Enhanced optical, electrical and charge transport properties of NaCaP₃O₉ ceramics for emerging advanced technologies

Abstract

Phosphate compounds are promising for next-generation optoelectronic and electronic applications due to their versatile structures and properties. In this work, NaCaP₃O₉ (NCPO) ceramics were synthesized by a conventional solid-state method and crystallize in a pure triclinic phase (space group P1̄), as confirmed by XRD and structural refinement. FTIR analysis verified the structural integrity through characteristic vibrational modes. Optical studies revealed a wide direct band gap of about 3.95 eV, highlighting the suitability of NCPO for ultraviolet optoelectronic applications. Dielectric and electrical investigations over wide temperature and frequency ranges demonstrated semiconducting behavior with a negative temperature coefficient of resistance. Impedance and electric modulus analyses indicated grain-dominated conduction and non-Debye relaxation behavior. The frequency-dependent conductivity follows Jonscher's law, and charge transport is governed by a thermally activated correlated barrier hopping mechanism with an activation energy of ∼0.36 eV. The estimated thermal sensitivity constant (β ≈ 3597 K) and low stability factor (SF ≈ 1.5) suggest strong thermistor performance and stable electrical properties. Overall, this study enhances the understanding of the electrical and dielectric behavior of NCPO and underscores its potential for advanced thermistor, sensor, and optoelectronic device technologies.

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1. Introduction

Over the past century, the growing demand for advanced technologies with efficient energy usage for modern electronic and industrial applications has become a major focus of scientific research.^1–3 Due to their remarkable performance, ceramic materials have emerged as promising alternatives, delivering solutions that lower environmental impact while increasing energy density, often in compact dimensions and at lower cost.^4,5 Recently, phosphate-based compounds, known for their structural versatility, have attracted considerable attention due to their exceptional physical properties such as high electrical conductivity, intriguing magnetic behavior, and elevated dielectric constants.^6–8 These features make them ideal candidates for various advanced technologies, including ceramic capacitors,⁹ microwave components,^10,11 next-generation 5G/6G communication networks,¹² and sensors applications.^6,13

The search for advanced sodium-based cationic conductors with favorable electrochemical characteristics has intensified in recent years, driven by their potential application in future Na-ion rechargeable battery technologies. Several candidates have been studied, including NaFe(PO₃)₃,¹⁴ Na₃V₂(PO₄)₃,¹⁵ NaZr₂(PO₄)₃,¹⁶ and Na₂CoP₂O₇.¹⁷ Structurally, these materials are composed of frameworks that combine PO₄ tetrahedra with MO_n polyhedra (where M denotes a transition metal and n ranges from 4 to 6). This results in complex connectivity patterns that form tunnels and cavities throughout the structure. The geometry and dimensions of these tunnels directly influence the movement of cations within the material. Therefore, exploring the crystal structures of such materials is essential for gaining insights into their ionic transport mechanisms and optimizing their performance.

In recent years, calcium-based phosphate materials have attracted increasing attention across a wide range of technological fields. Their structural versatility and intrinsic biocompatibility make them particularly suitable for advanced bio-materials and biomedical applications.^18–21 Beyond these areas, growing interest has emerged in the use of calcium phosphates for energy-related technologies, where they have shown promising performance in energy storage systems, such as batteries, as well as in energy conversion devices.²² Among the various synthesis approaches for phosphate-based materials, the conventional solid-state reaction technique remains the most prevalent. Its widespread adoption is attributed to its operational simplicity, low production cost, and suitability for large-scale synthesis.^6,23,24 Additionally, it supports the formation of phase-pure and highly crystalline compounds, which are key factors in achieving reliable and enhanced material performance. Within this context, considerable attention has been directed toward the NaMP₃O₉ family, where M denotes divalent or trivalent metal ions such as Fe, Mn, Ca, Sr, Co, Zn, and Mg.^11,25–29 Despite progress in their structural characterization, however, the electrical and dielectric behaviors of these materials remain insufficiently explored. This lack of data has spurred efforts to probe their ionic transport mechanisms at the microscopic level. Impedance spectroscopy has emerged as a powerful technique in this regard, offering detailed insights into the distinct roles of grains, grain boundaries, and electrode interfaces in governing charge transport. Moreover, these findings can be effectively linked to morphological and structural attributes, as identified by the impedance spectroscopy technique.³ Beyond this, impedance analysis enables evaluation of the negative temperature coefficient (NTC) of resistance observed in certain pyrophosphate compounds, positioning them as promising candidates for applications such as thermal regulation, temperature sensing, inrush current protection, and accurate thermal monitoring.^30,31 Among the NaMP₃O₉ family, NaCoP₃O₉ is the only compound synthesized via the solid-state method that has been investigated by impedance spectroscopy. It has demonstrated promising performance, exhibiting an electrical conductivity of 1.01 × 10⁻⁵ Ω⁻¹ cm⁻¹ at 753 K and an activation energy of 1.1 eV, as derived from Arrhenius analysis.³² This compound has also gained attention as a potential cathode material for sodium-ion battery applications.^25,33

NaCaP₃O₉, a member of the metaphosphate family, crystallizes in the triclinic system with space group P1̄.²⁹ It is isostructural with NaSrP₃O₉, and their triclinic forms represent the only polymorphs reported for these compounds, highlighting the unique stability of their linear polyphosphate chains. The relative ionic radii of M⁺ and M²⁺ cations strongly influence the degree of polymerization in these metaphosphates. To date, Eu³⁺-doped NaCaP₃O₉ exhibits strong photoluminescence with tunable emission, making it promising for high-performance white light-emitting diodes (WLEDs) and other luminescent devices.^34,35 However, its electrical and dielectric properties have not yet been explored, underscoring the novelty and significance of the present study.

The objective of this study is to provide the first comprehensive investigation of the structural, optical, and electrical properties of NaCaP₃O₉ (NCPO) ceramic using the conventional solid-state reaction technique. The relatively low sintering temperature (600 °C) and the affordability of the constituent materials make this ceramic particularly attractive. A multi-technique approach combining X-ray diffraction (XRD), infrared (IR) spectroscopy, UV-visible spectroscopy, and complex impedance spectroscopy is employed to characterize the material. Special emphasis is placed on evaluating the optical parameters relevant for optoelectronic applications. Furthermore, impedance spectroscopy confirms the NTCR behavior and reveals a thermally activated conduction process. To deepen our understanding of charge transport, frequency- and temperature-dependent conductivity data are analyzed. The interplay between conduction and relaxation processes is also explored using the Kohlrausch–Williams–Watts (KWW) model applied to the electric modulus formalism.

