crystallizes in a body-centered cubic structure known as sillenite 1. These crystals have many interesting properties, including piezoelectrical, electro- and elasto-optical, optical activity, and photoconductivity2-5. Of particular interest is the combination of electro-optical and photo-conductivity properties, which causes the so-called photorefractive effect that consists of a reversible light-induced change in the refractive index2. These properties render sillenite-type crystals useful in a variety of advanced and potentially promising applications such as reversible re-cording media for real-time holography or image processing applications.3 Depending on how well known the defects in these materials are, it is possible to gain an in-depth understanding of their optical and electrical behavior and, thus, of the optimization of their properties. According to [1] knowledge about the defect structures and processes of these crystals is still in a very preliminary stage, which is a serious shortcoming. Several studies have been published on the optical properties of sillenite-type crystals.1,3 However, few reports are available regarding their electric and dielectric properties,4 nor were any studies found on the characterization of these ceramics by impedance spectroscopy. A material's conductivity depends on its overall characteristics, such as its chemical composition, purity, and microstructure. Measurements taken with continuous currents provide only total conductivity with values failing. However, and on the material's electrode interface effects can be partially offset by impedance measurements. Impedance measurement is a flexible tool for simultaneous electrical and dielectrical characterization of materials, particularly for ceramics. The technique has been widely employed to characterize the dielectric behavior of single crystal, polycrystalline, and amorphous ceramic materials.5-7 Prior to analyzing impedance data, it is essential to use an adequate equivalent circuit analogous for each perceived physical mechanism.8 This equivalent circuit should, in particular, provide a physical representation of the relaxation mechanism. Thus, the main difficulty of impedance spectroscopy measurements lies in the interpretation of results. The experimental results of this study relate to the electric and dielectric behavior of the Bi12TiO20 ceramics, using impedance spectroscopy. The ac impedance data were presented on the complex plane by means of the Nyquist diagram. This representation allows for easy identification of the electric and dielectric parameters of the material such as, for instance, the relaxation mechanism. The frequency dependence of the complex permittivity e*, as well as conductivity at temperatures up to 700 °C, were also investigated.
THEORY
The impedance diagram recorded for polycrystalline materials usually composed of a sequence of semicircles. The high-frequency semicircle, which passes through the origin and describes the electrical properties of the grains (bulk contribution), is followed by an intermediary frequency semicircle relating to the conduction behavior of microstructural defects (grain boundaries, pores, second phases). The characteristics of the electrodes aredepicted at the lowest frequencies. In the case of single crystals, only the high and low-frequency contributions are present. As matrix conductivity is enhanced by rising operating temperature, the frequency distribution of the impedance diagram shifts toward higher frequencies, as it does in the other formalisms. When the depression angle of the bulk semicircle is low, the overall response of a polycrystalline sample can be interpreted, to a first approximation, in terms of a parallel resistance-capacitance (RC) element, i.e., a Debye-like behavior, regardless of the formalism chosen to analyze the impedance data. In the impedance or Nyquist plan, the low-frequency intercept of the bulk semicircle on the real axis corresponds to the matrix's specific resistance, R. The specific relaxation frequency, f. (which corresponds to the minimum of the imaginary part of the complex impedance), can be determined at the apex of the bulk semicircle. The specific capacitance, CS of the material's bulk (also called the geometric capacitance) can be calculated from the R, and f values, according to the basic relaxation equation:
2 p RsfsCs = 1 (1)
Then, the specific dielectric constant C can be calculated
according to
(2)
where e0 is the vacuum dielectric permittivity, l is the sample thickness, and A is the electrode area. It is worth noting that C can be determined only through the relaxation frequency (Eq. (1)) when more than half of the bulk semicircle is recorded. Otherwise, the frequency range available to define this contribution is too small and the resolution of the semicircle is unclear. In some low-conductivity materials, the dielectric constants deduced from the relaxation frequencies at temperatures above 300°C were found to be higher than the corresponding values measured at room temperature, when the materials behave as pure dielectrics (10). At high temperatures, the dielectric constant is overestimated because conduction processes within the grains affect the e. At low temperatures, when the response of the sample can be regarded as mostly dielectric, the recorded impedance is purely imaginary and represents that of the specific capacitance, Cs, which can be expressed as
-/m(Z) = jCs2rf (3)
whiere -lm(Z) is the opposite of the imaginary part of the impedance, j is the complex operator, and f is the ac measuring frequency. Based on Eq. (3), Cs can be determined from the slope of the straight line of the -Im(Z) versus 1/2pf plot. The specific dielectric constant, Cs, is then easily obtained from the Relation (2).
