ISRN CeramicsVolume 2012 (2012), Article ID 854831, 10 pagesdoi:10.5402/2012/854831

Research Article

Electrical Properties and AC Conductivity of (BiNa)BaTiO Ceramic

Ansu K. Roy,¹ Amrita Singh,¹ Karishma Kumari,¹ K. Amar Nath,¹ Ashutosh Prasad,¹ and K. Prasad²

¹University Department of Physics, T.M. Bhagalpur University, Bhagalpur 812007, India²Centre for Applied Physics, Central University of Jharkhand, Ranchi 835205, India

Received 9 August 2012; Accepted 29 August 2012

Academic Editors: H. Maiwa and J. Sheen

Copyright © 2012 Ansu K. Roy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Lead-free perovskite (BNBT06) was prepared by conventional ceramic fabrication technique at 1160°C/3h in air atmosphere. The crystal structure, microstructure, dielectric, polarization, piezoelectric properties, and ac conductivity of the sample were studied. X-ray diffraction data confirmed the formation of a single phase tetragonal unit cell. Williamson-Hall plot was used to calculate the lattice strain and the apparent particle size. The experimental relative density of BNBT06 was found to be ~96-97% of the theoretical one with an average grain size ~4 μm. Room temperature dielectric constant and loss factor at 1 kHz were found to be equal to 781 and 0.085, respectively. Longitudinal piezoelectric charge coefficient of the poled sample under 2.5 kV/mm at 80°C in silicone bath was found to be equal to 124 pC/N. Complex impedance and electric modulus spectroscopic analyses showed the dielectric relaxation in the material to be of non-Debye type. The Nyquist plots and conductivity studies showed the NTCR character of BNBT06. The correlated barrier hopping model (CBHM) as well as jump relaxation model (JRM) was found to successfully explain the mechanism of charge transport in BNBT06. The ac conductivity data were used to evaluate the minimum hopping length, apparent activation energy, and density of states at Fermi level.

1. Introduction

In recent years, there has been a global hunt of piezoelectric materials for their possible use in different electronic devices, and in near future a further increase in their demand is expected. In view of lead zirconium titanate, Pb(Zr,Ti)O₃ (PZT) containing 60–70% of lead which emanates the toxic lead oxide during calcination/sintering, attempts are being made nowadays by the European Union Governments to ban the use of lead-bearing substances harmful to the environment by way of directives like restriction of hazardous substances (RoHS) directive, the end-of-life vehicles (ELV) directive, and the waste electrical and electronic equipment (WEEE) directive [1], and so forth and other countries are following on similar lines. Therefore, there is an urgent need for novel lead-free piezoelectric solid solutions that can be used as alternatives to PZT [2]. Bismuth sodium titanate (Bi0.5Na0.5)TiO₃ (BNT), belonging to the ABO₃-type perovskite family, is considered to be an excellent lead-free piezoelectric material which shows strong ferroelectric properties with rhombohedral symmetry (R3c) at ambient temperature having the planar electromechanical coupling factor , Curie temperature () at 320°C, and a remnant polarization . However, dense BNT ceramic is difficult to be prepared, requiring sintering temperatures above 1200°C, resulting in significant loss of bismuth. Besides, high leakage currents and high coercive field () negatively affect the poling process and polarization saturation is difficult to achieve in conventionally fabricated BNT samples. In addition, BNT exhibits a low depolarization temperature ~200°C. Several workers [3–16] have investigated the electrical behavior, proposed more accurate measurement of the phase transition temperatures of BNT-based solid solutions, and determined their morphotropic phase boundary (MPB) compositions in order to improve dielectric and piezoelectric properties of BNT ceramic. Among these solid solutions, (Bi0.5Na0.5)1−xBaxTiO₃ (BNBTx) has attracted an enormous global attention owing to the existence of a rhombohedral-tetragonal morphotropic phase boundary (MPB) near in the system. Compared with pure BNT, the BNBTx ceramics have been reported to possess relatively high piezoelectric properties and low coercive field near the MPB. The study of electrical conductivity in the ferroelectric compounds is very important from the point of view that the associated physical properties like pyroelectricity, piezoelectricity, and strategy for poling are dependent on the nature and magnitude of conductivity in these materials. In this context, it would be worth mentioning that the information about the electrical properties of BNBTx is still not complete and consistent. Therefore, in order to have knowledge about the performance of BNBTx, it is important to know the carrier transport mechanism in this system and hence the system deserved further investigation. An extensive literature survey revealed that no attempt has so far been made to understand the conduction mechanism in BNBTx using impedance spectroscopy technique. It is with these views that the structural, microstructural, dielectric, polarization, piezoelectric, electric impedance, and ac conductivity studies of (Bi0.5Na0.5)0.94Ba0.06TiO₃ (abbreviated as BNBT06) ceramic near MPB composition have been undertaken in the present work. Further, an attempt has been made to explain the conduction mechanism in BNBT06 using complex impedance and electric modulus spectroscopy techniques.

