Dielectric behaviors of Pb Fe 2/3 W 1/3 -PbTiO 3 relaxors: Models comparison and numerical calculations

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1 JOURNAL OF APPLIED PHYSICS 101, Dielectric behaviors of Pb Fe 2/3 W 1/3 -PbTiO 3 relaxors: Models comparison and numerical calculations Cheng-Shong Hong Department of Electrical Engineering, National Cheng Kung University, Tainan Taiwan, Republic of China and Department of Electrical Engineering, Chienkuo Technology University, Changhua Taiwan, Republic of China Sheng-Yuan Chu a Department of Electrical Engineering and Center for Micro/Nano Science and Technology, National Cheng Kung University, Tainan 70101, Taiwan, Republic of China Wen-Chang Su and Ren-Chuan Chang Department of Electrical Engineering, National Cheng Kung University, Tainan Taiwan, Republic of China Hsiau-Hsian Nien Department of Electrical Engineering, Chienkuo Technology University, Changhua Taiwan, Republic of China Yung-Der Juang Department of Material Science, National University of Tainan, Tainan 70005, Taiwan, Republic of China Received 3 August 2006; accepted 30 December 2006; published online 15 March 2007 Regarding the dielectric characteristics of the diffusion phase transition DPTn ferroelectric relaxor materials, several researchers have provided similar but different laws to explain these phenomenon. These laws, presented by Burfoot et al. and Eiras et al., have been proven to provide a better explanation, especially the dielectric diffusion phenomenon, of the incomplete DPT materials if compared with the models presented by Smolensky, and Isupov. However, the differences in fitting adaptability and the physical and mathematical meanings between these two laws have never been discussed. In this paper, we analyze these two laws in the 1 xpbfe 2/3 W 1/3 -xpbtio 3 1 xpfw-xpt relaxor system using the statistical regression theory. We find that the laws of both Burfoot and Eiras demonstrate the same adaptability and provide smaller estimation bias on the samples than the models of Smolensky and Isupov, despite the complete or incomplete DPT characteristics. When x is smaller, the samples tend to be complete DPT phenomenon and the estimation bias of Smolensky and Isupovs models is relatively smaller. However, when the samples show ferroelectric characteristics, the estimation bias will be increased American Institute of Physics. DOI: / I. INTRODUCTION Complex perovskite type ferroelectric materials relaxors are commonly represented by A 1 A 2 B 1 B 2 O 3 in which A and B sites can be occupied by different cathode ions at the same time. Once A and B are occupied by different cathode ions, and the composition fluctuation will be triggered and capable of special dielectric physical characteristics. 1 7 Abnormal dielectric physical phenomenon described previously is known to be associated with the composition fluctuation. To describe this phenomenon, researchers have provided several kinds of models, such as the inhomogeneous micro region model, 8 15 the superparaelectric model, 1,2,16 the glass model, the order-disorder model, the random fields model, 24,25 the breath model, 26 the micro-macro domain transition model, 27,28 and the relaxation time distribution model. 6,29 However, none of these models has been successful in explaining it completely. Among them, the a Author to whom correspondence should be addressed. Electronic mail: chusy@mail.ncku.edu.tw inhomogeneous microregion model is suggested to be the preferable one to study the DTP phenomenon. Several researchers still adopt this model to discuss the DTP characteristics of the relaxors. 7,13,30 43 The inhomogeneous microregion model was first presented by Smolensky. 8 He hypothesized that there existed different phase transition temperatures T c, Curie temperature in the microregion due to the composition differences in the composition fluctuation status. According to the Gauss law, the phase transition temperature T c tends to be the normal distribution when it is near T m and the dielectric constant can be described as 8 = m exp T T m where m is the maximal dielectric constant, s the dielectric constant, T m is the temperature where m exists in other words, T m is the average value of the T c distribution, and is the standard deviation of the Gauss distribution as it also represents the standard deviation of the T c distribution and its /2007/1015/054120/7/$ , American Institute of Physics

