Electrical properties, conduction mechanism and modulus of diphosphate compounds

Abstract

Rubidium aluminium diphosphate was synthesized by a conventional solid-state technique and its conduction properties determined by impedance spectroscopy. X-ray diffraction analysis indicated the formation of a single-phase monoclinic structure (space group P2₁/c). The AC conductivity was determined using impedance spectroscopy over the frequency and temperature ranges of 40 Hz to 7 MHz and 400–700 K, respectively. The real and imaginary parts of the complex impedance were well-fitted to the equivalent circuit model and the frequency dependence of the AC conductivity followed Jonscher's law. The close values of the activation energies obtained from the relaxation time and the grain conductivity (σ_g) indicated that transport could be modelled by a hopping mechanism, probably dominated by the movement of Rb⁺ cations. The correlated barrier hopping model was used to describe the dominant conduction mechanism. A single relaxation peak was observed in the imaginary part of the modulus suggesting the grain response of the system.

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1. Introduction

Since the first characterization of KAlP₂O₇ by d'Yvoire,¹ much research has been devoted to the M^IM^IIIP₂O₇ type diphosphates. The interest in these compounds comes from both fundamental and applied fields of research and has included their structural characterization and potential technological applications. Some of these compounds have magnetic^2,3 and catalytic^4,5 properties and they have been proposed as solid electrolytes for rechargeable alkaline batteries^6,7 and as nanoparticles for the decontamination and remediation of water.⁸ The study of phosphates became very popular after the development of the NASICON group of fast ionic conductors.^9,10 We have previously reported the electrical properties of MAlP₂O₇ (M = K, Ag, Na) compounds.^11–13 In the study reported here, we investigated MAlP₂O₇, including its electrical and dielectric properties, using impedance spectroscopy to determine the conduction mechanism. MAlP₂O₇ can be distinguished by its crystal structure. The structural arrangement consists of corner-sharing AlO₆ octahedra and P₂O₇ diphosphate groups; the three-dimensional framework delimits the tunnels where the monovalent ions reside. RbAlP₂O₇ has a tunnel structure in which the Rb ions reside, which may explain the good conductivity of this compound.

We investigated the electrical properties and conduction mechanism of RbAlP₂O₇ using impedance spectroscopy to determine the conduction and dielectric relaxation mechanisms. There has been no previously published report of the DC and AC electrical conductivities of RbAlP₂O₇. The information on conductivity provided by this study may be important in practical applications.

2. Experimental

RbAlP₂O₇ was synthesized by a conventional solid-state reaction. Stoichiometric quantities of Rb₂CO₃, Al₂O₃ and NH₄H₂PO₄ were ground thoroughly and mixed. The homogenized powder was calcined at 573 K for 8 h to eliminate NH₃, CO₂ and H₂O. The resulting product was reground, pelletized into cylindrical pellets and heated from room temperature to 1073 K for 8 h.

The products were confirmed by XRD. The XRD pattern of the material was recorded at room temperature using a Phillips PW 1710 powder diffractometer with Cu Kα radiation at 1.5418 Å over a wide range of Bragg angles (10° ≤ 2α ≤ 65°). A pellet about 8 mm in diameter and 1.1 mm thick was used for electrical conductivity measurements with 3 T cm⁻² uniaxial pressures. The measurements were performed using an Agilent 4294A impedance analyser (two platinum electrodes and evaporation sputtering). The temperature range was 500–700 K.

3. Results and discussion

3.1. Powder X-ray analysis

3.1.1. Rietveld refinement. The data were analysed using the Rietveld method with the Fullprof program.²⁵ Fig. 1 shows the refinement of the XRD patterns of RbAlP₂O₇. The XRD pattern at room temperature clearly indicated a single phase with no detectable secondary phase.

Fig. 1 Rietveld refinement for RbAlP₂O₇: experimental data are shown in red; calculated data in black; difference between them in blue; and Bragg positions in green.

Profile refinement started with the scale and background parameters, followed by the unit cell parameters. All the reflection peaks were indexed to a monoclinic structure with the space group P2₁/c. The refined unit cell parameters of the compound were: a = 7.401 Å (3), b = 9.921 Å (4), c = 8.224 Å (3) and β = 105.611° (7).

3.1.2. Structural description. One of the best ways to describe the structure of the compound is to consider a framework built from AlP₂O₁₁ units (Fig. 3). AlP₂O₁₁ is built from corner-sharing AlO₆ octahedra and P₂O₇⁴⁻ groups. The diphosphate group is formed from two tetrahedral PO₄ groups sharing one common oxygen atom (Fig. 3).

Fig. 2 Crystal structure of the RbAlP₂O₇ diphosphates; view along the c-axis.

