Electrical properties, phase transition, and conduction mechanism of the novel pyrophosphate compound Li₂V_0.4P₂O₇ compared with Li₂Cu_(1−(5/2)x)V_xP₂O₇ (x = 0 and 0.2)

Abstract

The present study investigates the effect of aliovalent substitution of Cu²⁺ by V⁵⁺ on the electrical and dielectric properties of the Li₂Cu_(1−5/2)xV_xP₂O₇ series. Attention is mainly devoted to the synthesis and characterization of the fully substituted compound Li₂V_0.4P₂O₇ (x = 0.4), prepared by the ceramic method, and its behavior is compared to that previously reported for the compositions x = 0 and x = 0.2. Phase identification is confirmed by X-ray diffraction (XRD), revealing that the Li₂V_0.4P₂O₇ compound crystallizes in the monoclinic system with the P2₁/c space group, while infrared spectroscopy (IR) confirms the formation of the P₂O₇ group, and differential thermal analysis (DTA) shows a phase transition at 640 K. Electrical and dielectric performances are investigated over wide temperature (574–713 K) and frequency (209 Hz to 5 MHz) ranges using complex impedance spectroscopy. The DC conductivity exhibits two thermally activated regions separated by the phase transition at 640 K. For the Li₂V_0.4P₂O₇ compound, the total conductivity reaches 6.15 × 10⁻⁵ Ω⁻¹ cm⁻¹ at 603 K (below the transition temperature) and 66.3 × 10⁻⁵ Ω⁻¹ cm⁻¹ at 683 K (above the transition temperature), confirming the thermally activated nature of charge transport from the analysis of the frequency exponent s. It follows that the conduction mechanism changes from non-small polaron tunneling (NSPT) below the transition temperature to correlated barrier hopping (CBH) at higher temperatures. A comparison among the three compositions indicates that aliovalent V⁵⁺ substitution leads to a systematic increase in ionic conductivity with increasing vanadium content, which may be attributed to the enhanced mobility of Li⁺ ions within the crystal lattice, thereby promoting charge transport. The sample with x = 0.4 exhibits the best electrical performance.

Progressive aliovalent substitution of Cu²⁺ by V⁵⁺ in Li₂Cu_(1−(5/2)x) V_xP₂O₇ leads to structural changes and improved ionic conduction.

1. Introduction

Lithium ion-batteries (LIBs) dominate the current energy-storage technologies, setting increasingly severe requirements on the electrode material performance and stability.¹ In this frame, pyrophosphates are a technologically important class of compounds owing to their rich physico-chemical properties, enabling applications in electronic devices, solid electrolytes with high thermal stability, sensors, semiconductor lasers, luminescent materials, ceramics, catalysts, ionic conductors, dielectrics, and magnetic or fluorescent systems.² Lithium pyrophosphates of the type Li₂MP₂O₇ (M = Ni, Zn, Cu, Ba, and Sr) belong to the broader A₂BP₂O₇ family, characterized by the coexistence of an alkali cation (A⁺) and a divalent cation (B²⁺). Their structural diversity is outstanding, and these materials cannot be classified easily according to conventional criteria such as ionic size, coordination environment, or the type of bonding.^3–7

There are several lithium pyrophosphates reported in the existing literature that possess a vast range of structural as well as electrical characteristics, depending on the nature of the metal ion present and the type of crystal formed. It has been previously established that any alteration of the metal ion can lead to structural changes, thereby affecting the transport of lithium ions.⁸

Among these pyrophosphates, the Li₂CuP₂O₇ compound has already been studied and published,⁹ including its structural, vibrational, calorimetric, electrical, and dielectric properties. However, vanadium-substituted derivatives of this system remain less explored in the literature. In this context, an aliovalent substitution of the divalent Cu²⁺ cation by vanadium V⁵⁺ was performed to improve its electrical behavior. In a previously published work,¹⁰ the series of Li₂Cu_(1−5/2)xV_xP₂O₇ (x = 0 and 0.2) samples were synthesized and characterized, and the results demonstrated that vanadium incorporation enhances ionic conduction across the entire studied temperature range. In particular, vanadium was chosen for its high chemical activity and its several oxidation states, which make it one of the most commonly used doping elements in functional materials.¹¹

In addition, studies on lithium pyrophosphates containing vanadium have shown that the presence of vanadium may lead to structural modifications with a significant impact on lithium-ion diffusion properties, making it clear that aliovalent substitution is a powerful tool for controlling the physicochemical properties of lithium pyrophosphates.¹²

On the basis of previously obtained results, a higher substitution level (x = 0.4) was investigated to obtain the Li₂V_0.4P₂O₇ compound, with the aim of further enhancing ionic mobility and studying the effects of higher vanadium content on the structure and electrical properties. This composition can be considered the end member of complete substitution in the Li₂Cu_(1−5/2)xV_xP₂O₇ series. It offers a comparative study with the other two compositions, x = 0 and x = 0.2, giving us more insight into the effect of increasing vanadium content in this series of compounds. This fully substituted composition has not yet been reported, and its structural and electrical properties remain unexplored. The present paper reports, for the first time, the synthesis and structural, vibrational, and calorimetric characterization of Li₂V_0.4P₂O₇, together with the study of its electrical properties and conduction mechanism over wide temperature and frequency ranges using complex impedance spectroscopy.