2. Experimental work

2.1 Material preparation of NaCaP₃O₉ (NCPO)

The NaCaP₃O₉ sample was synthesized using a conventional solid-state reaction method. High-purity precursors NH₄H₂PO₄, CaCO₃, and Na₂CO₃ were accurately weighed according to the required stoichiometric ratios. The methodology of synthesis involved several steps, as depicted in Fig. 1.

Fig. 1 The synthesis steps for NCPO via the solid-state method.

The powders were thoroughly mixed in an agate mortar at room temperature to ensure uniformity. The resulting mixture underwent dry grinding for one hour, followed by wet grinding for an additional hour. This process yielded a well-homogenized powder. The blended powder was calcined in a programmable furnace, where the temperature was gradually raised to 300 °C and maintained for 6 hours. After cooling, the calcined material was finely reground for 2 hours to obtain a lump-free texture. Cylindrical pellets were then formed using a uniaxial press and subsequently sintered at 600 °C for 3 hours to promote densification and phase formation. The final product consisted of well-sintered pellets, ready for structural and functional characterization.

2.2 Analysis and characterization

2.2.1 X-ray diffraction (XRD). The phase composition and crystallographic structure of the NCPO compound were identified using a PANalytical X’Pert PRO X-ray diffractometer. The diffraction patterns were recorded over an angular range of 10°–70° with a step size of 0.02° and a dwell time of 1 s per step, using Cu Kα radiation (λ = 1.5406 Å). Structural parameters such as unit cell dimensions and phase purity, were extracted by performing Rietveld refinement using the FullProf Suite,³⁶ allowing for an accurate and detailed structural analysis of the material.

2.2.2 Spectroscopic measurements. Infrared absorption spectra were recorded using a PerkinElmer FTIR-100 spectrometer in the wavenumber range of 400–1300 cm⁻¹, with a spectral resolution of 4 cm⁻¹ and 32 accumulated scans. This technique was used to identify the vibrational modes of phosphate groups and confirm the molecular structure of the compound by analyzing characteristic bond vibrations.

The optical properties of the material were studied using a Shimadzu UV-Vis-NIR spectrophotometer (model 3100), operating in the spectral range of 200–800 nm.

Electrical measurements were conducted by coating the opposing faces of the sintered ceramic pellets with silver electrodes through vacuum thermal evaporation, ensuring effective electrical contact. The impedance data were collected using a Solartron 1260 analyzer, covering frequencies from 0.1 Hz to 1 MHz and temperatures ranging from 393 K to 633 K, within this interval, the electrical response exhibits thermally activated behavior with well-defined relaxation and conduction mechanisms. This setup enabled the investigation of electrical transport processes and dielectric relaxation behavior within the material. This allowed for the investigation of electrical transport processes and dielectric relaxation behavior within the material.

3. Results and discussion

3.1 Structural, SEM description, and vibrational characterization

The X-ray diffraction pattern of the synthesized NCPO compound is presented in Fig. 2(a). The presence of sharp, well-defined peaks confirms the material's high crystallinity. By applying Rietveld refinement, the diffraction peaks were successfully indexed to a triclinic crystal system with space group P1̄, indicating the successful formation of a single-phase NCPO structure. Table 1 summarizes the crystallographic parameters determined through structural refinement. The obtained goodness-of-fit value (χ² ≈ 1.01) reflects a strong correlation between the experimental data and the calculated diffraction pattern, confirming the reliability of the refinement process. Moreover, the extracted lattice parameters show excellent consistency with values previously reported in the literature.²⁹

Fig. 2 (a) X-ray diffraction (XRD) pattern of NCPO recorded at room temperature. (b) The unit cell of NCPO.

Table 1 Refined structural parameters of the NCPO compound at room temperature

Sample	NaCaP3O9
Structure	Triclinic
Space group	P1̄
DSC (nm)	56.886
Unit cell parameter
a (Å)	6.735 (6)
b (Å)	6.950 (6)
c (Å)	7.642 (4)
V (Å^3)	3160.820
α (°)	83.502
β (°)	81.260
γ (°)	82.804
Agreement factors
Rp (%)	29.40
Rexp (%)	37.77
Rwp (%)	38
RF (%)	1.665
RBragg (%)	2.187
χ^2	1.01

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According to JCPDS file No. 23-0669,³⁹ the unit cell structure of NCPO, shown in Fig. 2(b), features infinite (PO₃)ⁿ⁻_n chains formed by phosphate tetrahedra with two short and two long P–O bonds and an average P–O–P angle of ∼134.8°. These chains are folded to accommodate Ca²⁺ ions, which adopt distorted dodecahedral coordination with eight oxygen atoms, forming edge-sharing chains perpendicular to the phosphate backbone. Na⁺ ions occupy highly distorted octahedral (NaO₆) sites, each coordinated to six terminal oxygen atoms from adjacent phosphate chains. NaO₆ octahedra link parallel Ca–O chains, reinforcing the rigidity and stability of the triclinic NCPO framework.

The mean crystallite size was estimated using Scherrer's formula,³⁷ based on the full width at half maximum (FWHM) of the (2̄01) diffraction peak located at 2θ = 30.97° (refer to Fig. 2(a)):

In this equation, λ denotes the wavelength of the incident X-ray, β corresponds to the full width at half maximum (FWHM) of a prominent diffraction peak (expressed in radians), θ is the associated Bragg angle, and k is the Scherrer constant, generally taken as 0.9.³⁷ Based on this calculation, the crystallite size of NCPO is estimated to be around 56.886 nm, confirming that the synthesized phosphate material exhibits nanoscale dimensions.