2. Experimental

Polycrystalline (Bi0.5Na0.5)0.94Ba0.6TiO₃ ceramic was prepared by a standard high-temperature solid-state reaction technique using AR grade (purity more than 99.5%, Hi-Media) oxides and/or carbonates: Bi₂O₃, Na₂CO₃, BaCO₃, and TiO₂ in a suitable stoichiometry. The above ingredients were mixed thoroughly, first in air and then in methanol medium, using agate mortar and pestle. This mixture was calcined at an optimized temperature of 1160°C for about 3 hrs in an AR-grade alumina crucible. Then, by adding a small amount of polyvinyl alcohol (PVA) as binder to the calcined powder, circular and rectangular disc-shaped pellets were fabricated by applying uniaxial pressure of 6 tons/square inch. The pellets were subsequently sintered at an optimized temperature of 1180°C in air atmosphere for about 2 hrs to achieve maximum density (~97% of the theoretical density).

The X-ray diffraction (XRD) spectrum was observed on calcined powder of BNBT06 with an X-ray diffractometer (XPERT-PRO, Pan Analytical) at room temperature, using CuKα radiation (λ = 1.5405 Å) between 2θ values of and . The microstructure of the sintered pellet was studied at room temperature from the micrograph obtained by using a scanning electron microscope (JEOL-JSM840A). The dielectric constant (ε), loss tangent (tanδ), electrical impedance (Z), and phase angle at different frequencies (1 Hz–1 MHz) and temperatures (20°C–450°C) were measured using a computer-controlled LCR Hi-Tester (HIOKI 3532-50, Japan) on a symmetrical cell of type Ag|ceramic|Ag, where Ag is a conductive paint coated on either side of the pellet. AC conductivity data were obtained using a relation: . The P-E hysteresis loop at room temperature was traced using an automatic PE-loop tracer (Marine India Electrocom Ltd., New Delhi, India). Longitudinal piezoelectric charge coefficient () as well as transverse piezoelectric charge coefficient () of the poled ceramic sample (on which a dc electric field ~2.5 kV/mm was applied by keeping it in a silicone oil bath at 80°C) was measured using PM3500 meter (KCF Technologies, PA, USA).

3. Results and Discussion

3.1. Structural and Microstuctural Studies

Figure 1 shows the XRD spectrum of calcined BNBT06 powder at room temperature. A standard computer program (POWD) was utilized for the XRD-profile analysis. Good agreement between the observed and calculated interplanar spacing and no trace of any extra peaks due to the constituent oxides was found, thereby suggesting the formation of a single-phase compound having the tetragonal structure with lattice parameters and . The unit cell volume and tetragonality of lattice (c/a) were estimated to be and 1.0113, respectively. The occurrence of tetragonal nature in BNBT06, which has rhombohedral nature due to BNT phase and tetragonal nature due to barium titanate (BT), is confirmed by the splitting of peak between 46°-47° into two peaks (200) and (002). The apparent particle size and lattice strain were estimated by analyzing the X-ray diffraction peak broadening, using Williamson-Hall approach: , where is the apparent particle size, B is the diffraction peak width at half intensity, is the lattice strain, and is the Scherrer constant (= 0.89). The term Kλ/D represents the Scherrer particle size distribution. The lattice strain was estimated from the slope of the plot as a function of and the apparent particle size was estimated from the intersection of this line at . A linear least squares fitting to data provided the values of the intercept and slope of the plot. A Lorentzian model was applied to estimate the diffraction width at half peak intensity. Here and are the area and centre of the curve, respectively. The inset of Figure 1 illustrates the Williamson-Hall plot for BNBT06. The apparent particle size and lattice strain are estimated to be of the order of 25 nm and 0.0037, respectively.