2. Experimental procedure

2.1. Synthesis procedure

The Li₂V_0.4P₂O₇ compound was synthesized via the conventional solid-state reaction method. High-purity (99%, Sigma Aldrich) analytical-grade precursors, including lithium carbonate (Li₂CO₃), vanadium pentoxide (V₂O₅), and ammonium dihydrogen phosphate (NH₄H₂PO₄), were used. The starting materials were weighed in stoichiometric proportions according to the following reaction:

The powders were thoroughly mixed and ground using an agate mortar. The resulting homogeneous mixture was calcined at 573 K for 14 hours at a heating rate of 5 °C min⁻¹ in an electric furnace to remove volatile components. After calcination, the powder was reground and pressed into pellets using a uniaxial pressure of 3 T cm⁻² to facilitate the reaction. The pellets were then sintered at 813 K for 13 hours with a heating rate of 2 °C min⁻¹ to obtain the final Li₂V_0.4P₂O₇ phase. After sintering, the furnace was allowed to cool naturally to room temperature.

The dissociation of V₂O₅ into V₂O₄ and O₂ occurs only at temperatures above 973 K,¹³ as shown in the following reaction:

2V₂O₅ ↔ 2V₂O₄ + O₂ 1

Therefore, under our synthesis conditions, which involve temperatures below 973 K, vanadium is expected to remain in the +5 oxidation state.

2.2. Characterization

The synthesized compound was characterized to assess its phase purity and structural uniformity using X-ray powder diffraction. The measurements were carried out at ambient temperature within the angular interval of 4° < 2θ < 80°, employing a step increment of 0.02°, on a Philips PW 1710 diffractometer utilizing Cu Kα₁ radiation (λ = 1.5406 Å). Structural analysis and refinement of the identified phases were performed through the Rietveld method, using a combination of FullProf, Celeref 2, and FOX software. For vibrational characterization, infrared spectra were recorded with a SPECTRUM BK spectrophotometer across the wavenumber range of 1300–500 cm⁻¹.

A differential thermal analysis (DTA) measurement was carried out using a Linseis thermal analysis DTA-PT 1600 apparatus.

For electrical characterization, a pellet with a thickness of 1.1 mm and a diameter of 8 mm was prepared. Impedance measurements, including both real and imaginary components, were conducted over a broad temperature range (574–713 K) and frequency range (209 Hz to 5 MHz) using a TEGAM 3550 ALF automatic impedance analyzer, connected to a microcomputer and a compatible temperature controller. To ensure optimal electrical contact, both faces of the pellet were coated with a thin layer of gold and positioned between two copper electrodes within a dedicated sample holder. The impedance data were recorded while cooling from 713 K.

3. Results and discussion

3.1. X-ray analysis

Following our previous study¹⁰ that demonstrated the effective incorporation of vanadium into the Li₂CuP₂O₇ framework for x = 0.2, the present work advances this investigation by increasing the vanadium concentration to x = 0.4. The synthesized sample Li₂Cu_(1−5/2)xV_xP₂O₇ (x = 0.4) was characterized using powder X-ray diffraction. The diffraction patterns obtained at room temperature for compositions x = 0, 0.2, and 0.4 (Fig. 1(a)) clearly show structural modifications induced by vanadium substitution. In particular, the appearance of new diffraction peaks and the disappearance of others as x increases indicate a progressive rearrangement of the crystal structure. These spectral changes are associated with modifications in the diffraction profile and reflect significant alterations in the lattice symmetry and unit cell parameters due to the incorporation of vanadium into the structure.

Fig. 1. (a) X-ray diffractograms of Li₂CuP₂O₇ (x = 0), Li₂Cu_0.5V_0.2P₂O₇ (x = 0.2) and Li₂V_0.4P₂O₇ (x = 0.4) in the 2θ range 10°–80°. (b) Powder X-ray diffraction pattern and Rietveld refinement of the sample Li₂V_0.4P₂O₇ (circles correspond to the experimental data, and the calculated data are represented by the continuous line overlapping them; tick marks represent the positions of the allowed reflection, and a difference curve on the same scale is plotted at the bottom of the pattern).

Using several tools, including FullProf, CELREF, and FOX, Rietveld refinement of the powder XRD data was performed in order to better examine the structural characteristics of the prepared compound Li₂V_0.4P₂O₇. The structural models of comparable known compounds served as a reference for this refinement in order to guarantee precise fitting and structural interpretation.

