Conductivity of the PGT Synthesized by the High Energy Ball Milling ( HEBM )

Nanocrystalline Pb 1−3x/2 Gd x TiO 3 (where x = 0.01) abbreviated as PGT has been synthesised by high energy ball milling at room temperature. Milling was continuous and X-ray analysis shows that single phase tetragonal structure of nanocrystalline PGT was formed after 15 h milling. The average crystallite size was found to be 17 nm. The frequency dependent ac conductivity of the PGT ceramic was studied in the range 100–525C. Complex impedance analysis suggested the dielectric relaxation to be of non-Debye type. The activation energy was found to be 1.04 ev. The mechanism of charge transport in nanocrystalline PGT was successfully explained by correlated hopping model.


Introduction
Lead titanate (PbTiO 3 ) is a well-known ferroelectric ceramic having perovskite structure (ABO 3 ) with the Pb 2+ ions at the cell corner A-sites, Ti 4+ ions occupying the body centre B-sites, and the oxygen atoms at the face centre sites.The perovskite structure has an intrinsic capability to host ions of different sizes and a wide variety of elements can be accommodated into the structure considering several factors, namely, (a) charge neutrality, (b) tolerance factor, (c) ionic radius, and (d) solubility [1].It is well-known ferroelectrics which has several industrial applications like piezoelectric, capacitor, transducer, memory devices, and so forth [1].Detailed literature survey reveals that structural and electrical properties of PT ceramics can be modified by addition of Lanthanide ions [2].A high energy ball milling (HEBM) technique has successfully been used to synthesis nanocrystalline ferroelectrics and other alloys [3,4].Though nanocrystalline ceramics can be synthesised by a range of different physical, chemical, and mechanical methods, HEBM offers several advantages over the other methods [5][6][7].HEBM is very useful process to prepare lead based ceramics [8] and others because it takes place close to a room temperature, thus, effectively alleviating the loss of PbO.The study of electrical conductivity in the ferroelectric compounds is very important since the associated physical properties like dielectric, pyroelectric, and piezoelectric are dependent on the nature and magnitude of conductivity of the materials.Moreover complex impedance spectroscopy technique can be considered as a powerful experimental technique in order to correlate the electrical and structural characteristics in ceramics.This technique has been successfully used to understand the dielectric behaviour of single crystal, polycrystalline, and amorphous ceramic materials [9].It is the most commonly used experimental technique to analyse the dynamics of the ionic movement in ceramic materials.Contribution of various microscopic elements such as grain, grain boundary, and interfaces to total dielectric response in polycrystalline solids can be identified by an equivalent circuit which contains an array of parallel RC elements [10].A detailed literature survey suggests that not much work in this direction has been reported so far for nanocrystalline PGT.In this present work a high energy ball milling (HEBM) technique has successfully been used to synthesis nanocrystalline ferroelectrics and other alloys.Though nanocrystalline ceramics can be synthesised by a range of different physical, chemical, and mechanical methods, HEBM offers several advantages over the other methods.Pb 1−3/2 Gd  TiO 3 abbreviated as PGT nanocrystalline ceramics were prepared using HEBM at room temperature.The investigation of dielectric relaxation and electrical conduction properties of nanocrystalline PGT are reported.An attempt has also been made to explain the conduction mechanism in PGT using complex impedance spectroscopy.The correlated barrier hopping model has been applied to the ac conductivity data to understand the conduction mechanism of charge transport in the system.The ac conductivity data was used to estimate the apparent activation energy, density of states at Fermi level, and minimum hopping length.

