A comparative and correlation structure-Ac conductivity analysis of doped 1%Eu³⁺ and undoped ceramic of Li₂BaP₂O₇

Abstract

In this work, Li₂BaP₂O₇ doped 1% Eu³⁺ was prepared by a solid-state reaction method and characterized by X-ray diffraction technique, IR, and impedance spectroscopy. Rietveld refinement of the X-ray diffraction pattern suggests the formation of the single-phase desired compound with monoclinic structure at room temperature. Electrical properties were studied using complex impedance spectroscopy in the frequency range of 200 Hz–5 MHz and temperature range of 598–724 K. The temperature dependence of the bulk and grain boundaries’ conductivity was found to obey the Arrhenius law with activation energies E_g = 0.79 eV and E_gb = 1.85 eV respectively. Temperature dependence of the power law exponent s strongly suggests that tunneling of large polaron is the dominant transport process. The obtained results are compared with the undoped Li₂BaP₂O₇ sample and correlated with structural analysis. Doping pyrophosphate compound by Eu³⁺ presents better conductivity and electrical properties.

Introduction

Rare earth (RE) doped phosphate materials have attracted more and more attention due to their excellent thermal and chemical stability. They have an important role in the development of optical display systems using solid-state lighting such as field emission displays (FEDs), cathode ray tubes, plasma display panels (PDP), optoelectronic devices, biological fluorescence labeling, luminescent paints, and inks for security [1,2,3,4,5]. In order to understand the electrical properties of materials used for solid electrolytes in lithium-ion and to define their physical and chemical parameters, there are two general classes: inorganic ceramics and organic polymers. The pyrophosphates compounds Li₂MP₂O₇ (M = Fe, Mn, and Co), which contain P₂O₇ polyanions formed by two PO₄ units sharing one O–O edge, have also been identified as cathode materials for lithium-ion batteries. For this reason, we are interested in this family of compounds, lithium pyrophosphate Li₂MP₂O₇ (M = Ba, Zn, Cu …) are members of a large family of compounds of the A₂BP₂O₇ type which contain simultaneously an alkali metal ion (A⁺) and a divalent cation (B²⁺). Among these conductive materials, lithium pyrophosphate Li₂BaP₂O₇ is mentioned which exhibits interesting magnetic, optical, and electric properties. It is suggested that the Li₂BaP₂O₇:Eu³⁺ is a potential host material for laser application [6]. This material is applied in different fields such as energy and electricity, thanks to its high ion or proton mobility [7]. In 2002, Dridi et al. investigated the synthesis and crystal structure of dilithium barium diphosphate Li₂BaP₂O₇ [8]. Eu³⁺/Sm³⁺- and Ce³⁺-activated luminescence properties of Li₂BaP₂O₇ were reported by Wani et al. [9, 10]. Recently, the electrical properties of Li₂BaP₂O₇ were described by Krichen et al. [7].

In this paper, the AC conductivity of the ceramic compound Li₂BaP₂O₇:1% Eu³⁺ has been investigated to determine the mechanism for the AC conductivity. The results will be compared with the undoped Li₂BaP₂O₇ sample. Interpretation of all results has been correlated with structural analyses.

Experimental procedure

The pyrophosphate Li₂BaP₂O₇: 1%Eu³⁺ was obtained by the conventional solid-state reaction. Stoichiometric quantities of lithium carbonate (Li₂CO₃), barium carbonate (BaCO₃), erbium oxideEu₂O₃, and ammonium dihydrogen phosphate (NH₄H₂PO₄) were of high purity (≥ 99%). The reagents were firstly ground into fine powders using mortar and pestle, intimately mixed, and progressively heated first to 593 K for 8 h to expel gases. The obtained powder was then pressed into cylindrical pellets and heated once at 1023 K for 10 h. A pellet of about 8 mm diameter and about 1.5 mm thickness was used for the optical measurements.

X-ray powder diffraction (XRD) measurements were performed at room temperature by using a Phillips powder diffractometer PW 1710 with CuKα radiation (λ = 1.5405 Ǻ) at glancing angles between 4° and 94°.