A pseudo-Voigt function was used to model the peak profiles, while the background was estimated by linear interpolation between several points, whose heights were treated as adjustable parameters. The X-ray diffraction pattern of the synthesized Li₂V_0.4P₂O₇ compound is shown in Fig. 1(b). Rietveld refinement was performed to extract detailed structural information. The Li₂V_0.4P₂O₇ sample displays sharp and intense diffraction peaks, reflecting a high degree of crystallinity. All reflections are consistent with a monoclinic structure belonging to the P2₁/c space group. After several iterative refinement steps and optimizations, all peaks were successfully indexed. The refined lattice parameters for the compositions Li₂Cu_(1−5/2)xV_xP₂O₇ (x = 0, 0.2, and 0.4) are summarized in Table 1. Notably, the FullProf refinement did not reveal any secondary phases. The reliability factors—namely the profile R-factor (R_p), the weighted profile R-factor (R_wp), and the expected R-factor (R_exp)—were all below 10%, confirming the high phase purity of the sample and the absence of significant impurities.

Table 1. Crystal data of the Li₂Cu_(1−(5/2)x)V_xP₂O₇ (x = 0, 0.2 and 0.4) compound.

Sample	Li2CuP2O7	Li2Cu0.5V0.2P2O7	Li2V0.4P2O7
X	0	0.2	0.4
Color	Blue	Green blue	Green
2θ (°) range	10–60	6–82	7–80
System	Mono	Mono	Mono
a (Å)	14.084(9)	15.083(1)	15.875(1)
b (Å)	4.871(3)	7.872(6)	6.506(5)
c (Å)	8.622(5)	12.329(8)	13.749(1)
α, γ (°)	90.00	90.00	90.00
β (°)	98.97(0)	91.092(3)	94.770(4)
Volume (Å3)	584.26(6)	1463.79(2)	1415.24(1)
Space group	I2/a	P21/n	P21/c
Z	8	4	4
R p (%)	5.48	4.51	2.95
R wp (%)	7.40	6.55	5.16
R exp (%)	3.48	2.72	2.79
Bragg R-factor (%)	3.48	0.646	1.64
RF-factor (%)	4.06	0.318	1.12
χ 2	4.53	5.79	3.43

3.2. IR spectroscopic analysis

The infrared spectrum of the Li₂V_0.4P₂O₇ compound, recorded at room temperature, is presented in Fig. 2 and displays transmittance as a function of wavenumber. By comparison with similar phosphate-based compounds, the most prominent bands have been attributed to specific vibrational modes. According to Corbridge,¹⁴ the P₂O₇⁴⁻ group, represented as O₃P–O–PO₃, can be interpreted as a combination of the vibrational modes of the PO₃ units and the bridging P–O–P group. In addition to these characteristic modes, extra absorption bands are observed, which are attributed to VO₃ group vibrations, indicating the incorporation of vanadium into the structure. These assignments, summarized in Table 2, confirm the presence of diphosphate groups in the title compound.

Fig. 2. Infrared analysis spectrum of Li₂V_0.4P₂O₇ at room temperature.

Table 2. Band assignment (cm⁻¹) of Li₂V_0.4P₂O₇.

Bands (cm−1) Li2V0.4P2O7	Assignment
1294	ν as(PO3)
1256
1201
1178
1159
1135
1097
1064	ν s(PO3)
1052
1034
1005	ν as(PO3)
904	ν s(VO3)
783	ν as(VO3)
948	ν as(P–O–P)
925
755	ν s(P–O–P)
743
726
688	ν as(P–O–P)
605	δ(PO3) + ρ(PO3)
563
508
485
414

3.3. Calorimetric study

The differential thermal analysis (DTA) curve shown in Fig. 3 exhibits three well-defined endothermic events that characterize the thermal behavior of the material. The first endothermic peak, appearing near 400 K, is most likely associated with the release of physically adsorbed or weakly bound water within the sample.¹⁵

Fig. 3. Thermogravimetric analysis of the Li₂V_0.4P₂O₇ compound.

Two additional endothermic peaks are observed at 465 K and 640 K, suggesting the presence of thermally activated structural transformations within the material.

To identify the nature of these transitions, the associated entropy changes were evaluated using Boltzmann's relation:

ΔS = R Ln Ω = R Ln(N₂/N₁) 2

where Ω = N₂/N₁, with N₁ and N₂ representing the numbers of accessible configurations before and after the phase transition, respectively. When the entropy change exceeds 2 J mol⁻¹ K⁻¹, the transition is typically classified as an order–disorder process. In contrast, lower entropy values (ΔS < 2 J mol⁻¹ K⁻¹) generally indicate a displacive-type mechanism.^16,17

In our case, the entropy values obtained for the detected transition at 465 K and 640 K are ΔS = 0.299 J mol⁻¹ K⁻¹ (Ω = 1.036) and ΔS = 0.189 J mol⁻¹ K⁻¹ (Ω = 1.023), respectively, confirming that both transitions correspond to a displacive-type structural transition.

