Journal of Solid State Physics

Volume 2013 (2013), Article ID 451563, 9 pages

http://dx.doi.org/10.1155/2013/451563

Research Article

Dielectric Properties of Rhombohedral

Kriti Ranjan Sahu¹ and Udayan De²

¹Physics Department, Egra SSB College, Egra, West Bengal 721429, India

²Kendriya Vihar C-4/60, V.I.P. Road, Kolkata 700052, India

Received 30 April 2013; Revised 26 August 2013; Accepted 20 September 2013

Academic Editor: Hamdy Doweidar

Copyright © 2013 Kriti Ranjan Sahu and Udayan De. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Dielectric materials are needed in many electrical and electronic applications. So, basic characterizations need to be done for all dielectrics. (PN) is ferroelectric and piezoelectric only in its orthorhombic phase, with potential high temperature applications. So, its rhombohedral phase, frequently formed as an undesirable impurity in the preparation of orthorhombic PN, has been ignored with respect to possible dielectric characterizations. Here, essentially single phase rhombohedral PN has been prepared, checking structure from XRD Rietveld Analysis, and the real and imaginary parts of permittivity measured in an Impedance Spectrometer (IS) up to ~ and over 20 Hz to 5.5 MHz range, for heating and some cooling runs. Variations, with temperature, of relaxation time constant (), AC and DC conductivity, bulk resistance, activation energy and capacitance have been explored from our IS data.

1. Introduction

Commercial piezoelectric (PE) materials for applications in medicine, industry, and research have mostly been barium titanate (BaTiO₃), lead zirconate titanate (Pb[ZrxTi1-x]O₃, , or PZT), or materials based on one of these. But Curie temperature, the upper limit for piezoelectricity, is at best 130°C [1, 2] for the former and 390°C [2] for a specially modified PZT. Now, certain modern high temperature applications in industry need higher Curie temperature and, hence, newer materials with higher Curie temperature and suitable PE properties. This growing need for high temperature piezoelectric sensors and actuators [1, 2] has revived, in last few decades, a worldwide interest in lead niobate (PbNb₂O₆, to be shortened here as PN) in orthorhombic structure, which alone is piezoelectric. It was discovered [3] in 1953 but almost ignored for decades after a wave of pioneering work [3–5] of high quality. PN and PN-based ferroelectric samples involving chemical substitution &/or composite formation [6–8] are now being investigated by different groups, while present work is on pure PbNb₂O₆ in rhombohedral form.

PbNb₂O₆ has different structures as already indicated. The stable forms of PbNb₂O₆ are [3–5, 9] rhombohedral (at low temperature) and tetragonal (at high temperature). The latter is transformed, usually by quenching (), to PNQ, the metastable orthorhombic PbNb₂O₆, which alone can be made piezoelectric (fortunately with a high Curie temperature higher than 580°C for 5.5 MHz, e.g., [9]). Slow () cooling (from temperatures like 1270°C) leads [10, 11] to PNS, rhombohedral PbNb₂O₆, which is not ferroelectric; ruling out piezoelectric properties and piezoelectric applications. So, most R & D efforts have been concentrated on orthorhombic PbNb₂O₆ [1–5, 12–21]. Even the real part () and imaginary part () of the dielectric constant of the nonferroelectric PbNb₂O₆ with rhombohedral phase have rarely been reported [16, 22]. No analysis of its dielectric data has yet been made to study paraelectric and other physical properties in detail, although rhombohedral PbNb₂O₆ is a dielectric material in its own right. So, an analysis of its dielectric data [23] has been carried out in this work, after preparing rhombohedral PbNb₂O₆ in practically pure state, with X-ray Rietveld confirmation of the structure and impedance spectroscopy (IS) covering 20 Hz to 5.5 MHz at different values the sample temperature () up to 700°C.

2. Materials and Methods

Pellets from starting chemicals of PbO and Nb₂O₅ with 2% extra by weight of PbO (to compensate for presumed Pb loss during firing) have been calcined first for 3.5 h at 1050°C, then at 1290°C for 1 h, and finally at 1270°C for about 5 h. At the end of last firing, the samples have been slow-cooled at an average rate of 1.5°C per minute, to get the rhombohedral form. The samples have been characterized by XRD with Rietveld analysis [16]. Firing in compact pellet form and grinding plus repelletizing before each firing improved sample quality.