IR spectroscopic analyses were carried out with a FTIR-100 Perkin Elmer spectrophotometer in the wavenumber range of 1300–550 cm⁻¹.

To determine electrical and dielectric properties of the sample, the sintered pellets were sandwiched between two platinum electrodes, heated at 723 K for 1 h, and cooled to room temperature before measurements were performed as a function of both temperature (598–724 K) and frequency (200 Hz–5 MHz) using a TEGAM 3550 ALF impedance analyzer.

Results and discussions

X-ray analysis

The crystal structure of Li₂BaP₂O₇ was first reported by Liebertz and Stahr in 1983 [11] who found that it crystallizes in the orthorhombic crystal system with Cmcm space group. More detailed studies have revealed that Li₂BaP₂O₇ crystallizes in a monoclinic crystal system with C2/c space group [8].

The X-ray diffraction pattern of Eu³⁺ doped Li2BaP2O7 was recorded at room temperature, along with Rietveld refinement. Experimental, calculated, and shift powder XRD patterns were shown in Fig. 1. The obtained reliability factors are R_p = 12.5%, R_wp = 16.8%, R_exp = 11.9%, Bragg R factor = 13.7%, and χ² = 2.76. The sample crystallizes in the monoclinic space group C2_/c with the following unit cell parameters: a = 7.084 (2) Ǻ, b = 12.160 (3) Ǻ, c = 13.85 (2) Ǻ, and β = 90.69°. There is a very good agreement between all experiments realized in the same sample, confirming the crystal purity of the synthesized materials. In view of the effective ion radii of cations and disparity valence state, Li⁺ = 0.092 nm, Ba²⁺ = 0.142 nm, it is predicted that Eu³⁺ = 0.101 nm may perfectly occupy the different site of Ba²⁺ [9].

Fig. 1

Powder X-ray diffraction pattern and Rietveld refinement for the sample Li₂Ba_0.99Eu_0.01P₂O₇ (circle signs correspond to experimental data, and the calculated data are represented by the continuous line overlapping them: tick marks represent the positions of allowed reflection, and a difference curve on the same scale is plotted at the bottom of the pattern)

Crystal structure of the simple Li₂Ba_0.99Eu_0.01P₂O₇ found that Ba²⁺ is surrounded by 9 oxygen in irregular geometry. Each BaO₅ and EuO₅ polyhedron is surrounded by five PO₄ tetrahedral belonging to three different P₂O₇ groups. The Li⁺ ions are located in three different crystallographic sites. Li₁ is irregularly surrounded by five oxygen neighbors (site symmetry C₁), whereas Li₂ and Li₃ exhibit distorted tetrahedral geometry (site symmetry C₂) (Fig. 2). Atomic positions are summarized in Table 1. Interatomic distances and some selected angles are reported in Table 2 for both doped and undoped compound. The main values of P–O–P angles are 123.06° for Li₂BaP₂O₇ and 122.93° for Li₂Ba_0.99Eu_0.01P₂O₇.

Fig. 2

Crystal structure of Li₂Ba_0.99Eu_0.01P₂O₇

Table 1 Atomic coordinates of Li₂Ba_0.99Eu_0.01P₂O₇

Table 2 Comparison of angles and interatomic distances with Li₂BaP₂O₇ compound

IR spectroscopy analysis

The infrared bands for undoped Li₂BaP₂O₇ and doped Li₂Ba_0.99Eu_0.01P₂O₇ are displayed in Fig. 3. The assignment of the bands is made on the basis that ν_as (PO₃) > ν_s (PO₃) > ν_as(P–O–P) > ν_s(P–O–P) > δ (PO₃) > δ(P–O–P).ν_as and ν_s refer respectively to asymmetric and symmetric stretching vibrations of P–O bonds in (PO₃) groups or in (P–O–P) bridge; δ refers to deformation modes of (O–P–O) angles.

