Synthesis and electric characterization of rare earth double perovskite Ho₂CdZrO₆ ceramics

The X-ray diffraction pattern of HCZ measured at room temperature is shown in Fig. 1. The pattern is characteristic of a perovskite structure. All the reflection peaks of the X-ray profile are indexed and lattice parameters are determined using a least-squares method with the help of a standard computer programme (Crysfire). Good agreement between the observed and calculated interplanner spacing (d-values), Table 1, suggests that the compound crystallizes in monoclinic phase at room temperature. The cell parameters are a = 8.6134 ± 0.0028 Å, b = 6.4611 ± 0.0041 Å, c = 6.2778 ± 0.0018 Å and β = 94.0732 ± 0.0401° with cell volume V = 348.49 Å³. X-ray diffraction confirms that the specimen is single phase. A SEM image of the sample is shown in the inset of Fig. 1. The SEM image indicates the uniformity of the grains in the samples. The grain size of HCZ is found to be in the range 0.61–1.39 μm.

An alternative approach to investigate the dynamical aspects of electrical transport phenomena of materials is the complex electric modulus M*(ω). The electrode polarization effect is minimized in this representation. The complex electric modulus is represented by the following expression [16]:

Where C ₀ is the vacuum capacitance of the cell and ε* is the complex permittivity, M′ and M′′ represent the real and imaginary parts of complex electric modulus (M*).

M _∞  = 1/ε _∞ , M _∞ (high frequency asymptotic value of M′) and ε _∞ is the high frequency asymptotic value of the real part of the dielectric permittivity and φ(t) is the relaxation function, which gives the time evolution of the electric field within the materials. Figure 2(a) and (b) shows the frequency dependence of the real M′(ω) and imaginary M′′(ω) parts of the complex electric modulus (M*) of HCZ at various temperatures. The real part of electric modulus M′(ω) shows a dispersion tending towards M _∞ (high frequency asymptotic value of M′) i.e. at high frequencies, M′(ω) approaches a maximum M _∞ and at low frequencies M′ approaches to zero indicates that the electrode polarization makes a negligible low contribution to M′ [17]. The dispersion in between low and high frequencies is due to the conductivity relaxation. At higher temperatures and at higher frequencies, M _∞ levels off because the relaxation processes are spread over a range of frequencies. The peaks developed in the values of imaginary parts of electric modulus M′′(ω) indicate a relaxation process. The peak position shifts towards higher frequency side with an increase of temperature indicating the thermally activated nature of the relaxation time. Two apparent relaxation regions appear the region towards left of the peak associated with the conduction process where the charge carriers are mobile over a long distance; the region towards right of the peak associated with the relaxation polarization process where the charge carriers are spatially confined to the potential wells. We have fitted our experimental data to Eq. (2) as shown by solid lines in Fig. 2(b) with the Cole-Cole expression defined as [18]:

where, \( A = {{\left[ {1 + 2\left( {\omega {{\tau }^{{1 - \alpha }}}} \right)\sin \left( {\alpha \pi /2} \right) + {{{\left( {\omega \tau } \right)}}^{{2\left( {1 - \alpha } \right)}}}} \right]}^{{1/2}}} \) and \( \varphi = {{\tan }^{{ - 1}}}\left[ {{{{\left( {\omega \tau } \right)}}^{{1 - \alpha }}}\cos \left( {\alpha \pi /2} \right)/\left[ {1 + {{{\left( {\omega \tau } \right)}}^{{1 - \alpha }}}\sin \left( {\alpha \pi /2} \right)} \right]} \right] \).

The value of α is lying in between 0.10 and 0.18 for the temperatures ranging from 30 to 270 °C.

The temperature dependence of the characteristic relaxation time \( {{\tau }_m}\left( { = \omega_m^{{ - 1}}} \right) \) is shown in the inset of Fig. 3, which also obeys the Arrhenius law and the corresponding activation energy is E _a  = 0.23 eV (E _a represents activation energy obtained from the peak position of M′′ versus logω plots). Such a value of activation energy suggests the existence of a relaxation mechanism (a conductive process), which may be interpreted as polaron hopping between neighboring sites within the crystal lattice. In order to mobilize the localized electron, the aid of lattice oscillation is required. In these circumstances, electrons are considered not to move by themselves but by hopping motion activated by lattice oscillation i.e. the conduction mechanism is considered as phonon-assisted hopping of small polaron between localized states.