The dielectric constant, e, can also be expressed as a complex number:
e = e' -je' (4)
where e' and e " are the real and imaginary parts of the dielectric constant. The e' and e" values can be related to impedance data according to
(5)
(6)
where Z is the impedance modulus.
Strictly speaking, the two methods described to determine dielectric constant are not essentially different, because they are both based on a capacitance calculation. The latter method ensures that a possible conductivity contribution, if any. will not lead to the overestimation of the real dielectric constant of the material, resulting from the limited experimental conditions, in terms of the available frequency range.
EXPERIMENTAL PROCEDURE
The Bi12TiO20 single phase powder was obtained by solid-state reaction. The precursor oxides used in stoichiometric proportion were Bi2O3 and TiO2 (Alfa Aesar, grade purity 99,99%). These oxides were ball-milled in plastic recipients containing the powders, zircon balls and isopropyl alcohol in the volumetric proportion of 10:60:30. The as-dried material was calcined at 800°C for 3 hours in Pt crucible in an open atmosphere furnace. After the calcination, the powder was ground in an agate mortar, mixed with a binder solution of polyvinyl alcohol with a 0.1 g/ml concentration and conformed by uniaxial pressing in pellets with 6mm in diameter and 2mm in thickness.
Sintering in air for 5 h at 600, 700 and 750 °C (heating rate of 10°C/min) was used to produce samples with different densities and resulted, respectively, in pellets with relative densities equal to 63%, 88% and 95% of the theoretical density (8.952 g/cm3). These samples are referred to hereinafter as BTO63, BTO88 and BTO95. Densities were determined from the measurements of the samples' parameters of size and weight.
The electrodes required for impedance measurements were deposited on both faces of the ceramic pellets (thickness = 2 mm, diameter = 6 mm) by the application of a platinum paste and additional drying at 700°C for 30 min. Impedance data were recorded in the frequency range of 1 Hz-10MH (Solartron 1260) The amplitude of the measuring AC signal was 2000 mV. It should be noted that the impedance modulus was not found to depend on the amplitude of the sinusoidal signal at any measuring temperature. This indicates that the recorded impedances are governed mainly by intrinsic electrical properties of Bi12TiO20 and that electrode polarization (in terms of resistance) can be neglected within the investigated temperature range. The samples were kept under controlled at ± 1°C. Impedance measurements were taken on a two-electrode configuration cell, in dry air, from room temperature to 700°C, in 100°C steps.
Experimental impedance data were analyzed using software Zview, which provides the resistance and capacitance associated to each recorded contribution of the ceramic.
RESULTS
Figure 1 shows the Nyquist diagrams of dense and porous samples. It is well known that the dielectric and polar properties of single-phased ceramics are sensitive to the microstructures. Both Curie point and dielectric constant values, for instance, are generally functions of grain size and porosity 9. For a more in-depth investigation into the nature of the anomalies/transition, impedance spectroscopy measurements of porous bodies (BTO63 and BTO 88 samples) were taken under the same experimental conditions of the dense sample (BTO95). As we can see, the contribution from grain boundaries is more evident in the porous samples. From these results, it was possible to deconvolute the grain boundary contribution from the grain response. The R and C values obtained from the fitting, using equivalent circuits and brick-layer model 10.
Figure 1: Complex impedance diagrams of BTO63, BTO88 and BTO95, taken at 500 °C.
The ac conductivity (s'ac) of BTO was determined from the complex impedance diagrams, in which the semicircle diameter stands for the resistivity r = 1/s'ac of the system. From our results, it was verified that s'ac was thermally activated according to the Arrhenius law (see Fig. 2):
s = s0 exp (-Ea/kT) (7)
where s0 is a pre-exponential factor and Ea, k and T represent the apparent activation energy for conduction process, Boltzmann' constant and the absolute temperature, respectively. It was found the same activation energy (0.99 eV) either for grain and grain boundaries, evidencing the same transport charge mechanism in both regions.
Figure 5: Arrhenius plots for ac conductivities in BTO. The apparent activation energy indicated in the graphic was deduced from the slope of the linear regression of the data.
