- Email: [email protected]
2)O3–35 % PbTiO3: temperature and frequency dependences
Materials Science and Engineering B 117 (2005) 265–270
Dielectric behaviors of relaxor ferroelectric Pb(Mg1/2Nb1/2)O3–35 % PbTiO3: temperature and frequency dependences F. Yuana , Z. Pengc , J.-M. Liua,b,c,∗ a
Laboratory of Solid State Microstructures, Nanjing University, Hankou Road No. 22, Nanjing 210093, China b International Center for Materials Physics, Chinese Academy of Sciences, Shenyang, China c Department of Applied Physics, Hong Kong Polytechnic University, Hong Kong, China Received 17 August 2004; accepted 2 December 2004
Abstract The dielectric relaxation behaviors of relaxor ferroelectric Pb(Mg1/2 Nb1/2 )O3 –35 % PbTiO3 (PMN–PT) ceramics is studied by investigating the dielectric permittivity as a function of temperature and signal frequency. The concurrence of thermally activated relaxation at high temperature and resonance relaxation at low temperature in relaxor ferroelectrics is demonstrated. The measured dielectric relaxation behavior as a function of temperature and frequency is discussed based on the hybrid model, indicating that the two mechanisms have significant influence on the dielectric relaxation at high temperature and low temperature, respectively. © 2005 Elsevier B.V. All rights reserved. PACS: 77.22.Ch; 77.22.Gm; 77.84.Dy Keywords: Relaxor ferroelectrics; Dielectric permittivity; Resonance relaxation
1. Introduction Dielectric relaxation in solids represents one of the most intensely researched topics in condensed matter physics. In spite of the very long history of study on this problem and a number of theoretical models developed to explain the relaxation behavior of dielectrics and ferroelectrics, the conventional theoretical understanding has not yet been satisfactory. The present study is restricted to the dielectric relaxation of relaxor ferroelectrics (RFEs). Along this line intensive interest has increased since relaxor ferroelectrics have found a lot of useful properties, which lead to a multitude of technical applications ranging from piezoelectric sensors, actuators or transducers [1,2] to optical applications like phase conjugated mirrors [3,4], and also due to the possibility of investigating the lattice dynamics in partially disordered crystals. Recently, the relaxor ferroelectric behavior in organic materials (copolymer) has also been studied intensively [5,6]. ∗
Corresponding author. Tel.: +86 25 83596595; fax: +86 25 83595535. E-mail address: [email protected] (J.-M. Liu).
0921-5107/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mseb.2004.12.062
It is well known that normal ferroelectric materials show a sharp phase transition at the Curie temperature Tc and there is nearly no frequency dispersion of the dielectric constant up to the GHz region. Three major features set RFEs apart from normal ferroelectrics [7]: a broad maximum of the dielectric permittivity as a function of temperature, strong frequency dispersion of the dielectric constant in the low temperature range, and the existence of polar regions at temperature well above this maximum. The temperature with the maximum dielectric constant is always called Tm , which is no longer a well-defined phase transition temperature. In contrast to ferroelectrics, the diffuse phase transition of the relaxors is characterized by the fact that the macroscopic crystal symmetry does not change above and below Tm . There are various theoretical models (such as local compositional fluctuation model [8], superparaelectric model [7], dipolar glass model [9], etc.) that intend to describe the physical properties of relaxors, but none of them can explain all the properties of the RFE very well. What is clear from previous studies is that nanometer-scale structural inhomogeneity is mainly responsible for the manifestation of relaxor ferro-
266
F. Yuan et al. / Materials Science and Engineering B 117 (2005) 265–270
electricity. Recently, Cheng et al. proposed a model [10] in which there are two distinct microscopic mechanisms to be taken into account in order to describe the dielectric relaxation behaviors around Tm . One part is associated with the thermally activated flips of the polar regions and the other part is ascribed to a resonance phenomenon associated with the breathing of the frozen polar regions in the materials. Both of them were identified to contribute essentially to the relaxation in RFEs. The solid solution ceramics of lead magnesium niobate Pb(Mg1/2 Nb1/2 )O3 (PMN) with lead titanate PbTiO3 (PT) is the most intensively studied system in the family of relaxor ferroelectrics with perovskite structure. It exhibits significant frequency dispersion of the dielectric relaxation as a function of temperature. The present article is intended to show the characteristics of the relationships between dielectric relaxation and temperature and frequency, and to simulate the relations between temperature, frequency and dielectric constant of the ceramics, mainly using the model Cheng et al. presented. This model is validated by PMN with 10% PT [10] while the material in our experiment is PMN with 35 % PT. The differences between the PMN–35 % PT and PMN–10 % PT are that the PMN–35 % PT has a monoclinic phase [11] and also a clean rhombohedral phase in the low temperature [12]. And a higher PT content leads to a reduced band gap [13]. Therefore, it is necessary to examine whether the Cheng et al.’s model is only a special case, or it can describe the characteristics of RFE in a wider range.