2 Hong et al. J. Appl. Phys. 101, value can be used to describe the characteristics of the T c distribution. The higher the value is, the broader the T c distribution is and the diffusion characteristic is more obvious. s also regarded as the diffusion coefficient. Kirilov and Isupov derived the following equation by using Taylor s series expansion by ignoring the higher-order terms: 9 = m 1+ T T m 2 2 2, 2 where the physical and mathematical meanings of m, T m, and are the same as shown in Eq. 1. Researches used Eqs. 1 and 2 to describe the diffusion characteristics, however, it was only limited to complete DTP materials As this equation was applied to evaluate the dielectric diffusion characteristics in other incomplete DPT relaxors, a large bias was noted. Later, Martirena, Clarke, Burfoot, and Uchino modified Eq. 2 as = m 1+ T T m r It was shown that Eq. 3 was better than Eqs. 1 and 2 according to the experimental results ,37 However, the physical and mathematical meanings of and were not clearly explained. In Eq. 3, m and T m are the same with those in Eqs. 1 and 2; s related to the standard deviation of T c distribution. The larger s, the larger the standard deviation of the T c distribution. Therefore, it can be thought as a diffusion coefficient to evaluate the obvious degree of dielectric diffusion phenomenon. In Eq. 3, could be between 1 and 2 depending on the material property When =1, normal Curie-Weiss characteristics will be followed and will show normal ferroelectric properties; when =2, total DPT phenomenon exist and will deviate from the Curie- Weiss equation. Since n Eq. 3 has a quadratic relation, can have a reasonable unit temperature C or K physically only when =2. Furthermore, in order to describe the behaviors in the dispersion region near T m and the dielectric behaviors above T m, Santos and Eiras modified Eq. 3 as 13,14 = m 1+ T T m. Since Eq. 4s modified from Eq. 3, m and T m represent the maximal dielectric constant and the average value of T c, respectively. The meaning of s similar to n Eq. 3, the larger s, the larger the standard deviation of the T c distribution. n Eq. 4 has a similar meaning as and its value is between 1 and 2. 13,14,30,31 However, not many research groups investigate the relationship of and. Equation 4 not only improves the temperature dimension in Eq. 3, but provides a reasonable unit of the value to be the temperature C or K. 13,14 According to the experimental results of PMN and SBN samples, Eq. 4 gives a better explanation in the description of the dispersion region compared with Eq In our previous discussion, the values of and n Eqs. 3 and 4 are between 1 and 2. Bokov et al. reported it 4 to be 2, and the fitting adaptability of T m is found to be worse. 44 Until now, Eqs. 3 and 4 are constantly used to describe the dielectric diffusion characteristic of relaxors and the results fit very well. 7,13,30 43 However, the suitability of these laws Eqs. 3 and 4 has not been investigated. In this paper, we try to ignore the unit reasonability in Eq. 3 for convenience and examine the difference of the fitting curves in Eqs. 3 and 4 by using the statistical regression theories and find and are the same. The classic relaxor materials include PbSc 1/2 Nb 2/3 -PMN, 9,45,46 PbMg 2/3 Ta 2/3 -PMT, 34,47 PbSc 1/2 Ta 1/2 O 3 -PST, 21,48,49 PbSc 1/2 Nb 1/2 O 3 -PSN, 48 PbZn 1/3 Nb 2/3 O 3 -PZN, 5,50,51 and PbFe 2/3 W 1/3 O 3 -PFW. 52,53 Recently, many studies focus on PFW and 1 xpfw-xpt systems. 5,7,39,40,42,54 58 It has been shown that the different ratios of PT not only affect the lattice structures, but also influence the dielectric diffusion characteristics. When x is larger, the lattice structures tend to be tetragonal and the system shows the normal ferroelectric phenomenon. 