The SEM micrographs were employed to analyze the surface morphology of the synthesized phosphors. Fig. 3(a) displays the SEM image of the NCPO sample, revealing agglomerated particles with mixed sizes. The observed grain size lies in the micrometer range rather than the nanometer scale, which can be attributed to the aggregation of smaller particles during the annealing process. This agglomeration occurs because smaller particles, possessing higher surface energy, tend to merge into larger ones to minimize their overall surface free energy. The resulting particle sizes are thus on the order of a few micrometers. The morphology and particle dimensions of the obtained phosphors suggest their potential suitability for white LED (WLED) applications.³⁸ Fig. 3(b) presents the Lorentz-fitted grain size distribution histogram. Image analysis indicates an average grain size of approximately 1 µm. The grain size determined from SEM images is larger than that estimated from XRD analysis, a difference that can be explained by the fact that each grain observed in the SEM micrograph is composed of several crystallite domains.³⁹

Fig. 3 (a) SEM micrographs for NCPO, (b) particle size distribution.

FTIR spectroscopy plays a crucial role in the structural characterization of the NCPO compound by revealing detailed insights into its vibrational properties. The infrared spectrum of the synthesized sample is shown in Fig. 4, and the corresponding absorption bands, along with their tentative assignments, are summarized in Table 2. These assignments are based on previously reported data^29,40,41 and follow a typical order of metaphosphate vibrational modes, arranged by decreasing frequency. The FTIR spectrum exhibits characteristic absorption bands associated with polyphosphate groups, including asymmetric PO₂ stretching modes (ν_as) at 1288, 1262, and 1212 cm⁻¹, and symmetric PO₂ stretching modes (ν_s) at 1150, 1110, and 1085 cm⁻¹. Additionally, the bands observed within the 702–1045 cm⁻¹ range are attributed to symmetric and asymmetric P–O–P stretching vibrations. The lower frequency region, spanning from 446 to 580 cm⁻¹, corresponds to bending modes involving P–O and O–P–O linkages. These regions show medium to strong absorption intensities, confirming the presence of linear (PO₃)ⁿ⁻_n chains rather than cyclic phosphate units.

Fig. 4 Experimental infrared spectrum of NCPO at room temperature.

Table 2 IR vibrational frequencies (cm⁻¹) for NCPO compound at room temperature

Wave number (cm^−1)	Assignments
1288	υas (PO2)
1262
1212
1150	υs (PO2)
1110
1085
1045	υas (POP)
897
778	υs (POP)
745
702
580	P–O & O–P–O bend
561
531
509
476
454
446

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3.2 Optical characteristics

UV-Visible spectroscopy serves as a fundamental technique for investigating the optoelectronic properties of semiconducting materials, enabling detailed insights into their band structure and electronic transitions. The optical behavior of the synthesized sample was systematically analyzed through its absorbance spectra to elucidate its electronic structure. Prominent absorption features were identified, allowing precise determination of the optical band gap as well as the Urbach energy, which reflects the degree of structural disorder. Additionally, critical optical constants including the extinction coefficient, penetration depth, and refractive index were evaluated as functions of photon energy. This comprehensive analysis provides crucial information for assessing the material's potential suitability and performance in advanced optoelectronic applications.

3.2.1 Analyses of UV-VIS and optical energy band gap (E_g). The experimental UV-vis absorption spectrum of the NCPO compound, measured at room temperature over the wavelength range of 200–800 nm, is shown in Fig. 5(a). The absorption spectrum exhibits a broad band with a peak near 250 nm, which is attributed to a charge transfer transition from O²⁻ to Ca²⁺ ions.⁴² Additionally, the (PO₃)ⁿ⁻_n group shows a charge transfer from oxygen legends (O²⁻) to the central phosphor atom (P⁵⁺), consistent with previous findings.⁴³ The observed UV absorption suggests promising optical activity, attributed to charge-transfer-driven electronic transitions across the bandgap. Various techniques can be employed to precisely determine the bandgap energy (E_g).

Fig. 5 (a) Variation of the absorbance versus wavelength for NCPO. The derivative of absorbance in the inset. (b) Variation of (αhν)^1/2 and (αhν)² versus (hν) (E_gd and E_gi are respectively the direct and indirect bandgap). (c) Ln(αhν) versus Ln(hν − E_g).

The inset of Fig. 5(a) displays the derivative of the absorbance curve (dA/dλ) plotted against wavelength. The optical bandgap E_g of NCPO was estimated from the curve's minimum using the formula E_g = 1240/λ (with λ in nm). From the main peak at 300 nm, the bandgap is calculated to be about 4.13 eV, placing the material in the wide bandgap semiconductor range, suitable for UV photonics and high-power electronics devices.^44,45 However, this method does not allow for distinguishing between direct and indirect bandgaps. To overcome this limitation, Tauc's method, which is well-adapted for powdered materials, was applied using the following relation:

(αhν)^1/n = A(hν − E_g) (2)

In this framework, the nature of the bandgap is analyzed by examining the relationship between photon energy (hν) and the absorption coefficient (α) of the material. The exponent n characterizes the type of optical transition, where n = 1/2 corresponds to allowed direct transitions and n = 2 to allowed indirect transitions.⁴⁶

As illustrated in Fig. 5(b), the plots of (αhν)^1/2 and (αhν)² versus photon energy (hν) reveal linear regions. By extrapolating these linear portions to the energy axis, the optical bandgap energies E_g can be determined. The intercepts correspond to the direct and indirect bandgaps, denoted as E_gd and E_gi, respectively. The estimated values are 3.95 eV and 2.99 eV for the direct transition (n = 1/2) and the indirect transition (n = 2) respectively.

To determine whether the material exhibits a direct or indirect bandgap, a mathematical verification is necessary and should be systematically applied.⁴⁷

Ln(αhν) = ln(α₀) + n ln(hν − E_g) (3)

To confirm the direct nature of the bandgap in our NCPO sample, we utilized the previously determined bandgap energy (E_g = 4.13 eV). We analyzed the relationship between ln(αhν) and ln(hν − E_g), as presented in Fig. 5(c). The resulting linear plot yields a slope of approximately 0.4, consistent with a direct allowed transition. Direct bandgaps are preferred in optoelectronic applications due to their higher luminous efficiency, as they enable radiative recombination without phonon involvement, leading to enhanced light emission.^48,49 Consequently, the NCPO sample shows significant potential for light-emitting diode (LED) fabrication. Metaphosphate materials, such as NCPO, typically exhibit wide bandgaps and absorb light in the ultraviolet region, as reported by Abudoureheman et al.,⁵⁰ making them promising candidates for UV-emitting optoelectronic devices.

