# Synthesis, Structural, and Electrical Properties of Pure PbTiO3 Ferroelectric Ceramics

Authors: Vijendra A. Chaudhari; Govind K. Bichile
Journal: Smart Materials Research
 (2013)

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## Abstract

Single-phase polycrystalline samples of lead titanate with perovskite structure have been synthesized using solid-state reaction technique. The processing parameters have been optimized to obtain phase pure, dense, crack-free, and homogeneous samples. The sintering behavior of PT-powders has been investigated using X-ray diffraction patterns. The X-ray powder diffraction data have been analyzed to confirm the phase formation and phase purity, to obtain unit cell parameters and unit cell volume. The porosity of the samples has been obtained through X-ray density and bulk density. The average particle sizes of the phase pure samples were obtained from the X-ray peak width using Scherrer’s formula. The influence of sintering temperature and time on the microstructure of samples has also been studied by carrying out SEM investigations. The notable feature of this microstructure study shows that the samples sintered at 900°C for 12 hours possess a fairly uniform grain distribution. The electrical behavior (complex impedance Z*, complex permittivityε*, etc.) of the samples sintered at 900°C for 12 hours has been studied by complex impedance spectroscopy. The temperature variation of real permittivity gives evidence of the ferroelectric phase transition as well as of the relaxation behavior.