The impedance spectroscopy has been carried out using Solatron “SI 1260 Impedance Gain Phase Analyzer” with a high temperature attachment. Our slow-cooled PNS pellets (diameter ~12 mm and 2.2 to 2.4 mm thick) have not been poled before measurement. Silver paint put on the flat faces of the pellet under study has acted as the electrodes for IS measurements, using 100 mV AC excitation. This measurement of and (the real and imaginary components, resp., of the relative dielectric constant, i.e., permittivity) has been done for different measuring frequencies covering from 20 Hz to 5.5 MHz range at different values the sample temperature () while heating at the rate of 2°C per minute from room temperature to 700°C and also during many of the cooling runs.

3. Results and Discussion

X-ray Rietveld analysis shown in Figure 1 confirmed the rhombohedral PN structure with detected but unidentified impurity phase or phases, at a concentration of the order of 1% only. Impurity content can definitely be concluded to be lower than 5%. Due to factors like the unaccounted contribution of the unidentified impurities, the fit gives residue factors % and %, although it looks fairly good. Since this paper is finding the structure only to confirm the formation of rhombohedral PN and not trying to improve the well-established [22] structure of rhombohedral PN, somewhat poorer quality of fit can be tolerated. Still, this fit does show the correct space group, R3 m (hexagonal), no. 160, with lattice parameters  Å and  Å. From the cell volume defined by these lattice parameters and formula-guided mass associated with the cell, density has been calculated to be about 83.8% of the theoretical density on average. This lower than 100% density can affect the measurement of dielectric properties only to the extent of showing a lower value of dielectric constant.

Figure 1: XRD pattern of the slow-cooled PbNb₂O₆ sample (as symbols), Rietveld-refined fit (red line), and the difference. The starting parameters were from [22]. More in the text.

Figure 2 shows temperature () dependence of the real part of the dielectric constant () during heating from room temperature to 700°C, at various measurement frequencies (5.5 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 500 kHz, 50 kHz, 30 kHz, 10 kHz, 5 kHz, and 1 kHz) for the rhombohedral phase. Two insets offer enlarged view in the temperature range from 400°C to 700°C. There appear to be two broad but weak peaks (at 479°C to 509°C) and (at 571°C to 575°C) for different lower frequency (5 kHz to 50 kHz) results and only one broad but weak peak (at 572°C to 577°C) for higher frequencies (500 kHz to 5.5 MHz). Even a third peak appears at the lowest measuring frequency of 1 kHz. This peak (at ~684°C) is rather sharp. This matches, with respect to temperature, the ferroelectric to paraelectric transition [16] of orthorhombic PbNb₂O₆, confirming its presence as an impurity of much lower value of (height at peak2), compared to the value of [16] of the ferroelectric PbNb₂O₆, as detailed in Table 1, implying presence in low concentration. While the origin of other peaks or even of the peak is not very clear, their absence (Figure 4) in cooling runs confirms their origin to low concentration unstable phases or defect conditions that get annealed by the 700°C cycling. Further discussion can found in the section on Cooling Data.

Table 1: Values, for paraelectric PbNb₂O₆ (rhombohedral PbNb₂O₆), of and at room temperature and values of , , and with the corresponding temperatures , , and , as read from the experimental data during heating.

Frequency (Hz)

(room temperature)

(room temperature)

(°C)

(°C)

(°C)

1 k
64.3
0.044
253.4

495.8

711.4

5 k
64.0
0.013
204.0

281.0

10 k
63.9
0.005
186.5

238.8

30 k
63.8
0.003
165.6

195.9

50 k
63.8
0.006
160.9

182.0

500 k
64.1
0.051

131.3

1 M
64.9
0.106

117.8

2 M
65.1
−0.004

115.9

3 M
68.0
−0.004

115.2

4 M
71.7
0.002

118.7

5 M
75.3
−0.005

124.4

5.5 M
77.6
−0.026

128.5

Figure 2: Temperature () dependence of the real part of the dielectric constant () during heating runs at various frequencies (5.5 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 500 kHz, 50 kHz, 30 kHz, 10 kHz, 5 kHz, and 1 kHz) for the rhombohedral phase of PbNb₂O₆. Inset enlarges 400°C to 700°C view of the versus graph. After 700°C cycling in Figure 4, presently observed low intensity peaks, marked in the present figure, got removed.

