CHAPTER-5 COMPLEX IMPEDANCE SPECTROSCOPY

Transcription

1 CHAPTER-5 COMPLEX IMPEDANCE SPECTROSCOPY

2 Chapter Introduction The frequency dependent measurement of dielectric parameters of dielectric/ferroelectric ceramics, ionic solids etc., has some limitations on getting sufficient information regarding the proper characterization of their electrical microstructure. The limitations can be overcome by analyzing frequency-dependent electrical/dielectric data of the materials using the complex impedance technique [225, 242, 243].The first and most significant attempt in this regard was made by Cole and Cole [244], who made data analysis of real system by plotting the real and imaginary part of complex permittivity (ε) of dielectric materials in the complex plane, known as Cole-Cole plot. There after a lot of progress has been made in utilizing complex plane plots and frequency explicit plots of different parameters like complex permittivity (ε) [ ], complex impedance (Z) [ ], complex admittance (Y), complex electric modulus (M) [ ] and loss tangent (tan δ), to explain the dielectric behavior and electrical conductivity of materials. Complex impedance spectroscopy (CIS) is a helpful technique used for investigation, characterization of the electrical and electrochemical properties of polycrystalline materials in relation to their microstructure. Polycrystalline ceramics are inhomogeneous materials constituted by grains separated by grain boundaries. This method ensures proper separation among the bulk, grain, grain boundaries and electrode-interface properties. Some micro-structural properties of the material (i.e., charge transport, charge diffusion at the interface within the cell, dielectric relaxation) can also be investigated. In this chapter, the structure- electrical properties of all the studied compounds have been explored through the complex electric impedance formalism. 5.2 Experimental Some important electrical properties of the proposed materials were studied by the impedance measurement procedure using a computer-controlled PSM 1735: N4L 99

3 impedance analyzer in a wide temperature (room temperature-500 o C) and frequency ( khz) ranges in air atmosphere. 5.3 Complex impedance Ferroelectric ceramics are in general electrically heterogeneous. For characterization of these materials a proper relation between microstructure and electrical properties is essential. CIS technique is used for simultaneous electrical and dielectric characterization of samples. In impedance spectroscopy the impedance data are generally plotted in complex plane. The variation of real with imaginary part of the impedance is known as Nyquist plots. Similar to impedance spectroscopy, the modulus data plotted in complex plane are used to represent the response of dielectric systems [254]. Polycrystalline materials generally show inter-granular or grain-boundary impedance and capacitance. From micro-structural point of view, a ceramic sample is composed of both grains and grain boundaries which exhibit different resistivity and dielectric permittivity [255]. In order to establish a relation between the microstructure and electrical properties, a brick layer model [256] was proposed. In this model grains are assumed to be of cubic-shaped, and grain boundaries to exist as flat layers between grains. The advantage of adopting this is the determination of grain-boundary conductivity without detailed micro-structural and electrical information. To analyze the impedance spectra, data usually are modeled by an ideal equivalent circuit consisting of a resistor R and a capacitor C. The experimental impedance data points measured were fitted using software Zswimpwin, with an equivalent circuit. A circuit consists of a series collection of two sub-circuits (consisting of a resistor and capacitor connected in parallel), one representing grain effect and the other representing grain boundaries. If R g, R gb are the resistances of grains and grain boundary and C g, C gb are the capacitances of grains and grain boundaries respectively, the complex impedance for the equivalent circuit is given by: Z * (ω) = 1 R g 1 +iωc g iωc gb = Z' (ω) - iz'' (ω) (1) R gb Z' (ω) = R g 1+(ωR g C g ) 2 + R gb 1+(ωR gb C gb ) 2.. (2) 100