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## Body

## 1. Introduction

Crystals of the perovskitefamily, such as PbTiO3, BaTiO3, SrTiO3 and, have been of constant interest because some ofthese materials show ferroelectric behavior and undergo structural phase transitions [1]. PbTiO3 has been considered to be one of the most important members of this family. It has a high Curie temperature, high pyroelectric coefficient, low dielectric constant, and high spontaneous polarization [2]. Lead titanate (PbTiO3, PT) is a ferroelectric ceramic that has not been proved to be a technologically important material by itself but is a significant component material in electronics such as capacitors, ultrasonic transducers, thermistors, and optoelectronics [3–7]. It is also a promising material for pyroelectric infrared detector applications because of its large pyroelectric coefficient and relatively low permittivity [8, 9]. PbTiO3 (PT) has also been extensively used in a range of piezoelectric applications, as well as being the end member of technologically significant ferroelectric perovskite series such as PbZr(x)Ti(1−x)O3 (PZT), Pb(x)Ca(1−x)TiO3, Pb(Zn1/3Nb2/3)O3-(x) PbTiO3 (PZN-PT), and Pb (Mg1/3Nb2/3) O3-(x) PbTiO3 (PMN-PT) and so forth. At ambient temperature, the material has a strong anisotropy which develops during cooling through the cubic-tetragonal phase transition of approximately 490°C. The anisotropy as measured by the tetragonality of the unit cell, c/a, may be as high as ~1.06. A large c/a ratio is considered favorable for enhanced electrical properties [10, 11].The conventional method of synthesizing PbTiO3 relies on the solid-state reaction between TiO2 and PbCO3 at high temperature. The conventional solid-state reaction has a tendency to produce a coarse PbTiO3 powder with compositional inhomogeneity and a degree of particle agglomeration if the processing parameters are not carefully optimized. Therefore, many chemistry-based processing routes, including coprecipitation [12], sol-gel synthesis [13], hydrothermal [14], and citrate routes [15], have been devised for the preparation of an ultrafine, sintering-reactive PbTiO3 powder. However, almost all these chemistry routes require calcination of the precursors at an elevated temperature to develop the desired PbTiO3 phase. Furthermore, most of these chemistry-based processing routes require high-purity inorganic or organometallic chemicals as the starting materials, which are many times more expensive than the widely available oxides and carbonates. It has been observed that the synthesis of a mechanically robust, high density, monolithic ceramic pure PbTiO3 is not simple [16]. Problems typically encountered include loss of lead (Pb) due to the volatility of PbO at elevated temperatures, porosity, and microcracking, in extreme cases leading to spontaneous fracture. The main hurdle in the PT fabrication is the synthesis of a single phase with required perovskite structure. The primary difficulty is due to the volatility of PbO at elevated temperatures. The PbTiO3 structure can only tolerate minor loss of lead (Pb), higher levels of which effectively promote second phase formation and the degradation of piezoelectric properties. An example is the formation of a PT phase with the pyrochlore structure observed during a coprecipitation synthesis experiment [12]. A more common occurrence is the formation of a two-phase mixture with TiO2 [17]. The volatilization of PbO is known to increase markedly at temperatures above 800°C though the critical temperature is debated. Kim et al. [18] have observed that a PbO-rich PT in liquid phase is formed above 838°C, Algueró et al. [19] found that at 650°C an excess of 20% PbO was required because of Pb-loss during thermal treatments of sol-gel prepared La-modified PT thin films, whereas Ananta and Thomas [20] found that Pb volatility in PMN-PT could be minimized by careful sintering up to 1250°C. What is clear is that the loss of lead depends on particle size of constituent oxides, processing conditions, and chemical stability. The degree of Pb incorporation into the presintered crystal structure affects the volatility enormously [17]. Many groups report that loss of lead (Pb) may be minimized by sintering compacted powders in a surrounding lead-based powder or a PbO vapor atmosphere [18, 21] although this may lead to a Pb gradient in the final sintered product [17]. The problem of porosity has largely been addressed through particle size control of the starting powder. Therefore, one of the aim of the present work is to optimize the processing parameters to obtain homogeneous, dense, crack-free samples using the conventional solid-state reaction method. A careful and systematic study optimization of the synthesis parameters provides an alternative means of minimizing the problem of cracking, porosity, and Pb volatility. Careful sintering at lower temperatures may restrict grain growth and on cooling internal strains are reduced. In addition, sintering at low temperatures, the problem of volatility of lead is minimized. It has been shown that homogenous powders with a fine grain size can be produced by the ceramic process on the optimization of synthesis parameters [22]. The electrical properties result from the different contributions made from various components and processes present in the material, and complex impedance spectroscopy has been commonly used to evaluate and separate the contribution of the overall electrical properties in the frequency domain due to electrode reactions at the electrode/sample interface and migration of ions through the grains and across the grain boundaries in a polycrystalline material. In view of this, we report our studies on electrical properties of PT samples using complex impedance spectroscopy technique. The frequency dependent properties of a material can be described via the complex permittivity (ε*), complex impedance (Z*), and dielectric loss or dissipation factor (tan δ). These complex parameters are interrelated to one another as follows:
(1)Z*=Z′-jZ′′=1jωC0ε*,(2)ε*=ε′-jε′′,(3)ε′=-Z′′ωC0(Z′2+Z′′2),(4)ε′′=Z′ωC0(Z′2+Z′′2),(5)tanδ=ε′′ε′=Z′Z′′,
where ω=2πf is the angular frequency, Co is the geometrical capacitance, and j=-1. The peak of the semicircle in the complex plane plot enables us to evaluate the relaxation frequency (fmax) of the bulk material in accordance with the relation
(6)ωτ=1,(2πf)(RbCb)=1,f=12πRbCb,
where Rb and Cb are the bulk resistance and capacitance, and f is the relaxation frequency. Relaxation frequency and relaxation time (τ) depend on the intrinsic properties of the material and not on the sample geometrical factors. The term intrinsic properties of the material referred to the properties attributed to structure/microstructure (i.e., grain interior or bulk, grain boundary, etc.). These properties govern the distribution of resistive and capacitive components in the material on which the relaxation time ultimately depends. In short, here we have made an attempt to optimize the synthesis parameters and further study the structural, microstructural, and electrical properties of the PT compound.In this paper, we have investigated the effect of processing parameters and optimized them to obtain single-phase, crack-free samples. The sintering behavior of the powders at various temperatures has been investigated using X-ray diffraction (XRD) patterns and scanning electron microscopy. The electrical behavior (complex impedance, complex permittivity, etc.) has been studied by complex impedance spectroscopy. The temperature and frequency variation of the electrical properties has also been investigated.