Both parent compounds, Li₂Cu_(1−5/2)xV_xP₂O₇ (x = 0 and x = 0.2), have already been reported in the literature. When comparing our prepared sample to these references, we observe that the intermediate composition (x = 0.2) exhibits a significantly higher value of ΔS, as well as an increased value of Ω. Furthermore, the compound Li₂V_0.4P₂O₇ shows improved values compared to Li₂CuP₂O₇.^9,10

These results clearly indicate that vanadium substitution enhances the configurational disorder within the structure, especially for the intermediate composition Li₂Cu_0.5V_0.2P₂O₇. This conclusion is supported by the marked increase in ΔS and by the corresponding rise in the configurational parameter Ω.

3.4. Electrical properties

The electrical properties were studied by complex impedance spectroscopy (CIS), which is a method that links theoretical circuit elements to the real electrical behavior of the material. CIS provides significant information on microstructural components such as grains, grain boundaries, and electrode interfaces. Furthermore, this method allows for the extraction of important electrical parameters from the complex impedance over a wide range of frequencies and temperatures.^18–20

The complex impedance is expressed as:²¹

Z* = Z′ + jZ″ 3

where Z′ and Z″ represent the real and imaginary components, respectively. This decomposition offers a detailed and accurate representation of the material's overall electrical response.

Fig. 4(a) shows the Nyquist diagrams recorded at low temperatures, while Fig. 4(b) displays the corresponding spectra obtained at higher temperatures. The complex impedance plots exhibit two semicircular arcs, followed at very low frequencies by a short tail attributed to electrode polarization effects. For each temperature, the semicircles appear depressed below the real axis, indicating a distribution of relaxation times involved in the different conduction processes. This means that the electrical response does not follow the Debye model, but is consistent with the Cole–Cole formalism. In addition, the decreasing diameter of the semicircles with increasing temperature confirms the thermally activated nature of ionic conduction.

Fig. 4. (a) Complex impedance spectra at low temperatures (phase I) and (b) high temperatures (phase II) of Li₂V_0.4P₂O₇ with (inset) the electrical equivalent circuit, complex impedance plots of Li₂CuP₂O₇ (x = 0), Li₂Cu_0.5V_0.2P₂O₇ (x = 0.2) and Li₂V_0.4P₂O₇ (x = 0.4) (c) at T = 603 K (phase I) and (d) at T = 683 K (phase II) and (e) variation of the Ln(σT) as a function of temperature for the Li₂V_0.4P₂O₇ sample.

Using the ZView software, the experimental impedance data were fitted to the continuous lines shown in the figure. The obtained spectra were successfully modeled using a standard equivalent circuit presented in the inset of Fig. 4(b).²² This equivalent circuit consists of two cells connected in series. The first cell is composed of a resistance R₁ in parallel with a capacitance C₁, while the second cell includes a resistance R₂ in parallel with a capacitance C₂ and a constant phase element (CPE₁). These two cells correspond to the bulk (grain) and grain boundary contributions, respectively. In addition, a fractal capacitance element (CPE₂) is introduced in series to account for electrode polarization effects.

A constant phase element (CPE) is characterized by two parameters, Q and α, and its impedance is expressed as:²³

4

where Q has units of capacitance and ω is the angular frequency. The exponent α takes the values 1, 0, and 0.5 for ideal capacitive behavior, ideal resistive behavior, and Warburg-type diffusion, respectively.

Fig. 4(c and d) demonstrate a strong consistency between the experimental data (scatter) and the fitted theoretical curves (solid lines) for both the real and imaginary components of the impedance, confirming that the selected equivalent circuit provides an accurate representation of the electrical response of the material across the investigated temperature range.

It is also observed that the grain resistance (R₁) is considerably lower than the grain boundary resistance (R₂), indicating that ionic transport within the material is predominantly governed by grain boundary contributions. Moreover, the decrease of R₂ with increasing temperature reflects the thermally activated nature of ionic conduction, likely dominated by Li⁺ mobility.²⁴

To show the effect of substituting copper with vanadium, Fig. 5(c) and (d) present the Nyquist plots for the three compositions (x = 0, 0.2, and 0.4) at 603 K and 683 K. All samples display a similar overall impedance behavior. However, as the vanadium content increases, the radii of the semicircles become smaller. This decrease clearly indicates that both grain and grain boundary resistances are reduced when copper is replaced by vanadium. Therefore, vanadium substitution improves ionic transport in the material by lowering the resistive barriers to ion migration.