4 Z'' (ω) =R g ωr g C g 1+(ωR g C g ) 2 + R gb ωr gb C gb 1+(ωR gb C gb ) 2. (3) Based on the above equations, the response peaks of the grains and grain boundaries are represented by 1/ (2πR g C g ) and 1/ (2πR gb C gb ) respectively, and the peak values are proportional to the associated resistance. Therefore, in the impedance spectra (Nyquist plot), the higher frequency response corresponds to the grains and the lower one to the grain boundaries [257]. The high- frequency semicircle is due to bulk effect that is the parallel combination of bulk resistance (R b ) and bulk capacitance (C b ) along with a constant phase element (CPE). The admittance Y of CPE is normally expressed as, Y (CPE) = A o (jω) n = A ω n + j B ω n.(4) Where A=A o Cos (nπ/2), B = A o Sin (nπ/2), A o gives the magnitude of dispersion and 0 n 1 [258]. For ideal capacitor the vaue of n=1 and for ideal resistor n=0. In order to analyze and interpret experimental data, it is essential to have an equivalent circuit model that gives more information of the electrical properties. The circuit model provides (i) the kind of impedances and their arrangement (series/parallel) in the sample, (ii) confirmation of the experimental data to see the consistency of experimental value with the proposed circuit, and (iii) compares the temperature dependence of the resistance and capacitance values to that of simulated values [242]. According to Debye s model, a material having single relaxation time gives rise to a semicircle whose centre lies on Z' axis but for non-debye type of relaxation, the centre lies below Z' axis. The complex impedance in such case is given by Z*(ω) = Z'(ω) - iz'' (ω) = R / [1+ (iω/ω o ) 1-α ] (5) Where α represents the magnitude of departure of the electrical response from ideal condition and this can be found out from the location of the centre of the semicircles. The value of α increases with rise in temperature. If α approaches 0 then Eqn. 5 gives rise to Debye s formalism. But in practice, an ideal Debye-like response is not generally realized. Instead of getting a perfect semicircle, depressed semicircles are observed with their center lying below the real axis. A non-ideal Debye type behavior is represented by introducing a 101

5 constant phase element (CPE) with resistors and capacitors. A CPE has impedance which is given by [259], Z * CPE = [A o (jω) n ] -1 (6) Where A o = A/cos(nπ/2), j= -1. A and n are independent of frequency but depend on temperature. Thus the CPE impedance has a Joncher s power law dependence. Figure 5.1 (i) Figure 5.1 (ii) Figure 5.1 (iii) Figure 5.1 (iv) Fig. 5.1 (i) and (ii) represent the ideal type of RC circuits for bulk and grain- boundary contributions, whereas the actual data modeled circuits are shown in Fig. 5.1 (iii) and (iv) with constant phase element (CPE) in parallel with RC-circuit Nyquist Plots (a) (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (BLFN): The complex impedance spectra (Nyquist plots) of (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (x= ) at selected temperatures (400, 420, 440 and 460 o C) are shown in Fig At lower temperatures (< 200 o C) these plots represent a straight line parallel to Y-axis indicating insulating behavior of the samples at low temperatures. It is observed that the slope of the lines decreases as temperature increases and then gradually bend towards real (Z') axis. 102

6 Above 200 o C, the tendency of formation of semicircular arcs is seen. The intercept of the semicircular arc along Z' axis gives the value of bulk and grain boundary resistance which decreases on increasing temperature (given in Table 5.1) showing increase in conductivity. This shows negative temperature coefficient of resistance (NTCR) property of the material which is normal behavior of semiconductors. These plots suggest that the dielectric relaxation is of non-debye type. Fig. 5.2: Complex impedance spectra of (Bi 1-x Li x )(Fe 1-x Nb x )O 3 at selected temperatures. On increasing temperature the plots consist of depressed semicircular arcs. At high temperatures two semicircular arcs could be seen which means at low frequencies a small tail appears. The effect of temperature on the impedance behavior of the samples is clearly visible from these plots. This additional semicircular arc may be attributed to the inter or intra granular (bulk and grain boundary) phenomenon. The two semicircles in the figures represent two types of relaxations: one corresponding to the grain (high frequency range) 103

7 and the other grain boundary (low frequency range). In BFO, above 340 o C, a second semicircular arc starts to appear showing the grain boundary effect whereas this effect in x=0.1and x=0.3 is seen above 400 o C.Thus the electrical properties of all the samples above 400 o C can be represented by two parallel RC series connected in series. For all the samples except x=0.2, the bulk property of the material dominates in the total value of the electrical response. For x=0.2 it is clear that the low frequency peaks are with high R gb values attributing to the insulating grain boundaries and oxidized insulating surface layers [260]. More depressed semicircular arcs are observed for x=0.4. The values of R b, R gb, C b and C gb at different temperatures are compared in Table 5.1. The bulk resistance of BFO is least suggesting the most cation defects or oxygen vacancies present in the grains [261]. Table 5.1: Comparison of impedance parameters-r b, R gb, C b and C gb at different temperatures of (Bi 1-x Li x )(Fe 1-x Nb x )O 3. Temperature R b (Ω) C b (F) R gb (Ω) C gb (F) ( o C) x= x x x x x x x x x x x x x x x x10 10 x= x x x x x x x x x x x x x x10 7 x= x x x x x x x x x x x x x x x x10 10 x= x x x x x x x x x x x x x x10 10 x= x x x x x x x x x x x x x x x x10 10 x= x x x x x x x x x x x x x x x x