Fig. 1. Three-dimensional plotting of the dielectric constant as a function of temperature T (ranging from 20 to 250 ◦ C) and frequency ω (ranging from 100 Hz to 1 MHz).
of temperature T and frequency ω is shown in Fig. 1. In order to make the plotting clearer, we show the measured dielectric constant ε , dielectric absorption ε and loss tangent tan δ = ε /ε versus T in Fig. 2(a)–(c), respectively, in which we randomly plot the curves at ω = 1, 3.162, 10, 31.62, 100 and 316.2 kHz. First of all, let us look at ε (real part) as a function of T measured at different frequencies, as shown in Fig. 2(a) where ε decreases with increasing frequency. The temperature at the maximum dielectric constant (εmax ), Tm , increases with increasing ω. It is seen that the dielectric dispersion at
2. Experimental procedure The material used in the experiment is the solid solution of 65% lead magnesium niobate (PMN) with 35% lead titanate (PT). The bulk PMN–35% PT ceramics were prepared using a conventional mixed-oxide method. The detailed method can be found elsewhere [14]. Disk-like PMN–35% PT samples were coated by silver electrodes on the two major surfaces. The X-ray diffraction checking of the sample structure was performed, confirming that the sample is of perovskite phase with high quality. The complex dielectric permittivity of the samples was measured by a computer-controlled HP4192A system. The temperature of the oven containing the sample can be controlled from 20 to 250 ◦ C, with a heating rate of 0.5 ◦ C/min. The measurement was performed isothermally by a frequency-scanning mode of the HP4192A with an acvoltage of 1.0 V applied to the sample, and the temperature step is 2 ◦ C. The frequencies in the measurement range from 100 Hz to 1 MHz.
3. Results of measurement and discussions Using tens of thousands of data we measured, the threedimensional plotting of the dielectric constant as a function
Fig. 2. Dielectric behavior as a function of temperature at different frequencies. The measuring frequencies are 1, 3.162, 10, 31.62, 100 and 316.2 kHz and the arrow indicates the increasing of frequency: (a) dielectric constant, (b) dielectric absorption and (c) loss tangent.
F. Yuan et al. / Materials Science and Engineering B 117 (2005) 265–270
T < Tm is significant. For the dielectric absorption (imaginary part ε ) as a function of T, a peak is also seen, as shown in Fig. 2(b). ε increases with increasing frequency. The temperature at the maximum dielectric absorption (Tm ) increases with increasing frequency. Given a fixed frequency, the value of Tm is lower than the value of Tm . We also evaluate the dielectric loss (tan δ) as a function of T at different frequencies, as shown in Fig. 2(c). Although there is some fine relationship regarding tan δ as a function of T, generally the loss increases with increasing frequency, and at T Tm (Tm ), the loss becomes very small. All of these demonstrate the typical characteristics of dielectric behavior in RFEs. In order to understand the fundamental mechanisms of above phenomena, we perform an analysis on the data from the point of view of dielectric relaxation. According to the Debye relation [15], which is the basic equation describing the dielectric relaxation and can also be the start of our analysis, the complex dielectric permittivity ε* can be expressed as εs − ε∞ ε∗ = ε∞ + (1) 1 + iωτ(T ) where ε* = ε − iε , ε∞ is the dielectric constant at the ultrahigh frequency and is independent of T and ω, εs is the static dielectric constant and is independent of frequency, and τ is the relaxation time. In Fig. 2(a), it is obviously shown that the dependences of ε on T and ω over T > Tm and T < Tm are different from each other. It indicates that there are two mechanisms in the material and they have different influence on the dielectric relaxation in RFEs at different T. It was experimentally found that the relation between ε and T at T Tm can be expressed as follows [16]: εH (T ) = exp(α − βT )
(2)
where εH stands for the dielectric constant at T Tm , α and β are positive constants, T is the absolute temperature. The fitting gives α = 15.15 and β = 16.61 × 10−3 K−1 . The value of β is associated with the production rate of the polar regions in the material, and it describes the degree of the dielectric relaxation of the RFE [16]. The fitted results and measured results are shown in Fig. 3(a). In the figure, the fitted curve is almost the same as the measured curves and no difference between them can be identified, indicating that the expression above describes very well the relation between T and ε at T Tm . According to the general dielectric relaxation theory, the relaxation time of RFEs deceases with increasing temperature [17]. Therefore, in the high temperature range, the static dielectric constant εs almost equals the measured dielectric constant and can also be described by Eq. (2). To understand the relaxation characteristics from Eq. (1), knowing the expression of τ(T) is necessary. Because the volume of the polar region has its upper limit and lower limit, the relaxation time also has its own upper limit (τ 1 ) and lower limit (τ 0 ). Assume that the lower limit of the relaxation time is τ 0 = ω0−1 (ω0 is the Debye frequency independent of T), so
267
Fig. 3. (a) Dielectric constant (ω = 1, 3.162, 10, 31.62, 100 and 316.2 kHz) at T Tm and the fitted curve. (b) The value of b(ω) and −c(ω) as a function of frequency, respectively.