Fig. 3

Infrared analysis spectrum of Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇

Since these absorption bands generally overlap, it is not possible to give more precise assignment than that proposed in Table 3. Band assignments for the fundamental modes of Li₂Ba_0.99Eu_0.01P₂O₇ and Li₂BaP₂O₇, given in Table 3, confirm the presence of the diphosphate groups in the title compounds [12].

Table 3 Assignment of IR bands for Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇

The bands due to the symmetric and antisymmetric stretching frequencies of P–O–P are generally observed in the regions 970–910 cm⁻¹ and 770–670 cm⁻¹.

The bands due to δ (PO₃) and δ (P–O–P) deformations are observed in the regions 501–640 cm⁻¹and 554–656 cm⁻¹ respectively for Li₂Ba_0.99Eu_0.01P₂O₇ and Li₂BaP₂O₇.

Electrical impedance analysis

Equivalent circuit study

Equivalent circuit models are described to analyze the electrical response of the sample. Diverse equivalent circuit models have been reported in the literature. Usually, electrical responses, obtained in ceramic materials, are attributed to grain, grain boundary, and electrode effect. Each contribution can be described by a parallel combination of a resistor (R) and a pure capacitor (C) [13] for the ideal response (Debye’s model).

These contributions can conventionally be displayed in a complex plane plots (Nyquist diagram) in terms of the same formalism:

Complex impedance:

$$ {Z}^{\ast}\left(\upomega \right)={Z}^{\prime }+j{Z}^{\prime \prime } $$

(1)

Complex permittivity:

$$ {\upvarepsilon}^{\ast}\left(\upomega \right)={\upvarepsilon}^{\prime }-j{\upvarepsilon}^{\prime \prime } $$

(2)

Complex modulus:

$$ {M}^{\ast}\left(\upomega \right)=\frac{1}{\ {\upvarepsilon}^{\ast}\left(\upomega \right)}={M}^{\prime }+{jM}^{\prime \prime } $$

(3)

Experimental impedance spectra for the ceramic compound Li₂Ba_0,99Eu_0,01P₂O₇, measured at different temperatures from 598 to 724 K, are shown in Fig. 4a. All diagrams are composed of two successive semicircles which indicate some degree of decentralization. In this case, the centers of semicircles that compose total electric response are centered below of real axis (Z′) which confirms the presence of non-Debye type of relaxation in the materials.

Fig. 4

a Complex impedance spectra at various temperatures of Li₂Ba_0.99Eu_0.01P₂O₇ with electrical equivalent circuit. b Frequency dependence of the imaginary part M′′ at 724 K of Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇. c Complex impedance plots for Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇ at 609 K and the equivalent circuit model

To verify the presence of different types of contributions in the studied material, analysis of all relaxation properties can be tested by the modulus representation. Imaginary parts of modulus M′′ are calculated by the following equation:

$$ {M}^{\prime \prime }=\omega {C}_0{Z}^{\prime } $$

(4)

where ω, C₀, and Z′ are angular frequency, vacuum capacitance of cell, and the real parts of the complex impedance respectively. The obtained modulus spectra M′′(ω) versus frequency for Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇ at 724 K is depicted in Fig. 4b. There are two relaxation peaks. Thus, the smaller peak at a lower frequency is associated to grain boundary effects, and the well-defined one at higher frequency is correlated to bulk effects. Thereby, a typical equivalent circuit, consisting of a series of combinations of grains and grain boundary elements, was used to fit the measured data. The first consists of a parallel combination of resistance (R_g) and fractal capacitance CPE₁, whereas the second consists of a parallel combination of resistance (R_gb), capacitance, and fractal capacitance CPE₂ (inset Fig. 4c).