8 5.3.1 (b) (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (BNFN): The complex impedance spectra (Nyquist plots) of the (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (x= ) at selected temperatures (300, 320, 340 and 360 o C) are shown in Fig For x=0.0 (BFO) incomplete single semicircular arcs appear up to 300 o C. At 340 o C, a second semi-circular arc starts to appear showing the grain boundary activity. As shown in the figure, above 340 o C the impedance of BFO decreases by several orders which have also been given in Table 5.2. At elevated temperatures though the second semicircle appears, it is poorly resolved. For x=0.1 above 280 o C these arcs take the shape of depressed semicircles. At more elevated temperatures (i.e., above 340 o C) poorly resolved second semicircular tails starts to appear. Fig. 5.3: Complex impedance spectra of (Bi 1-x Na x )(Fe 1-x Nb x )O 3 at selected temperatures. For x=0.2 the second semicircular arc starts to appear at relatively lower temperature (280 o C onwards) indicating increased conductivity at this concentration. For x=0.3 the 105

9 second semicircular arc starts appearing 290 o C onwards. This suggests that for x=0.2 and 0.3 grain boundary resistance becomes dominant at lower temperatures. For x=0.4 grain boundary effect is seen from 320 o C onwards. It is seen that for this compound with rise in temperature the bulk resistances decreases gradually, and the grain boundary resistance increases. For x=0.5 the grain boundary contribution to the overall impedance is seen from 320 o C onwards. For all other compounds the grain and grain boundary resistances are found to be decreasing with increase in temperature. Above 320 o C (i.e., at higher temperatures) it is seen that grain boundary resistance increases with increase in NaNbO 3 content up to x=0.4 but decreases for x=0.5. Table 5.2: Comparison of impedance parameters -R b, R gb, C b and C gb at different temperatures of (Bi 1-x Na x )(Fe 1-x Nb x )O 3. Temperature R b (Ω) C b (F) R gb (Ω) C gb (F) ( o C) x= x x x x x x x x x x x x10-10 x= x x x x x x x x x x10-9 x= x x x x x x x x x x x x x x x x10-8 x= x x x x x x x x x x x x x x x x10-8 x= X x x x x x x x x x x10-10 x= x x x x x x x x x x x x x

10 5.3.1 (c) (Bi 1-x K x )(Fe 1-x Nb x )O 3 (BKFN): Fig. 5.4 shows the complex impedance spectra of (Bi 1-x K x )(Fe 1-x Nb x )O 3 (x= ) at selected temperatures (300, 320, 340 and 360 o C). On increasing temperature the semicircles make smaller intercepts on the real Z axis showing decrease in impedance and supporting NTCR behavior which has been clearly given in Table 5.3. With increase in temperature the peak maxima of the plots decreases and the frequency shifts towards higher frequency side. The poly-dispersive nature of dielectric relaxation can be explained using complex impedance plots. Fig. 5.4: Complex impedance spectra of (Bi 1-x K x )(Fe 1-x Nb x )O 3 at selected temperatures. The low-frequency arcs at high temperatures are due to the presence of grain boundary that is due to a parallel combination of grain boundary resistance (R gb ) and grain boundary capacitance (C gb ). It is found that the measured and fitted data are in good agreement along with an equivalent circuit. For x=0.1 the grain boundary effect is seen at 340 o C and above 107

11 similar to that of BFO (x=0.0). For x=0.2 depressed semicircles with a tendency of formation of second semicircle is clearly seen. However, the decreasing value of impedance for some compounds (x=0.2 and 0.4) as shown in Figure 5.4 indicates the increase in conductivity of the compounds. Moreover, for these two compounds well developed semicircular arcs (starting at 260 o C) indicate the contribution of bulk to the electrical property of the compounds and the grain boundary effect for were seen at 300 and 320 o C respectively. The decrease in impedance value for these two compositions may be due to their increased grain size which can be seen in the SEM micrograph. For x=0.1, 0.3 and 0.5 bulk resistances decreases with increase in KNbO 3 content. Table 5.3: Comparison of impedance parameters -R b, R gb, C b and C gb at different temperatures of (Bi 1-x K x )(Fe 1-x Nb x )O 3. Temperature R b (Ω) C b (F) R gb (Ω) C gb (F) ( o C) x= x x x x x x x x x x x x10-10 x= x x x x x x x x x x x x10-7 x= x x X X x x x x x x x x x x x x10-10 x= x x x x x x x x x x10-10 x= x x x x x x x x x x x x x x10-10 x= x x x x x x x x x X