the distribution of the relaxation time is [18]: 0 τ < τ0 , τ > τ1 f (τ) = 1 τ0 ≤ τ ≤ τ1 τ1 −τ0
(3)
Using Eqs. (1)–(3), and neglecting ε∞ (much smaller than the measured dielectric constant), we obtain [18]: τ12 1 + ω2 τ02 εs (4) ln 2 ε = 2 ln(τ1 /τ0 ) τ0 1 + ω2 τ12 ε =
εs [arctg(ωτ1 ) − arctg(ωτ0 )] ln(τ1 /τ0 )
(5)
In the high temperature range, Eq. (4) becomes ε = εs , and Eq. (5) leads to the conclusion that ε = 0. In the low temperature range, Eqs. (4) and (5) can be simplified as ω εs 0 (6) ln ε = ln(τ1 /τ0 ) ω ε =
π εs 2 ln(τ1 /τ0 )
(7)
So ε in the low temperature range depends on both T and ω, and ε is independent of frequency. In Fig. 2, the characteristics of the dielectric behavior of the material coincide with the above results. From Eq. (6), we get expression that describes the relation between T, ω and dielectric constant in the low temperature range: εL (ω, T ) = A(T )(ln ω0 − ln ω)
(8)
268
F. Yuan et al. / Materials Science and Engineering B 117 (2005) 265–270
and
Table 1 The fitted values of b and c at different frequencies
l T A(T ) = exp γ + δ
(9)
where εL stands for the dielectric constant at T Tm , γ, δ and l are positive constants, and l is associated with the frozen rate of polar regions in the material [10]. Fitting the measured data, we obtain ω0 = 2.9 × 1011 Hz, γ = 3.15, δ = 149.9 K and l = 1.972. It is noted that Eqs. (2) and (8) are completely different, indicating that there are indeed two different polarization mechanisms in the material. Then the dielectric constant can be expressed as the sum of two dielectric constants which are derived from the two different polarization mechanisms: εm (ω, T ) = ε1 (ω, T ) + ε2 (ω, T )
(10)
εm
stands for the measured dielectric constant in the where experiment, and ε1 , ε2 is the dielectric constant derived from the two mechanisms respectively. ε1 and ε2 become Eq. (2) and Eq. (8) at the extreme conditions (T Tm and T Tm ). Therefore, ε1 ε2 , ε1 ≈ εm at T Tm and ε1 ε2 , ε2 ≈ εm at T Tm . From the above analysis, we can describe ε1 , ε2 as [10]: ε1 (ω, T ) =
εH (ω, T ) 1 + B(ω, T )
(11)
ε2 (ω, T ) =
εL (ω, T ) 1 + C(ω, T )
(12)
As discussed above, B(ω, T) is very small in the high temperature range and increases with decreasing temperature. C(ω, T) is very large in the low temperature range and increases with increasing temperature. Following Cheng et al. [10], we assume
m εH (T ) B(ω, T ) = b (13) A(T ) ln ω0
n εH (T ) C(ω, T ) = c (14) A(T ) ln ω0 where b, c are independent of temperature, but change with frequency, and m, n are constants. It is easy to find that B(ω, T), C(ω, T) satisfy the above conditions when m > 0 and n < 0. Because b and c are variable, it is difficult to determine all parameters in Eqs. (10)–(14). To solve this problem, we take a temperature where B(ω, T) and C(ω, T) are independent of m and n. At this temperature, we can easily fit the data of dielectric constant and frequency. The fitted results of b(ω) and c(ω) at various frequencies are listed in Table 1 and also shown in Fig. 3(b). One sees that b(ω) and c(ω) are strongly dependent on frequency. Both b(ω) and c(ω) increase with increasing frequency. With the fitted values of b(ω) and c(ω), we use Eq. (10) to fit m and n. The fitted values are m = 3.166 and n = −2.622. Consequently, we obtain all of the parameters
Frequency (Hz)
b
c
316.2 1000 1738 3162 10000 17380 31620 100000 173800 316200
0.6659 0.9937 1.1781 1.4863 1.9550 2.2274 2.6681 3.3142 3.6117 4.4866
−5.7121 −3.5212 −2.8818 −2.1694 −1.7711 −1.6106 −1.3785 −1.2521 −1.2249 −1.0215
we set, from which we can make further discussion about the dielectric relaxation behavior of PMN–35% PT, used for our experiment. Firstly, we take a look at the relation between temperature, frequency and the dielectric constant contributed from the first mechanism. Fig. 4(a) is drawn by Eqs. (11) and (13) with the parameters derived out before. We find that there are peaks in the curves and the maxima of ε1 increase with decreasing frequencies. The temperature corresponding to the maximum of ε1 (Tm1 ) increases with increasing frequency. At T Tm1 , the values of ε1 are independent of frequency. At temperature around and lower than Tm1 , there is a strong dispersion and ε1 decreases rapidly with decreasing temperature. In RFEs, the density of the polar regions, which determine the behaviors of RFEs, increases with decreasing temperature. At high temperature, almost all of the polar regions
Fig. 4. Temperature dependence of the fitted dielectric constant contributed from the two different mechanisms at frequencies 3.162, 10, 31.62, 100 and 316.2 kHz. (a) Temperature dependence of ε1 (associated with the thermally activated relaxation process). (b) Temperature dependence of ε2 (associated with the resonance relaxation process).