The expressions of real (Z′) and imaginary (Z″) components of impedance related to the equivalent circuit are:

$$ {Z}^{\prime }=\frac{R_g^2{Q}_1{\omega}^{\alpha_1}\mathit{\cos}\left(\frac{\alpha_1\pi }{2}\right)+{R}_g}{\left[1+{R}_g{Q}_1{\omega}^{\alpha_1}\mathit{\cos}\left(\frac{\alpha_1\pi }{2}\right)\right]2+{\left[{R}_g{Q}_1{\omega}^{\alpha_1}\mathit{\sin}\left(\frac{\alpha_1\pi }{2}\right)\right]}^2}+\frac{R_{jg}^{-1}+{Q}_2{\omega}^{\alpha_2}\cos \left(\frac{\alpha_2\pi }{2}\right)}{{\left[{R}_{jg}^{-1}+{Q}_2{\omega}^{\alpha_2}\cos \left(\frac{\alpha_2\pi }{2}\right)\right]}^2+{\left[ C\omega +{Q}_2{\omega}^{\alpha_2}\sin \left(\frac{\alpha_2\pi }{2}\right)\right]}^2} $$

(5)

$$ -{Z}^{\prime \prime }=\frac{R_1^2{Q}_1{\omega}^{\alpha_1}\mathit{\sin}\left(\frac{\alpha_1\pi }{2}\right)}{{\left[1+{R}_1{Q}_1{\omega}^{\alpha_1}\mathit{\cos}\left(\frac{\alpha_1\pi }{2}\right)\right]}^{{}^2}+{\left[{R}_1{Q}_1{\omega}^{\alpha_1}\mathit{\sin}\left(\frac{\alpha_1\pi }{2}\right)\right]}^2}+\frac{C\omega +{Q}_2{\omega}^{\alpha_2}\sin \left(\frac{\alpha_2\pi }{2}\right)}{{\left[{R}_2^{-1}+{Q}_2{\omega}^{\alpha_2}\cos \left(\frac{\alpha_2\pi }{2}\right)\right]}^2+{\left[ C\omega +{Q}_2{\omega}^{\alpha_2}\sin \left(\frac{\alpha_2\pi }{2}\right)\right]}^2} $$

(6)

As shown in Fig. 4a, those diagrams clearly present a good agreement between theoretical (line) and experimental (scatter) data. Consequently, the suggested equivalent circuit describes reasonably well the crystal-electrolyte interface. Fitted values of grain and grain boundary’s parameters, for all temperatures, are summarized in Table 4.

Table 4 Extracted parameters of the equivalent circuit for the Li₂Ba_0.99Eu_0.01P₂O₇ sample

The capacitance values of the high- and the low-frequency semicircles are found to be respectively in the range of pF and nF proving that the observed semicircles represent the bulk and grain boundary response of the system respectively [14, 15]. In addition, we note that α₁ values are very close to unity indicating that the interactions between the dipoles are low.

Figure 4c shows the complex impedance plots for Li₂BaP₂O₇ and Li₂Ba_0,99Eu_0,01P₂O₇ powder at 609 K and the equivalent circuit model. The modeling of these is carried out with the same equivalent circuit (inset Fig. 4c), whose refinement parameters are summarized in Table 5.

Table 5 Comparison of the extracted parameters of the equivalent circuit at 609 K for the Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇

Investigation of DC conductivity

The values of resistance R_g, R_gb, and total resistance R_tot (R_tot = R_g + R_gb), extracted by equivalent circuit, for each semicircle are used to determine the direct conductivity DC. The DC conductivity (σ_DC) has been calculated for each temperature by means of the following relations:

$$ {\sigma}_g=\frac{e}{R_gS} $$

(7)

$$ {\sigma}_{gb}=\frac{e}{R_{gb}S}\times \frac{Q_g}{Q_{gb}} $$

(8)

$$ {\sigma}_{tot}=\frac{e}{R_{tot}S}=\frac{e}{\left({R}_g+{R}_{gb}\right)S} $$

(9)

Where e is the thickness (e = 1 mm), S is the surface of the sample(S = 0.503 cm²), and Q_g and Q_gb are respectively the capacitance of grain and grain boundary.