12 5.3.1 (d) Ceramic-polymer composites: The ionic conductivity has been determined from ac impedance analysis. Fig. 5.5 exhibits a typical impedance spectrum of pure PVDF, BFOP, BLFNP, BNFNP and BKFNP composites at different temperatures in the frequency range of 1 khz-1 MHz. At room temperature these plots represent straight lines parallel to the ordinate indicating a high order insulating behavior of the samples. With increase in temperature the curves show a tendency to bend towards the abscissa to form semicircles with their centers lying below real axis. This indicates a distribution of relaxation time with a deviation from ideal Debyetype behavior. It can be clearly noticed that the values of R g decreases with rise in temperature for all samples which indicates the NTCR character of the samples. Fig. 5.5: Complex impedance plots of pure PVDF, BFOP (BFO+PVDF), BLFNP (BLFN+PVDF), BNFNP (BNFN+PVDF) and BKFNP (BKFN+PVDF) composites at different temperatures. It is seen that the R g value is maximum for BLFNP. By fitting the impedance response with one given by an appropriate equivalent circuit we can obtain information about the resistive 109

13 and capacitive responses of the components. The component in the complex impedance spectra can be assigned to a RC-parallel circuit response which indicates the contribution of grains in pure PVDF as well as its composites. No other relaxation mechanism such as grain boundary or electrode effect could be identified in the samples studied in this frequency range. PVDF shows more insulating nature as compared to other compounds. At 120 o C there is a tendency of forming a semicircular arc for all the composites Variation of Z' with frequency (a) (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (BLFN): The temperature and frequency dependent ac conductivity of the materials can be explained in terms of the variation of the real part of impedance with frequency. Fig. 5.6 shows the variation of Z' with frequency at different temperatures (400, 420, 440 and 460 o C) of (Bi 1- xli x )(Fe 1-x Nb x )O 3 (x= ). Fig. 5.6: Variation of Z' with frequency of (Bi 1-x Li x )(Fe 1-x Nb x )O 3 at selected temperatures. 110

14 It is seen that in the low-frequency region Z' shows sigmoidal variation as a function of frequency followed by a saturation in the high-frequency region (> 10 khz) irrespective of temperature for all the plots. With rise in temperature spreading of the dispersion region in the high-frequency region is observed. The decrease in value of Z' with increase in frequency may be due to a slow dynamics relaxation process in the material which may be attributed to space charges [262]. At higher temperatures in the low-frequency region plateau is observed which may be related to frequency invariant electrical property of the materials. At higher frequencies the real part of impedance merges suggesting a possible release of space charge, and consequently lowering the barrier in the ceramic samples [263, 264]. Z' decreases with rise in temperature. This indicates enhancement of ac conductivity exhibiting negative temperature coefficient of resistance (NTCR) behavior similar to that of semiconductors. For x=0.0 a low-frequency dispersion followed by a plateau region is seen and these curves finally merge above 1000 khz. But for x=0.2 plateau region was not observed. The low-frequency dispersion decreases from x=0.1 to x=0.4 but then again increases for x=0.5. With increasing temperature defects interacted and had a significant influence on the conducting process (b) (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (BNFN):Fig. 5.7 shows the frequency dependence of Z' at different temperatures (300, 320, 340 and 360 o C) of (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (x= ). From the plots it is clear that the value of Z' decreases with rise in both temperature and frequency. These plots indicate an increase in conduction with temperature (NTCR behavior). The plateau in the low frequency region indicates the presence of relaxation process in the materials. More dispersion in low-frequency region is seen for x=0.0 (BFO) and least for x=0.4. Above 300 o C, this low-frequency plateau and shifting of merger of Z' towards high frequency side are seen. The low-frequency dispersion gradually decreases with increases in NaNbO 3 content up to x=

15 Fig. 5.7: Variation of Z' with frequency of (Bi 1-x Na x )(Fe 1-x Nb x )O 3 at selected temperatures (b) (Bi 1-x K x )(Fe 1-x Nb x )O 3 (BKFN): Fig. 5.8 shows the variation of Z with frequency at different temperatures (300, 320, 340 and 360 o C) of (Bi 1-x K x )(Fe 1-x Nb x )O 3 (x= ). They show a monotonous decrease of Z with rise in frequency and then attains a constant value irrespective of temperature for all the plots which may possibly be due to increase in the ac conductivity with rise in frequency. The space charge has less time to relax and so recombination is faster. Hence the space charge polarization reduces in the high frequency region leading to a merger of curves at higher frequency. Low frequency plateau and shifting of merger of Z' towards high frequency side is seen is seen for all the samples above 300 o C except for x=