7 The aim of this study is to investigate the fitting results using Eqs. 1 4, and to study the fitting degree, the variation, and the estimation bias based on the experimental results from 1 xpfw-xpt relaxor system. II. REGRESSION THEORY AND EVIDENCE A. Estimation of the parameters in Eq. 3 by using the regression theory In 3 = m 1+ T T m r 2 2 m 1=T T m r 2 2. Taking the natural logarithm of both sides ln m 1 = r lnt T m ln2 2., The total errors are calculated according to the following equation: 59 n E = ln m i=1 1 lnt i T m ln2 2 2, where E is the total error between the measurement and the estimation; m and T m represent the maximal dielectric constant and the corresponding temperature; T i represents the individual temperature measurement point, and represents the individual dielectric constant in T i. n is the amount of measurement points resulted from repeated measurements with the same or different samples; and are the fitting parameters. In order to minimize the total errors, it can be differentiated as

3 Hong et al. J. Appl. Phys. 101, E ln =2 m 1 2 r lnt i T m ln2 lnt i T m =0, 5 E =2 ln m 1 r lnt i T m ln2 2 2 =0. 6 After arranging 5 and 6, we can get ln m 1lnT i T m r lnt i T m 2 +ln2 2 lnt i T m =0, ln m 1 r lnt i T m + n ln 2 2 =0. From 8 ln 2 2 = r n lnt i T m 1 n ln m 1. Substituting 7 and 9, weget r = n ln m 1lnT i T m ln m 1 lnt i T m Substituting 9 and 10, weget ln 2 2 = n lnt i T m 2 lnt i T m ln m 1lnT i T m lnt i T m ln m 1 lnt i T m 2 n lnt i T m 2 lnt i T m B. Estimation of the parameters in Eq. 4 by using the regression theory In 4 = m 1+ T T m, 1= m T T m. Taking the natural logarithm of both sides ln m 1 = lnt T m ln. The total errors are calculated by the following equation: 59 n E = ln m i=1 1 lnt i T m ln 2, where the meanings of E, m, T m,, T i, and n are the same with those in Sec. II A. In order to minimize the total errors, it can be differentiated as follows: E ln =2 m 1 lnt i T m ln lnt i T m +ln =0, 12 E =2 ln m 1 lnt i T m ln =0, 13 Arranging 12 and 13, weget ln m 1lnT i T m lnt i T m 2 +2 ln lnt i T m ln ln m 1 nln 2 =0, ln m 1 lnt i T m + n ln =0. From 15, weget ln = 1 n ln m n lnt i T m. Substituting 14nto 16, weget

4 Hong et al. J. Appl. Phys. 101, = n ln m 1lnT i T m ln m 1 lnt i T m n lnt i T m 2 lnt i T m Substituting 16nto 17, weget ln = ln m 1lnT i T m lnt i T m ln m 1 lnt i T m 2 n lnt i T m 2 lnt i T m In summary, we find that = upon comparing 10 with 17, and ln 2 2 =ln upon comparing 11 and 18. Therefore, and are the same in 3 and 4. Since ln 2 2 =ln, there should be a specific relation between and ; the fitting curves should be coincident. III. EXPERIMENTAL PROCEDURES Raw materials were mixed using pure reagent PbO, Fe 2 O 3,WO 3, and TiO 2 powders 99.5% purity. The materials 1 xpbfe 2/3 W 1/3 O 3 -xpbtio 3, x= were synthesized by calcining at 800 C for 2 h followed by pulverization. After that, the powders were dried and milled with 8wt% of a 5% PVA solution. The samples were pressed into a disk of 12 mm diameter and 2 mm thickness at a pressure of 25 kg/cm 2. Specimens were sintered isothermally at a heating rate of 5 C/min at 900 C for 2 h. A PbO-rich atmosphere was maintained to minimize the lead loss during sintering. In order to measure the electrical properties, silver paste was coated to form electrodes on both sides of the sample, followed by a firing at 750 C for 25 min. The dielectric properties of the samples were measured using an impedance analyzer HP4294An the temperaturecontrolled container. The phase relations for the sintered samples were identified using an x-ray diffractometer XRD and a second phase was not found. IV. EXPERIMENTAL RESULTS AND DISCUSSION The relaxor s behaviors and the DPT characteristics depend on the x ratio in the 1 xpfw-xpt system. 7 The dielectric constant decreases with increasing the measurement frequency. Figure 1 shows the dielectric constant as a function of temperature for 0.9PFW-0.1PT ceramics measured at a different frequency. Beside the maximum dielectric constant peak which is related to the ferroelectric and paraelectric phase transitions, the second relaxation peak dielectric constant is found at a higher temperature and obviously observed at a lower frequency. It is similar to other reports and is attributed to the dielectric relaxation. 