T′ and ε₃₀′ are ε′ at T °C and 30°C, respectively.

Over a wide temperature range (−30–125°C), the Δε′(T) value of the GdCa₂Cu₃O_δ sample is less than ±15% (Figure 18(b)). Therefore, it is recommended that the GdCa₂Cu₃O_δ ceramic has potential use for the EIA code Y5R and Y6R capacitors. Y5R and Y6R capacitor codes have been designated by the American Electronic Industries Alliance (EIA) Standards (Table 3). ‘Y’ specifies a minimum operating temperature of – 30°C, whereas the ‘5’ and ‘6’ specify maximum temperatures of 105°C and 125°C, respectively [55].

6. Conclusion

To conclude, studying the correlation between polaron conduction and colossal permittivity in GdCa₂Cu₃O_δ (ε′ = 10⁴) has provided valuable insights into the electrical properties of the material. We have established a clear link between these two phenomena using a modified equivalent circuit based on the internal barrier layer (IBL) model. This correlation highlights the complex interaction between the structure and the dielectric properties.

To comprehend the dielectric polarization of GdCa₂Cu₃O_δ, it is necessary to examine the various mechanisms that contribute to its high dielectric constant and its behavior at various temperatures. The defect chemistry of the compound can explain electronic heterogeneity in GdCa₂Cu₃O_δ ceramics, which consist of insulating GB and semiconducting grains regions. Two distinct defect mechanisms may explain the giant dielectric response. According to the Maxwell-Wagner behavior observed, the high dielectric constant is caused by enhanced interfacial polarization rather than relaxation polarization.

By using the distribution function, a hopping conduction model of localized electrons can successfully explain DC conductivity (by variations in Cu content). It has been proven that dielectric relaxation is strongly linked to the hopping electrons in the localized states. The small polaron hopping processes (SPH) at high temperatures and variable range hopping (VRH) processes at low temperatures were responsible for the electrical transport. The double Jonscher power law has been utilized to analyze the conductivity spectrum. The frequency exponent’s variances are responsible for describing the AC electrical characteristics of the studied system by activating correlated barrier hopping (CBH) and quantum mechanical tunneling (QMT) conduction processes.

The findings contribute to both fundamental understanding and practical applications of GdCa₂Cu₃O_δ in advanced electronic devices, sensors, capacitors, and related technologies. Future research should continue to explore how these insights can be harnessed to optimize material design and enhance performance in various electrical and energy storage applications. By further refining our understanding of polaron dynamics and permittivity mechanisms in GdCa₂Cu₃O_δ, we can pave the way for innovative solutions that leverage its unique electrical properties. Astonishingly, GdCa₂Cu₃O_δ ceramic specimens sintered at 950°C exhibited high ε_r′ with temperature stability of <±15% over a wide range of −30–105°C. Therefore, this sample is proposed to be a promising material for the EIA code Y5R, and Y6R capacitors.

Acknowledgments

This work was supported by Faculty of Science of Bizerte, University of Carthage, Tunisia. Laboratory of Physics of Materials: Structure and Property.

Conflict of interest

The authors whose names are listed immediately below certify that they have no affiliations with or involvement in any organization or entity with any financial interest in the subject matter or materials discussed in this manuscript.