F. Yuan et al. / Materials Science and Engineering B 117 (2005) 265–270
are thermally activated. Therefore, the dielectric constant increases with decreasing temperature as the density of polar regions increases. However, the relaxation time of the polar regions increases with decreasing temperature [18], so the interaction between the polar regions also increases. This leads some of the polar regions not to change their orientations under the external field. We call these polar regions the frozen ones. With the temperature continuing to decrease, the density of unfrozen polar regions decreases. That is why at low temperature the value of ε1 decreases with decreasing temperature. All of these features indicate that the first mechanism is attributed to the relaxation polarization process, which is associated with the thermally activated flips of polar regions in relaxor ferroelectrics. Secondly, we can also obtain the relation between temperature, frequency and ε2 with the parameters we derived out earlier. The results are shown in Fig. 4(b). We find that there are peaks too, and the maxima of ε2 increase with decreasing frequencies. The temperature at the maximum of ε2 (Tm2 ) increases with increasing frequencies. At T Tm2 , the value of ε2 decreases rapidly with increasing temperature. At the whole temperature range, there is a strong dispersion and ε2 increases with increasing frequency in the high temperature range and decreases with increasing frequency in the low temperature range. It is obvious that not all of these characteristics can be explained only by the relaxation polarization process. Although the external field does not change the orientations of polar regions, it can change the sizes and surfaces of the polar regions [10]. With variable external conditions, the frozen polar regions change as a breathing mode [19]. The breathing of the polar regions leads to change of the dipole moments, which determine the behaviors of the dielectrics. This is the second polarization process in the material of PMN–35% PT. As we discussed above, the density of the frozen polar regions increases with decreasing temperature. Therefore, the dielectric constant increases with decreasing temperature. But at the extreme low temperatures, the increasing frozen strength makes it is difficult to change the sizes and surfaces of the polar regions. That is why ε2 decreases at T Tm . In fact, the breathing of the frozen polar regions is the oscillation of the ions around its equilibrium position in the materials. It is known that the dielectric response of an electric particle oscillating around an equilibrium position is a resonance [19]. Therefore, the dielectric behavior of ε2 resembles the dielectric response of a resonance polarization. To confirm this model, we compare the value of the measured dielectric constant and the fitted dielectric constant. The results are shown in Fig. 5(a) and (b). The measured data and the fitting data at the different frequencies are almost the same curve. This allows us to argue that the model proposed by Cheng et al. [10] can fit the measured data well and the argument of two types of relaxation mechanisms contributing to the dielectric relaxation in PMN–35% PT is also physically reasonable.
269
Fig. 5. Measured dielectric constant (εm ) and the fitted results (ε1 , ε2 and εf = ε1 + ε2 ) at frequencies (a) 1 kHz and (b) 31.62 kHz. The values of εm at temperatures lower than 20 ◦ C are absent because of limitation of the experiment.
4. Conclusion In conclusion, the dielectric relaxation behaviors of PMN–35% PT ceramic relaxor have been measured and we use the hybrid model proposed by Cheng et al. [10] to explain the measured results. It has been confirmed by the excellent consistence between the measured data and the hybrid model of Cheng et al. that for PMN–35% PT the dielectric relaxation is ascribed to the concurrence of the thermally activated dielectric relaxation mechanism (Debye-type) and the resonance polarization process resulting from the breathing of the frozen polar regions of the RFE. At low temperature, the resonance polarization process may dominate the dielectric relaxation, while at high temperature, the thermally activated mechanism is more significant.
Acknowledgements The authors acknowledge the financial support from the Natural Science Foundation of China (innovative group project 10021001 and project 50332020), the National Key Projects for Basic Research of China (2002CB613303, 2004CB619000). References [1] K. Uchino, Piezoelectric Actuators and Ultrasonic Motors, Kluwer Academic, Boston, 1996. [2] K. Uchino, Ferroelectrics 151 (1994) 321. [3] M.J. Miller, E.J. Sharp, G.L. Wood, W.W. Clark III, G.J. Salamo, R.R. Neurgaonkar, Opt. Lett. 12 (1987) 340.
270
F. Yuan et al. / Materials Science and Engineering B 117 (2005) 265–270
[4] X. Yue, Z. Shao, J. Chen, X. Lu, D. Sun, Q. Jiang, Y. Song, H. Chen, Opt. Lett. 17 (1992) 142. [5] V. Bobnar, A. Levstik, C. Huang, Q.M. Zhang, Phys. Rev. Lett. 92 (2004) 047604. [6] S. Guo, X.-Z. Zhao, Q. Zhou, H.L.W. Chan, C.L. Choy, Appl. Phys. Lett. 84 (2004) 3349. [7] L.E. Cross, Ferroelectrics 76 (1987) 241. [8] G.A. Smolensky, J. Phys. Soc. Jpn. 28 (1970) 26. [9] D. Viehland, S.J. Jang, L.E. Cross, M. Wutting, J. Appl. Phys. 68 (1990) 2916. [10] Z.-Y. Cheng, R.S. Katiyar, X. Yao, A.S. Bhalla, Phys. Rev. B 57 (1998) 8166. [11] J.-M. Kiat, Y. Uesu, B. Dkhil, M. Matsuda, C. Malibert, G. Calvarin, Phys. Rev. B 65 (2002) 064106.
[12] G. Xu, D. Viehland, J.F. Li, P.M. Gehring, G. Shirane, Phys. Rev. B 68 (2003) 212410. [13] K.Y. Chan, W.S. Tsang, C.L. Mak, K.H. Wong, P.M. Hui, Phys. Rev. B 69 (2004) 144111. [14] S.L. Swartz, T.R. Shrout, Mater. Res. Bull. 17 (1982) 1245. [15] A.K. Jonscher, J. Phys. D: Appl. Phys. 32 (1999) 57. [16] Z.-Y. Cheng, L.Y. Zhang, X. Yao, J. Appl. Phys. 79 (1996) 8615. [17] I. Bunget, M. Popescu, Physics of Solid Dielectrics, Elsevier, Amsterdam, 1984. [18] Z.-Y. Cheng, R.S. Katiyar, X. Yao, A. Guo, Phys. Rev. B 55 (1997) 8165. [19] A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectrics Press, London, 1983.