Considering that the analyzed NCPO compound exhibits semiconducting behavior with a direct bandgap, we attempted to theoretically estimate the positions of the conduction band minimum (E_CB) and valence band maximum (E_VB) using the following empirical formula:⁵¹

E_CB = χ − E_e − 0.5 × E_g (4)

E_VB = E_CB + E_g (5)

In this equation, χ represents the absolute electronegativity of the compound, while E_e is a fundamental constant corresponding to the free electron energy on the hydrogen scale, typically taken as 4.5 eV. The electronegativity of the sample can be estimated using the following expression:⁵²

In this context, α, β, and γ denote the multiplicities of atoms Na, Ca, and oxygen positions, respectively. Based on our calculations, the absolute electronegativity of the sample was determined to be 5.7 eV. Using this value along with the calculated E_VB, E_CB, and E_g, we constructed a schematic representation of the band structure, as illustrated in Fig. 6. These properties further support NCPO's potential for UV-emitting LED applications, as the wide bandgap (E_g = 3.95 eV) enables emission in the ultraviolet region, critical for applications such as sterilization, water purification, and UV sensing. The direct allowed transition enhances radiative recombination efficiency, ensuring high quantum yield, while the calculated E_CB and E_VB align favorably with common electrode materials, facilitating efficient charge carrier injection in LED device structures.

Fig. 6 Descriptive scheme of the band structure.

3.2.2 Urbach energy and threshold wavelength. To quantify structural disorder, defect density, and impurity levels in materials, The Urbach energy E_u, a crucial optical parameter, must be resolute.^1,53 It provides insight into the exponential absorption edge and the formation of localized states within the bandgap. Theoretical estimation of E_u is commonly performed using the Urbach–Martienssen model, defined by the following relation:⁵³

In this equation, α₀ is a constant and hν is the photon energy. As shown in Fig. 7(a), the Urbach energy (E_u) for NCPO is estimated to be approximately 0.58 eV, corresponding to 14.68%. This relatively low E_u value suggests high crystallinity and minimal structural disorder, further supporting the material's suitability for efficient UV-emitting LEDs, as low defect densities enhance carrier mobility and radiative recombination efficiency.

Fig. 7 (a) Variation of Ln(α) as a function of incident photon energy. (b) (A/λ)² vs. 1/λ plots for the NaCaP₃O₉ compound.

Extending Urbach's rule, Skettrup proposed the steepness factor (σ) as an additional optical parameter, which can be used to further evaluate the Urbach energy through the following empirical relation:⁴⁷

where k_B is the Boltzmann constant, T is the absolute temperature (typically taken as 300 K). In addition, the steepness parameter (s) describes how sharply the absorption edge broadens, reflecting the influence of electron–phonon interactions within the bandgap. Furthermore, σ is directly linked to the electron–phonon coupling strength (E_e–ph), and this correlation can be expressed using the following equation:^47,54

The calculated values for the steepness parameter and the electron–phonon interaction strength are approximately 0.0446 eV⁻¹ and 14.94 eV, respectively. Electron–phonon interaction refers to the coupling between charge carriers and lattice vibrations (phonons) in solids, playing a crucial role in governing fundamental properties such as electrical resistivity, superconductivity, and thermal conductivity.⁴⁷

In the context of optoelectronic devices, the maximum wavelength of incident radiation denoted as λ_T or threshold wavelength is a key parameter for assessing a material's suitability for such applications. It represents the shortest wavelength capable of initiating optoelectronic processes, such as absorption or emission, within the material. The λ_T value in our study was estimated using the following relation:⁵⁵

In this equation, α represents the absorption coefficient, λ is the wavelength of the incident radiation, and C is a constant. From the fitted curves, the λ_T value for our sample was determined to be 325 nm, as shown in Fig. 7(b).

3.2.3 Penetration depth, extinction coefficient and refractive index. The skin depth (δ) is a key parameter that indicates the extent to which incident radiation or light can penetrate into the material. Using the absorption coefficient α(λ), δ was calculated according to the relation provided in the referenced study:⁵⁶

The δ(λ) values for the NCPO sample exhibit distinct regions that correspond to variations in the absorption coefficient, as shown in Fig. 8(a). In the 200–300 nm range, the material effectively blocks UV-vis radiation, indicating its potential as a UV filter.⁵⁷ As the wavelength shifts toward the visible region, the δ values increase, with a noticeable sharp peak observed at 365 nm. Within the visible spectrum, the δ values then decrease progressively, accompanied by a prominent absorption band.

Fig. 8 (a) Variation of the penetration depth δ against λ for the studied NCPO material. (b) Variation of the extinction coefficient and refractive index as a function of wavelength (λ).

The extinction coefficient (k) can be calculated from the absorption coefficient α(λ) using the relation:⁵⁷

which describes the material's ability to absorb and scatter light, indicating its effectiveness in attenuating incident radiation. Similarly, the refractive index (n) is a fundamental optical parameter influenced by electronic polarization and internal fields, and is critical for designing optical devices such as filters, switches, and modulators.⁵⁸ It can be determined using the following expression:

Fig. 8(b) shows the variation of the extinction coefficient (k) and refractive index (n) as functions of the incident wavelength. Both n and k decrease with increasing wavelength, exhibiting high values in the UV region and declining to relatively low levels in the visible range. This behavior reflects the excellent optical quality and compactness of the synthesized samples. The gradual decrease in optical opacity from the UV to the visible region confirms strong UV absorption. Based on these observations, the material shows great potential for use in UV sensing applications.^59,60 Consequently, it can be considered a promising candidate for integration into optoelectronic devices.