## 2. Experimental Procedures

PbTiO3 (PT) samples were prepared by solid-state reaction processing using high purity oxide and carbonate as the starting materials, involving several steps such as mixing, wet grinding, calcination, and sintering along with intermediate grinding at each stage. Calcination at temperatures ranging from 450 to 550°C was carried out to synthesize the PT compound before sintering at different temperatures. Commercially available PbCO3 (99.9% purity; Aldrich Chemicals) and TiO2 (99.9% purity; Loba Chemie were used as starting materials. In order to compensate the loss of PbO during the high temperature sintering, 5% by weight excess of PbO was added.The excess PbO helps to compensate for the lead evaporation during sintering process. The polycrystalline samples of PbTiO3 were prepared using the previous high purity ingredient materials in the required stoichiometry, and the powders were wet-milled. The sieved fine powders were then calcined in the temperature range 450–550°C for 8 hrsders were again finely ground and sieved through a 75 μm. The sieved powders were pressed in to pellets using polyvinyl alcohol (PVA) as a binder. The PT powders were then cold pressed into pellets of diameter 10 mm and thickness 1-2 mm at a pressure of 50 MPa using an isostatic hydraulic press. The pellets were sintered at four different temperatures 700°C, 750°C, 800°C, and 900°C for 12 hr. The heating program was carried out using Carbolite programmable high temperature furnace with Eurotherm temperature controller. The phase structure of sintered specimens was investigated with X-ray diffraction.The X-ray diffraction data was collected over the scattering angle range 10°≤2θ≤80° at 2θ step 0.02° using CuKα (λ=1.5418A) radiations recorded at room temperature using X-ray powder diffractometer (Xpert Pro-PAN Philips). The morphological studies of all the sintered samples were carried out by using scanning electron microscopy (SEM). Samples were examined using a JEOL JSM-6360 A analytical scanning electron microscopy operated at 15 kV. For determining the average particle size from full width at half maximum (FWHM) the intensity of Bragg peak (101) was used. The density of the PT ceramic was measured using single pan balance and Archimedes principle. For electrical characterization, the sintered disks (900°C, 12 hrs) of PT samples were polished to make both the faces flat and parallel and electroded with high-purity air-drying conducting silver paste. The impedance measurements were carried out using a computer-controlled impedance analyzer (HP 4192 A LF HEWLETT) over a wide range of temperature (33°C–560°C) and frequency (5 Hz–13 MHz) with an applied voltage of 1.3 V. Measurements provided the real and imaginary parts of the impedance and dielectric as frequency-dependent values.


## 3. Results and Discussion

### 3.1. Structural and Microstructural Characterization

X-ray diffraction patterns collected at room temperature for the samples sintered at 700°C, 750°C, 800°C, and 900°C are shown in Figure1. All the observed peaks/reflections are indexed in tetragonal crystals systems on the basis of the best agreement between observed and calculated interplaner (d) values. The values of unit cell parameters (a, c) and axial ratio (c/a) and density are given in Table 1. The comparative behavior of the tetragonality and unit cell volume with sintering temperature is shown in Figure 2, and the density variation with sintering temperature is shown in Figure 3. Figure 3 shows the measured and X-ray density (ρm, ρx) as a function of a sintering temperature. A slight increase in density is observed as sintering temperature increases up to 900°C. However, a density jump occurs at 750°C, and almost dense ceramics are achieved thereafter. The results are shown in Figure 3.Table 1
Unit cell parameters andc/a ratio of PT powder samples sintered at different temperatures.
Sinteringtemperature (°C)
a (Å)
c (Å)
c/a ratio
Density (X-ray) (g/cm3)
ρ
x
ρ
m
700
3.9008
4.067
1.0426
6.91
6.09
750
3.9013
4.0673
1.0425
7.85
7.18
800
3.9032
4.0679
1.0422
7.91
7.35
900
3.9023
4.0657
1.0419
7.97
7.43Figure 1
X-ray diffraction patterns of PbTiO3 sample sintered at different temperatures ((a) = 700°C, (b) = 750°C, (c) = 800°C, and (d) = 900°C).Figure 2
Variation ofc/a and unit cell volume (V) with sintering temperature (°C).Figure 3
X-ray and measured density of the PbTiO3 ceramics as a function of sintering temperature.The powder diffraction peaks from sample sintered at different temperatures showed that all the samples possess single-phase perovskite structure. The diffraction peaks have narrowed substantially with the increasing sintering temperature and the PT phase has persisted without evidence of second phase formation. Sintering athigher temperatures does not appear to substantially alter the nature of patterns. However, the XRD pattern collected at the highest temperature 900°C, Figure1(d), shows additional minor diffraction peaks, suggesting the beginning of the evolution of second phase (indicated by arrow). Figure 4 shows the XRD patterns for the samples that have been sintered at 900°C for various time periods (4, 8, and 12 hrs). Fine perovskite PbTiO3 crystallites, as indicated by the peaks at 2θ angle of 21.8°, 22.5°, 31.6°, 39.1°, 44.5°, 46.0°, 50.5°, 51.5°, 55.9°, and 56.9°, were observed, together with much weakened and broadened peaks of TiO2 when the samples were subjected to for 4 hr sintering. This observation indicatesthat sintering at 900°C for the first 4 hrs triggers the formation of perovskites PbTiO3 phase, and at the same time, the peak broadening of TiO2 implies that the first sintering has led to a significant refinement in the particle and crystallite sizes. The PbTiO3 phase was observed further when the sintering was extended to 8 hrs, which indicates that the formation of PbTiO3 phase occurred at the expense of constituent oxides with increasing sintering time. PbTiO3 was the only phase that was observed in the X-ray diffraction pattern in the sample that was sintered for 12 hrs The average particle size has been determined from the full width at half maximum (FWHM) of 101 peak using Scherrer’s equation [23]
(7)t=0.9λBcosθB,
where t is the particle diameter, λ the X-ray wavelength, B the FWHM of a diffraction peak, and θB the diffraction angle. The value of FWHM includes errors from the apparatus conditions such as the slit width of the X-ray diffractometer and these errors were corrected. The results were supported by the SEM micrograph studies. Figure 5 shows the variation of average particle sizes as a function of sintering temperatures determined from the peaks in the inset figures. Figure 6 shows the typical SEM pattern of PbTiO3 sample sintered at 900°C for 12 hrs. The pattern shows that the PbTiO3 particles have a more or less spherical morphology and consist of particle agglomerate of varying size, and the sizes of the grains are in the range 500 to 600 nm. The particle size (XRD), grain size (SEM), and phase and space group are listed in Table 2.Table 2
Particle size (XRD), grain size (SEM) and phase and space group of PbTiO3.
Sinteringtemperature(°C)
Particle size(nm)
SEM(nm)
Phase and space group
700
371
450
T (P4 mm)
750
446
540
T (P4 mm)
800
527
670
T (P4 mm)
900
608
883
T (P4 mm)Figure 4
X-ray diffraction patterns of PbTiO3 sample sintered at 900°C for various time durations (4, 8, and 12 hrs).Figure 5
Particle sizes as a function of sintering temperatures. Figure in the inset shows the (101) peak. FWHM values were calculated from the Gaussian fits (solid lines).SEM micrograph for sample sintered at (a) = 750°C, (b) = 800°C, and (c) = 900°C for 12 hrs.
(a)
(b)
(c)