Dielectric behaviors of relaxor ferroelectric Pb(Mg1/2Nb1/2)O3–35 % PbTiO3: temperature and frequency dependences F. Yuana , Z. Pengc , J.-M. Liua,b,c,∗ a
Laboratory of Solid State Microstructures, Nanjing University, Hankou Road No. 22, Nanjing 210093, China b International Center for Materials Physics, Chinese Academy of Sciences, Shenyang, China c Department of Applied Physics, Hong Kong Polytechnic University, Hong Kong, China Received 17 August 2004; accepted 2 December 2004
Abstract The dielectric relaxation behaviors of relaxor ferroelectric Pb(Mg1/2 Nb1/2 )O3 –35 % PbTiO3 (PMN–PT) ceramics is studied by investigating the dielectric permittivity as a function of temperature and signal frequency. The concurrence of thermally activated relaxation at high temperature and resonance relaxation at low temperature in relaxor ferroelectrics is demonstrated. The measured dielectric relaxation behavior as a function of temperature and frequency is discussed based on the hybrid model, indicating that the two mechanisms have significant influence on the dielectric relaxation at high temperature and low temperature, respectively. © 2005 Elsevier B.V. All rights reserved. PACS: 77.22.Ch; 77.22.Gm; 77.84.Dy Keywords: Relaxor ferroelectrics; Dielectric permittivity; Resonance relaxation
1. Introduction Dielectric relaxation in solids represents one of the most intensely researched topics in condensed matter physics. In spite of the very long history of study on this problem and a number of theoretical models developed to explain the relaxation behavior of dielectrics and ferroelectrics, the conventional theoretical understanding has not yet been satisfactory. The present study is restricted to the dielectric relaxation of relaxor ferroelectrics (RFEs). Along this line intensive interest has increased since relaxor ferroelectrics have found a lot of useful properties, which lead to a multitude of technical applications ranging from piezoelectric sensors, actuators or transducers [1,2] to optical applications like phase conjugated mirrors [3,4], and also due to the possibility of investigating the lattice dynamics in partially disordered crystals. Recently, the relaxor ferroelectric behavior in organic materials (copolymer) has also been studied intensively [5,6]. ∗
Corresponding author. Tel.: +86 25 83596595; fax: +86 25 83595535. E-mail address: [email protected] (J.-M. Liu).
0921-5107/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mseb.2004.12.062
It is well known that normal ferroelectric materials show a sharp phase transition at the Curie temperature Tc and there is nearly no frequency dispersion of the dielectric constant up to the GHz region. Three major features set RFEs apart from normal ferroelectrics [7]: a broad maximum of the dielectric permittivity as a function of temperature, strong frequency dispersion of the dielectric constant in the low temperature range, and the existence of polar regions at temperature well above this maximum. The temperature with the maximum dielectric constant is always called Tm , which is no longer a well-defined phase transition temperature. In contrast to ferroelectrics, the diffuse phase transition of the relaxors is characterized by the fact that the macroscopic crystal symmetry does not change above and below Tm . There are various theoretical models (such as local compositional fluctuation model [8], superparaelectric model [7], dipolar glass model [9], etc.) that intend to describe the physical properties of relaxors, but none of them can explain all the properties of the RFE very well. What is clear from previous studies is that nanometer-scale structural inhomogeneity is mainly responsible for the manifestation of relaxor ferro-
266
F. Yuan et al. / Materials Science and Engineering B 117 (2005) 265–270
electricity. Recently, Cheng et al. proposed a model [10] in which there are two distinct microscopic mechanisms to be taken into account in order to describe the dielectric relaxation behaviors around Tm . One part is associated with the thermally activated flips of the polar regions and the other part is ascribed to a resonance phenomenon associated with the breathing of the frozen polar regions in the materials. Both of them were identified to contribute essentially to the relaxation in RFEs. The solid solution ceramics of lead magnesium niobate Pb(Mg1/2 Nb1/2 )O3 (PMN) with lead titanate PbTiO3 (PT) is the most intensively studied system in the family of relaxor ferroelectrics with perovskite structure. It exhibits significant frequency dispersion of the dielectric relaxation as a function of temperature. The present article is intended to show the characteristics of the relationships between dielectric relaxation and temperature and frequency, and to simulate the relations between temperature, frequency and dielectric constant of the ceramics, mainly using the model Cheng et al. presented. This model is validated by PMN with 10% PT [10] while the material in our experiment is PMN with 35 % PT. The differences between the PMN–35 % PT and PMN–10 % PT are that the PMN–35 % PT has a monoclinic phase [11] and also a clean rhombohedral phase in the low temperature [12]. And a higher PT content leads to a reduced band gap [13]. Therefore, it is necessary to examine whether the Cheng et al.’s model is only a special case, or it can describe the characteristics of RFE in a wider range.