16 Fig. 5.8: Variation of Z' with frequency of (Bi 1-x K x )(Fe 1-x Nb x )O 3 at selected temperatures Variation of Z'' with frequency (a) (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (BLFN): In order to make a deeper understanding of the space charge effect and the relaxation processes, the frequency dependence of imaginary part of impedance with frequency at different temperatures (300, 320, 340 and 360 o C) have been studied and the results are shown in Fig. 5.9 for (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (x= ). At high temperatures the curves exhibit peaks. A single peak (Z'' max ) which is temperature dependent is seen above 10 khz for all the samples. These peaks shift towards higher frequencies on increasing temperature and a broadening in the curves is observed with the decrease in peak height. This broadening suggests spreading of relaxation time (i.e., the existence of a temperature dependent electrical relaxation phenomenon in the compound) [265]. 113

17 Fig. 5.9: Variation of Z'' with frequency of (Bi 1-x Li x )(Fe 1-x Nb x )O 3 at selected temperatures. It indicates a thermally activated dielectric relaxation process in the materials and shows that with temperature bulk resistance reduces. But at low temperatures (not shown) these peaks have not been found. This may be due to the weak current dissipation in the material or may be beyond the experimental range of frequency [266]. The dispersion curves appear to merge at higher frequencies irrespective of LiNbO 3 content in BFO due to release of space charges [267, 268]. This plot suggests an enhancement in the net impedance of LiNbO 3 modified BFO thereby increasing the barrier to the mobility of charge carrier in the materials [261] (b) (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (BNFN): The loss spectrum of (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (x= ) at different temperatures (300, 320, 340 and 360 o C) are shown in Fig For x=0.0 no peak appears before 280 o C and 114

18 it appears in terms of very broad and diffused peak at elevated temperatures. For x=0.2 another small peak appears on low-frequency side at 300 o C which gradually becomes prominent at elevated temperatures. The first peak in the low-frequency region is correlated to the grain boundary contribution while the second one in the high-frequency region is correlated with the bulk response. The shifting of the peaks towards higher temperature may be due to reduction in the bulk resistance. Fig. 5.10: Variation of Z'' with frequency of (Bi 1-x Na x )(Fe 1-x Nb x )O 3 at selected temperatures. The merger of the entire high-frequency end (>100 khz) indicates the depletion of space charges at those frequencies, since these curves basically denote the ac losses of the samples. But for x=0.4 the curves at high frequency merge at >10 khz. For x=0.4 the dispersion in low frequency region is least and maximum for x=

19 5.3.3 (c) (Bi 1-x K x )(Fe 1-x Nb x )O 3 (BKFN): The loss spectrum of (Bi 1-x K x )(Fe 1-x Nb x )O 3 (x= ) is shown in Fig for selected temperatures (300, 320, 340 and 360 o C). At lower temperatures ( 250 o C), the value of Z'' falls monotonically on increasing frequency without any peak in the investigated frequency range. It indicates that the samples may not relax at lower temperatures due to presence of polarization field in the lattice. Above 250 o C, the broad and asymmetric peaks start to appear in the low-frequency region. Fig. 5.11: Variation of Z'' with frequency of (Bi 1-x K x )(Fe 1-x Nb x )O 3 at selected temperatures. For x=0.2 onset of Z peak (Z'' max ) has been observed above 250 o C. But for other compounds these peaks are seen above 280 o C. The peak position shifts to higher frequencies as temperature increases. This shift occurs at maximum frequency for all the samples indicating an active conduction associated with dipole reorientation. This offset is characteristic of high-permittivity systems as well as of localized-conduction electronics 116

20 due to grains (bulk) and grain-boundary (interface) effects for all the compounds. The value of Z'' max shows a decreasing trend on increasing temperature indicating an increasing loss in the resistive property of the samples. Moreover the peaks in the Z'' spectra occur in the region of frequency dispersion in Z' spectra. For x=0.4 the dispersion in low frequency region is least. The peaks are more sharp and intense in comparison to the rest of the compounds in this series. 5.4 Complex modulus Complex modulus analysis is a convenient technique which determines analyzes and interprets the dynamical aspects of transport phenomena (i.e., parameters such as carrier/ion hopping rate, conductivity relaxation time etc.). Another advantage of this technique is that it can discriminate against electrode polarization and grain boundary conduction processes. The combined analysis of impedance and modulus spectroscopic plots to rationalize the dielectric properties was suggested by Sinclair and West [242, 269]. Complex impedance plots are useful in determining the dominant resistance in the sample whereas complex modulus plots are useful in determining the smallest capacitance. Hence the modulus plots are used to separate the components with similar resistance but different capacitance. The Nyquist plot (M'' vs. M') gives rise to a semicircle, and the smallest one corresponds to the highest capacitance. Also the absence of subsequent semicircles in the modulus plots neglects the electrode effects [270]. The electrical properties of materials showing a single circular arc in complex modulus plane are defined by the parallel combination of grain capacitance (C) and resistance (R). Complex electric modulus can be calculated from the impedance data using the following relation: M * (ω) = 1 ε = M'(ω) + i M''(ω) 0 = M 1 exp ( iωt) dφ (t) dt dt where, M = 1/ε, ε is the limiting high-frequency real part of permittivity, and the function φ(t) is a relaxation function or Kohlrausch-Williams-Watts (KWW) function [242]. 117