### 3.2. Complex Impedance Study

Figure7 shows the complex impedance spectrum of the PT compound at five different temperatures. From the nature of these plots, it is evident that the change in temperature ensures a distinct effect on the impedance spectrum of the material. The appearance of low temperature single circular arcs is due to the bulk properties of the material, and appearance of double arcs at high temperatures (inset Figure 7) is due to the consequence of bulk and grain boundary contribution to the conduction. These arcs appear in distinct frequency ranges, one at a higher frequency followed by another lower frequency arc. This feature is almost similar at different temperatures, also with a difference in radii of curvature of the arcs, which reduces with rise in temperature. The pattern in the impedance spectrum is a representative of the electrical processes taking place in the material which can be expressed as an equivalent electrical circuit comprising a parallel combination of resistive and capacitive elements. The presence of two arcs accordingly can be thought of as resulting from cascading effect of parallel combination of resistive and capacitive elements arising due to the contribution of bulk property of the material and grain boundary effect. The high frequency is attributed to the bulk property of the material (parallel combination of bulk resistance and bulk capacitance), and low frequency is due to the grain boundary effects in the material (parallel combination of grain boundary resistance and capacitance). The electrical processes taking place within the material have been modeled on for a polycrystalline system and are shown in terms of the equivalent electrical circuit in the Figure 7 (inset). Figure 8 shows the variation Z′′ with frequency at different temperatures. The average peak position regularly shifts towards higher frequency as the temperature increases. This insures the temperature dependent relaxation process in the sample. Furthermore, as evident from the plots, as the temperature increases, the magnitude of Z′′ decreases, the effect being more pronounced at the peak position. The shift of the peak towards higher frequency in raising the temperature is possibly due to the reduction in the bulk resistivity. The asymmetric peaks suggest the presence of electrical processes in the material with spread of relaxation time. The relaxation species may possibly be electrons or immobile species at lower temperature and defects at higher temperature that may be responsible for electrical conduction in the material.Figure 7
Nyquist plot of PbTiO3 material at five different temperatures. Figure in the inset shows the equivalent electrical circuit.Figure 8
Frequency variation of imaginary impedance (Z′′) of PbTiO3 at different temperatures.Effectively large values ofZ′ and Z′′ at low frequencies or temperatures indicate a predominant effect of the polarizations which consists of large ε values of PTs. The intercepts of the two semicircles are used to calculate the bulk resistance (Rb) and grain boundary resistance (Rgb) while the corresponding frequency values evaluated from the apex of the semicircles have been used to calculate the bulk and grain boundary capacitance (Cb) using the relation, for a parallel combination of R and C(8)ωmaxCbRb=1orωmaxCgbRgb=1.
As temperature increases, both the grain resistance (Rb) and grain boundary resistance (Rgb) are found to decrease with rise in temperature indicated by a shift in the radius of the arcs towards left side on the real (Z′) axis with increase in temperature. This provides convincing evidence that the electrical properties of PT are dependent on microstructure as well as temperature. A single nondegenarate process involves a single nondegenerate relaxation time for a given set of (Rb) and (Cb) at a temperature
(9)τ=CbRb=1ω.
The value of τ determined at selected temperatures in the region of measuring temperature using the values of (Rb) and (Cb) in (9) is listed in Table 3. The τ-value is found to be decreasing linearly on increasing value of the temperature (~ above 600 K) suggesting a typical semiconductor behavior. The variation of τ with temperature (Figure 9) implies that the relaxation process is temperature dependent. The activation energy Ea evaluated from the slope of the curve by the relation
(10)τ=τ0e-Ea/kT,
where τ0 is preexponential factor, Ea activation energy, k the Boltzmann constant and T the absolute temperature, is found to be 0.43 eV below 600 K and 2.1 eV above 600 K.Table 3
Relaxation time calculated from impedance data for PbTiO3.
Temperature (°C)
R
g (Ω)
C
g (nF)
τ (s)
240
8.37× 105
0.213
1.41× 10−1
320
2.47× 105
0.202
3.42× 10−2
400
4.51× 104
0.223
1.10× 10−2
440
9.87× 103
0.204
5.92× 10−3
480
2.87× 103
0.248
2.40× 10−3
500
2.00× 103
0.316
1.53× 10−3
520
9.90× 102
0.808
6.78× 10−4
540
6.44× 102
0.554
2.46× 10−4Figure 9
Plot of relaxation time with103/T(K-1) of PbTiO3.