3.3 Frequency-dependent electrical properties assessed by impedance spectroscopy

Impedance spectroscopy is a versatile and powerful technique widely employed to investigate the electrical and electrochemical properties of materials, offering valuable information about charge transport mechanisms and electronic structure in electrochemical and semiconducting systems. By analyzing a material's response to an applied alternating electric field over a broad frequency range, this method allows one to assess the processes governing energy storage and dissipation, which are directly linked to the intrinsic properties of the material.^61,62 The electrical response in the high-frequency domain is primarily attributed to the bulk (grain) contribution, where charge carriers move through the crystalline lattice with relatively low resistance. Conversely, the low-frequency region is mainly governed by grain boundary effects, characterized by higher resistive behavior arising from structural disorder, defect accumulation, and the presence of potential barriers at grain interfaces. These grain boundaries function as charge carrier trapping sites, resulting in extended relaxation times compared to those associated with the grain interiors. The observed relaxation phenomena are mainly driven by thermally activated hopping of charge carriers between localized states and across interfacial barriers, and are strongly affected by microstructural heterogeneity. Comparable grain and grain boundary contributions, as well as similar relaxation behavior, have been reported for phosphate- and oxide-based ceramic materials, confirming the effectiveness of impedance spectroscopy in distinguishing bulk and interfacial transport processes.^63–66 The complex impedance is expressed as the sum of a real part, representing the resistive component, and an imaginary part, corresponding to the capacitive component, according to the relation Z* = Z′ + jZ″.

Fig. 9(a) illustrates how the real part of the complex impedance (Z′) varies with angular frequency across a temperature range of 393 K to 633 K for the studied NCPO system. The curves exhibit two characteristic zones. At low frequencies, Z′ shows a nearly flat response, indicating DC conduction dominated by long-range charge transport, where carriers move between neighboring sites through successful hopping.^6,67,68 At higher frequencies, a significant dispersion in Z′ is observed, which is attributed to AC conduction caused by localized motion of carriers, reflecting limited or incomplete hopping processes. This dual behavior highlights two distinct conduction mechanisms: DC conduction is evident at lower frequencies due to extended charge migration, while at higher frequencies, the influence of localized relaxation and short-range motion becomes dominant, contributing to AC conductivity. With rising temperature, the transition point where Z′ begins to depend on frequency shifts to higher frequencies (from 10 rad s⁻¹ at 393 K to 312 rad s⁻¹ at 633 K), suggesting that the system undergoes a thermally driven relaxation process.

Fig. 9 (a) Variation of the real part of the complex impedance (Z′) as a function of angular frequency at different temperatures ranging from 393 K to 633 K. (b) Variation of the imaginary part of the complex impedance (Z″) as a function of angular frequency at different temperatures ranging from 393 K to 633 K.

At high frequencies, Z′ levels off and reaches a minimum value regardless of temperature, implying that thermal energy helps reduce the number of barriers impeding charge flow. In the low-frequency domain, a clear decrease in Z′ with increasing temperature is noted, which reflects a negative temperature coefficient of resistance (NTCR). This trend is a typical indicator of semiconducting behavior and confirms that the material exhibits NTC thermistor characteristics.⁶⁹

The imaginary component of the impedance spectrum provides valuable insight into the dielectric relaxation behavior of the NCPO compound, which is influenced by the movement of charge carriers or the reorientation of dipolar entities under an applied electric field. As shown in Fig. 9(b), the −Z″ values are plotted against angular frequency across various temperatures (393–633 K), revealing the evolution of relaxation dynamics with thermal excitation. Each spectrum exhibits a prominent peak, approximately corresponding to the transition region between DC and AC conduction, as previously identified in the real part of the impedance (see Fig. 9(a)). This peak, referred to as the relaxation frequency, signifies the shift from long-range charge transport to localized carrier dynamics. The noticeable broadening of these peaks indicates a departure from the ideal Debye relaxation model,⁶² reflecting the complex nature of polarization and conduction processes in the material. Below the relaxation peak, charge transport is primarily governed by long-range carrier motion, whereas above this frequency, the conduction behavior is mainly influenced by localized relaxation phenomena.

To get additional information on charge carrier dynamics can be obtained from the analysis of Nyquist plots, which are especially effective in differentiating the electrical responses originating from the bulk (grain) and interfacial (grain boundary) regions of the material. In this work, Fig. 10(a) and (b) displays the Nyquist plots (Z″ vs. Z′) for the NCPO compound. The impedance spectra exhibit clear semi-circular arcs, primarily attributed to the bulk (grain) contribution, with negligible evidence of grain boundary effects.⁷⁰ These arcs are notably depressed, with centers lying below the real axis (Z′), suggesting a non-ideal relaxation process that deviates from classical Debye behavior and is more accurately described by the Cole–Cole model. As the temperature increases, the diameter of the arcs decreases, reflecting a decline in bulk resistance, which is characteristic of thermally activated conduction.

Fig. 10 (a) and (b) Nyquist plots (Z″ vs. Z′) of the NCPO compound recorded at various temperatures; (c) equivalent circuit model used for impedance fitting; (d) Arrhenius plot of ln(R × T) as a function of inverse temperature (1000/T), illustrating thermally activated conduction behavior.

To model the impedance data accurately, fitting was performed using ZView software,⁷¹ which identified the most appropriate equivalent circuit consisting of a parallel combination of a resistor (R), a constant phase element (CPE), and a capacitor (C), as shown in the Fig. 10(c). The corresponding fitting parameters are summarized in Table 3.

Table 3 Electrical values of the equivalent circuit parameters calculated for NCPO compound at different temperatures

T (K)	R (MΩ)	C (10^−11 F)	Q (10^−10 F)	α
633	42.66	3.341	1.644	0.61925
613	50.379	3.342	1.593	0.61851
593	63.594	3.337	1.331	0.63248
573	80.868	3.324	0.8607	0.67241
553	109.8	3.354	1.194	0.62405
533	145	3.287	0.6215	0.71270
513	204.2	3.295	0.6148	0.70946
493	276.8	3.280	0.4424	0.74262
473	378.8	3.304	0.3108	0.75810
453	535.9	3.289	0.3576	0.75905
433	721.8	3.332	0.3309	0.74483
413	1085	3.466	0.8024	0.50460
393	1364	3.356	0.1786	0.78068

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The results summarized in Table 3 reveal a noticeable reduction in resistivity as the temperature rises, indicating the sample exhibits semiconducting behavior. To extract key parameters such as activation energy, sensitivity constant, and stability factor, Fig. 10(d) illustrates the variation of ln(R × T) as a function of the reciprocal temperature (1000/T). The resulting plot exhibits a clear linear trend, indicating that resistivity decreases steadily with increasing temperature an expected behavior for materials exhibiting excellent negative temperature coefficient resistance (NTCR) characteristics. Within the investigated temperature range, the data align well with the Arrhenius equation:

In this expression, “A” denotes the pre-exponential factor, “k_B” is Boltzmann's constant, and “E_a” represents the activation energy.