Figure 3 shows temperature dependence of the imaginary part of the dielectric constant (), during heating from room temperature to 700°C for different measuring frequencies in the range 1 kHz to 5.5 MHz for PNS. For 1 kHz graph, increases sharply in the temperature range of 450°C to 650°C and then decreases with the increase of temperature, in our study being up to 700°C. Inset shows to be practically constant with temperature up to 450°C at high frequencies (1 MHz to 5.5 MHz). But for low frequencies (1 kHz to 500 kHz) and temperatures above 300°C, increases significantly with temperature. Cooling data (Figure 4), taken after cycling to 700°C, shows the temperature () dependence of the real part of dielectric constant () as Figure 4(a). Inset Figures 4(b) and 4(c) show the temperature dependence of imaginary parts of dielectric constant () and temperature dependence of the dielectric loss () at various measurement frequencies (5.5 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 500 kHz, 50 kHz, 30 kHz, 10 kHz, 5 kHz, and 1 kHz) during cooling (about 2°C/min) from 700°C to 410°C. Here, scale of plotting is grossly different from that for earlier heating data. We observe negative values of and at high frequencies (5.5 MHz, 5 MHz, 4 MHz, 3 MHz, and 2 MHz). The values of , , and of PNS sample are much smaller and less affected by temperature in cooling runs than in heating runs, as has been shown in Table 2, as well. The nonconclusive jitter in data for 1 kHz may be due to the enlarged view of the signal, and hence not significant. It is clear that the 700°C cycling has removed the unstable phases or conditions (Figure 2) and provided a clearer view of the rhombohedral phase. Figure 5 shows frequency dependence of the real part () of dielectric constant and (inset view) of the PNS sample. In these heating runs at different temperatures (Figure 5), both quantities ( and tan) decrease with the increase of the measuring frequency.

Table 2: Real part of dielectric constant () and dielectric loss () of PbNb₂O6 in rhombohedral phase, PNS sample, for different frequencies at two different temperatures for heating and cooling graphs. Here, and indicate the real part of dielectric constant at temperature 410°C and 575°C, respectively, and and indicate the dielectric loss values at 410°C and 575°C, respectively.

Frequency (Hz)
Heating run
Cooling run

1 k
171.3
2.839
483.0
120.0
13.89
−0.003
16.80
0.215

5 k
112.1
1.123
278.7
41.80
14.41
0.024
15.74
0.074

10 k
97.7
0.745
236.8
24.71
14.17
−0.028
15.61
0.043

30 k
83.6
0.394
195.5
10.11
14.20
−0.003
15.07
0.036

50 k
79.6
0.287
181.3
6.616
14.26
0.006
15.04
0.038

500 k
70.5
0.125
131.2
1.181
14.24
0.047
14.32
0.072

1 M
70.1
0.157
117.5
0.862
14.32
0.103
14.36
0.114

2 M
70.0
0.036
115.6
0.477
13.72
−0.029
13.66
−0.025

3 M
72.9
0.021
115.3
0.383
14.34
−0.105
14.35
−0.102

4 M
76.9
0.024
118.8
0.342
14.36
−0.117
14.35
−0.116

5 M
80.8
0.012
124.3
0.313
12.64
−0.276
12.65
−0.270

5.5 M
83.4
−0.016
128.4
0.290
11.68
−0.562
11.52
−0.558

Figure 3: Temperature dependence of the imaginary part of the dielectric constant () during heating runs for different measuring frequencies in the range 1 kHz to 5.5 MHz for the paraelectric PbNb₂O₆. The inset shows details.

Figure 4: Cooling data for the rhombohedral phase of PbNb₂O₆, from 700°C to 410°C: (a) temperature dependence of the real part of dielectric constant (), inset (b) temperature dependence of the imaginary part of dielectric constant (), and (c) temperature dependence of the dielectric loss (), at various frequencies (1 kHz to 5.5 MHz).