The scaling behaviour of M′′ i.e. \( M\prime \prime /M\prime {{\prime }_{{\max }}} \)versus log (ω/ω _max) plot of HCZ at various temperatures is shown in Fig. 3. The overlapping of the curves for all the temperatures into a single master curve indicates that the dynamical processes are nearly temperature independent.

The electrical properties of the material were determined by impedance spectroscopy. Experimental complex impedance data may well be approximated by the impedance of an equivalent circuit consists of resistors, capacitors and possibly various distributed circuit elements. These equivalent circuit can be physically interpreted and assign them to appropriate process. The display of impedance data in complex plane plot appears in the form of a succession of semicircles attributed to relaxation phenomena with different time constants due to the contribution of grain (bulk), grain-boundary and interface/electrode polarization in the polycrystalline materials. Hence the contribution to the overall electrical properties by various components in the material is separated out easily. Figure 4 shows a typical complex impedance plane plot and the corresponding equivalent circuit of HCZ at 400, 420 and 450 °C. The impedance spectrum shows only one high frequency semicircular arc, and the semicircular arc becomes smaller with increasing temperature. The high frequency semicircular arc corresponds to bulk property of the material arising due to parallel combination of bulk resistance (R _b ) and bulk capacitance (C _b ) of the material (inset of Fig. 4). The complex impedance (Z*) for the equivalent circuit in this system is

Where R is the series resistance, Z′ and Z′′ are the real and imaginary parts of complex impedance given by:and

We have fitted our experimental data with Eqs. (3) and (4) as shown by solid lines in Fig. 4. The fitted parameters for R _b are 2.24 MΩ, 1.27 MΩ and 0.75 MΩat temperatures 400, 420 and 450 °C respectively, and the corresponding bulk capacitance (C _b ) values are 4.87 × 10^-11 F, 4.95 × 10^-11 F, 5.81 × 10^-11 F. A good agreement between the experimental and fitted data indicates that the polarization mechanism correspond to the bulk effect arising from the semiconductive grains.

AC conductivity measurement is an important tool for studying the transport properties of materials. The conductivity is found to be frequency independent in low frequency region and is illustrated as dc conductivity. The temperature dependence of the dc conductivity gives information about the long time charge carrier dynamics and it is well described by the Arrhenius law, reflecting the activated nature of hopping processes. At high frequencies, the conductivity becomes strongly frequency dependent, varying approximately as a fractional power of frequency. Full analysis of this dispersion not only required accurate data but also appropriate model to represent the response. We have taken Jonscher’s power law for ac conductivity analysis. The frequency dependence of real part of ac conductivity σ′(ω) at various temperatures for HCZ is shown in Fig. 5. The real part of conductivity σ′(ω) shows dispersion that shifts to the higher frequency side with increasing temperature. From Fig. 5 it is clear that σ′(ω) decreases with decreasing frequency and becomes frequency independent after a certain value (i.e. in low frequency region) giving dc conductivity (σ _dc ). The value of σ _dc is found to increase with an increase of temperature. For higher frequencies power-law behaviour, σ′(ω) ~ Aω ⁿ is observed. The ac conductivity spectra then can be described by the so-called Jonscher’s universal power law [19] defined as:

Where σ′(ω), σ _dc , σ _ac (ω) have their usual meanings, A is a constant and n is the frequency exponent lies in the range 0 < n < 1. The term Aω ⁿ comprises the ac dependence and characterizes all dispersion phenomena. The experimental conductivity data of HCZ are fitted to Eq. (5) keeping in mind that the values of parameter n are weakly temperature dependent. The best fit of conductivity spectra is shown by solid lines in Fig. 5 with values of n are 0.88–0.92 for temperatures 350–450 °C. There is a vast number of theoretical approaches to deduce this behavior from the microscopic transport properties of various classes of materials [20]. The quantum-mechanical tunneling model, which corresponds to the variable range hopping (VRH) model for nonzero frequencies, predicts a temperature-independent frequency exponent n near 0.8 [20].