Figure 5 shows that the conductivity obeys the Arrhenius expression of the form:

$$ {\sigma}_{DC}T={\sigma}_0\exp \left(\frac{-{E}_a}{k_BT}\right) $$

(10)

Fig. 5

Variation of the ln (σT) as function of temperature of the Li₂Ba_0.99Eu_0.01P₂O₇ sample

Where T is absolute temperature, σ₀ is pre-exponential factor, k_B is Boltzmann constant, and E_a is activation energy of the transport process. This plot indicates that electrical conductivity is thermally activated. The value of activation energy, estimated from the Arrhenius plot of σ_g, σ_gb, and σ_tot, with respect to 1000/T are respectively E_g = 0.79 eV, E_gb = 1.85 eV, and E_tot = 1.23 eV (Fig. 5).

These later are summarized in Table 6 with some other data such as the grain’s conductivity, total conductivity, and the average inter-atomic distances. This table compares the various values obtained by the undoped compound Li₂BaP₂O₇ [7]. The conductivity of these compounds can be influenced by factors such as tunnel size, polarizability, and charge carrier mobility. The ease with which an ion can reach a neighboring site is controlled by the activation energy. This latter indicates the barrier of free energy that an ion must overcome to succeed its jump between sites. We note that the studied compound is characterized by the higher activation energies and better conductivity. This result can be explained by the structure of the studied compound. Indeed, the decreasing of the Li–O and Ba–O bond length results to an increase of the attraction force between Li⁺, Ba²⁺, and Eu³⁺ and O²⁻, and the ions become more attached to the crystal. Structural analyses show that the size of the charge carriers Li⁺, inserted in a tunnel, is much wider of the undoped structure. Added to that, the decrease of the number of oxygen forming Barium polyhedron from BaO₆ for the undoped compound to BaO₅/EuO₅ for Li₂Ba_0.99Eu_0.01P₂O₇ may explain the obtained result of conductivity. Consequently, this deformation of structure can prove the highest conductivity of our compound. It produces the increase of potential barrier and requires higher activation energy.

Table 6 Activations energies and conductivities at 683 K of grain, grains boundaries, total conductivity, and inter-atomic distances in Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇

AC conductivity analysis

AC measurements are very important for any dielectric material as it gives a lot of information about dynamic properties such as capacitance, conductivity, and loss factor. AC measurements are also helpful in identifying the nature of conduction mechanism.

The complex conductivity can be given as:

$$ {\sigma}_{AC}^{\ast }={\sigma}^{\prime }+j\ {\sigma}^{\prime \prime } $$

(11)

In the present work, the real and imaginary of complex conductivity were calculated in accordance with the following relation:

$$ {\sigma}^{\prime }=\frac{e}{s}\left(\frac{Z^{\prime }}{Z^{\prime 2}+{Z}^{\prime \prime 2}}\right) $$

(12)

$$ {\sigma}^{\prime \prime }=\frac{e}{s}\left(\frac{Z^{\prime \prime }}{Z^{\prime 2}+{Z}^{\prime \prime 2}}\right) $$

(13)

where Z′ and Z′′ are the real and imaginary part of complex impedance, and e and s are respectively the thickness and the area of the present pellet.

The real part of AC conductivity can be written by following the power law [16]:

$$ {\sigma}_{AC}={\sigma}_{DC}+A{\omega}^S $$

(14)

Where σ_DC is the direct conductivity in a particular range of temperature, A is a temperature dependent parameter, and s is the temperature-dependent exponent in the range of 0 ≤ s ≤ 1.

This later parameter s presents the degree of interaction between mobile ions with the environments surrounding them, and A determines the strength of polarizability.

Figure 6 presents the frequency dependence of the AC conductivity (σ_AC) of Li₂Ba_0.99Eu_0.01P₂O₇ at different temperatures (586–724 K). The AC conductivity increases with the increase of frequency. As illustrated in Fig. 6, these curve increases with increasing of both temperature and frequency.