Fig. 5. (a) Frequency dependence of the real part of the AC conductivity of Li₂V_0.4P₂O₇ at various temperatures, variation in the real part of the AC conductivity as a function of the angular frequency for Li₂Cu_(1−(5/2)x)V_xP₂O₇ (x = 0, 0.2 and 0.4) (b) at T = 603 K (phase I) and (c) at T = 683 K (phase II), (d) variation of the universal exponent s as a function of temperature and frequency dependence of the imaginary part of the AC conductivity of Li₂V_0.4P₂O₇ (e) at low temperatures (phase I: T < 624 K) and (f) at high temperatures (phase II: T > 624 K).

3.5. DC conductivity

The fitting of the Nyquist diagrams of the Li₂V_0.4P₂O₇ compound using the proposed equivalent circuit enables the extraction of the electrical parameters R₁, C₁, R₂, C₂, CPE₁(Q), CPE₁(α), CPE₂(Q) and CPE₂(α).

These parameters then allow the calculation of the dc conductivity (σ_DC) of the grains, the grain boundaries, and the total conductivity at each temperature using the following expressions:²⁵

5

6

7

Where e and l, respectively, represent the thickness and the surface area of the sample-electrode contact. C_g and C_gb are the capacitances of the grains and grain boundaries, respectively, which are determined by the following equation:²⁶

C = R^{((1−α)/α)}Q^1/α 8

The temperature evolution of the dc ionic conductivities corresponding to the various contributions is shown in Fig. 4(e). The curves display a linear variation with a noticeable change in slope around the transition temperature identified by the DSC analysis. Such linear behavior over the studied temperature range indicates that the conductivities obey an Arrhenius-type dependence:²⁷

9

where σ₀ is the pre-exponential factor, E_a is the activation energy for conduction, T is the absolute temperature, and k_B is the Boltzmann constant.

The activation energies and corresponding pre-exponential factors obtained from the Arrhenius fits are summarized in Table 3.

Table 3. Activation energies, pre-exponential factor and conductivities at 603 K (phase I) and 683 K (phase II) of grain, grain boundaries, and total conductivity for Li₂V_0.4P₂O₇.

Li2V0.4P2O7 (x = 0.4)		Phase I	Phase II
Activation energy (eV)	Grain	0.81	1.04
	Grain boundary	1.20	1.23
	Total conductivity	0.98	1.08
Pre-exponential factor σ0 (Ω−1 cm−1 K)	Grain	2.47 × 106	2.47 × 108
	Grain boundaries	8.27 × 108	2.95 × 109
	Total conductivity	5.61 × 106	3.94 × 107
Conductivity (T = 603 K) (Ω−1 cm−1)	Grain	1.30 × 10−3
	Grain boundary	12.77 × 10−5
	Total conductivity	6.15 × 10−5
Conductivity (T = 683 K) (Ω−1 cm−1)	Grain	8.11 × 10−3
	Grain boundary	354 × 10−5
	Total conductivity	66.3 × 10−5

The conductivity at two selected temperatures—one in phase I (603 K) and another in phase II (683 K)—was calculated and reported in Table 3. By comparing these values with those previously published for the compounds Li₂CuP₂O₇ and Li₂Cu_0.5V_0.2P₂O₇,^9,10 it is observed that increasing the vanadium content leads to an enhancement of the ionic conductivity.

This improvement may be associated with the small size of V⁵⁺, which allows better incorporation into the host lattice and facilitates ion mobility. The aliovalent V⁵⁺ substitution likely increases the concentration of mobile charge carriers through the formation of defect states, thereby enhancing ionic transport. Moreover, the conductivity increases with temperature (σ_DC (phase I) < σ_DC (phase II)) since thermal activation provides energy to ionize the defects and assist the migration of Li⁺ ions between the available lattice sites.

In order to gain additional insight into how the results obtained in the present study compare with those reported for lithium pyrophosphates, Table 4 summarizes the literature data for conductivity, activation energy, temperature range, and mechanism of conduction for a number of different Li₂MP₂O₇ compounds. From this comparative analysis, it is clear that the cation of the metal plays a prominent role in determining the conduction characteristics of lithium pyrophosphates.

Table 4. Electrical conductivity of the Li₂MP₂O₇-type compounds.