Fig. 1. Three-dimensional plotting of the dielectric constant as a function of temperature T (ranging from 20 to 250 ◦ C) and frequency ω (ranging from 100 Hz to 1 MHz).
of temperature T and frequency ω is shown in Fig. 1. In order to make the plotting clearer, we show the measured dielectric constant ε , dielectric absorption ε and loss tangent tan δ = ε /ε versus T in Fig. 2(a)–(c), respectively, in which we randomly plot the curves at ω = 1, 3.162, 10, 31.62, 100 and 316.2 kHz. First of all, let us look at ε (real part) as a function of T measured at different frequencies, as shown in Fig. 2(a) where ε decreases with increasing frequency. The temperature at the maximum dielectric constant (εmax ), Tm , increases with increasing ω. It is seen that the dielectric dispersion at
2. Experimental procedure The material used in the experiment is the solid solution of 65% lead magnesium niobate (PMN) with 35% lead titanate (PT). The bulk PMN–35% PT ceramics were prepared using a conventional mixed-oxide method. The detailed method can be found elsewhere [14]. Disk-like PMN–35% PT samples were coated by silver electrodes on the two major surfaces. The X-ray diffraction checking of the sample structure was performed, confirming that the sample is of perovskite phase with high quality. The complex dielectric permittivity of the samples was measured by a computer-controlled HP4192A system. The temperature of the oven containing the sample can be controlled from 20 to 250 ◦ C, with a heating rate of 0.5 ◦ C/min. The measurement was performed isothermally by a frequency-scanning mode of the HP4192A with an acvoltage of 1.0 V applied to the sample, and the temperature step is 2 ◦ C. The frequencies in the measurement range from 100 Hz to 1 MHz.
3. Results of measurement and discussions Using tens of thousands of data we measured, the threedimensional plotting of the dielectric constant as a function
Fig. 2. Dielectric behavior as a function of temperature at different frequencies. The measuring frequencies are 1, 3.162, 10, 31.62, 100 and 316.2 kHz and the arrow indicates the increasing of frequency: (a) dielectric constant, (b) dielectric absorption and (c) loss tangent.
F. Yuan et al. / Materials Science and Engineering B 117 (2005) 265–270
T < Tm is significant. For the dielectric absorption (imaginary part ε ) as a function of T, a peak is also seen, as shown in Fig. 2(b). ε increases with increasing frequency. The temperature at the maximum dielectric absorption (Tm ) increases with increasing frequency. Given a fixed frequency, the value of Tm is lower than the value of Tm . We also evaluate the dielectric loss (tan δ) as a function of T at different frequencies, as shown in Fig. 2(c). Although there is some fine relationship regarding tan δ as a function of T, generally the loss increases with increasing frequency, and at T Tm (Tm ), the loss becomes very small. All of these demonstrate the typical characteristics of dielectric behavior in RFEs. In order to understand the fundamental mechanisms of above phenomena, we perform an analysis on the data from the point of view of dielectric relaxation. According to the Debye relation [15], which is the basic equation describing the dielectric relaxation and can also be the start of our analysis, the complex dielectric permittivity ε* can be expressed as εs − ε∞ ε∗ = ε∞ + (1) 1 + iωτ(T ) where ε* = ε − iε , ε∞ is the dielectric constant at the ultrahigh frequency and is independent of T and ω, εs is the static dielectric constant and is independent of frequency, and τ is the relaxation time. In Fig. 2(a), it is obviously shown that the dependences of ε on T and ω over T > Tm and T < Tm are different from each other. It indicates that there are two mechanisms in the material and they have different influence on the dielectric relaxation in RFEs at different T. It was experimentally found that the relation between ε and T at T Tm can be expressed as follows [16]: εH (T ) = exp(α − βT )
(2)
where εH stands for the dielectric constant at T Tm , α and β are positive constants, T is the absolute temperature. The fitting gives α = 15.15 and β = 16.61 × 10−3 K−1 . The value of β is associated with the production rate of the polar regions in the material, and it describes the degree of the dielectric relaxation of the RFE [16]. The fitted results and measured results are shown in Fig. 3(a). In the figure, the fitted curve is almost the same as the measured curves and no difference between them can be identified, indicating that the expression above describes very well the relation between T and ε at T Tm . According to the general dielectric relaxation theory, the relaxation time of RFEs deceases with increasing temperature [17]. Therefore, in the high temperature range, the static dielectric constant εs almost equals the measured dielectric constant and can also be described by Eq. (2). To understand the relaxation characteristics from Eq. (1), knowing the expression of τ(T) is necessary. Because the volume of the polar region has its upper limit and lower limit, the relaxation time also has its own upper limit (τ 1 ) and lower limit (τ 0 ). Assume that the lower limit of the relaxation time is τ 0 = ω0−1 (ω0 is the Debye frequency independent of T), so
267
Fig. 3. (a) Dielectric constant (ω = 1, 3.162, 10, 31.62, 100 and 316.2 kHz) at T Tm and the fitted curve. (b) The value of b(ω) and −c(ω) as a function of frequency, respectively.