Fig. 6

Frequency dependence of AC conductivity at various temperatures of Li₂Ba_0.99Eu_0.01P₂O₇

The classic Jonscher equation (Eq. (14)) does not allow us to explain the behavior of our experimental data. For this, we tried to use the developed Jonscher’s equation which gives a better agreement with the experimental values [17]:

$$ {\sigma}_{AC}\left(\omega \right)=\frac{\sigma_s+{\sigma}_{\infty}\tau 2\omega 2}{1+\tau 2\omega 2}+A{\omega}^s $$

(15)

In this above equation, σ_s and σ_∞ are respectively an estimate of conductivity at low and high frequencies; ω is angular frequency; and τ represents the characteristic relaxation time.

Frequency dependence of AC conductivity at 683 K of Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇ is shown in Fig. 7. In high-frequency domain, we note that AC conductivity for Li₂Ba_0.99Eu_0.01P₂O₇ is greater than that obtained for the undoped compound.

Fig. 7

Frequency dependence of AC conductivity at 683 K of Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇

As presented in Fig. 6, solid line shows the best modeling of the experimental data by Eq. (15) at different temperatures. This fitting allows us to determine several parameters such as the exponent s. Variation of this parameter s with temperature is illustrated in Fig. 8 for both doped and undoped compound Li₂Ba_0.99Eu_0.01P₂O₇. It seems from this figure that values of s decrease to a minimum value, at about 657 K for Li₂Ba_0.99Eu_0.01P₂O₇, and then increase again with increasing temperature. In addition, thermal variation of this parameter shows a slight evolution in our compound Li₂Ba_0.99Eu_0.01P₂O₇ relative to that of Li₂BaP₂O₇. This reflects a weak interaction between the different sites of the free-charge carriers upon insertion of the rare earth ion Eu³⁺ in the matrix Li₂BaP₂O₇.

Fig. 8

Temperature dependence of the exponent s for both Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇

Theory investigation of conduction mechanism

The temperature dependence of s is very useful for determining the type of conduction mechanism in different materials. In the literature, various models have been predicted to explain the temperature dependence of s behavior such as quantum mechanical tunneling (QMT) model, correlated barrier hopping (CBH) model, and the overlapping large-polaron tunneling (OLPT) model [18, 19].

It is seen from Fig. 8 that the values of s, obtained at a different temperature, are less than 0.8 and temperature dependent. Consequently, QMT model is non-applicable to the present electrolyte system. Comparing the results obtained (Fig. 8) with the predicted exponent (s) temperature dependence behavior for CBH and OLPT models, the conduction mechanism for the Li₂Ba_0.99Eu_0.01P₂O₇ compound can be most probably interpreted based on the overlapping large polaron tunneling (OLPT) model, where the exponent s decreases with increasing temperature, reaches a minimum at 657 K and, thereafter, increases with temperature.

If overlapping large polaron is formed, the expression for AC conductivity and frequency exponent is given as [20, 21]:

$$ {\sigma}_{AC}\left(\omega \right)=\frac{\pi^4}{12}{e}^2\left({k}_BT\right)2N2\frac{\omega {R}_{\omega}^4}{2\alpha {k}_BT+\left({W}_{H0}{r}_0/{R}_{\omega}^2\right)} $$

(16)

The exponent s is given by:

$$ s=1-\frac{8\alpha {R}_{\upomega}+\frac{6{W}_{HO}{r}_0}{R_{\upomega}{k}_BT}}{{\left[2\alpha {R}_{\upomega}+\frac{W_{HO}{r}_0}{R_{\upomega}{k}_BT}\right]}^2} $$

(17)

where R_ω is the hopping length at angular frequency ω (is the tunneling distance), r_p is the polaron radius, ε_p is the effective dielectric constant of the material, α is inverse localization length, and N is the density of defect states.

Dielectric study

Studies of the electrical conductivity’s frequency-dependent of materials are important to explain the mechanisms of conduction in these materials. Furthermore, dielectric relaxation studies are important too. In the first place, it reveals significant information about the chemical as well as physical behavior. This, in turn, may be useful in the determination of the structure and defects in solids [22, 23]. Real ε′ and imaginary parts ε″ of dielectric are calculated using the formula:

$$ {\varepsilon}^{\ast }=\frac{1}{j\omega C{Z}^{\ast }}={\varepsilon}^{\prime }+j{\varepsilon}^{\prime \prime } $$

(18)

where Z* is the complex impedance, $C=\frac{\varepsilon_0A}{d}$, A is the effective area of the electrodes, d is the thickness of the pellet, ε₀ is the vacuum permittivity, and ω is angular frequency.