7 The electric hole conductivity and the space charge are expected to lead to the second relaxation peak dielectric constant. 7 Since the polarization due to the electric hole conductivity and the space charge can be filtered at higher frequency, the dielectric behavior caused by the sintering temperature is discussed only at 1 MHz. The dielectric constant of 1 xpfw-xpt system at 1 MHz is shown in Fig. 2. T c Curie temperaturencreases when x increases. It is reasonable since T c of PT is 490 C, and that of PFW is 90 C, and T c increases upon the increasing of the PT ratio. It is similar with those in other studies. 7,46,53 The broad peak dielectric constant-temperature FIG. 1. The dielectric constant as a function of temperature for 0.9PFW-0.1PT ceramics at different frequencies. FIG. 2. Dielectric constant-temperature measurement and the fitting curves using different diffusion phase transition models in the 1 xpfw-xpt x = system.

5 Hong et al. J. Appl. Phys. 101, FIG. 3. The dependence of the diffusion parameters,,, and on the x ratios in the 1 xpfw-xpt system. picture is one of the main features of relaxor ferroelectrics and the DPT characteristics. 7,44 From our experimental results, it is worth noting that when x is smaller, the variation of dielectric constant changing with the temperature near T m is smaller. It means that the dielectric diffusion is more obvious. Comparing the experimental results with the theoretical results presented in Sec. II and using the least-squares method to fit Eqs. 1 4, we can obtain,, and, asa function of the x ratio and the results are shown in Fig. 3; the dependence of and on the x ratio are shown in Fig. 4. The theoretical results derived from Eqs. 1 4 separately and the experimental results are all shown in Fig. 2. In Fig. 2, the region TT m is well fitted by Eqs. 1 4, but the range of temperatures TT m is not well described by the above equations. 7 Although Eqs. 1 4 have good fitting adaptability as TT m ; however Eqs. 3 and 4 have better fitting adaptability as compared to Eqs. 1 and 2. FIG. 5. The total estimation errors between the theoretical calculations and the experimental results in Eqs. 1 4 as a function of x ratios in the 1 xpfw-xpt system. From Fig. 4 and the discussion in the previous section, we find and are the same and they decrease when x increases and get closer to one. Therefore, the characteristics of the 1 xpfw-xpt material are closer to be the normal ferroelectric. 7 In Fig. 3, the diffusion coefficients,,, and, decrease upon an increasing x, which means the dielectric diffusion characteristic decreases as x increases. and show similar tendency, however, s different from and and the reason is under investigation. Although there are different and values for Eqs. 3 and 4 as shown in Fig. 3; as,,, and are replaced into Eqs. 3 and 4, the same fitting curve will be obtained as shown in Fig. 2. According to the fitting results and the statistical regression theory, there is a coincident curve between Eqs. 3 and 4. To evaluate the inaccuracy of the fitting results, the mean square root error can be described as The mean square root error = 1 n est meas 2, FIG. 4. The dependence of the diffusion parameters, and on the x ratios in the 1 xpfw-xpt system. where n represents the measure point number; est is the dielectric constant predictive value from the fitting result; and meas is the dielectric constant from the actual measurement. Figure 5 illustrates the estimation error with different x ratios derived from Eqs We have shown that Eqs. 3 and 4 have the same fitting curve, therefore, there is no doubt that they also have the same estimation error. Equations 3 and 4 have a smaller estimation error than Eqs. 1 and 2n any x ratio, which means that the fitting ability of Eqs. 3 and 4s better than that of Eqs. 1 and 2. When x is smaller, Eqs. 1 and 2 have a smaller estimation error, not much different from that in Eqs. 3 and 4. When gets closer to 2 the characteristic of the samples get closer to the total DPT, using Eqs. 1 and 2 to describe the dielectric diffusion characteristic will not cause a large error. However, when the behavior of the sample gets closer to be normal ferroelectric characteristic, it is not suitable to use Eqs.