Figure 5: Frequency dependence of the real part () of dielectric constant in the frequency range 1 kHz to 5.5 MHz, compiled from heating runs at different temperatures, for the rhombohedral phase of PbNb₂O₆. The inset is the variation of in the same frequency and temperature range.

In this section, proper analysis the dielectric data of rhombohedral PbNb₂O₆ (PNS samples) has been carried out to calculate the values of temperature has dependent relaxation time (), bulk resistance (), and capacitance () using Nyquist diagram/Cole-Cole plot as well as frequency dependent AC conductivity () and frequency independent DC conductivity () for different temperatures. We also calculate the value of corresponding activation energy () for different temperature ranges.

The plot of (imaginary part of the impedance) versus (real part of the impedance) is called Nyquist diagram [23]. It is based on the well-known Cole-Cole plot of versus . The versus plot for our rhombohedral PNS sample is shown in Figure 6. The data points have been joined by a standard polynomial fit, indicated by open circles in Figure 6. Single semicircular nature of the plots is clearer in the higher temperature plots (for 450°C to 700°C), as the lower temperature plots, like that for 400°C in the insert of Figure 6, show large semicircular arcs. An important observation is that the Nyquist diagram here shows only one semicircular arc [9, 23] in higher temperature region over the studied frequency range 20 Hz to 5.5 MHz. One semicircle implies only one contribution (here from the sample grains) and not from any second source like grain boundaries or electrode effect). A sample with one semicircle Nyquist diagram or Cole-Cole plot is equivalent [9, 23] to a bulk resistance and a capacitance in parallel. These elements give rise to a time constant , the dielectric relaxation time of the basic material. The frequency of the peak of the Nyquist diagram can find the relaxation time as for this peak point [9, 23]. At this point, has its maximum value. So, many authors use the frequency of peak (i.e., maximum) of versus frequency plot to find the relaxation time . Often the centre of the semicircular arc of the Nyquist diagram is not on the -axis, so that the semicircle is seen to be depressed by an angle () below the -axis. This is the case, if there is a distribution of relaxation times. This is determined by the width of the distribution of relaxation time. Dielectric relaxation time ( in μs) is a temperature-dependent parameter and it decreases with temperature, as has been shown in Figure 10.

Figure 6: Plot of the imaginary part, in kΩ, of impedance versus its real part, in kΩ, for the paraelectric PbNb₂O₆ with rhombohedral phase, at temperatures (a) 400°C and 450°C (inset) (b) 560°C, 565°C, 580°C, and 590°C, and (c) 500°C, 550°C, 570°C, 600°C, 650°C, and 700°C (inset), since the complex impedance , quantity () has actually been plotted in the figure.

If the semicircle starts near the origin at and intercepts the -axis again at , gives [9, 23] the bulk resistance. Fitting the impedance spectroscopy ( versus ) data, usually forming an arch, to the equation of a circle, its radius and and hence can be determined. Bulk resistance has thus been calculated from our data in Figure 6 and plotted in Figure 7. It is found that the semicircle of data points for a particular temperature in Figure 6 reduces in size with increase of temperature, implying smaller bulk resistance of the sample at higher temperatures as documented in Figure 7 (left side, indicated by black solid sphere). Using the relation , we calculate the value of capacitance ( in nF) and plot in Figure 7 (right side, indicated by blue solid stars). In Figure 7, it is found that capacitance does not change much with temperature in region 500°C to 700°C except at the temperature ~550°C that gives a sharp peak.

Figure 7: Bulk resistance in (in left side, black solid sphere) and capacitance in nF (in right side, blue solid star) of the equivalent circuit of slow cooled PbNb₂O₆ with rhombohedral phase, plotted against temperature, . Here, and have been calculated from Nyquist diagram of and in 20 Hz to 5.5 MHz range and the corresponding time constant , related to the dielectric relaxation time.