21 5.4.1 Nyquist plots (a) (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (BLFN): Fig shows the complex modulus spectrum (M' ~ M'') of (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (x= ) at selected temperatures (300, 320, 340 and 360 o C). These plots do not form exact semicircles rather they form depressed semicircles with their centers positioned below the x-axis. This indicates the spreading of relaxation time and hence non-debye type of relaxation in these compounds. Fig. 5.12: complex modulus spectra of (Bi 1-x Li x )(Fe 1-x Nb x )O 3 at some selected temperatures. The Nyquist plots of electric modulus justify the poly-dispersive nature for the dielectric relaxation at lower frequencies. The appearance of asymmetric semicircular arcs indicates the electrical relaxation phenomenon in the materials. The plots show a semicircle with a 118

22 tendency of formation of another semicircle for all the compounds except x=0.4. The intercept on the real axis indicates the total capacitance contributed by the grain and grain boundaries. Appearance of single arc for x=0.4 indicates negligible contribution of the grain boundary effect to the polarization in the temperature range. The grain boundary effect is more prominent for x=0.2. The modulus loss profiles are collapsed into one master curve suggesting temperature independent relaxation time (with different mean time constants) (b) (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (BNFN): Fig shows the complex modulus spectrum (M' ~ M'') of (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (x= ) at selected temperatures (300, 320, 340 and 360 o C). For x=0.0 and 0.1 the total capacitances increases with increase in temperature. Fig. 5.13: complex modulus spectra of (Bi 1-x Na x )(Fe 1-x Nb x )O 3 at some selected temperatures. 119

23 For these compounds depressed semicircular arcs appear up to 340 o C and above this temperature, they are incomplete at the high frequency side. For x=0.2, 0.3 and 0.5, the arc/semicircles overlap with each other at various temperatures implying the presence of relaxation phenomenon in these compounds. For x=0.4 up to 340 o C the semicircles overlap but at 360 o C there is a tendency of formation of another semicircle. The intercept of first semicircle on the real axis gives the capacitance contributed by grain, and the second semicircle to the contribution from grain boundary (c) (Bi 1-x K x )(Fe 1-x Nb x )O 3 (BKFN): The complex modulus spectra of (Bi 1-x K x )(Fe 1-x Nb x )O 3 (x= ) at some selected temperatures (300, 320, 340 and 360 o C) are shown in Fig For x=0.1 and 0.2 depressed semicircular arcs are seen with intercepts of the arcs on M' axis which decreases with rise in temperature indicating increase in capacitance. Fig. 5.14: complex modulus spectra of (Bi 1-x K x )(Fe 1-x Nb x )O 3 at some selected temperatures. 120

24 This indicates the spreading of relaxation time with different mean time constant and non- Debye type of relaxation in the materials. For x=0.1 there is a tendency of formation of second semicircle confirming the active role of grain boundary capacitance in the conduction mechanism. The broadening observed for all the samples in the semicircular arcs in the complex modulus plots suggest the involvement of both the grain and grain boundary towards electrical capacitance in the ceramic samples. For x=0.3, 0.4 and 0.5 the arcs perfectly overlap which indicates several relaxations occurring in these materials Variation of M' with frequency (a) (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (BLFN): Another formalism of data presentation is the complex electric modulus, M* formalism. The frequency dependence of real part of electric modulus (M') at selected temperatures (400, 420, 440, and 460 o C) of (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (x= ) is shown in Fig It is found that at low frequencies the value of M' is found to be very low (or nearly equal to zero). A continuous dispersion with increase in frequency is observed and finally these curves have a tendency to saturate at a maximum asymptotic value designated at M in the high-frequency region irrespective of temperature. These phenomena can be related to lack of restoring force governing the mobility of charge carriers under the action of an induced electric field [271]. But this saturation of M' at high frequency region is not seen for x=0.0 and 0.4. This confirms elimination of electrode effect in the materials. M' is also found to decrease with the increase in temperature which indicates a temperature dependent relaxation process in the materials. It is also found that the dispersion region shifts towards higher frequency side suggesting the long-range mobility of charge carriers. The plateau region (or its tendency to saturate) observed at higher frequencies suggests about the frequency invariant electrical properties of the materials. 121