Compound	Sys	Temperature range (K)	E a (eV)	Conductivity (Ω−1 cm−1)		Conduction mechanism	Ref.
Li2BaP2O7	Mono	589–724	1.08	σ 675K = 3.05 × 10−7	σ 683K = 2.13 × 10−6	OLPT	4
Li2SrP2O7	Ortho	519–628	1.03	—	—	CBH	5
Li2CaP2O7	Tri	373–673	0.49	—	—	—	7
Li2CuP2O7	Mono	576–710	1.44	σ 603K = 0.213 × 10−5	σ 683K = 3.49 × 10−5	CBH (T < 622 K)	9
			1.26			OLPT (T > 622 K)
Li2Cu0.5V0.2P2O7	Mono	574–713	0.87	σ 603K = 3.19 × 10−5	σ 683K = 38.9 × 10−5	NSPT (T < 624 K)	10
			1.19			OLPT (T > 624 K)

3.6. AC conductivity and conduction mechanism

AC conductivity studies help to understand the conduction mechanism in the studied compound over the explored temperature and frequency range. Impedance spectroscopy measurements were used to determine the AC conductivity values. The values are calculated using impedance data through the following relation:²⁸

10

The real and the imaginary parts of the total AC conductivity, are calculated using the following expressions:

11

12

where e, l, Z′, and Z″, respectively, are the thickness, the cross-sectional area, and the real and the imaginary parts of complex impedance.

The frequency dependence of the real part of the AC conductivity, at different temperatures is shown in Fig. 5(a). At low frequencies, the curves exhibit an almost frequency-independent plateau corresponding to the dc conductivity. This plateau reflects the long-range transport of charge carriers through hopping mechanisms. At higher frequencies, a dispersive region appears, which is attributed to complex interactions between mobile charge carriers and the lattice. This overall behavior is consistent with Jonscher's universal power law:²⁹

13

where σ_s is the conductivity at low frequencies, σ_∞ is an estimate of the conductivity at high frequencies, ω = 2πf is the angular frequency, τ represents the characteristic relaxation time, A is a constant that is temperature-dependent, and s is the power law exponent, where 0 < s < 1.

The fitting of the experimental data, using Jonscher's equation, shows excellent agreement between the experimental points (scatter) and the theoretical fits (solid lines). This good agreement confirms that Jonscher's model accurately describes the AC conduction behavior in the compound under investigation.

Fig. 5(b) presents the evolution of for the different compositions Li₂Cu_(1−5/2)xV_xP₂O₇ (x = 0, 0.2, and 0.4) at T = 683 K (phase I), while Fig. 5(c) shows the corresponding behavior at T = 603 K (phase II). The obtained curves reveal that increasing the vanadium content leads to a noticeable enlargement of the low-frequency plateau associated with σ_DC, reflecting an improvement in the long-range charge transport. Also, this plateau is shifted upward for higher values of x, which implies an evident enhancement of the dc conductivity. Such an increase in conductivity can be explained by the increase in the mobile charge carriers' concentration and, correspondingly, in their mobility, facilitated by the structural and defect transformations caused by V⁵⁺ incorporation. In this context, the improvement in conductivity may be associated with enhanced Li⁺ mobility within the lattice.

The simulation of the slope by eqn (13) for the Li₂V_0.4P₂O₇ compound at different temperatures enables the extraction of the frequency exponent s. Its thermal evolution, shown in Fig. 5(d), makes it possible to determine the most probable conduction mechanism in the material and to follow its changes as the temperature increases.

According to the literature, there are different models explaining the behavior of this exponent, such as overlapping large-polaron tunneling (OLPT),³⁰ non-overlapping small polaron tunneling (NSPT),³¹ quantum mechanical tunneling (QMT)³² and correlated barrier hopping (CBH).³³

The variation of s reflects changes in the conduction mechanism. In phase I, T < 640 K, the s values increase with increasing temperature (note that the inset of Fig. 5(d) provides an enlarged view of this region), indicating that the non-overlapping small polaron tunneling (NSPT) model is consistent with the observed behavior. In phase II, T > 640 K, s decreases as the temperature increases, which suggests that the correlated barrier hopping (CBH) model governs the conduction process.

The frequency dependence of the imaginary part of the total AC conductivity, obtained from eqn (12) for some temperatures below and above 640 K, is presented in Fig. 5(e) and (f), respectively. In the high-frequency region, during phase I (T < 640 K), an almost linear variation is observed, which remains independent of the temperature. The corresponding slope:³⁴

14

stays close to unity, indicating that dipolar reorientation mechanisms dominate the dielectric response in this regime.

In phase II (T > 640 K), a deviation from linearity appears in the low-frequency region, where increases markedly with temperature. This behavior reflects the increasing contribution of thermally activated charge carriers, which may be associated with ionic species such as Li⁺.

For the Li₂V_0.4P₂O₇ compound, a characteristic dispersive behavior is observed in the intermediate-frequency range: while the curves remain linear at high frequencies, they become increasingly dispersed toward the low-frequency side in phase II. This trend contrasts with that observed for the copper-based compounds Li₂CuP₂O₇ and Li₂Cu_0.5V_0.2P₂O₇, which exhibit dispersion extending across both low- and high-frequency regions.