Figure 1: X-ray diffraction pattern of BNBT06 at room temperature. Inset: enlarged view of XRD pattern between and the Williamson-Hall plot.

The theoretical density for (Bi0.5Na0.5)0.94Ba0.06TiO3 (BNT) was calculated using the equation: ρ = (+)/(/+ /), where and are the weight percentages of the BNT and BT in the mixture, respectively, and and are the densities of the (Bi0.5Na0.5)TiO₃(=5.32 g/cm³) and BaTiO₃(=6.02 g/cm³), respectively. The theoretical density of BNBT06 was found to be equal to 5.357 g/cm³. Using Archimedes principle, the apparent density of sintered ceramic was found to be equal to 5.234 g/cm³, thereby giving the relative density of BNBT06 ~96-97% of the theoretical one. Figure 2 shows the SEM micrograph of BNBT06 with 10 μm scale. Grains of unequal sizes appear to be distributed throughout the sample. The average grain size was estimated to be about 4 μm. The ratio of average crystallite size to the grain size of BNBT06 is found to be . Also, the SEM micrograph does not seem to contain any pores or voids, thereby clearly indicating the high density of the sample.

Figure 2: SEM micrograph of BNBT06 with 10 m scale.

3.2. Dielectric Studies

The frequency dependence of the dielectric constant (ε) and loss tangent (tanδ) at different temperatures are shown in Figure 3. It is observed that ε follows an inverse dependence on frequency, normally followed by almost all dielectric and/or ferroelectric materials. Dispersion with relatively high dielectric constant can be seen in the ε-f graph in the lower frequency region and dielectric constant drops at higher frequencies. The modified Debye equation related to a free dipole oscillating in an alternating field is expressed as , where and are the low- and high-frequency values of ε; ω (=2π f) is the cyclic frequency f being the frequency of measurement; τ the relaxation time; α a measure of the distribution of relaxation time. A relatively high dielectric constant at low frequencies is a characteristic of a dielectric material. At very low frequencies (ω≪ 1/τ), dipoles follow the field and we have ε ≈ (the value of dielectric constant at quasistatic fields). As the frequency increases (), dipoles begin to lag behind the field and hence ε slightly decreases. When the frequency reaches the characteristic frequency (ω = 1/τ), the dielectric constant drops (relaxation process) and at high frequencies (ω≫ 1/τ), dipoles are no longer able to follow the field and hence . This behavior has been observed to be followed in case of BNBT06, at least qualitatively. The value of ε at room temperature at 1 kHz was found to be equal to 781. Further, the value of tanδ decreases with increasing frequency in the high temperature region. On the other hand, at a lower temperature, it reaches a minimum that shifts to the lower frequency side. The frequency-dependent dielectric loss of the material implies that the hopping of charge carriers plays an important role in modifying transport processes occurring in it because a loss peak is an essential feature of the charge carrier hopping transport [17]. Figure 4 shows and tanδ for BNBT06 at different frequencies. It is seen from the plots that the position of the dielectric loss peak shifts to higher frequency side with increase in temperature, thereby suggesting the relaxation to be thermally activated. Further, all the plots show broad maxima (i.e., diffuse phase transition (DPT) near  (~350°C), where anti-ferroelectric to paraelectric phase transition takes place, and another maxima near depolarization temperature (where ferroelectric to antiferroelectric phase transition takes place) at around 125°C. The broadening in the dielectric peak is a common feature in solid solutions, thereby indicating the presence of more than one cation in the sublattice that can produce some kind of heterogeneity. Also, almost stationary temperature-dependent loss tangent curves retain their shapes upto the transition temperature () beyond which the tanδ values sharply increase, possibly due to release of space charge polarization. Room temperature value of tanδ at 1 kHz was found to be equal to 0.085.

(a)

(b)

Figure 3: Variation of dielectric constant and loss tangent with frequency at different temperatures for BNBT06.

Figure 4: Temperature-dependent variation of dielectric constant and loss tangent at different indicated frequencies for BNBT06.