25 Fig. 5.15: Variation of M' with frequency at selected temperatures of (Bi 1-x Li x )(Fe 1- xnb x )O (b) (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (BNFN): Fig shows the frequency response of real part of electric modulus (M') of (Bi 1- xna x )(Fe 1-x Nb x )O 3 (x= ) at different temperatures (300, 320, 340, and 360 o C). The sigmodial increase in the value of M' with frequency approaches ultimately to a value of M for all temperatures which indicates short-range mobility of carriers (especially ions). Well dispersed curves are observed in all the compounds except for x=0.4. This dispersive nature of the compounds implies that a well defined relaxation mechanism occurs over several decades of frequency at all these temperatures. 122

26 Fig. 5.16: Variation of M' with frequency at selected temperatures of (Bi 1-x Na x )(Fe 1- xnb x )O (c) (Bi 1-x K x )(Fe 1-x Nb x )O 3 (BKFN): Fig shows the frequency response of real part of electric modulus (M') of (Bi 1- xk x )(Fe 1-x Nb x )O 3 (x= ) at different temperatures (300, 320, 340, and 360 o C). The plots clearly show very low value of M' in the low-frequency region with a continuous dispersion in the high-frequency region for all temperatures. These curves have a tendency to saturate at a maximum asymptotic value designated as M in the high-frequency region. With increase in frequency each ion moves a shorter path of electric field till the electric field changes so rapidly that the ions only rattle within the confinement of their potential energy wells. This indicates the long-range mobility of charge carriers [272]. 123

27 Fig. 5.17: Variation of M' with frequency at selected temperatures of (Bi 1-x K x )(Fe 1-x Nb x )O 3. As temperature increases the value of M' decreases and the dispersion region shifts to higher frequency side indicating a thermally activated relaxation process. Dispersion decreases with increase in KNbO 3 content except for x=0.4. For x=0.3 at very high frequency the curves tend to merge Variation of M'' with frequency (a) (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (BLFN): Fig shows the frequency dependence of imaginary part of electric modulus (M ) at selected temperatures (400, 420, 440, and 460 o C) of (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (x= ). The modulus spectra exhibit well-resolved asymmetric peaks. On decreasing frequency these peaks indicate that there is a transition from short range to long-range mobility. 124

28 Fig. 5.18: Variation of M'' with frequency at selected temperatures of (Bi 1-x Li x )(Fe 1- xnb x )O 3. The lower-frequency side (below M'' max ) of the peak represents the range of frequencies in which the ions are capable of moving long distances from one site to the neighboring site by hopping. In the high-frequency region (above M'' max ), the ions are confined to their potential wells and can execute only localized motion [273, 274]. On increasing the temperature the peaks shifts towards higher frequencies side confirming the thermally activated nature of relaxation time. Some of the main reasons for such broadness in the spectra are: (i) the random orientation of anisotropically conducting species, (ii) the presence of phases of more than one composition or structure [275] (iii) distribution of relaxation times due to local defects. For compounds except 0.4, the height of M'' max increases gradually with rise in temperature. 125

29 5.4.3 (b) (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (BNFN): Fig shows the frequency dependence of imaginary part of modulus (M'') at various temperatures (300, 320, 340, and 360 o C) of (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (x= ). These asymmetric peaks shift towards higher frequency which indicates correlation between motions of mobile ion charges [266]. The broadening of peak indicates the distribution of relaxation times indicating relaxation of non- Debye type. Fig. 5.19: Variation of M'' with frequency at selected temperatures of (Bi 1-x Na x )(Fe 1- xnb x )O 3. For x=0.1, 0.2 and 0.5 the height of decreases with rise in temperature. The constancy of peak height in the modulus plot for x=0.3 at different temperatures suggests the invariance of the dielectric constant and distribution of relaxation times with temperature [276]. The distribution is due to irregularities in the lattice structure near the defect sites. For 126

30 compounds except x=0.0 and 0.4, the height of M'' max decreases slightly with rise in temperature (c) (Bi 1-x K x )(Fe 1-x Nb x )O 3 (BKFN): Fig shows the frequency dependence of imaginary part of electric modulus (M'') of (Bi 1-x K x )(Fe 1-x Nb x )O 3 (x= ). The peaks are clearly resolved and appear at unique frequency at various temperatures. It is clearly shown that the M'' max shifts towards higher relaxation frequency with the temperatures increases. Fig. 5.20: Variation of M'' with frequency at selected temperatures of (Bi 1-x K x )(Fe 1- xnb x )O 3. This behavior (non-debye type) suggests that the relaxation process is thermally activated in which hopping of charge carriers with small polarons is dominated intrinsically [277]. The low-frequency side of the M'' max represents the range of frequencies in which charge carriers can move over a long distance and the high frequency represents localized motion. 127