A view of the structure along the [110] direction is shown in Fig. 2. Rubidium aluminium diphosphate is composed of [P₂O₇] and [AlO₆] groups. The [P₂O₇]⁴⁻ anion consists of a pair of corner-sharing PO₄ groups and the Al³⁺ cations are bonded to six oxygen atoms. Each octahedron is linked to five different P₂O₇ groups (Fig. 2). The AlO₆ octahedra bridge the diphosphate anions, forming a three-dimensional network with a tunnel filled by Rb⁺ cations (Fig. 2). This arrangement is expected to promote ionic mobility.

Fig. 3 Geometry of the (P₂O₇)⁴⁻ anion in RbAlP₂O₇, the AlP₂O₁₁ unit and AlO₆ octahedra.

The P₂O₇ diphosphate anion is characterized by a P–O–P angle of 127.8° (10) and the P–O bond ranges from 1.49 (2) to 1.68 (2) Å. The AlO₆ octahedra are characterized by an Al–O bond ranging from 1.93 (1) Å to 1.98 (1) Å.

3.2. Impedance analysis and equivalent circuit

Impedance spectroscopy is the most reliable technique with which to study the electrical properties and processes of materials. It gives a direct correlation between the response of a real system and an idealized model circuit composed of discrete electrical components.¹⁴ It helps to separate grain and grain boundaries contributions in the transport properties of the materials.

Fig. 4 shows the complex impedance plane plots of RbAlP₂O₇ at different temperatures. In the temperature range explored, the complex plot consists of a depressed semi-circle. The decentralization is indicative of a non-Debye-type relaxation process and the material obeys Cole–Cole formalism.¹⁵ The point of intercept on the real axis shifts towards the origin of the complex impedance spectrum, indicating a decrease in the resistive properties of this material. The presence of single semi-circular arcs indicates that the electrical processes in the material are a result of contributions from the bulk material¹⁶ and thus it can be modelled as an equivalent electrical circuit consisting of a parallel combination of a bulk resistance (R_g) and a bulk capacitance (CPE_g). The impedance of the constant phase element is

where Q indicates the value of the capacitance of the CPE element and α is the fractal exponent. CPE is used to describe the distribution (dispersion) of the value of some physical property of the system and the microscopic material process. A more general non-linear least-squares fit was used to extract the impedance parameters using Origin 8 software. Table 1 gives the magnitudes of R_g and CPE at different temperatures. It is obvious that all the capacitance values are in the pF range, showing that the observed semi-circle represents the grain response of the system.

Fig. 4 Nyquist plots and equivalent circuit model of RbAlP₂O₇. The solid curves are fitted to the equivalent circuit elements.

Table 1 Extract parameters for the circuit elements

T (K)	R (kΩ)	Q (μF)	α
500	856	0.145	0.944
520	696	0.163	0.932
540	549	0.147	0.944
560	445	0.142	0.949
580	365	0.172	0.933
600	300	0.172	0.933
620	256	0.164	0.939
640	214	0.172	0.936
660	168	0.188	0.931
680	137	0.187	0.932
700	107	0.201	0.928

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The expressions of the real and imaginary components of the impedance related to the equivalent circuit are:

Fig. 5 shows the variation of Z′ as a function of frequency at different temperatures. The solid line represents best fit to experimental data according to eqn (2). The value of Z′ decreases with frequency followed by plateau-type behaviour in the high-frequency region. The decrease in Z′ with increasing temperature and frequency indicates the possibility of an increase in the AC conductivity with increasing temperature and frequency.¹⁷

Fig. 5 Variation of real part of the impedance as a function of frequency and temperature.

The coincidence of the Z′ values at high frequencies for all temperatures indicates a possible release of the space charge as a result of a reduction in the resistive behaviour of the material with increasing temperature.¹⁸

Fig. 6a shows the variation in the imaginary part of the impedance Z′′ as a function of frequency at different temperatures. The solid line represents the best fit to the experimental data according to eqn (3). The nature of the variation in Z′′ with frequency is characterized by: (1) the appearance of peaks at a particular frequency; (2) a decrease in the height of the peaks with increasing temperature; (3) a marked asymmetry in the pattern of the peaks; and (4) merging of the spectra at higher frequencies. In addition, the frequency ω_max (corresponding to the maximum value of Z′′) gives the most probable relaxation time τ from the condition ω_maxτ = 1. Such behaviour therefore indicates the presence of relaxation in the system.¹⁹ A similar observation was made by Ben Taher et al.²⁰ for AgAlP₂O₇. The excellent agreement between the experimental (scatter) and theoretical curve (line) indicates that the equivalent circuit describes the compound–electrolyte interface reasonably well.

Fig. 6 (a) Variation of imaginary part of the impedance as a function of frequency at various temperatures. (b) Variation of ln(τ) versus 1000/T.