This difference originates from the combined effects of structural symmetry and cation substitution. Li₂CuP₂O₇ crystallizes in the I2/a space group, Li₂Cu_0.5V_0.2P₂O₇ in P2₁/n, whereas Li₂V_0.4P₂O₇ adopts the P2₁/c symmetry. Such changes in the space-group symmetry modify the connectivity of the VO₆/PO₄ polyhedral and alter the local potential landscape experienced by charge carriers, thereby influencing their relaxation dynamics. Moreover, replacing Cu²⁺ (Jahn–Teller active, 3d⁹) with V⁵⁺ (non-degenerate, 3d⁰) reduces local distortions and can contribute to improved ionic mobility. This accounts for both the more regular high-frequency behavior and the distinct low-frequency dispersion observed in the vanadium-rich compound.

3.6.1. The NSPT model (phase I)

In order to interpret the behavior of the frequency exponent s in region I for Li₂V_0.4P₂O₇, one notices that s remains below unity and increases progressively with temperature. Such a trend is characteristic of the Non-Overlapping Small Polaron Tunneling (NSPT) model, which therefore provides the most suitable description of the conduction mechanism in this regime.

In this model, the AC conductivity is given by:³⁵

15

with

16

From the above equations, α⁻¹ is the spatial extension of the polaron (=1 × 10¹⁰ m), N (E_F) is the density of the states at the Fermi level, R_ω is the tunneling distance and W_H represents the polaron hopping energy.

Within the tunneling model, the frequency exponent s is given by:

17

where W_H is the polaron hopping energy, k_B is the Boltzmann constant, and τ₀ is the characteristic relaxation time, typically on the order of 10⁻¹³ s.

For large values of W_H/k_BT, s simplifies to

18

These relations highlight how NSPT conduction arises from thermally assisted tunneling of small polarons between localized states, with the exponent s capturing the sensitivity of the process to both temperature and frequency.

The value of W_H was obtained from the slope of the (1 − s) versus temperature plot, leading to an activation energy of approximately 0.63 eV for Li₂V_0.4P₂O₇ (x = 0.4) (Fig. 6(a)).

Fig. 6. (a) Variation of 1 − s as a function of temperature, (b) fitting of the real part of the AC conductivity at different frequencies using the NSPT model, (c) variation of the parameter N (cm⁻³) as a function of frequency in region I and (d) temperature dependence of the tunneling distance R_ω (Å) at different frequencies for Li₂V_0.4P₂O₇.

The fitting of the versus temperature curves using eqn (15) at different frequencies exhibits excellent agreement between the experimental data and the theoretical predictions (Fig. 6(b)), further confirming that the non-overlapping small polaron tunneling model accurately describes the conduction in phase I.

Moreover, the modeling allows the extraction of the key physical parameters-namely, the density of localized states N(E_F) and the tunneling distance R_ω. As shown in Fig. 6(c), N(E_F) increases with increasing frequency, reflecting the greater availability of localized states participating in tunneling. In contrast, R_ω decreases with both temperature and frequency (Fig. 6(d)), indicating that charge carriers require shorter hopping distances as thermal activation and frequency rise. The obtained values of R_ω fall within the range of 1.2–2.2 Å, which is fully consistent with short-range polaron hopping.

3.6.2. The CBH model (phase II)

According to the CBH model, conduction occurs through successive hops over the potential barrier separating two nearby defect centers. This model proposes that the AC conductivity arises from either single-polaron or bipolaron hopping across the Coulomb barrier between adjacent localized sites. The AC conductivity is influenced by several factors, including:³⁶

19

where n is the number of polarons involved in the hopping process (n = 1 and 2 for the single- and double–polaron processes, respectively), NN_p is proportional to the square of the density of states, ε′ is the dielectric constant, and R_ω is the hopping distance of the charge carriers responsible for conduction.

The hopping distance, defined by R_ω for a given frequency and temperature within the framework of the CBH model, is given by:³⁷

20

where e is the electron charge, ε₀ is the vacuum permittivity, k_B is the Boltzmann constant, T is the temperature, and τ₀ is the characteristic relaxation time.

In the bipolaron hopping process, the parameter NN_p can be related to the density of localized states N_T through:NN_p = N_T²In the case of single-polaron hopping, the expression is:

21

where U_eff is the effective correlation energy due to the electron–phonon interactions, which might become negative when the electron–phonon coupling is strong enough.

Fig. 7(a) depicts the evolution of Ln σ_ac as a function of the inverse absolute temperature at various applied frequencies. The presented curves are characterized by a linear increase with a rise in temperature, which confirms that the AC conductivity follows the thermally activated mechanism that involves localized charge carriers. The lines in the figure are the best fits of the experimental data. This consequently denotes that AC conduction in lithium-vanadate pyrophosphate is mainly controlled by a bipolaron hopping process.

Fig. 7. (a) Fitting of the real part of the AC conductivity at different frequencies using the CBH model and (b) temperature dependence of the tunneling distance R_ω (Å) at different frequencies for Li₂V_0.4P₂O₇.