31 The region where peak occurs indicates transition from long-range to short-range mobility with increase in frequency [278]. The peak height in the modulus plot decreases with rise in temperature for x=0.1, 0.2, 0.3 and 0.4. For x=0.5 the height of M'' max increases with rise in temperature. This type of effect has also been seen in some real ionic conductors [279]. This steady increase in the values of M'' max as a function of temperature indicates decrease in capacitance [280] Normalization of modulus spectra (a) (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (BLFN): Fig shows the normalized plot of (Bi 1-x Li x )(Fe 1-x Nb x )O 3 (x= ) versus at various temperatures (400, 420, 440 and 460 o C). This plot is known as modulus master curve which enables us to have an insight into the dielectric process occurring in the materials as a function of temperature. It is seen that all the curves irrespective of temperature coalesced into a single master curve. This coincidence indicates temperature independent behavior of the dynamical processes occurring in the material. This indicates that the distribution function for relaxation times is nearly temperature independent with non-exponential type of conductivity relaxation. These plots may be analyzed in terms of non-exponential decay function or Kohlrauseh Williams Watts (KWW) parameter by the expression: Φ (t) = exp [(-t/τ m ) β ]; (0 < β < 1) Where Φ (t) is the time evolution of an electric field, and τ m is the characteristic relaxation time. For an ideal Debye single relaxation, β = 1, which indicates that the interaction between the ions is maximum [281]. A non-exponential type relaxation suggests the possibility of ion migration that takes place via hopping [282]. The comparison of impedance and modulus plots are helpful in rationalizing the bulk response in terms of dielectric (localized) relaxation and conductivity (non-localized) relaxation process. 128

32 Fig. 5.21: Plot of M''/ M'' max with f/f max of (Bi 1-x Li x )(Fe 1-x Nb x )O 3 at selected temperatures (b) (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (BNFN): Fig shows the normalized plot of (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (x= ) versus at various temperatures (300, 320, 340 and 360 o C). All the peaks collapse into one master curve at different temperatures suggesting temperature independent distribution of relaxation time. For x=0.0, 0.1, 0.4 and 0.5 a small deviation from scaling curve both at higher and lower frequencies have been observed. But for x=0.0 this deviation in the scaling curve is not observed above 380 o C (Figure 5.21). The observed deviation at higher frequency indicates a change in the dynamic properties of the materials. The low-frequency deviation mostly originates from some interfacial effects. 129

33 Fig. 5.22: Plot of M''/ M'' max with f/f max of (Bi 1-x Na x )(Fe 1-x Nb x )O 3 at selected temperatures. It indicates that the failure of merging into a single master curve for these compounds may be due to the change of concentration of charge carriers of the materials [283]. The above results show that the relaxation dynamics of oxide ions are temperature independent but they depend on the structure and/or the concentration of charge carriers (c) (Bi 1-x K x )(Fe 1-x Nb x )O 3 (BKFN): Fig shows scaling behavior of imaginary part of modulus (M'') with frequency at different temperatures (300, 320, 340 and 360 o C) of (Bi 1-x Na x )(Fe 1-x Nb x )O 3 (x= ). All the curves superimpose into a single master curve indicating that all the dynamic processes occur at different frequencies. 130

34 Fig. 5.23: Plot of M''/ M'' max with f/f max of (Bi 1-x K x )(Fe 1-x Nb x )O 3 at selected temperatures. The dielectric processes occurring in the material can be investigated via these plots [263]. The coincidence of all the peaks at different temperatures exhibits temperature independent behavior of the dynamic processes occurring in the materials [279]. It is observed that all the peaks of the pattern appear at unique frequency for different temperatures. Small deviation from scaling at low and high frequency is seen for x=0.1, 0.3 and Summary On the basis of above results the following conclusions have been drawn: The impedance spectroscopy data provides the contribution of both grain and grain boundary on the electrical properties of the materials. But for PVDF and ceramic- 131

35 polymer composites no grain boundary effect was observed in the studied temperature and frequency range. The impedance pattern suggests a decrease in bulk resistance with rise in temperature. Negative temperature coefficient of resistance (NTCR) behavior of the materials indicates semiconducting nature of the materials. The equivalent circuit models provide an insight of the structure-property relationship of materials. The combined analysis of impedance with modulus spectroscopy provides important information about the contribution to the relaxation process of different micro-regions in the poly-crystalline ceramics, such as grains and grain boundaries. 132