6 Hong et al. J. Appl. Phys. 101, V. CONCLUSIONS FIG. 6. The relations between the dielectric constant variance at different x ratios and the deviated temperature region in the 1 xpfw-xpt system. 1 and 2 to describe the dielectric diffusion characteristic. The estimation error in Eq. 1s larger than that of Eq. 2 especially when the x ratio is larger. As and become smaller with a larger x ratio, the dimension difference becomes larger and the estimation error increases upon decreasing both and. Since the higher order terms of the power series are ignored in Eq. 2, the error due to the dimension difference also decreases accordingly. To clarify the dielectric diffusion characteristic of the relaxor materials, we can examine the variation of the dielectric constant. Variance can be the mean-squares sum of the difference between all dielectric constant values at all measured points and m and can be expressed as 59 vart = 1 n m 2, where vart means that the dielectric constant variance has deviated from T m in a specific temperature region, n is all measured points; is the dielectric constant values at all relative measured temperature points; m is the maximal dielectric constant during the measurement, and is also the dielectric constant at the average transition temperature T m. The dielectric constant-temperature relation of the 1 xpfw-xpt relaxor system at x=0.1, 0.2, 0.3, and 0.4 evaluated by the dielectric constant variance is shown in Fig. 6. Irrespective of the x ratio, the dielectric constant variance becomes larger when the deviation of temperature from T m is larger. The dielectric constant-temperature curve of the samples shows a monotonous change. When x is smaller, the dielectric constant variance is smaller no matter what the temperature region is. It means that the diffusion characteristic is better. This result is in accordance with the parameters previously described in Eqs When the temperature deviation region is smaller, the change in variance with the x ratio is smaller; on the other hand, when the temperature deviation region is larger, the change in variance with the x ratio becomes relatively obvious. Regarding the characteristic of diffusion phase transitions of the relaxor materials, Smolensky and Isupov hypothesized T c as the Gauss distribution and proposed Eqs. 1 an 2 to explain this characteristic. However, these two models become biased when applied to incomplete DPT materials. Equations 3 and 4 proposed by Burfoot et al. and Eiras et al. show better fitting adaptability both in total DPT and incomplete DPT materials. In this study, we not only show that Eqs. 3 and 4 have the same fitting result by the statistical regression theory, but also verify the 1 xpfw-xpt relaxor system and obtains the following conclusions: 1. Equations 3 and 4 have the same fitting curve; 2. when x is smaller, the relaxor material show total DPT characteristic; there will not be a large error if Eqs. 1 and 2 are used; 3. no matter whether it tends to be the total DPT or normal ferroelectric characteristic, Eqs. 3 and 4 are able to describe the DTP behavior and are more accurate than Eqs. 1 and 2; 4. the dielectric constant variance in 1 xpfw-xpt relaxor materials decreases upon the decreasing x; it suggests that the dielectric materials have a better dielectric diffusion phenomenon. This result is in accordance with that obtained from Equations 3 and 4 When x is smaller, and are larger.. 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