Fig. 10(d) reveals a clear linear trend in the temperature-dependent electrical behavior, supported by a strong correlation coefficient (R² = 0.99587), which confirms the presence of thermally activated conduction. The slope of the linear fit allowed for the determination of the activation energy, calculated to be 0.31 eV. Using this value, the thermal sensitivity constant β defined by the relation ,¹ was found to be approximately 3597.4 K. This value is notably higher than that reported for similar thermistor materials, such as Cu_0.2Ni_0.5ZnMn_1.3O₄ (β ≈ 3356 K),⁷² indicating the superior thermal response of the NCPO system. In addition, the material's electrical stability over temperature was assessed by calculating the stability factor (SF), which is the logarithmic ratio between the maximum and minimum resistivity recorded within the measured temperature range.⁷³ The resulting SF ≈ 1.5 suggests minimal resistivity fluctuations, which is in the same order of the value reported for ZnO nanoceramics (SF ≈ 1.796),⁷³ indicating stable electrical behavior.

Together, these results affirm that NCPO exhibits both high thermal sensitivity and electrical stability, making it a strong candidate for use in thermistor and thermal sensing technologies.

3.4 Study of modulus formalism and relaxation behavior

The complex electric modulus formalism is a powerful tool used to examine charge transport mechanisms and relaxation dynamics in ceramic materials, particularly those involving charge carrier accumulation and mobility.⁷⁴ These phenomena are fundamental contributors to both dielectric polarization and electrical conductivity, which are critical factors in evaluating the dielectric performance and multifunctional behavior of the NCPO ceramic system. The complex electric modulus M*(ω) is mathematically defined in terms of the complex impedance Z*(ω) as follows:⁷⁵

M*(ω) = jωC₀(Z′ + jZ″) = jωC₀Z′ − ωC₀Z″ (15)

Based on this relation, the real and imaginary components of the modulus can be expressed as follows:

M′(ω) = ωC₀Z″ (16)

M″(ω) = ωC₀Z′ (17)

where C₀ = ε₀A/d is the vacuum capacitance of the sample geometry (with ε₀ the vacuum permittivity, A the electrode area, and d the sample thickness).

Fig. 11 displays the M″ versus M′ plots of the complex electric modulus for the studied compound across a range of temperatures. These diagrams consistently exhibit single, well-shaped semicircular arcs, which are indicative of a uniform, single-phase system.⁷⁴ Utilizing the electric modulus formalism proves advantageous, as it minimizes the effects of electrode polarization and enhances the resolution between bulk (grain) and interfacial (grain boundary) contributions. The consistent presence of one semicircular arc across all measured temperatures suggests that electrical conduction in the material is primarily driven by grain (bulk) processes. In contrast, a multiphase material would typically exhibit multiple arcs or features, each associated with distinct conductive regions. Thus, this analysis not only provides insight into the relaxation dynamics but also reinforces the conclusion that the NCPO sample exhibits single-phase behavior and conduction dominated by the bulk region.

Fig. 11 Imaginary versus real electric modulus (M″ vs. M′) plot.

Fig. 12(a) illustrates the evolution of the imaginary part of the complex electric modulus (M″) with angular frequency at various temperatures (393–633 K) for the NCPO compound. At lower frequencies, the M″ values remain nearly negligible, indicating the absence of significant electrode polarization effects in this spectral region.^47,76 As the frequency increases, M″ exhibits a notable rise, reaching a distinct peak that signals the onset of a relaxation process. This peak progressively shifts to higher frequencies as the temperature increases, confirming the thermally activated character of the dynamic relaxation behavior. The observed broad and asymmetric nature of the M″ peaks indicate a deviation from ideal Debye relaxation, pointing toward a distribution of relaxation times within the system. Analyzing the curve in two segments, it is evident that frequencies below the characteristic relaxation frequency (ω_p) correspond to long-range charge displacement within the bulk, while frequencies above ω_p reflect localized carrier dynamics. This crossover underscores the coexistence of extended and confined charge transport processes within the material.

Fig. 12 (a) Variation of the imaginary part of the electric modulus (M″) as a function of angular frequency at temperatures ranging from 393 K to 633 K; (b) evolution of the β parameter with temperature; (c) temperature dependence of the characteristic angular frequency (ω_p); (d) Arrhenius plot of ln(ω_p × T) versus inverse temperature (1000/T).

To better capture the asymmetric nature of the modulus spectra, the Bergman model an extension of the modified Kohlrausch–Williams–Watts (KWW) function is commonly used and is expressed as follows:⁷⁷

In this formulation, represents the maximum of the imaginary part of the electric modulus, occurring at the characteristic angular frequency ω_p. The parameter β, which lies between 0 and 1, quantifies the extent of relaxation dispersion. A “β” value of 1 signifies ideal Debye-type relaxation, typically associated with uniform dipole–dipole interactions. Conversely, values of β less than 1 reveal deviations from Debye behavior, indicating the involvement of more distributed or complex relaxation mechanisms.⁷⁴ As illustrated in Fig. 12(b), the β parameter exhibits a temperature-dependent trend. Its values remain consistently below one, indicating a deviation from ideal Debye-type relaxation. This behavior implies a wide distribution of relaxation times, likely arising from irregular dipole–dipole interactions in the NCPO system.