3.3. Hysteresis and Piezoelectric Studies

Attempts have been made to measure the electric field-induced polarization through (P-E) hysteresis loop for the virgin sample kept in silicone oil at room temperature (Figure 5). The study revealed that no saturation in polarization-electric field (P-E) curve could be obtained for the ceramic up to the maximum applied electric field due to the low resistivity of the sample. The sample could not withstand electric field beyond 1.5 kV/mm and further increase in electric field led to electrical breakdown, hence an unsaturated P-E loop for the bulk material sample. For maximum applied electric field ~1.223 kV/mm, coercivity was found to be equal to 0.815 kV/mm.

Figure 5: Room temperature P-E hysteresis loop of BNBT06 sample.

Longitudinal piezoelectric charge coefficient () of the poled ceramic sample was found to be equal to 124 pC/N. Such a high value of confirmed the presence of morphotropic phase boundary (coexistence of rhombohedral and tetragonal phases) in the BNBT06 composition. However, the value of transverse piezoelectric charge coefficient was found to be low (~12 pC/N), giving thereby the value of hydrostatic charge coefficient (= + ) equal to 148 pC/N. Due to the limiting range of measurement (which is only 5 kV) of our poling instrument and coercivity of the material used, we had to take the thickness of the samples small (~1 mm or less). Consequently, we could get only a small amount of charge accumulation on the comparatively small perpendicular surface area, thereby yielding a smaller value of in comparison with that of , thus causing a discrepancy. On a naïve view, this anomaly may possibly be ascribed to the leakage of charge from the surface during manual adjustment of gap between the tips meant for measurement in the instrument (meter). However, there is every chance of creeping up of other dominating factors causing the anomaly, not known to the authors at this point of time. Things may come out only after repeating the experiments under different conditions of poling such as changing the temperature of silicone oil (normally not exceeding 100°C) and/or time of poling (normally not exceeding 30 minutes).

3.4. Electrical Impedance, Electric Modulus, and AC Conductivity Studies

3.4.1. Electrical Impedance Analysis

Complex impedance spectroscopy has been considered as a powerful non-destructive tool to study the microstructure and the electrical properties of solids, where the electrical properties are often represented in terms of some complex parameters like complex impedance ; complex permittivity ; complex electric modulus ; complex admittance . They are interrelated as and the loss tangent, , where , are the series resistance and capacitance; , are the parallel resistance and capacitance, respectively. It is known that for Debye-type relaxation, one expects semicircular Argand (Cole-Cole or Nyquist) plots with the centre located on the . On the other hand for a non-Debye-type relaxation, these Argand complex plane plots are close to semicircular arcs with endpoints on the real axis and the center lying below the abscissa. The complex impedance in such a case can be described as , where α, as referred to earlier, represents the magnitude of the departure of the electrical response from an ideal condition and this can be determined from the location of the centre of the semicircles. Further, it is known that when α approaches zero, that is, , this expression gives rise to classical Debye’s formalism. Figure 6 shows the variation of real and imaginary parts of impedance (, resp.) with frequency at different temperatures. It is observed that the magnitudes of both and decrease with increase in frequency, thus indicating an increase in ac conductivity with rise in frequency. At temperatures <400°C, the plots (not shown in the plot for the sake of brevity) almost resemble with straight lines having large slopes, thereby indicating insulating behavior of the samples. As the temperature is increased, the slopes of the curves decrease and the curves bend down towards the abscissa, thereby indicating an increase in conductivity of the material. As shown in Figure 7(a), two semicircular arcs or depressed circular arcs are seen in the complex plane plot corresponding to a temperature of which can be explained on the basis of two parallel RC elements connected in series, one branch is associated with the grain and other with the grain boundary of the sample. The arc of grain generally lies on a frequency range higher than that of grain boundary, since the relaxation time for the grain boundary is much larger than that of the bulk crystal. This trend indicated negative temperature coefficient of resistance (NTCR) behavior of BNBT06 like that for a semiconducting material. The centers of the semicircles lie below the -axis at an angle “” (not shown in the plots for the sake of brevity), thereby indicating non-Debye-type relaxation process in BNBT06. The observed data indicated that the conduction in BNBT06 is predominant through grain boundary and thus it gives a scope for variety of device applications for this material.

(a)

(b)

Figure 6: Frequency dependence of real and imaginary parts of the impedance of BNBT06 ceramic at the different indicated temperatures.

(a)

(b)

Figure 7: (a) and (b): Nyquist plots in the complex impedance and electric modulus planes, respectively, for BNBT06.