AC conductivity () has been calculated from the dielectric loss () data using the empirical formula: [9, 24], where  F·m⁻¹ is the vacuum permittivity and is the speed of light in vacuum. Figure 8(a) presents this result on AC conductivity (as , in Ω⁻¹·m⁻¹) as a function of 1000/T (in Kelvin inverse, K⁻¹), inset Figure 8(b) shows versus in the temperature range 373 to 973 K (100 to 700°C), and Figure 8(c) shows versus in the temperature range 100 to 450°C. AC activation energy ( in eV) has been calculated using the empirical Arrhenius relation , where is the preexponential factor, is the Boltzmann constant (= 8.617343 × 10−5 eV·K⁻¹), and is the temperature of the sample in Kelvin,and has listed the values of frequency-dependent in different temperature regions in Table 3, and variation of has been shown in Figure 9. Figure 9 shows the activation energy () in frequency range 20 Hz to 50 kHz for different temperature ranges.

Table 3: Calculated activation energy using the empirical formula exp(/) = = , where is the activation energy, is the  eV·, and is the F·. Values of and for different measuring frequencies and temperatures ranges have been given in (a) 50 to 520°C and (b) 540 to 700°C.

(a)  

Frequency (Hz)
Temperature range

(50 to 380°C)
(380 to 420°C)
(420 to 520°C)

(meV)
  Ω·m
(meV)
  Ω·m
(meV)
  Ω·m

1 k
—
—
766.7
119.8
1910.4
2223.8

5 k
—
—
715.1
64.9
1795.8
401.2

10 k
—
—
704.1
62.3
1727.0
144.7

30 k
—
—
644.2
29.2
1597.7
21.8

50 k
101.61
2.29
610.2
19.6
1519.4
7.26

500 k
17.96
15.37
272.5
0.239
1094.2
0.023

1 M
9.07
49.02
122.4
0.046
872.6
0.001

2 M
75.45
25.22
503.0
13.9
1188.5
0.145

3 M
—
—
1196.2
1.85 M
1276.0
0.642

4 M
—
—
570.4
63.3
1153.8
0.124

(b)  

Frequency (Hz)
Temperature range

(540 to 565°C)
(565 to 650°C)
(675 to 700°C)

(meV)
  Ω·m
(meV)
  Ω·m
(meV)
  Ω·m

1 k
829.6
303.6
324.0
0.273
−915.4
59.7

5 k
830.6
309.4
322.9
0.270
−904.2
68.9

10 k
830.6
310.5
320.9
0.264
−897.6
74.9

30 k
824.3
319.3
316.1
0.250
−889.3
83.6

50 k
813.0
248.3
311.6
0.238
−878.3
96.2

500 k
610.3
19.9
224.3
0.093
−795.3
308.5

1 M
525.4
7.84
171.2
0.059
−741.0
736.4

2 M
554.2
12.8
161.3
0.056
−839.9
233.6

3 M
561.8
17.0
145.7
0.054
−885.5
156.6

4 M
561.0
20.5
131.8
0.055
−890.9
176.2

5 M
588.0
35.6
123.3
0.059
−916.2
152.7

Figure 8: Frequency-dependent (a) AC conductivity (, in Ω⁻¹·m⁻¹) as a function of (1000/T, in K⁻¹) over the temperature range 373 to 973 K (100 to 700°C) and insets (b) and (c) showing frequency-dependent AC conductivity as a function of , in °C, in different temperature ranges for the rhombohedral PbNb₂O₆.

Figure 9: Frequency dependence of activation energy () in different temperature ranges (50–380°C, 380–420°C, 420–520°C, 540–565°C, 565–655°C, and 675–700°C). Value of calculated using the Arrhenius relation , where is the pre-exponential factor, is the Boltzmann constant that equals 8.617343 × 10⁻⁵ eV·K⁻¹ and is the temperature in Kelvin.

Figure 10: Temperature dependent DC conductivity () of the PbNb₂O₆ with rhombohedral phase, in the temperature range 400 to 700°C. Inset: temperature dependent relaxation time (, in μs) of the same sample in the same temperature range.