the distribution of the relaxation time is [18]: 0 τ < τ0 , τ > τ1 f (τ) = 1 τ0 ≤ τ ≤ τ1 τ1 −τ0
(3)
Using Eqs. (1)–(3), and neglecting ε∞ (much smaller than the measured dielectric constant), we obtain [18]: τ12 1 + ω2 τ02 εs (4) ln 2 ε = 2 ln(τ1 /τ0 ) τ0 1 + ω2 τ12 ε =
εs [arctg(ωτ1 ) − arctg(ωτ0 )] ln(τ1 /τ0 )
(5)
In the high temperature range, Eq. (4) becomes ε = εs , and Eq. (5) leads to the conclusion that ε = 0. In the low temperature range, Eqs. (4) and (5) can be simplified as ω εs 0 (6) ln ε = ln(τ1 /τ0 ) ω ε =
π εs 2 ln(τ1 /τ0 )
(7)
So ε in the low temperature range depends on both T and ω, and ε is independent of frequency. In Fig. 2, the characteristics of the dielectric behavior of the material coincide with the above results. From Eq. (6), we get expression that describes the relation between T, ω and dielectric constant in the low temperature range: εL (ω, T ) = A(T )(ln ω0 − ln ω)
(8)
268
F. Yuan et al. / Materials Science and Engineering B 117 (2005) 265–270
and
Table 1 The fitted values of b and c at different frequencies
l T A(T ) = exp γ + δ
(9)
where εL stands for the dielectric constant at T Tm , γ, δ and l are positive constants, and l is associated with the frozen rate of polar regions in the material [10]. Fitting the measured data, we obtain ω0 = 2.9 × 1011 Hz, γ = 3.15, δ = 149.9 K and l = 1.972. It is noted that Eqs. (2) and (8) are completely different, indicating that there are indeed two different polarization mechanisms in the material. Then the dielectric constant can be expressed as the sum of two dielectric constants which are derived from the two different polarization mechanisms: εm (ω, T ) = ε1 (ω, T ) + ε2 (ω, T )
(10)
εm
stands for the measured dielectric constant in the where experiment, and ε1 , ε2 is the dielectric constant derived from the two mechanisms respectively. ε1 and ε2 become Eq. (2) and Eq. (8) at the extreme conditions (T Tm and T Tm ). Therefore, ε1 ε2 , ε1 ≈ εm at T Tm and ε1 ε2 , ε2 ≈ εm at T Tm . From the above analysis, we can describe ε1 , ε2 as [10]: ε1 (ω, T ) =
εH (ω, T ) 1 + B(ω, T )
(11)
ε2 (ω, T ) =
εL (ω, T ) 1 + C(ω, T )
(12)
As discussed above, B(ω, T) is very small in the high temperature range and increases with decreasing temperature. C(ω, T) is very large in the low temperature range and increases with increasing temperature. Following Cheng et al. [10], we assume
m εH (T ) B(ω, T ) = b (13) A(T ) ln ω0
n εH (T ) C(ω, T ) = c (14) A(T ) ln ω0 where b, c are independent of temperature, but change with frequency, and m, n are constants. It is easy to find that B(ω, T), C(ω, T) satisfy the above conditions when m > 0 and n < 0. Because b and c are variable, it is difficult to determine all parameters in Eqs. (10)–(14). To solve this problem, we take a temperature where B(ω, T) and C(ω, T) are independent of m and n. At this temperature, we can easily fit the data of dielectric constant and frequency. The fitted results of b(ω) and c(ω) at various frequencies are listed in Table 1 and also shown in Fig. 3(b). One sees that b(ω) and c(ω) are strongly dependent on frequency. Both b(ω) and c(ω) increase with increasing frequency. With the fitted values of b(ω) and c(ω), we use Eq. (10) to fit m and n. The fitted values are m = 3.166 and n = −2.622. Consequently, we obtain all of the parameters
Frequency (Hz)
b
c
316.2 1000 1738 3162 10000 17380 31620 100000 173800 316200
0.6659 0.9937 1.1781 1.4863 1.9550 2.2274 2.6681 3.3142 3.6117 4.4866
−5.7121 −3.5212 −2.8818 −2.1694 −1.7711 −1.6106 −1.3785 −1.2521 −1.2249 −1.0215
we set, from which we can make further discussion about the dielectric relaxation behavior of PMN–35% PT, used for our experiment. Firstly, we take a look at the relation between temperature, frequency and the dielectric constant contributed from the first mechanism. Fig. 4(a) is drawn by Eqs. (11) and (13) with the parameters derived out before. We find that there are peaks in the curves and the maxima of ε1 increase with decreasing frequencies. The temperature corresponding to the maximum of ε1 (Tm1 ) increases with increasing frequency. At T Tm1 , the values of ε1 are independent of frequency. At temperature around and lower than Tm1 , there is a strong dispersion and ε1 decreases rapidly with decreasing temperature. In RFEs, the density of the polar regions, which determine the behaviors of RFEs, increases with decreasing temperature. At high temperature, almost all of the polar regions
Fig. 4. Temperature dependence of the fitted dielectric constant contributed from the two different mechanisms at frequencies 3.162, 10, 31.62, 100 and 316.2 kHz. (a) Temperature dependence of ε1 (associated with the thermally activated relaxation process). (b) Temperature dependence of ε2 (associated with the resonance relaxation process).