### 3.3. Complex Permittivity Study

The variation of real permittivity with temperature is shown in Figure10. It is observed that the temperature dependence of real dielectric constant (ε′) at different frequencies has typical features of relaxor ferroelectric. (i) the value of ε′ increases gradually with rise in temperature and reaches a maximum value εmax′ at a particular temperature known as Curie temperature (Tc). This general feature is similar at all frequencies. (ii) The value of εmax′ at Tc decreases with increasing frequency. Further its value is larger at lower frequency (0.1 kHz). (iii) The value of Tc shifts towards the higher temperature side with increases in frequency. (iv) The Dielectric dispersion is absent at high temperatures beyond Tc whereas the dielectric constant has been observed to decrease with increasing frequencies at temperatures around and lower than Tc. This indicates strong dielectric dispersion at temperatures around and below Tc. The Tc value shifts from 750 to 850 K with the increased frequency value 0.1 kHz to 1000 kHz.Figure 10
Temperature dependence of dielectric constant of PbTiO3 at different frequencies.Figure11 shows tan (δ) versus temperature (K) for PbTiO3 at different frequencies. Dispersion in dielectric loss and shifting of dielectric loss peaks towards higher temperature side were observed with increase of frequency.Figure 11
Temperature dependence of dissipation factor of PbTiO3 at different frequencies.For a normal ferroelectric above the Curie temperature, the dielectric constant follows the Curie-Weiss law:(11)ε=cT-T0,
where c is the Curie constant and T0 is the Curie-Weiss temperature [24–26]. For a ferroelectric with a diffuse phase transition (broad peak), the following equation:
(12)1ε=(T-Tm)2
has been shown to be valid over a wide temperature range instead of the normal Curie-Weiss law (11) [27, 28]. In (2), Tm is the temperature at which the dielectric constant is maximum. The degree of disorderness was evaluated using the relation [28, 29]
(13)1ε=1εmax+(T-Tmax)γ2εmaxδ2,
where εmax is the maximum permittivity, γ the diffusivity, and δ is diffuseness parameter. The quantitative assessment of the diffusivity (γ) of the broadened peaks in the paraelectric phase was evaluated using the expression [30]
(14)ln(1ε-1εmax)versus(T-Tc)γ.
The plot (Figure 12) of ln(1/ε-1/εmax) versus ln(T-Tc) for all frequencies was use to obtain the values of γ. For value of γ=1, the material exhibits Curie-Weiss behavior and for γ=2, it is completely disordered. The value of γ lies between 1 to 2 and this confirm the diffuse phase transition. Table 4 gives the value of dielectric parameters for different frequencies.Table 4
Dielectric parameters of PbTiO3 with different frequencies.
Frequency (kHz)
ε
RT
TanδRT
T
c (°C)
ε
max
tanδ at Tc
γ
0.1
1400
2.10
480
9140
1.78
1.84
1
1100
2.20
490
4200
1.63
1.65
100
700
2.40
496
2300
1.56
1.68
1000
300
2.00
510
1170
0.51
1.80Figure 12
Plots ofln(1/ε-1/εmax) versus ln(T-Tc) for PbTiO3 ceramics at different frequencies.