Fig. 12(c) illustrates how the relaxation angular frequency (ω_p) varies with temperature, showing an upward trend as temperature rises. This increase indicates that the relaxation process shifts to higher frequencies, consistent with thermally activated dynamics. To evaluate the activation energy governing this relaxation, Fig. 12(d) plots ln(ω_p × T) against the inverse temperature (1000/T). The linear behavior confirms that the relaxation process follows an Arrhenius-type dependence. From the slope of the fitted curve, the activation energy was calculated to be 0.33 eV. This value closely matches that obtained from electrical resistivity analysis, indicating that both the relaxation dynamics and the charge transport mechanism are governed by similar thermally activated processes.^1,52,60

3.5 Modeling the conduction mechanism in NCPO

This section focuses on the detailed investigation of charge transport mechanisms in the NCPO compound by examining the variation of electrical conductivity as a function of both angular frequency and temperature (393–633 K). Fig. 13 illustrates how conductivity evolves with frequency, highlighting complex conduction behavior. In the low-frequency range (region I), particularly below 100 rad s⁻¹, the conductivity spectrum σ(ω) exhibits a plateau corresponding to the DC conductivity (σ_dc), reflecting frequency-independent behavior. As the frequency increases, the system enters region II, characterized by a dispersive increase in conductivity. The parameter ω_h (the hopping frequency) precisely characterizes the onset of the AC conductivity regime (see Table 4). This transition is well-described by Jonscher's empirical universal power law, which characterizes the frequency dependence of AC conductivity as follows:⁷⁸

σ(ω,T) = σ_dc(T) + Aω^s(T) (19)

Fig. 13 Frequency-dependent electrical conductivity spectra of the NCPO compound recorded over the temperature range of 393 K to 633 K.

Table 4 Jonscher law parameters for NaCaP₃O₉ at different temperatures

T (K)	σdc × 10^−9	s	A × 10^−12	ωh
393	0.123	0.83503	0.83235	396.486
413	0.176	0.83377	0.86384	586.488
433	0.238	0.83073	0.91021	812.860
453	0.331	0.82955	0.93414	1183.724
473	0.473	0.82070	1.05725	1697.398
493	0.610	0.80747	1.26256	2108.914
513	0.827	0.80431	1.32857	2978.074
533	1.18	0.78828	1.67268	4106.998
553	1.52	0.77388	2.07059	5047.692
573	2.06	0.76367	2.44065	6791.591
593	2.66	0.74176	3.33203	8176.163
613	3.35	0.73796	3.63234	10 414.561
633	4.10	0.72678	4.33603	12 426.529

---

In this expression, the parameter A serves as a scaling factor that determines the magnitude of the frequency-dependent component of the conductivity. It reflects the degree of charge carrier polarization in response to an alternating field.⁷⁹ The exponent s, often referred to as the frequency exponent, characterizes the variation of conductivity with frequency. The first term in the equation corresponds to the DC conductivity, which remains constant regardless of frequency, while the second term accounts for the AC conductivity, which increases with frequency due to dynamic charge transport mechanisms. The “s” parameter is especially informative, as it provides insight into the nature and strength of interactions between mobile charge carriers and the crystal lattice, offering a deeper understanding of the material's conduction behavior.⁸⁰ The dc conductivity (σ_dc), pre-exponential factor (A), and frequency exponent (s) were evaluated using eqn (19), and the resulting values are listed in Table 4.

Several models have been proposed to describe the frequency exponent behavior, including the correlated barrier hopping (CBH) model introduced by Elliott.⁸¹ In addition, tunneling-based mechanisms such as quantum mechanical tunneling (QMT),⁸² overlapping large-polaron tunneling (OLPT), and non-overlapping small-polaron tunneling (NSPT) have been considered, all of which involve charge transport between localized states near the Fermi level.⁸³

Fig. 14(a) illustrates the temperature dependence of the frequency exponent s and the parameter ln(A). Both quantities decrease systematically with increasing temperature, while the exponent s remains below unity, indicating non-Debye relaxation behavior.⁸⁴ This behavior is in good agreement with the correlated barrier hopping (CBH) model,⁸¹ which suggests that charge transport in the NCPO system is primarily governed by thermally activated hopping of localized charge carriers. Recent studies on ceramics show that ac conduction is dominated by thermally activated hopping of localized charge carriers, well described by the correlated barrier hopping (CBH) model. This approach accounts for charge transport between defect-related states over potential barriers and explains the temperature-dependent, non-Debye relaxation behavior observed in these materials.^84–87

Fig. 14 (a) Temperature dependence of the frequency exponent “s”; (b) variation of DC conductivity (σ_dc) with temperature.

To further probe the conduction mechanism, we analyzed the variation of DC electrical conductivity (σ_dc) with temperature across a broad range (393–633 K). This investigation helps elucidate both the nature of electrical conduction and the dominant hopping and scattering processes that influence the transport response of NCPO. Fig. 13(b) presents the evolution of σ_dc with temperature, revealing an increasing trend indicative of a thermally activated process. This increase is attributed to the activation of a small polaron hopping (SPH) mechanism, as described by Mott and Davis.⁸⁸ In this framework, the temperature dependence of DC conductivity is expressed by:

Here, E_a denotes the activation energy required for a charge carrier to hop between localized states, σ₀ is the pre-exponential factor, and k_B is Boltzmann's constant.

The plot of ln(σ_dc × T) versus 1/k_BT, shown in Fig. 15(a), yields a linear fit with a correlation coefficient R² = 0.99333. This confirms that the SPH mechanism is the dominant conduction process. From the slope of the Arrhenius plot, the activation energy was calculated to be 0.36 eV. The activation energy corresponds to the minimum energy required for charge carriers to overcome potential barriers or hop between localized states during electrical conduction. It reflects the degree of carrier localization and the ease of charge transport within the material, where a lower activation energy indicates enhanced mobility and more efficient conduction. Consequently, the activation energy provides important insight into the dominant charge transport mechanism governing the electrical behavior of the system.

Fig. 15 (a) Arrhenius plot of ln(σ_dc × T) versus inverse temperature (1/k_BT). (b) Scaled conductivity spectra at various temperatures for the NCPO compound.

In addition, various scaling models have been proposed in the literature to further analyze frequency-dependent conductivity behavior.^89–91 Among these, the Ghosh model has gained considerable attention due to its effectiveness in describing hopping conduction mechanisms. It is expressed as:⁹¹

where the hopping frequency ω_h is defined as:

Fig. 15(b) displays the temperature-dependent evolution of the scaled conductivity, represented as plotted against the reduced frequency . The fact that all curves collapse onto a single master curve across the entire temperature range strongly suggests that the underlying charge transport mechanism does not change with temperature.⁹² This superposition behavior supports the applicability of the time–temperature superposition principle (TTSP) and indicates that the same fundamental conduction dynamics are maintained throughout the measured conditions.⁹³

To establish a correlation between the electrical behavior and structural characteristics and to identify the charge carriers responsible for conduction within the framework of the CBH (correlated barrier hopping) model. According to this model, the frequency-dependent AC conductivity can be expressed using the following equation:⁹⁴

In this expression, the polaron number n (typically 1 or 2) reflects the number of polarons participating in the hopping mechanism, while NN_p is proportional to the square of the density of available states (NN_p = N_T² for the bipolaron case; (NN_p = N_T² exp(−U_eff/2k_BT) (for the single polaron case). The term ω represents the angular frequency, and ε′ corresponds to the real part of the dielectric permittivity.