Fig. 6b shows the plot of the relaxation time τ = 1/ω_max as a function of the inverse of the absolute temperature. The linear behaviour of the relaxation time is in close agreement with the Arrhenius relation τ = τ₀ exp(−E_a/k_BT), where E_a is the activation energy, k_B is the Boltzmann constant and T is the absolute temperature. The obtained activation energy (E_a) estimated from the Arrhenius plot is 0.85 eV.

The grain conductivity was obtained from (R_g) by the relation:

where e and S are the thickness and area of the pellet, respectively.

The thermal evolution of the conductivity of grains, ln(σ_gT) versus 1000/T, is shown in Fig. 7, indicating an Arrhenius-type behaviour. The activation energy was 0.87 eV. This value is similar to the activation energy deduced from the relaxation time plot, suggesting that the relaxation process and the conductivity can be ascribed to the same effect and that the charge carriers have to overcome the same energy barrier in both conduction and relaxation processes.²¹ This close resemblance indicates that the mobility of the charge carriers is probably due to a hopping mechanism dominated by the motion of the Rb ions.

Fig. 7 Variation of ln(σT) versus 1000/T.

The only conduction pathway is along tunnels parallel to the c-direction and the modest ionic conductivity of this material is determined by the dimensions of the tunnel sections. The bottleneck widths of the tunnels vary from 5.055 to 6.988 Å (Fig. 8) and thus they are smaller than the geometrical size of Rb, corresponding to (2 × r_Rb⁺) = 2 × 1.52 = 3.04 Å. We concluded that the transport properties in this material are a result of the movement of Rb⁺ ions along the c-direction.

Fig. 8 Shape and size of the (110) tunnel sections.

3.3. Conductivity study

AC measurements are important in any study of an electrical compound as they give relevant information about properties such as conductivity. The AC conduction is also helpful in identifying the nature of the conduction mechanism. Fig. 9 shows the frequency variation of AC conductivity at different temperatures for RbAlP₂O₇. The conductivity pattern can be divided into two parts. The conductivity is almost independent of frequency in the low frequency region and shows a plateau region that characterizes the direct conductivity resulting from the displacement of charge carriers.²² At higher frequencies the AC conductivity increases in parallel with the increase in frequency and obeys the power law Aω^s. The point at which the change in slope occurs is known as the hopping frequency¹⁹ and this shifts towards higher frequencies with increasing temperature. This indicates the presence of a hopping type mechanism for conduction in this material.²¹

Fig. 9 Variation of σ_ac with angular frequency at different temperatures.

The observed conductivity curves can be described by Jonscher's law:²³

σ_ac = σ_dc + Aω^s (4)

where σ_dc is the direct conductivity at low frequencies, A is a factor dependent on temperature and s is an exponent function of the temperature (0 < s < 1). The exponent s represents the degree of interaction between the mobile ions and the environments surrounding them and is used to suggest an appropriate model for the conduction mechanism.²⁴ Eqn (4) was used to fit the AC conductivity data.

3.4. Investigation of the theory of the conduction mechanism

To determine the most probable conduction mechanism for the AC conductivity of the sample, a model is proposed that takes into account the various theoretical models correlating the conduction mechanism of AC conductivity with the s(T) behaviour.²⁶

According to previously published work, the following theoretical models have been proposed to correlate the conduction mechanism of AC conductivity with the s(T) behaviour.

• The non-overlapping small polaron tunnelling conduction mechanism, in which the exponent s increases with increasing temperature.^11,25

• The correlated barrier hopping (CBH) model, in which s decreases with increasing temperature.^20,26

• The overlapping large polaron tunnelling conduction mechanism, in which the exponent s is both temperature and frequency dependent and s decreases with increasing temperature to a minimum value and then increases as the temperature increases further.²⁷

• The quantum mechanical tunnelling conduction mechanism,²⁸ in which the exponent s is almost equal to 0.8 and increases slightly with increasing temperature or is independent of temperature.

In this instance, s decreases with increasing temperature. The temperature dependence of s is plotted in Fig. 10. This suggests that the CBH model is the most appropriate model to characterize the conduction mechanism in RbAlP₂O₇. The CBH model has been used for many diphosphate materials.^20,29 In the CBH model, the conduction occurs via a single polaron (or bipolaron) hopping process over the Coulomb barrier separating two defect centres. Using the CBH model, the binding energy was computed from the following relation:²⁶

where k_B is the Boltzmann constant, T is the absolute temperature, W_m is the binding energy, ω is the angular frequency and τ₀ is the characteristic relaxation time, which is of the order of the vibrational period of atoms, τ₀ = 10⁻¹³ s. A first approximation of eqn (5) gives the simple expression:²⁶

Fig. 10 Temperature dependence of the exponent s and (1 − s) for RbAlP₂O₇.