Typical plot of temperature variations of log of dc conductivity σ _dc for HCZ is shown in Fig. 6. It is obvious that the σ _dc increases with increasing temperature showing the semiconducting behaviour of the material. This semiconducting behavior can also be observed in Fig. 5 at lower frequencies. To understand the mechanism of electrical conduction, we have analyzed the conductivity data in terms of small polaron hopping and variable range hopping (VRH) models [21, 22].

The temperature dependence of the dc conductivity σ _dc is shown in Fig. 7. The results indicate that σ _dc (T) cannot be fitted assuming a thermally activated conduction mechanism, which is commonly used to characterize band conduction. The dc conductivity [22] in semiconductors, in which the main conduction mechanism is related to excited carriers beyond the mobility edge into the non-localized or extended states, is expressed as:

where, σ ₀ is the pre-exponential factor, E _a the activation energy for conduction, k _B the Boltzmann constant and T is the Kelvin temperature. The Arrhenius plot shown in the inset of Fig. 7 does not lead to straight line over significant temperature ranges. The slight curvature in the conductivity log σ _dc vs. T ⁻¹ plot occurred within the low temperature range. Such a curvature in the plot suggested that the variable range hopping (VRH) of small polaron model could be concerned in such a case. The variable range hopping of small polaron conductivity model described the conductivity with the formula [21‐25]: where N(E _F ) is the density of localized states at the Fermi level, L the localization length and T ₀ is the Mott’s characteristic temperature. The experimental data were fitted satisfactorily by solid line to Eq. (7) in the temperature range (~463–723 K) with T ₀ = 5.2 × 10⁷  K was obtained from fit as shown in Fig. 7. The reported value of N(E _F )(=5.68 × 10¹⁷ eV^-1 cm^-3) [21, 22, 24, 25] yields a location length L of the order of 18 Å for HCZ as calculated from Eq. (8). This value of location length L is found to be in the same order of magnitude as reported by Mahato et al. [6, 24] for double perovskite oxides. The average hopping distance, R_h(T) between two localized states and hopping energy, E_h(T), are given by the following equations [22]:and,

The calculated values of R_h(300 K) and E_h(300 K) are 13.8 nm and 0.132 eV respectively. It is evident from Fig. 7 that the plot log σ _dc vs. 1/T ^1/4 leads to straight line over a considerable temperature range (463–723 K) i.e. Mott’s law is obeyed over a temperature range 463–723 K and indicates 3D charge transport in HCZ compound. Below 463 K, it was observed that conductivity data deviate from the straight line behaviour because in low temperature region charge conduction is mainly dominated by the thermally stimulated tunneling through the localized sites. Therefore the observed dc conductivity is contributed as the sum of hopping conduction and tunneling conduction. Band conduction is absent because the extended states are far away from the Fermi level. Such a deviation from linearity at lower temperature has also been observed by Molak et al. [26] and explained with the help of variable range hopping of small polaron model.

Figure 7 also suggests that a change in conductivity mechanism likely occurs near 463 K. Above 463 K the conductivity is associated with polaron transport in the extended states at the mobility edge. At temperatures nearing 463 K the probability of thermal release becomes rapidly smaller, which makes polaron hops to a neighbouring localized state more likely to take place below 463 K.

We have performed a scaling process of the a.c. conductivity as a function of frequency as shown in Fig. 8, which shows the conductivity master curves of the sample HCZ. In Fig. 8 we have used log (σ_ac/σ_dc) as the y-axis scaling parameter and log (ω/σ_dcT) as the x-axis parameter at different temperatures. This scaling was proposed by Rolling et al. [27] where it was found that the product Tσ_dc obeys an Arrhenius relation. From Fig. 8, it is clear that a quite satisfying overlap of the data at different temperatures on a single master curve illustrates well the dynamic processes occurring at different frequencies need almost the same thermal activation energy. Another indication of those scaled master curves is that all Arrhenius temperature dependence of conductivity is embedded in the d.c. conductivity term [28].