F. Yuan et al. / Materials Science and Engineering B 117 (2005) 265–270
are thermally activated. Therefore, the dielectric constant increases with decreasing temperature as the density of polar regions increases. However, the relaxation time of the polar regions increases with decreasing temperature [18], so the interaction between the polar regions also increases. This leads some of the polar regions not to change their orientations under the external field. We call these polar regions the frozen ones. With the temperature continuing to decrease, the density of unfrozen polar regions decreases. That is why at low temperature the value of ε1 decreases with decreasing temperature. All of these features indicate that the first mechanism is attributed to the relaxation polarization process, which is associated with the thermally activated flips of polar regions in relaxor ferroelectrics. Secondly, we can also obtain the relation between temperature, frequency and ε2 with the parameters we derived out earlier. The results are shown in Fig. 4(b). We find that there are peaks too, and the maxima of ε2 increase with decreasing frequencies. The temperature at the maximum of ε2 (Tm2 ) increases with increasing frequencies. At T Tm2 , the value of ε2 decreases rapidly with increasing temperature. At the whole temperature range, there is a strong dispersion and ε2 increases with increasing frequency in the high temperature range and decreases with increasing frequency in the low temperature range. It is obvious that not all of these characteristics can be explained only by the relaxation polarization process. Although the external field does not change the orientations of polar regions, it can change the sizes and surfaces of the polar regions [10]. With variable external conditions, the frozen polar regions change as a breathing mode [19]. The breathing of the polar regions leads to change of the dipole moments, which determine the behaviors of the dielectrics. This is the second polarization process in the material of PMN–35% PT. As we discussed above, the density of the frozen polar regions increases with decreasing temperature. Therefore, the dielectric constant increases with decreasing temperature. But at the extreme low temperatures, the increasing frozen strength makes it is difficult to change the sizes and surfaces of the polar regions. That is why ε2 decreases at T Tm . In fact, the breathing of the frozen polar regions is the oscillation of the ions around its equilibrium position in the materials. It is known that the dielectric response of an electric particle oscillating around an equilibrium position is a resonance [19]. Therefore, the dielectric behavior of ε2 resembles the dielectric response of a resonance polarization. To confirm this model, we compare the value of the measured dielectric constant and the fitted dielectric constant. The results are shown in Fig. 5(a) and (b). The measured data and the fitting data at the different frequencies are almost the same curve. This allows us to argue that the model proposed by Cheng et al. [10] can fit the measured data well and the argument of two types of relaxation mechanisms contributing to the dielectric relaxation in PMN–35% PT is also physically reasonable.
269
Fig. 5. Measured dielectric constant (εm ) and the fitted results (ε1 , ε2 and εf = ε1 + ε2 ) at frequencies (a) 1 kHz and (b) 31.62 kHz. The values of εm at temperatures lower than 20 ◦ C are absent because of limitation of the experiment.
4. Conclusion In conclusion, the dielectric relaxation behaviors of PMN–35% PT ceramic relaxor have been measured and we use the hybrid model proposed by Cheng et al. [10] to explain the measured results. It has been confirmed by the excellent consistence between the measured data and the hybrid model of Cheng et al. that for PMN–35% PT the dielectric relaxation is ascribed to the concurrence of the thermally activated dielectric relaxation mechanism (Debye-type) and the resonance polarization process resulting from the breathing of the frozen polar regions of the RFE. At low temperature, the resonance polarization process may dominate the dielectric relaxation, while at high temperature, the thermally activated mechanism is more significant.
Acknowledgements The authors acknowledge the financial support from the Natural Science Foundation of China (innovative group project 10021001 and project 50332020), the National Key Projects for Basic Research of China (2002CB613303, 2004CB619000). References [1] K. Uchino, Piezoelectric Actuators and Ultrasonic Motors, Kluwer Academic, Boston, 1996. [2] K. Uchino, Ferroelectrics 151 (1994) 321. [3] M.J. Miller, E.J. Sharp, G.L. Wood, W.W. Clark III, G.J. Salamo, R.R. Neurgaonkar, Opt. Lett. 12 (1987) 340.
270
F. Yuan et al. / Materials Science and Engineering B 117 (2005) 265–270
[4] X. Yue, Z. Shao, J. Chen, X. Lu, D. Sun, Q. Jiang, Y. Song, H. Chen, Opt. Lett. 17 (1992) 142. [5] V. Bobnar, A. Levstik, C. Huang, Q.M. Zhang, Phys. Rev. Lett. 92 (2004) 047604. [6] S. Guo, X.-Z. Zhao, Q. Zhou, H.L.W. Chan, C.L. Choy, Appl. Phys. Lett. 84 (2004) 3349. [7] L.E. Cross, Ferroelectrics 76 (1987) 241. [8] G.A. Smolensky, J. Phys. Soc. Jpn. 28 (1970) 26. [9] D. Viehland, S.J. Jang, L.E. Cross, M. Wutting, J. Appl. Phys. 68 (1990) 2916. [10] Z.-Y. Cheng, R.S. Katiyar, X. Yao, A.S. Bhalla, Phys. Rev. B 57 (1998) 8166. [11] J.-M. Kiat, Y. Uesu, B. Dkhil, M. Matsuda, C. Malibert, G. Calvarin, Phys. Rev. B 65 (2002) 064106.
[12] G. Xu, D. Viehland, J.F. Li, P.M. Gehring, G. Shirane, Phys. Rev. B 68 (2003) 212410. [13] K.Y. Chan, W.S. Tsang, C.L. Mak, K.H. Wong, P.M. Hui, Phys. Rev. B 69 (2004) 144111. [14] S.L. Swartz, T.R. Shrout, Mater. Res. Bull. 17 (1982) 1245. [15] A.K. Jonscher, J. Phys. D: Appl. Phys. 32 (1999) 57. [16] Z.-Y. Cheng, L.Y. Zhang, X. Yao, J. Appl. Phys. 79 (1996) 8615. [17] I. Bunget, M. Popescu, Physics of Solid Dielectrics, Elsevier, Amsterdam, 1984. [18] Z.-Y. Cheng, R.S. Katiyar, X. Yao, A. Guo, Phys. Rev. B 55 (1997) 8165. [19] A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectrics Press, London, 1983.