The relaxation frequency was extracted from impedance data >400°C, which allowed the dielectric constant to be calculated using Eqs. (I) and (3). Because of the limitations of the available frequency range below 400°C, the bulk semicircle was incomplete; thus, only Eq. (3) was applied. For a thorough analysis of the dielectric constant and its temperature dependence, the contributions of the bulk and grain-boundary components must be separated11. By combining impedance and electric modulus formalisms, Dyre 10 clearly evidenced the influence of the ever-present gain-boundary capacitance on fixed-frequency Curie-Weiss plots. At high temperatures, this capacitance causes large departures from linearity and, at lower temperatures, its influence leads to incorrect values of the Curie temperature11. Whatever the measuring temperature, the electric modulus plots recorded on the BTO95 sample give only one arc, which passes through the origin of the complex plane on the low-frequency side (Fig. 1). Moreover, the single peaks' maxima recorded on both the impedance and electric modulus spectroscopic plots (imaginary components of Z and M versus the logarithm of the AC measuring frequency, not shown here) are frequency-coincident, as expected for a nearly ideal Debye-like response. This clearly proves that the recorded high-frequency response refers to the bulk properties with no contribution from the grain boundaries' dielectric constant. Accordingly, the dielectric constant values deduced from the high-frequency intercept of the electric modulus plot on the real axis are in agreement with those determined from Eq. (3). This analysis is also valid for porous samples within the frequency range describing the specific response of BTO grains.
Data measured as complex impedance Z* were converted to the complex permittivity formalism by the relation
(8)
The real and imaginary parts of complex permittivity were used to evaluate its dependence on frequency. Figure 3a shows a log-log plot of the real parts of permittivity as a function of frequency at several temperatures. With increasing temperature, a high degree of dispersion in the permittivity occurs at low frequencies. This behavior is generally found in dielectrics, in which a conduction hopping-type mechanism is present12. The degree of dispersion decreases at frequencies up to 10 kHz, although nonfrequency dependent regions were observed between 10 and 1000 kHz depending on the temperature. The imaginary part as a function of the frequency, which plot is given in Fig. 3b, decreases with increasing frequency.This behavior can be considered a normal dielectric response13. Considering the behavior as a function of frequency and temperature, the BTO response acts as an equivalent circuit comprising a parallel combination of capacitance and resistance, as shown in the impedance fitting diagrams given in Fig. 1.
Figure 3: (a) Real part of permittivity e , (b) imaginary part of permittivity e as a function of frequency at several temperatures for BTO ceramics.
The behavior at a low frequency may be interpreted in terms of dielectric loss using the relation
tg a = e´/ e´´
where tg a is the dissipation factor. Figure 4 shows tg a for the samples BTO63 and BTO95 at 500 °C. The BTO showed a high dissipation factor on the order of 102 over a frequency range from 102 to 104 Hz. This substantial conduction loss observed may be attributed to the increase in conductivity at high temperature.
Figure 4: Loss tangent of BTO63 and BTO95 as a function of the signal frequency, taken at 500 °C.
The imaginary part of the permittivity as a function of frequency is presented on a double logarithmic scale at 500 °C, as shown in Fig. 5. The straight linear line represents the theoretical fit based on the assumption of the universal dynamic law 14-16, expressed by
e´´ = Kwn-1
where K is a proportionality constant and v the angular frequency. As Fig. 7 illustrates, expression has exponent n equal to 0.04. The magnitude of the exponent can be considered very small, physically reflecting the level of non ideality for the capacitance. Furthermore, the deviation of the capacitance from an ideal behavior is clear evidence of the existence of intrinsic defects 16.
Figure 5: Plot of the universal dynamic response, considering n = 0.04.
IV. CONCLUSIONS
The complex impedance diagram displays a shape similar to the ones observed experimentally and described by the empirical function of Cole-Cole. Thus, the electrical properties of single crystal Bi12TiO20 can be described as parallel RC circuits. Porous samples allowed the separation of grain and grain-boundaries contributions, which are too convoluted in the dense sample. The electrical conductivity follows the Arrhenius law with an activation energy of 0.99 eV. A very similar activation energy is found for the Arrhenius plot of relaxation frequencies. The use of high temperature measurements leads to an overestimation of permittivity values.