Experimental Details
Rare earth Gadolinium modified lead titanate (PGT) nanoceramic with a general formula Pb 1−3/2 Gd  TiO 3 (where  was fixed at 0.01) was synthesised from high purity ingredients, PbO, Gd 2 O 3 , and TiO 2 .The starting oxides were mixed in stoichiometric ratio and to compensate for the expected Pb loss at high sintering temperatures 2% extra PbO was used throughout the experiment.
The synthesis of PGT was carried out by high energy ball milling technique.The milling was performed in a planetary ball mill (Retsch PM 200) at room temperature for different milling times (0 to 45 h).The milling was carried out with WC (Tungsten carbide) vial and WC balls (with 10 mm dia.) at a speed of 300 rpm and ball to powder weight ratio of 15 : 1. Milling was stopped for 30 minutes after every 1 hour of milling.The phase identification was carried out using an X-ray diffractometer (XRD) (Rigaku Miniflex II with Cu K radiation  = 0.15418 nm).The powder obtained after 45 hrs of milling was mixed thoroughly with a 3%PVA and then uniaxially compacted into disk samples.Disks were formed using a hydraulic pellet press at a pressure of 4 MPa to form pellets of the size of 2 mm in thickness and 10 mm in diameter.The green pellets were sintered at 900 ∘ C for 2 hrs in air atmosphere.The flat polished surfaces of the sintered pellets were electroded with high purity silver paste and then dried at 700 ∘ C for 15 mins before making any electrical measurements.The electrical parameters were measured using a HIOKI-3532 LCR Hitester under a weak electric field (with a maximum magnitude of 1 V) in the temperature range of 100-525 ∘ C at different frequencies varying from 10 2 to 10 6 Hz at a heating/cooling rate of 2 ∘ C/min.

Results and Discussion
3.1.Structures and Microstructures.Figure 1 shows the XRD pattern of PGT subjected to different durations of milling.Most of the XRD peaks of PGT were identified and indexed in tetragonal crystal system using X-pert High Score Plus.A unit cell of PGT was selected and its Lattice parameters were refined.The tetragonality ratio (c/a) was found to be 1.0467.The average crystallite size and lattice strain determined using Scherrer's formula [7] were 17 nm and 0.012, respectively.10 KHz, 100 KHz, and 1 MHz.The sample exhibits  maxima in its curves which correspond to a phase transition from ferroelectric to paraelectric state.The dielectric constant has a maximum value at 470 ∘ C for each measured frequency and a temperature lower than 490 ∘ C for undoped lead titanate (PT) as reported by Kong et al. [11].It was observed from Figure 2 that phase transition temperature was independent of frequency which suggests a nonrelaxor behaviour of PGT.

Dielectric Studies.
The lowering of the transition temperature and the increase in dielectric constant values confirm the incorporation of Gd to the perovskite structure, in correspondence with other reports about the effect of small cation substitution of the Pb 2+ in the A-site with Ca 2+ , Sr 2+ , and La 3+ ions [11][12][13].
In low frequency range dielectric constant has a high value Advances in Materials Science and Engineering 3 which can be attributed to various polarization effects, namely, ionic and dipolar.At high frequency only electronic polarization contribution dominates and hence the dielectric values are less.