3.4.2. Modulus Spectrum Analysis

In polycrystalline materials, impedance formalism emphasizes grain boundary conduction process, while bulk effects on frequency domain dominate in the electric modulus formalism. The use of modulus spectroscopy plot is particularly useful for separating the components with similar resistance but different capacitance. The other advantage of electric modulus formalism is that the electrode effect is suppressed. Due to the above reasons, complex electric modulus formalism was also adopted in the present case in order to see the different effects separately. Dielectric relaxation studies were carried out in the complex modulus formalism. Variation of real and imaginary parts of the electric modulus ( and as a function of frequency at various temperatures) was studied. From the study, it transpired that the value of increases from the low frequency towards a high-frequency limit and the dispersion shifts to high frequency as temperature increases. The Nyquist plot of versus at 450°C (Figure 7(b)) clearly revealed grain and grain-boundary peaks, thereby giving different values of , , , and . These data yield the grain and grain-boundary relaxation times equal to 0.1365 ms and 1.170 ms, respectively. Figure 8 shows the log-log plot of ac electrical conductivity versus frequency at different temperatures. The plots show dispersion throughout the chosen frequency range. Further, the ac conductivity, in most of the materials due to localized states, is expressed by Jonscher’s power law: where is the frequency-independent (electronic or dc) part of ac conductivity, is the index, is angular frequency of applied ac field and is a constant, is the electronic charge, is the temperature, α is the polarizability of a pair of sites, and is the number of sites per unit volume among which hopping takes place. Such a variation is associated with displacement of carriers which move within the sample by discrete hops of length between randomly distributed localized sites. The term can often be explained on the basis of two distinct mechanisms for carrier conduction: quantum mechanical tunneling (QMT) through the barrier separating the localized sites and correlated barrier hopping (CBH) over the same barrier. In these models, the exponent is found to have two different trends with temperature and frequency. If the ac conductivity is assumed to originate from QMT, s is predicted to be temperature independent but is expected to show a decreasing trend with ω, while for CBH the value of should show a decreasing trend with an increase in temperature. The exponent has been found to behave in a variety of forms [18–20]. In general, the frequency dependence of conductivity does not follow the simple power relation as given above but follows a double power law [18–21] given as where is the frequency-independent (electronic or dc) part of ac conductivity. The exponent characterizes the low-frequency region, corresponding to translational ion hopping and the exponent characterizes the high-frequency region, indicating the existence of well-localized relaxation/reorientational process [21], the activation energy of which is ascribed to reorientation ionic hopping. Further, it may be inferred that the slope is associated with grain-boundary conductivity whereas depends on grain conductivity [22]. In the jump relaxation model (JRM) introduced by Funke [18] to account for ionic conduction in solids, there is a high probability for a jumping ion to jump back (unsuccessful hop). However, if the neighborhood becomes relaxed with respect to the ion’s position, the ion stays in the new site. The conductivity in the low-frequency region is associated with successful hops. Beyond the low-frequency region, many hops are unsuccessful, and as the frequency increases, more hops are unsuccessful. The change in the ratio of successful to unsuccessful hops results in dispersive conductivity. In the perovskite-type oxide materials, presence of charge traps in the band gap of the insulator is expected. The JRM suggests that different activation energies are associated with unsuccessful and successful hopping processes. The frequency and temperature dependence of ac conductivity supports hopping-type conduction in the test material. Applying JRM to the frequency response of the conductivity for the present material, it was possible to fit the data to a double power law as given in (2). The temperature-dependent variations of the exponents, and , are shown in Figure 9. From the plots, it is manifested that assumes a maximum value ~1 while assumes maximum values ~1.4–1.5 with peaks appearing near and . The recent literature also endorsed that the exponent (or ) is not limited to values below 1 [23].

Figure 8: Variation of ac conductivity with frequency at different temperatures for BNBT06 ceramic.

Figure 9: Temperature dependence of low- and high-frequency hopping parameters ( and ) of BNBT06 ceramic.