Additionally, Rω represents the hopping distance. These parameters can be determined using the following expressions:⁹⁴

In this relation, e = charge of electron, ε′ = dielectric constant, k = Boltzmann constant, T = temperature, W denotes the binding energy required to move a charge carrier from one site to another and τ₀ = characteristic relaxation time.

Fig. 16 presents the variation of ln(σ_ac) as a function of 1000/T. A strong agreement is observed between the experimental data points (represented as scatter) and the theoretical predictions (depicted by the fitted line). The various parameters employed in the fitting procedure are compiled in Table 5. Furthermore, the analysis indicates that the frequency-dependent increase in the density of localized states N in NCPO suggests reduced structural disorder, enabling more efficient ionic hopping. Concurrently, the decline in effective barrier energy U_eff with frequency supports a CBH-type conduction mechanism. These results highlight the key role of localized states in governing AC conductivity, reinforcing the potential of NCPO for solid-state ionic applications.^95,96

Fig. 16 Variation of ln(σ_ac) with inverse of temperature at various frequencies.

Table 5 Parameters obtained from the fitting of experimental data of total Ac conductivity with CBH model for NCPO compound

Frequency (Hz)	N (10^18 eV^−1 cm^−3)	Ueff (eV)
100	2.538	0.1817
1000	2.563	0.1425
10 000	5.456	0.1225

---

Fig. 17(a) illustrates the temperature dependence of the hopping distance Rω. According to the correlated barrier hopping (CBH) model, this distance tends to increase with temperature. As thermal energy rises, charge carriers (polarons) gain sufficient energy to hop over longer distances, facilitating improved interchain interactions and enhancing charge transport.⁹⁵

Fig. 17 (a) The temperature dependence of Rω (Å) at different indicated frequencies. (b) The variation of Rω (Å) with frequency.

On the other hand, Fig. 17(b) presents the evolution of the hopping distance Rω as a function of different frequencies. It is evident that Rω exhibits a stronger temperature dependence in the high-frequency range, whereas this sensitivity progressively diminishes at lower frequencies. This behavior suggests that thermal activation has a more pronounced effect on short-range hopping processes dominating at higher frequencies.

Fig. 18 presents a schematic representation of the CBH conduction model in NCPO. To validate this mechanism, the hopping distance Rω was compared to the Na–Na interatomic spacing. The close agreement between the calculated Rω values (∼4.18 Å) and the average Na–Na distance (∼4.16 Å)²⁹ strongly suggests that charge transport is primarily facilitated by localized Na⁺ ion migration through a small polaron hopping process, consistent with the CBH model.

Fig. 18 Schematic model of the AC conduction mechanism in NCPO.

4. Conclusion

This comprehensive study presents a detailed investigation of the structural, optoelectronic, and electrical properties of NaCaP₃O₉ (NCPO) metaphosphate synthesized via a straightforward and scalable solid-state method. The successful structural refinement confirmed the pure triclinic phase crystallizing in the P1̄ space group, providing a robust platform for multifunctional properties. The identified wide direct band gap (∼3.95 eV) positions NCPO as a compelling candidate for ultraviolet (UV) optoelectronic devices, including UV photodetectors and transparent electronic components, where strong UV absorption and high optical transparency are essential.

Our extensive impedance spectroscopy analysis revealed NCPO's intrinsic semiconducting behavior marked by a negative temperature coefficient of resistance (NTCR), a key attribute for thermistor and sensor applications demanding reliable temperature sensitivity and stability.

The thermally activated conduction with an activation energy ≈0.36 eV and compliance with Jonscher's power law confirm that charge transport is governed by localized Na⁺ ion hopping within the correlated barrier hopping (CBH) framework, a key factor in tailoring the material's electronic performance.

Based on the activation energy obtained, the thermal sensitivity constant β was estimated to be around 3597.4 K. This relatively high value, compared with those reported for comparable thermistor materials, reflects the enhanced thermal response of the NCPO system. Moreover, the stability factor SF, close to 1.5, indicates low resistivity fluctuations when compared to typical literature values (SF ≈ 1.796), confirming the good electrical stability of the material.

Furthermore, modulus spectroscopy unveiled complex dielectric relaxation behavior with both short- and long-range ionic mobility, captured effectively by the Kohlrausch–Williams–Watts (KWW) function. The close agreement of activation energies obtained from modulus (0.33 eV) and conductivity analyses underscores the interconnected nature of dielectric and electrical processes in NCPO, reflecting a unified thermally activated mechanism.

Collectively, these findings underscore the material's exceptional multifunctionality combining structural stability, favorable optical absorption, and robust electrical conduction making NaCaP₃O₉ metaphosphate not only a scientifically intriguing system but also a highly promising candidate for next-generation optoelectronic devices, temperature sensors, and energy storage technologies. Our study significantly expands the understanding of phosphate-based metaphosphates and lays a critical foundation for future exploration and optimization of these materials in advanced electronics and photonics.

Author contributions

Mayssa Karray: investigation, data curation, formal analysis, visualization, writing – original draft, writing – review & editing. Iheb Garoui: investigation, data curation, formal analysis, visualization, writing – original draft, writing – review & editing. Saber Nasri: investigation, data curation, formal analysis, visualization, writing – original draft, writing – review & editing. Nourah A. Alsobai: investigation, formal analysis, visualization, writing – original draft, writing. Noweir Ahmad AlGhamdi: investigation, formal analysis, visualization, writing – original draft, writing. Abderrazek Oueslati: conceptualization, methodology, formal analysis, validation, writing review & editing.

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The authors confirm that the data used to support the findings of this study are included within the article and are available from the corresponding author upon reasonable request.

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