## 3.1. Structural and Microstructural Characterization

X-ray diffraction patterns collected at room temperature for the samples sintered at 700°C, 750°C, 800°C, and 900°C are shown in Figure1. All the observed peaks/reflections are indexed in tetragonal crystals systems on the basis of the best agreement between observed and calculated interplaner (d) values. The values of unit cell parameters (a, c) and axial ratio (c/a) and density are given in Table 1. The comparative behavior of the tetragonality and unit cell volume with sintering temperature is shown in Figure 2, and the density variation with sintering temperature is shown in Figure 3. Figure 3 shows the measured and X-ray density (ρm, ρx) as a function of a sintering temperature. A slight increase in density is observed as sintering temperature increases up to 900°C. However, a density jump occurs at 750°C, and almost dense ceramics are achieved thereafter. The results are shown in Figure 3.Table 1
Unit cell parameters andc/a ratio of PT powder samples sintered at different temperatures.
Sinteringtemperature (°C)
a (Å)
c (Å)
c/a ratio
Density (X-ray) (g/cm3)
ρ
x
ρ
m
700
3.9008
4.067
1.0426
6.91
6.09
750
3.9013
4.0673
1.0425
7.85
7.18
800
3.9032
4.0679
1.0422
7.91
7.35
900
3.9023
4.0657
1.0419
7.97
7.43Figure 1
X-ray diffraction patterns of PbTiO3 sample sintered at different temperatures ((a) = 700°C, (b) = 750°C, (c) = 800°C, and (d) = 900°C).Figure 2
Variation ofc/a and unit cell volume (V) with sintering temperature (°C).Figure 3
X-ray and measured density of the PbTiO3 ceramics as a function of sintering temperature.The powder diffraction peaks from sample sintered at different temperatures showed that all the samples possess single-phase perovskite structure. The diffraction peaks have narrowed substantially with the increasing sintering temperature and the PT phase has persisted without evidence of second phase formation. Sintering athigher temperatures does not appear to substantially alter the nature of patterns. However, the XRD pattern collected at the highest temperature 900°C, Figure1(d), shows additional minor diffraction peaks, suggesting the beginning of the evolution of second phase (indicated by arrow). Figure 4 shows the XRD patterns for the samples that have been sintered at 900°C for various time periods (4, 8, and 12 hrs). Fine perovskite PbTiO3 crystallites, as indicated by the peaks at 2θ angle of 21.8°, 22.5°, 31.6°, 39.1°, 44.5°, 46.0°, 50.5°, 51.5°, 55.9°, and 56.9°, were observed, together with much weakened and broadened peaks of TiO2 when the samples were subjected to for 4 hr sintering. This observation indicatesthat sintering at 900°C for the first 4 hrs triggers the formation of perovskites PbTiO3 phase, and at the same time, the peak broadening of TiO2 implies that the first sintering has led to a significant refinement in the particle and crystallite sizes. The PbTiO3 phase was observed further when the sintering was extended to 8 hrs, which indicates that the formation of PbTiO3 phase occurred at the expense of constituent oxides with increasing sintering time. PbTiO3 was the only phase that was observed in the X-ray diffraction pattern in the sample that was sintered for 12 hrs The average particle size has been determined from the full width at half maximum (FWHM) of 101 peak using Scherrer’s equation [23]
(7)t=0.9λBcosθB,
where t is the particle diameter, λ the X-ray wavelength, B the FWHM of a diffraction peak, and θB the diffraction angle. The value of FWHM includes errors from the apparatus conditions such as the slit width of the X-ray diffractometer and these errors were corrected. The results were supported by the SEM micrograph studies. Figure 5 shows the variation of average particle sizes as a function of sintering temperatures determined from the peaks in the inset figures. Figure 6 shows the typical SEM pattern of PbTiO3 sample sintered at 900°C for 12 hrs. The pattern shows that the PbTiO3 particles have a more or less spherical morphology and consist of particle agglomerate of varying size, and the sizes of the grains are in the range 500 to 600 nm. The particle size (XRD), grain size (SEM), and phase and space group are listed in Table 2.Table 2
Particle size (XRD), grain size (SEM) and phase and space group of PbTiO3.
Sinteringtemperature(°C)
Particle size(nm)
SEM(nm)
Phase and space group
700
371
450
T (P4 mm)
750
446
540
T (P4 mm)
800
527
670
T (P4 mm)
900
608
883
T (P4 mm)Figure 4
X-ray diffraction patterns of PbTiO3 sample sintered at 900°C for various time durations (4, 8, and 12 hrs).Figure 5
Particle sizes as a function of sintering temperatures. Figure in the inset shows the (101) peak. FWHM values were calculated from the Gaussian fits (solid lines).SEM micrograph for sample sintered at (a) = 750°C, (b) = 800°C, and (c) = 900°C for 12 hrs.
(a)
(b)
(c)