The different parameter values used in the fitting procedure are summarized in Table 5. We can conclude that the density of localized states decreases with an increase in frequency. Following Mott's explanation, such a decrease in the number of localized states with an increase in frequency can thus be considered to be due to an increase in the structural disorder, which reduces the stability of the localized states and favors their partial delocalization.³⁸

Table 5. Experimental parameters extracted for different frequencies for the CBH model.

Frequency (Hz)	N (eV−1 m−1)	W m (eV)
209	6.48 × 1025	1.79
1060	5.98 × 1023	1.67
11 400	2.44 × 1021	1.49
102 000	1.35 × 1020	1.37
1 010 000	1.06 × 1019	1.36

The frequency variation of the hopping distance (R_ω) for the Li₂V_0.4P₂O₇ compound is presented in Fig. 7(b). It is observed that R_ω increases monotonically with increasing frequency. This behavior is consistent with a decrease in the density of localized states (N), as shown in Table 5. Moreover, the potential barrier height W_m shows a slight decrease with frequency. This suggests that the energy barrier separating neighboring sites becomes lower at higher frequencies, meaning that charge carriers require less energy to perform short-range hopping. Consequently, the conduction process becomes increasingly localized as the frequency increases.

3.7. Thermodynamics constants

The relaxation time may be represented as follows using Eyring's theory:³⁹

22

where h is Planck's constant, R is the universal gas constant, ΔS represents the migration entropy, and ΔH is the enthalpy of the system.

The thermodynamic parameters ΔH and ΔS can be extracted from the linear plot of Ln(τ*T) as a function of the inverse of temperature (Fig. 8(a)).

Fig. 8. (a) Temperature dependence of Ln(τT) and (b) variation in ΔG as a function of Li₂Cu_(1−(5/2)x)V_xP₂O₇ (x = 0, 0.2 and 0.4) at T = 603 K (phase I) and T = 683 K (phase II).

The curve shows Arrhenius-type behavior, and the slope clearly changes near the transition temperature that was identified by thermal analysis. This deviation allows the determination of the thermodynamic parameters separately for the two regions.

During phase I, the change in enthalpy of activation ΔH was determined to be 17.33 kcal mol⁻¹, while the corresponding entropy (ΔS/R) was −0.959. The change in enthalpy ΔH is 12.721 kcal mol⁻¹ during phase II, and entropy (ΔS/R) is −4.582.

Instead of reporting ΔG at single temperatures, the Gibbs free activation energy was calculated as a function of the vanadium substitution level using the following relation:

ΔG = ΔH − TΔS 23

Fig. 8(b) shows the evolution of ΔG as a function of the vanadium content x, as computed at the characteristic temperatures of each phase, 603 K for phase I and 683 K for phase II.

A clear decreasing trend is observed: ΔG decreases with increasing vanadium content. This reduction in the activation free energy suggests that V⁵⁺ incorporation lowers the energy barrier for charge transport, which may be associated with enhanced Li⁺ mobility within the structure.

It should be noted that ΔS values are not necessarily positive. Negative values of ΔS correspond in general to dipole–dipole interactions, which indicate that the molecular entities are more ordered or more closely packed in the activated state.

4. Conclusion

The Li₂V_0.4P₂O₇ compound was prepared by the conventional ceramic method. X-ray diffraction analysis confirmed the formation of a single phase compound crystallizing in the monoclinic system with space group P2₁/c. The replacement of Cu²⁺ by V⁵⁺ induces a significant structural evolution as compared to the end compositions (x = 0 and x = 0.2). Such an evolution is attested by the variation of the unit cell parameters and symmetry. IR spectroscopy confirms the presence of characteristic entities of the P₂O₇⁴⁻ anionic group, while the examination of the thermogram reveals a thermal transition at 640 K.

Impedance spectroscopy reveals thermally activated charge transport and the systematic enhancement of electrical conductivity with an increase in the vanadium content. An equivalent circuit was used to model the Nyquist plots, which allowed the refinement of parameters and the extraction of the DC conductivity at any specific temperature, σ_DC. The thermal variation of σ_DC follows an Arrhenius-type dependence, although it shows a noticeable deviation around the transition temperature identified by thermal analysis. In addition, the thermal variation of the frequency exponent s reflects a variation in the predominant conduction mechanism: NSPT below the transition temperature gives way to a CBH at higher temperatures.

Overall, the comparison in the three studied compositions shows that aliovalent V⁵⁺ substitution significantly enhances charge mobility. This improvement may be associated with lattice distortions and increased structural disorder induced by the Cu²⁺ → V⁵⁺ substitution, which facilitates hopping transport within the pyrophosphate framework and may contribute to Li⁺ mobility. The x = 0.4 composition thus presents the best electrical performance within this series. These results highlight the potential of this material for solid-state functional applications involving thermally activated charge transport.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.