Impedance Analysis.
The real and imaginary parts of dielectric constant were obtained from the impedance data in a conventional way using the following relations [14]: where The ac conductivity data were obtained using a relation and the real and imaginary part of  * () were obtained as In the conductivity representation for electronic conduction, the real part   () should be constant and imaginary part   () increases linearly with frequency.The ac electrical conductivity in most of the materials due to localised states is given by where  0 is the frequency independent part of ac conductivity,  (0 ≤  ≤ 1), is the index,  = angular frequency of the applied ac field,  =  2  2 /6  (2) is a constant,  is the electronic charge,  is the temperature,  is the polarizability of a pair of sites, and  is the number of sites per unit volume among which hopping takes place.Such variation is associated with the displacement of carriers which move within the sample by discrete hops of length  between randomly distributed localised sites.The term   can often be explained on the basis of two distinct mechanisms for carrier conduction: (a) quantum mechanical tunnelling (QMT) through the barrier separating the localised sites and (b) correlated barrier hopping (CBH) over the same barrier.In these models the exponent  is found to have two different trends of variations with temperature and frequency.In QMT,  is predicted to be temperature independent and is expected to show a decreasing trend with : where  0 is the characteristic relaxation time.The ac conductivity is expected to be [15] where  −1 is the spatial decay parameter for the localised wave function and   is the tunnelling length at frequency : Further, if ac conductivity occurs from CBH [16], where (  ) is the density of states at Fermi level,  0 is the photon frequency, and  is the localised wave function.
The exponent  and minimum hopping length  min can be expressed as [17,18] where   is the binding energy, which is defined as the energy required to remove an electron completely from one site to another site and  is the dielectric constant of PGT.
The frequency dependence of real permittivity   and imaginary permittivity   is shown in Figures 3(a) and 3(b).Both curves show a decreasing trend with increase in frequency.Dielectric materials have a high dielectric constant at lower frequencies due to space charge contribution.This is evident from the graph of   and frequency where the dispersive nature with relatively high dielectric constant is observed at lower frequency site.
The dipoles follow the field at low frequencies and as the frequency increases dipoles begin to lag behind the field; at very high frequencies the dipoles can no longer follow the field and the value of   decreases.
The variation of real   and imaginary part of impedance   versus frequency plot are given in Figures 4(a) and 4(b).It can be seen that the curves are temperature dependent and the sharpness decreases with increase in temperature.The decrease in magnitude of   with increase in frequency for all temperatures indicates an increase in ac conductivity with rise in frequency.The   values for all temperatures merge at high frequency due to the release of space charge as a result of reduction in barrier properties of the material with rise in temperature [10] and may be responsible factor for the enhancement of ac conductivity of the material with temperature at high frequencies.The decrease in   values at low frequency with increase in temperature shows negative temperature coefficient of resistance (NTCR) similar to that of semiconductors.It can also be seen from Figure 4(a) that the   value gives a dip at high frequency and that decreases with increase in temperature which may be due to charge carrier hopping (correlated barrier hoping CBH). Figure 4(b) shows the variation of imaginary part of impedance (  ) with frequency.The frequency corresponding to the maximum impedance (complex part) shifts towards right with increase in temperature.This shift is mainly due to the reduction in bulk resistance with temperature.The broadening of the peaks suggests that there is a spread of relaxation times.The significant broadening of peaks on increasing temperature suggests the presence of a temperature dependent relaxation process in the materials.The relaxation species may be possibly electrons at low temperature and defects at higher temperature.In the case of dielectric materials the localised relaxation dominates [19] (i.e., defect relaxation) because of a low dielectric ratio  ( =   / ∞ ), where   and  ∞ are the dielectric constants at low and high frequencies, respectively.
For the sample under study the value of  at   is calculated to be 1.52 at frequencies 1 KHz and 1 MHz.
Figure 5 shows the variation of real (  ) and imaginary (  ) parts of impedance at different temperatures.It is also a powerful technique for the characterization of grain and grain boundaries in ceramics.
It is well known that complex impedance analysis helps in understanding the nature of dielectric relaxation in the material [20].For Debye type relaxation, the centre of the semicircular plots should be located on the   axis, whereas for a non-Debye type relaxation the argand plane plots are close to semicircular arcs with the centre below the real axis.The impedance spectrum for PGT sample consists of two semicircular arcs (Figure 5), which exhibit some degree of decentralization.This decentralisation or non-Debye relaxation obeys the Cole-Cole formalism [21], where the depressed semicircle represents typically a phenomenon with a spread of relaxation time.The Cole-Cole formalism is given by where  represents the magnitude of deviation of the electrical response from the ideal condition and this can be determined from the location of the centre of the semicircles and  0 is the frequency at the maximum of the semicircle.The exponent  → 0 (i.e., 1− = 1) in the above equation gives the classical Debye formalism.This nonideal behaviour may happen due to the presence of distributed elements in the material electrode system [13].The semicircular pattern in the impedance spectrum is representative of the electrical processes taking place in the material which can be thought of as resulting from the cascading effect of a parallel combination of resistive and capacitive elements arising due to the contribution of the bulk property of the material and the grain boundary effects.The high frequency semicircle is due to the bulk property of the material (parallel combination of bulk resistance and bulk capacitance) and the low frequency semicircle to be due to the grain boundary effects (parallel combination of grain boundary resistance and capacitance).The absence of a third semicircle shows that the electrode material interface contribution to impedance is negligible.
It is clear from Figure 5 that with increase in temperatures the radius of the semicircular arc shifts towards the left side and thus the bulk resistance   decreases with an increase in temperature.It is observed that at temperature 375 ∘ C and above, two semicircles can be traced with different values of grain (  ) and grain boundary (  ) and their values can be obtained from the intercept of the traced semicircles with   axis.The decrease in the values of   with temperature indicates negative temperature coefficient of resistance and is in confirmation with Figure 4(a).Frequency (Hz)  conductivity.The plots of   show dispersion throughout the chosen frequency range and with the increment in temperature the lines get flattened.Translation from long range hopping to short range ion motion is evident from the switchover from frequency independent to the frequency dependent region which also shows beginning of the conductivity relaxation.The values of  were obtained from the slopes of   −  plots in the low frequency region.