## 3.2. Complex Impedance Study

Figure7 shows the complex impedance spectrum of the PT compound at five different temperatures. From the nature of these plots, it is evident that the change in temperature ensures a distinct effect on the impedance spectrum of the material. The appearance of low temperature single circular arcs is due to the bulk properties of the material, and appearance of double arcs at high temperatures (inset Figure 7) is due to the consequence of bulk and grain boundary contribution to the conduction. These arcs appear in distinct frequency ranges, one at a higher frequency followed by another lower frequency arc. This feature is almost similar at different temperatures, also with a difference in radii of curvature of the arcs, which reduces with rise in temperature. The pattern in the impedance spectrum is a representative of the electrical processes taking place in the material which can be expressed as an equivalent electrical circuit comprising a parallel combination of resistive and capacitive elements. The presence of two arcs accordingly can be thought of as resulting from cascading effect of parallel combination of resistive and capacitive elements arising due to the contribution of bulk property of the material and grain boundary effect. The high frequency is attributed to the bulk property of the material (parallel combination of bulk resistance and bulk capacitance), and low frequency is due to the grain boundary effects in the material (parallel combination of grain boundary resistance and capacitance). The electrical processes taking place within the material have been modeled on for a polycrystalline system and are shown in terms of the equivalent electrical circuit in the Figure 7 (inset). Figure 8 shows the variation Z′′ with frequency at different temperatures. The average peak position regularly shifts towards higher frequency as the temperature increases. This insures the temperature dependent relaxation process in the sample. Furthermore, as evident from the plots, as the temperature increases, the magnitude of Z′′ decreases, the effect being more pronounced at the peak position. The shift of the peak towards higher frequency in raising the temperature is possibly due to the reduction in the bulk resistivity. The asymmetric peaks suggest the presence of electrical processes in the material with spread of relaxation time. The relaxation species may possibly be electrons or immobile species at lower temperature and defects at higher temperature that may be responsible for electrical conduction in the material.Figure 7
Nyquist plot of PbTiO3 material at five different temperatures. Figure in the inset shows the equivalent electrical circuit.Figure 8
Frequency variation of imaginary impedance (Z′′) of PbTiO3 at different temperatures.Effectively large values ofZ′ and Z′′ at low frequencies or temperatures indicate a predominant effect of the polarizations which consists of large ε values of PTs. The intercepts of the two semicircles are used to calculate the bulk resistance (Rb) and grain boundary resistance (Rgb) while the corresponding frequency values evaluated from the apex of the semicircles have been used to calculate the bulk and grain boundary capacitance (Cb) using the relation, for a parallel combination of R and C(8)ωmaxCbRb=1orωmaxCgbRgb=1.
As temperature increases, both the grain resistance (Rb) and grain boundary resistance (Rgb) are found to decrease with rise in temperature indicated by a shift in the radius of the arcs towards left side on the real (Z′) axis with increase in temperature. This provides convincing evidence that the electrical properties of PT are dependent on microstructure as well as temperature. A single nondegenarate process involves a single nondegenerate relaxation time for a given set of (Rb) and (Cb) at a temperature
(9)τ=CbRb=1ω.
The value of τ determined at selected temperatures in the region of measuring temperature using the values of (Rb) and (Cb) in (9) is listed in Table 3. The τ-value is found to be decreasing linearly on increasing value of the temperature (~ above 600 K) suggesting a typical semiconductor behavior. The variation of τ with temperature (Figure 9) implies that the relaxation process is temperature dependent. The activation energy Ea evaluated from the slope of the curve by the relation
(10)τ=τ0e-Ea/kT,
where τ0 is preexponential factor, Ea activation energy, k the Boltzmann constant and T the absolute temperature, is found to be 0.43 eV below 600 K and 2.1 eV above 600 K.Table 3
Relaxation time calculated from impedance data for PbTiO3.
Temperature (°C)
R
g (Ω)
C
g (nF)
τ (s)
240
8.37× 105
0.213
1.41× 10−1
320
2.47× 105
0.202
3.42× 10−2
400
4.51× 104
0.223
1.10× 10−2
440
9.87× 103
0.204
5.92× 10−3
480
2.87× 103
0.248
2.40× 10−3
500
2.00× 103
0.316
1.53× 10−3
520
9.90× 102
0.808
6.78× 10−4
540
6.44× 102
0.554
2.46× 10−4Figure 9
Plot of relaxation time with103/T(K-1) of PbTiO3.