The frequency-independent and temperature-dependent DC conductivity () have been calculated using empirical formula [6, 25] for slow-cooled PN sample (paraelectric PbNb₂O₆) with rhombohedral structure, as shown in Figure 10. Here, we observe that the DC conductivity of the slow-cooled PN sample increases with temperature up to ~580°C, then remains practically constant up to 600°C, and then increases again giving a wide peak at 650°C. Data points have been joined by a continuous line through a polynomial fit of the equation , where  Ω⁻¹·m⁻¹.°C⁻¹, and  Ω⁻¹·m⁻¹·°C⁻². Here, (relative permittivity in vacuum) = 8.854187 × 10⁻¹² F·m⁻¹, and is the real part of dielectric constant for high frequency. Here, is the time constant of the equivalent circuit (also called relaxation time), and this has been calculated using Nyquist diagram [9, 23] as detailed previously. Value of DC activation energy (), due to DC conductivity [9], has been calculated using the Arrhenius relation [26], where is the preexponential factor, is the Boltzmann constant that equals 8.617343 × 10−5 eV·K⁻¹, and is the temperature in Kelvin. The values of and have been summarized in Table 4.

Table 4: Calculated activation energy in different temperature ranges, using the empirical formula = exp/) = /, where is the activation energy, =  eV· (Boltzmann constant), (relative permittivity in vacuum) = F·, is the real part of dielectric constant for high frequency and is the time constant of the CR equivalent circuit, also called the relaxation time constant.

Temperature range (°C)
(eV)
  Ω·m

450–530
2.172

530–570
0.813

570–600
0.124

650–700
−0.578

Comparing the AC and DC conductivity data of PNS sample, we observe that (for frequencies from 1 MHz to 1 kHz) and (for frequencies 1 MHz to 5.5 MHz). Intriguing negative resistance can be noticed in Figures 8(b) and 8(c), although its confirmation by more experiments is desired. Negative resistance is, however, known in electronic devices and certain materials [27–30], and the granular ceramic PNS sample is a good candidate for such effects at high frequencies. This has indeed been observed here only at highest frequencies used. We plan to do further studies on this controversial issue.

Activation energy, calculated from DC conductivity, decreases with increase of temperature (Table 4). Negative activation energy for highest temperatures in Table 4 is linked to decrease of with increase of temperature for . However, negative values of DC and AC activation energy have been reported earlier [9, 31] without discussion. Presently observed AC activation energy is negative in the temperature range of 675–700°C, similar to positive and negative values of activation energy on two sides of a peak in [31] in their Figure 11 over 1 kHz to 1 MHz range studied there. This simply implies that thermal activation of carriers in a semiconductor, increases the resulting current with temperature up to a certain temperature. If no more carriers are available, further increase of temperature cannot lead to more carriers. But factors like scattering by increasing lattice vibration and recombination lead to decrease of current and hence of conductivity. Already energetic carriers lose energy in such processes. Application of activation equation to such cases gave negative activation energy.

4. Conclusion

Presently prepared PbNb₂O₆ samples have practically pure rhombohedral phase as revealed by X-ray diffraction. The samples are further characterized by impedance spectroscopy (IS) over 20 Hz to 5.5 MHz and up to 700°C. Cooling data of IS measurement appear to reveal better the real features of this paraelectric material; unexplained features (small broad peak/s in versus graph in Figure 2), appearing in the first heating run in IS measurement on our PNS sample, are conjectured to be related to minor phases at low concentration. 700°C cycling removes unexplained features, and no such unexplained peaks have been observed in the cooling data.

Present analysis of the impedance spectroscopy data on rhombohedral PNS has been very detailed over wide ranges of temperature and frequency, with DC and AC conductivities, capacitance (), and resistance () of the equivalent circuit, relaxation time, and related activation energies evaluated and discussed either more extensively than the literature or for the first time. Here, we are the first to present the Nyquist diagram and the plot of versus , for rhombohedral lead meta-niobate. For the AC and DC conductivity of PNS sample we observe for frequencies in 1 MHz to 1 kHz range and for frequencies in 1 MHz to 5.5 MHz range. Rhombohedral PbNb₂O₆, frequently an undesirable impurity in orthorhombic PbNb₂O₆, is, however, a dielectric in its own right. The present work is its first detailed study from that viewpoint.

This pelletized sample appears to show negative resistance under certain conditions.

Acknowledgments

The impedance spectroscopy setup at the Department of Ceramic Engineering, NIT, Rourkela 769008, India, has been used with the active encouragement of Professor Swadesh K Pratihar. Authors thank Professor Hamdy Doweidar for helpful comments.

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