In this case, the dielectric relaxation is described by a non-Debye model which gives the frequency-dependent complex permittivity in the form [22]:

$$ {\varepsilon}^{\ast}\left(\omega \right)={\varepsilon}_{\infty }+\frac{\varepsilon_s-{\varepsilon}_{\infty }}{1+{\left(j\frac{\omega }{\omega_1}\right)}^{1-\alpha }}+\frac{\sigma_{DC}}{j{\varepsilon}_0\omega } $$

(19)

Where σ_DC represents the specific conductivity, ε_s is the static permittivity, ε₀ is the permittivity of free space, ε_∞ is the high-frequency value of ε′, α is the distribution of relaxation time (0 < α < 1), and ω₁ is the frequency of relaxation of the Debye process.

Real and imaginary parts of ε* are given by the following expression:

$$ {\varepsilon}^{\prime}\left(\omega \right)=\frac{\left({\varepsilon}_s-{\varepsilon}_{\infty}\right)\left[1+{\left(\frac{\omega }{\omega_1}\right)}^{1-\alpha}\cos \frac{\left(1-\alpha \right)\pi }{2}\right]}{1+2{\left(\frac{\omega }{\omega_1}\right)}^{1-\alpha}\cos \frac{\left(1-\alpha \right)\pi }{2}+{\left(\frac{\omega }{\omega_1}\right)}^{2\left(1-\alpha \right)}} $$

(20)

$$ {\varepsilon}^{{\prime\prime}}\left(\omega \right)=\frac{\left({\varepsilon}_s-{\varepsilon}_{\infty}\right){\left(\frac{\omega }{\omega_1}\right)}^{1-\alpha}\sin \left(\frac{\left(1-\alpha \right)\pi }{2}\right)}{1+2{\left(\frac{\omega }{\omega_1}\right)}^{1-\alpha}\cos \frac{\left(1-\alpha \right)\pi }{2}+{\left(\frac{\omega }{\omega_1}\right)}^{2\left(1-\alpha \right)}}+\frac{\sigma_{DC}}{\varepsilon_0\omega } $$

(21)

As far as illustrated in Fig. 9a and Fig. 9b, it shows the frequency dependence of real and imaginary part of the dielectric constant at 683 K for both studied compound, Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇, respectively in Fig. 9a and Fig. 9b. The best fit, using Eq. (21), generates a suitable fitting of the curves resulting from experimental data (solid line in Fig. 9b).

Fig. 9

a Frequency dependence of the real part of the dielectric constant at 683 K for Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇. b Frequency dependence of the imaginary part of the dielectric constant at 683 K for Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇

From these curves, we note that the imaginary part of permittivity ε′′ decreases rapidly at low frequency without any relaxation; this confirms that the dielectric behavior of this material does not obey the law of Debye. It is important to mention that there are no substantial relaxation peaks in the frequency range employed in this study. Furthermore, the dielectric loss rises sharply at low frequency indicating that electrode polarization and space charge effects have occurred confirming non-Debye dependence [24].

Modulus study

An alternative approach to investigate the electrical response of materials that present some degree of ionic conductivity is to use the complex electric modulus:

$$ {M}^{\ast}\left(\omega \right)=\frac{1}{\varepsilon^{\ast}\left(\omega \right)}={M}^{\prime }+{jM}^{\prime \prime } $$

(22)

This formalism is particularly suitable to detect phenomena as electrode polarization [25] and bulk phenomenon property as average conductivity relaxation times τ [26,27,28].