Further, hopping conduction mechanism is generally consistent with the existence of a high density of states in the materials having band gap like that of a semiconductor. Due to localization of charge carriers, formation of polarons takes place and the hopping conduction may occur between the nearest neighboring sites. The activation energy for conduction was obtained using the Arrhenius relationship: A linear least squares fitting of the conductivity data to (3) gives the value of the apparent activation energy, . Figure 10 shows the temperature-dependent ac conductivity at different frequencies and the inset shows the low- and high-temperature frequency-dependent conductivity-based activation energies of the sample. The values of are found to be ~0.156 eV and ~0.512 eV in the high- and low-temperature regions, respectively, at 1 kHz and conductivity-based activation energies show a decreasing trend of variation with increasing frequency. The low value of may be due to the carrier transport through hopping between localized states in a disordered manner [24, 25]. The enhancement in conductivity with temperature may be considered on the basis that within the bulk, the oxygen vacancies due to the loss of oxygen are usually created during sintering and the charge compensation follows the Kröger and Vink equation [26]: which shows that free electrons are left behind in the ionization process, making the materials n-type.

Figure 10: Temperature-dependent variation of ac conductivity at the different indicated frequencies for BNBT06 ceramic. Inset shows the low- and high-temperature frequency-dependent conductivity-based activation energies.

Based on CBH model, the minimum hopping length, was estimated using the relation [27, 28]: , where ; (i = 1 or 2) is the corresponding binding energy. Figure 11 shows the variation of with frequency at various temperatures. It is characterized by very low value of in the low-frequency region, a continuous dispersion with the increase in frequency having a tendency to saturate in the high-frequency region. Such observations may possibly be related to a lack of restoring force governing the mobility of charge carriers under the action of an induced electric field. This behavior supports long-range mobility of charge carriers. On the other hand, a sigmoidal increase in the value of with the frequency approaching ultimately to a saturation value may be attributed to the conduction phenomena due to short-range mobility of charge carriers. The Inset to Figure 11 presents the variation of with temperature at the two limiting frequencies (=1 kHz and 1 MHz), showing almost exponential decay type of trends. In terms of evaluated up to 54 kHz and evaluated above 60 kHz, the values of () were found out in the two frequency ranges. It can also be seen that the values of decrease with increase in temperature. All the values of were found to be  m at room temperature. Thus, on an average, the value of was found to be times smaller in comparison to the grain size of BNBT06.

Figure 11: Frequency dependence of of BNBT06 ceramic at different indicated temperatures. Inset shows the temperature dependence of at 1 kHz and at 1 MHz.

Lastly, the ac conductivity data have been used to evaluate the density of states at Fermi level using the relation [29]: where is the electronic charge; is the phonon frequency ; is the localized wavenumber. Figure 12 shows the frequency dependence of at different temperatures. It can be seen that each of the plots shows a minimum up to after which the minimum starts disappearing and that the value of decreases with the increasing frequency. Further, it is noted that the minima shift towards higher frequency side with the increasing value of due to rise in temperature. The inset to Figure 12 shows temperature-dependent variation of at different frequencies.The plots show peaks at and near and a sharp increase beyond and the anomaly is most pronounced at 100 Hz. The reasonably high values of suggest that the hopping between the pairs of sites dominates the mechanism of charge transport in BNBT06.

Figure 12: Frequency dependence of of BNBT06 ceramic at different temperatures. Inset: temperature-dependent variation of at different indicated frequencies.

4. Conclusion

Morphotropic phase boundary composition ceramic (Bi0.5Na0.5)0.94Ba0.6TiO₃ prepared through a high-temperature solid-state reaction method was found to have a perovskite-type structure in which both rhombohedral and tetragonal phases coexist. SEM micrograph of the sintered ceramic pellet shows dense and homogeneous packing of grains. Room temperature dielectric constant and loss factor are found to be equal to 781 and 0.085, respectively, at 1 kHz. Longitudinal piezoelectric charge coefficient () of the poled ceramic sample is found to be equal to 124 pC/N. The high value of dielectric constant, relatively low loss factor, and high piezoelectric charge coefficient of the test ceramic sample make this compound useful for different electronic and other sensor/actuator applications. The dielectric relaxation was found to be of non-Debye type. The frequency-dependent ac conductivity is found to obey the universal double power law. Both the pair approximation type correlated barrier hopping model and jump relaxation model are found to successfully explain the mechanism of charge transport in this substance. The minimum hopping length was found to be times smaller in comparison to the grain size. The ac conductivity data were used to evaluate the minimum hopping length, apparent activation energy, and density of states at Fermi level.

Acknowledgment

The present work was supported by the Department of Science and Technology, New Delhi under Grant no. SR/S2/CMP-017/2008.

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