AC and DC Conductivity.
Figure 7 shows the temperature dependence of  and it is also evident from the slopes that the values of  are always less than 1 and decrease with the rise in temperature.The value of  approaches 0 at higher temperatures indicating the domination of dc conductivity at higher temperatures in the low frequency region following (1).
The model based on correlated hopping of electrons over barrier predicts a decrease in the value of the index  with the increase in temperature and so it is consistent with the experimental results.Thus the conduction in the system may be considered to be due to the short-range translational type hopping of charge carriers.This indicates that the conduction process is a thermally activated process.Materials with high density of states a band gap like that of the semiconductors are generally consistent with hopping conduction mechanism.Due to localisation of charge carriers, formation of polarons takes place and the hopping conduction may occur between the nearest neighbouring sites.
Figure 8 shows the variation of ln   versus 10 3 / at 1 KHz and 1 MHz.The activation energy for conduction using Arrhenius relationship  =  0 exp(−  /) was obtained as 1.09 eV and 0.81 eV, respectively, for 1 KHz and 1 MHz.The low value of activation energy suggests a possibility of carrier transport through hopping between localised states in a disordered manner [21].
The values of (  ) were calculated using (9) by assuming  0 = 10 13 Hz and  = 10 10 m −1 at various operating frequencies and temperatures.It can be seen from Figure 9 that the values of (  ) decrease with increase in operating frequency and almost merge above 100 KHz. Figure 10 shows variation of density of states at Fermi level of PGT ceramic with temperature.It is evident from the figure that the density of states increase with temperature for all frequencies.Comparing the two figures (Figures 9 and 10) one can conclude that at low frequencies the conduction is effected by both frequency as well as temperature and at high frequencies it is effected by thermal excitations since the charges are localised.The high values of (  ) are suggestive of hopping between the pair of sites which dominates the conduction mechanism in PGT.

Conclusion
Nanocrystalline Gd doped lead titanate (PGT) powders were successfully synthesized by high energy ball milling techniques at room temperature.The frequency dependent ac conductivity at different temperatures indicated that the conduction process is thermally activated process.The activation energy calculated from the impedance analysis and conductivity data are comparable.At higher temperature PGT behaves as a semiconductor.The semiconductivity is attributed to the extra positive charge in the conduction band caused by the substitution of Pb +2 ions by trivalent Gd +3 ions.With rise in temperature the donor cations are the major contributors to the conduction process.The donors create an energy level closer to the conduction band thus reducing the amount of energy required to activate the donors to the conduction band.The correlated barrier hopping model is found to successfully explain the mechanism of charge transport in PGT.The results are well supported by density of states at Fermi level.

Figure 3 :
Figure 3: (a) Temperature dependence of dielectric constant (real) at different frequencies.(b) Temperature dependence of dielectric constant (imaginary) at various frequencies.Insets show the temperature dependence of real and imaginary parts of the dielectric constant.
Figures 6(a) and 6(b) show the log-log plot of real and imaginary parts of ac electrical

Figure 4 :
Figure 4: (a) Plot of real part of impedance (  ) with frequency.(b) Plot of imaginary part of impedance (  ) with frequency.

Figure 5 :
Figure 5: Plot of real part of impedance (  ) with imaginary part of impedance (  ) of PGT.

Figure 6 :
Figure 6: Variation of (a) real and (b) imaginary parts of ac conductivity with frequency at different temperatures for PGT ceramic.

Figure 7 :
Figure 7: Variation of index  with temperature.

Figure 8 :
Figure 8: Variation of real part of ac conductivity with inverse of temperature at 1 KHz and 1 MHz for PGT ceramic.

3 )Figure 9 :− 3 )Figure 10 :
Figure 9: Variation of density of states at Fermi level of PGT ceramic with frequency.