## 3.3. Complex Permittivity Study

The variation of real permittivity with temperature is shown in Figure10. It is observed that the temperature dependence of real dielectric constant (ε′) at different frequencies has typical features of relaxor ferroelectric. (i) the value of ε′ increases gradually with rise in temperature and reaches a maximum value εmax′ at a particular temperature known as Curie temperature (Tc). This general feature is similar at all frequencies. (ii) The value of εmax′ at Tc decreases with increasing frequency. Further its value is larger at lower frequency (0.1 kHz). (iii) The value of Tc shifts towards the higher temperature side with increases in frequency. (iv) The Dielectric dispersion is absent at high temperatures beyond Tc whereas the dielectric constant has been observed to decrease with increasing frequencies at temperatures around and lower than Tc. This indicates strong dielectric dispersion at temperatures around and below Tc. The Tc value shifts from 750 to 850 K with the increased frequency value 0.1 kHz to 1000 kHz.Figure 10
Temperature dependence of dielectric constant of PbTiO3 at different frequencies.Figure11 shows tan (δ) versus temperature (K) for PbTiO3 at different frequencies. Dispersion in dielectric loss and shifting of dielectric loss peaks towards higher temperature side were observed with increase of frequency.Figure 11
Temperature dependence of dissipation factor of PbTiO3 at different frequencies.For a normal ferroelectric above the Curie temperature, the dielectric constant follows the Curie-Weiss law:(11)ε=cT-T0,
where c is the Curie constant and T0 is the Curie-Weiss temperature [24–26]. For a ferroelectric with a diffuse phase transition (broad peak), the following equation:
(12)1ε=(T-Tm)2
has been shown to be valid over a wide temperature range instead of the normal Curie-Weiss law (11) [27, 28]. In (2), Tm is the temperature at which the dielectric constant is maximum. The degree of disorderness was evaluated using the relation [28, 29]
(13)1ε=1εmax+(T-Tmax)γ2εmaxδ2,
where εmax is the maximum permittivity, γ the diffusivity, and δ is diffuseness parameter. The quantitative assessment of the diffusivity (γ) of the broadened peaks in the paraelectric phase was evaluated using the expression [30]
(14)ln(1ε-1εmax)versus(T-Tc)γ.
The plot (Figure 12) of ln(1/ε-1/εmax) versus ln(T-Tc) for all frequencies was use to obtain the values of γ. For value of γ=1, the material exhibits Curie-Weiss behavior and for γ=2, it is completely disordered. The value of γ lies between 1 to 2 and this confirm the diffuse phase transition. Table 4 gives the value of dielectric parameters for different frequencies.Table 4
Dielectric parameters of PbTiO3 with different frequencies.
Frequency (kHz)
ε
RT
TanδRT
T
c (°C)
ε
max
tanδ at Tc
γ
0.1
1400
2.10
480
9140
1.78
1.84
1
1100
2.20
490
4200
1.63
1.65
100
700
2.40
496
2300
1.56
1.68
1000
300
2.00
510
1170
0.51
1.80Figure 12
Plots ofln(1/ε-1/εmax) versus ln(T-Tc) for PbTiO3 ceramics at different frequencies.


## 4. Conclusions

The following interesting trends were revealed in the present study.The unit cell dimensions are not significantly influenced by the sintering temperature. Thec/a ratio does not vary significantly with sintering temperature, and its value is nearly 1.04. Gaussian’s FWHM values are all small suggesting that the interparticle strains may contribute little to the diffraction peak shapes in these samples. The SEM micrographs do not exhibit the fracture surfaces of sample sintered at 900°C, and the sample has developed a relative uniform microstructure. PbTiO3 powder can be synthesized directly from the constituent oxide using conventional ceramic technique by optimizing sintering temperature and time and by using starting oxides having fine uniform particle size. There is strong corelation between the crystallite sizes on the sintering temperature and supporting to the notion that crystallite size determines the propensity of microcracking. The c/a ratio does not differ much for the samples sintered at different temperatures. The experimental results on electrical properties indicate that the material exhibits conduction due to bulk and grain boundary effect at high temperature. The nature of loss spectrum (Figure 8) shows that the peaks shift towards the high frequency side with increase in temperature indicating relaxation process in the system. The broadening of the peaks on increasing temperature suggests the presence of temperature-dependant relaxation process in the system.



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