The variation of the frequency dependence of M″ (ω) for Li₂Ba_0.99Eu_0.01P₂O₇ is shown in Fig. 10. Two peaks are observed in these curves for all temperature: one at low frequency which corresponds the grain boundary relaxation, while the second one, detected at high frequency, is correlated to grain interior.

Fig. 10

Frequency dependence of the imaginary part of electric modulus at several temperatures

As already established for the compound Li₂BaP₂O₇, the imaginary part of the electric modulus of our compound has been fitted for different temperatures with an approximate frequency representation of the Kohlrausch–Williams–Watts (KWW) function, proposed by Bergman [7]:

$$ {M}^{{\prime\prime} }=\frac{M_{1\ \max}^{{\prime\prime} }}{\left(\left(1-{\beta}_1\right)+\left(\frac{\beta_1}{1+{\beta}_1}\right)\right)\left[\left(\frac{\omega_{1\ \max }}{\omega}\right)+{\left(\frac{\omega }{\omega_{1\ \max }}\right)}^{\beta_1}\right]}+\frac{M_{2\ \max}^{{\prime\prime} }}{\left(\left(1-{\beta}_2\right)+\left(\frac{\beta_2}{1+{\beta}_2}\right)\right)\left[\left(\frac{\omega_{2\ \max }}{\omega}\right)+{\left(\frac{\omega }{\omega_{2\max }}\right)}^{\beta_2}\right]} $$

(23)

where M_max and ω_max are respectively the modulus and the frequency maximum. The value of β is positioned in the 0–1 range, which reflects the importance of coupling between mobile ions in the conduction process.

The relaxation frequency ω_max, extracted from the fit of the modulus for both the doped and undoped compound, is presented as function of temperature for grain and grain boundary respectively in Figs. 11 and 12. These curves follow the Arrhenius relation:

$$ {\omega}_{\mathrm{max}}={\omega}_0\exp \left(\frac{-{E}_a}{k_BT}\right) $$

(24)

Fig. 11

Temperature dependence of the grain relaxation frequency for Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇

Fig. 12

Temperature dependence of the grain boundary relaxation frequency for Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇

Where ω₀ is the frequency at infinite temperature and E_a is the activation energy. The activation energy determined of grain and grain boundary, for both studied compound Li₂Ba_0.99Eu_0.01P₂O₇ and Li₂BaP₂O₇, is summarized in Table 7.

Table 7 Activation energy of Li₂BaP₂O₇ and Li₂Ba_0.99Eu_0.01P₂O₇ from modulus

The obtained values are different from that obtained by conductivity σ_DC which proves that conduction and relaxation do not follow the same mechanism. Added to that, frequency variation of M″ at the temperature T = 724 K, as shown in Fig. 4b, we notice a faster relaxation of dipole moments in the doped compound Li₂Ba_0.99Eu_0.01P₂O₇; indeed, the maximum angular frequency of relaxation of the modulus M″ of Li₂Ba_0.99Eu_0.01P₂O₇ is higher than that of Li₂BaP₂O₇ (ωτ = 1).

Conclusion

In this work, we have synthesized the Li₂BaP₂O₇ doped with 1% Eu³⁺ compound by solid-state method. Rietveld refinement indicates that the sample is single-phase. The impedance plots show two semicircles, which confirm the presence of two relaxation processes associated with the grain and grain boundaries. The OLPT model was found to explain successfully the mechanism of charge transport in Li₂BaP₂O₇:Eu³⁺. A comparative study of structural, vibrational, electric, and dielectric aspects was carried out with the undoped compound Li₂BaP₂O₇. The AC conductivity study shows that the doped compound has a better electrical property. This result can be interpreted on the basis of structural analyses which shows that the charge carriers responsible of conduction for Li⁺ inserted in a tunnel whose size is much wider of undoped structure, the interaction of Li⁺ ion with oxygen atoms decreases (s_doped < s_undoped: the s represents the degree of interaction between mobile ions with the environments surrounding them) which increases their polarizability factor. Consequently, doping of 1% Eu³⁺ in the Ba²⁺ sites improves the electrical conductivity and dielectric properties.