Fig. 10 shows the variation of (1 − s) as a function of temperature; the fit of this curve was used to calculate the value of W_m (the binding energy of the carrier in its localized sites), which was equal to 0.52 eV.

These above results can be easily explained using a modified CBH model. According to this model:³⁰

where n is the number of polarons involved in the hopping process, NN_P is proportional to the square of the concentration of states and ε′ is the dielectric constant. NN_P is given by:³¹

NN_P = N_T

for bipolaron hopping and
for single polaron hopping, where U_eff is the effective correlation energy (correlation between electrons and phonons).

The AC conductivity of this sample can be satisfactorily explained by considering only one conduction mechanism (single polaron). Fig. 11 shows the temperature dependence of the AC conductivity in RbAlP₂O₇. It is clear that the AC conductivity varied exponentially with temperature because the ln(σ_ac) vs. 1000/T plots are straight lines. Fig. 11 clearly shows that the theoretical calculations fit well with the experimental data. The values of the CBH model parameters have been adjusted to fit the calculated graphs of ln(σ_ac) versus 1000/T to the experimental curves.

Fig. 11 Fitting of AC conductivity at different frequencies using the CBH model.

The hopping distance R_ω is given by:³²

Fig. 12 shows that R_ω increases with increasing temperature. For a fixed temperature, R_ω decreases with the increase in angular frequency. This behaviour is in harmony with the increase in the conductivity as the frequency increases. The increase in temperature contributes thermal energy to the polaron, which then moves and facilitates the jump because of the inter-chain interaction that occurs.

Fig. 12 Frequency dependence of R_ω (Å) at different temperatures.

The values of R_ω are of the order of the interatomic spacing. The distance Rb–Rb in the direction (001) was 4.3 Å and therefore we can deduce that the conduction process occurs by the movement of Li⁺ along c-axis tunnels.

The results of this experimental study are compared with previous work in Table 2. Good ionic conduction was obtained compared with AB^IIIP₂O₇ studies.

Table 2 Electrical conductivity of the studied compound type A^IM^IIIP₂O₇^a

Compound [reference]	σ600 K (Ω cm)^−1	Conduction mechanism
a CBH, correlated barrier hopping; NSPT, non-overlapping small polaron tunnelling.
NaAlP2O7 (ref. 12)	7.16 × 10^−6	Simple hopping
KAlP2O7 (ref. 11)	8.67 × 10^−7	NSPT
AgAlP2O7 (ref. 20)	8.73 × 10^−5	CBH
LiAlP2O7 (ref. 7)	8.76 × 10^−6	NSPT and CBH
RbAlP2O7 [this work]	5.24 × 10^−6	CBH

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3.5. Modulus study

Formalism of the electric modulus is used to study the electrical relaxation process and is suitable for extracting electrode polarization. The complex electric modulus M is defined by the reciprocal of the complex permittivity ε (M = 1/ε), which can be described as:¹³

M = M′ + jM′′

where M′ and M′′ are the real and imaginary parts of the complex modulus, respectively.

Fig. 13 and 14 shows the real and imaginary parts of the modulus, respectively, at different temperatures RbAlP₂O₇. Fig. 13 shows that the real modulus M′ is dispersed as the frequency increases and tends to saturate at M_∞ at higher frequencies. The small value of M′ in the low frequency region facilitates the migration of ion conduction.³³

Fig. 13 Variation in the real part of the modulus as a function of angular frequency at different temperatures.

Fig. 14 Variation in the imaginary part of the modulus as a function of angular frequency at various temperatures.

Fig. 14 shows that M′′ has a single relaxation peak centred at the dispersion region of M′ and is associated with the grain effect. The region to the left of the peak is the place at which the Rb⁺ ions are mobile over long distances. The region to the right is the place at which the ions are spatially confined to their potential wells.³⁴

The movement of the charge carriers becomes faster as the temperature is increased, leading to a decreased relaxation time and a consequent shift in the peak value of M′′ towards higher frequencies. This behaviour suggests that the relaxation is thermally activated and charge carrier hopping is taking place.³³

An alternative method has been proposed by Bergman et al.³⁵ and the imaginary part of the complex electric modulus in the frequency domain was simulated as:

where M′′_max is the peak maximum, ω_max is the peak frequency of the imaginary part of the modulus and β is an exponent.

4. Conclusion

RbAlP₂O₇ was synthesized using a solid-state technique. Structural investigation using XRD confirmed that the powder had a monoclinic structure of space group P2₁/c. It was concluded from the impedance data that this compound can be modelled by the simple equivalent circuit with R parallel to CPE. The AC conductivity spectra obeyed Jonscher's universal power law, whereas the DC conductivity showed a typical Arrhenius conductivity. An investigation of the ionic conductivity resulting from the mobility of Rb ions within the structure was carried out. A CBH-type model successfully explained the behaviour of the exponent.

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