Analysis of the dielectric characteristics for polycrystalline Ba0.65Sr0.35TiO3 (II) - d.c. field dependence with a modified bias equation

J O U R N A L O F M AT E R I A L S S C IE N C E: M ATE R IA LS I N EL E C T RO N I C S 1 1 (2 00 0 ) 64 5± 65 1

Analysis of the dielectric characteristics for

polycrystalline Ba

_0:65

Sr

_0:35

TiO

₃

(II) ± d.c. ®eld

dependence with a modi®ed bias equation

JIH-WEI LIOU, BI-SHIOU CHIOU

Department of Electronics Engineering & Institute of Electronics,

National Chiao Tung University, Hsinchu 300, Taiwan

E-mail: bschiou@cc.nctu.edu.tw

The effect of direct-current (d.c.) biasing on the dielectric properties of Ba

0:65

Sr

0:35

TiO

3

(BST)

with various grain sizes (0.87 mm to 5.43 mm) have been studied in the paraelectric state.

The leakage current of BST samples is proportional to the biasing ®eld. Specimens with

larger grains exhibit higher leakage current under biasing. The bias equation used in the

literature to predict the d.c. ®eld dependence of permittivity is modi®ed by introducing a bias

exponent parameter b. This modi®ed bias equation gives a better ®t to the experimental

results both in this study and in the literature. Similar trends of a (representing the

distribution of relaxation times), b, and the variance of lattice distortion with grain sizes

suggests that b is a material parameter. It is concluded that the temperature dependence, the

frequency response and d.c. ®eld effect for the dielectric characteristics of Ba

0:65

Sr

0:35

TiO

3

are

all suppressed by the transition broadening.

# 2000 Kluwer Academic Publishers

1. Introduction

Ferroelectric barium strontium titanate (BST) is of great interest for many practical applications. The dielectric characteristics of polycrystalline BST depend strongly both on its microstructure and the environment during usage. The effects of d.c. ®eld on the dielectric behavior of BST have been reported [1, 2] and applied in various devices such as tunable capacitors [3] and microwave phase shifters [4, 5]. However, only a few studies [3, 6, 7] have been conducted on the d.c. ®eld related properties of paraelectric BST, especially the BST system with various grain sizes. The non-linearity of dielectric permittivity with d.c. biasing ®eld in the paraelectric state is believed to originate from the anharmonic interaction of Ti ions in the perovskite cubic structure. Johnson [2] started from Devonshire's theory and proposed a bias equation to correlate the dielectric constant and d.c. ®eld. Outzourhit et al. [3] ®tted this phenomenological equation to their experimental data and argued that deviations from the theoretical prediction were attributed to the dielectric dispersion caused by non-uniform grain sizes. In our previous work [7], it was observed that the d.c. effect is suppressed for BST with ®ner grains due to the smaller dielectric constant caused by transition broadening.

In this study, the d.c. ®eld dependence of dielectric constant and dielectric loss for Ba_0:65Sr_0:35TiO₃ with various grain sizes are investigated in the paraelectric

state. Dopants MgO and MnO₂ are used to obtain

uniform grained, low-loss dielectrics [8, 9]. Samples with various grain sizes are obtained by varying sintering temperatures and times. The purpose is to analyze the

frequency response and d.c. effect for Ba_0:65Sr_0:35TiO₃ and to ®nd the correlation between these characteristics and the grain size. The frequency response, reported previously [10], was analyzed with two power laws which are obtained from the Cole±Cole expression at low frequencies. The d.c. ®eld dependence of the dielectric permittivity is analyzed with a modi®ed bias equation which is proposed in this study.

2. Experimental procedures

The Ba_0:65Sr_0:35TiO₃ polycrystalline samples with var-ious grain sizes were prepared by the conventional solid state reaction method with commercial powders of BaCo₃, SrCo₃, TiO₂, MnO₂, and MgO. Powder mixtures containing additives of 1.0 mol % MgO and 0.05 mol % MnO₂were calcined at 1100_{C for 2 h in air and pressed}

to form disk-shaped samples at 180 MPa. Samples with various grain sizes were prepared with a fast-®ring, two-stage, sintering pro®le described previously [7]. Phase identi®cation and average grain size were analyzed by powder X-ray diffraction and scanning electron micro-scopy (SEM) examination, respectively. Lattice distortion in polycrystalline BST is calculated by the X-ray diffraction peak broadening of a single peak pro®le. For dielectric characteristic measurement, the bulk sample was polished to 0.4 mm in thickness before making electrodes. The current±voltage curve was obtained with an HP4140 pA meter (Hewlett Packard). The capacitance and tangent loss were measured with an HP4275A LCR meter (Hewlett Packard). A 200 V d.c.

measuring frequency range in this study is from 10 kHz to 4 MHz.

3. Results and discussion

When the electrical signal passes through dielectrics, the transmitted information is deteriorated by leakage currents, especially when a d.c. ®eld is applied. Therefore, it is crucial to understand the d.c. ®eld dependence of leakage current. The current densities for each samples at various applied d.c. ®elds are shown in Fig. 1. A larger leakage current for Ba_0:65Sr_0:35TiO₃ is observed for samples with larger grams. Dopants MgO and MnO₂used in this study are known to serve as grain growth inhibitors barium titanate because they segregate preferentially at grain surfaces. However, there is a small fraction of these dopants which may dissolve in the grains and become p-type acceptors [11, 12]. Lewis et al. [11] calculated the surface defect energies for BaTiO3

with various dopants, such as Mg and Mn. They reported some accompanied oxygen defects with the acceptors substituting on Ti sites. Recently, Langhammer et al. [13] found that the defect concentration in Mn-doped BaTiO₃ increased as more Mn was added. In this study, samples with larger grains were obtained with longer sintering time or higher sintering temperature, the acceptor-type dopants were believed to dissolve more into the grains, thus more oxygen defects were produced due to valence charge compensation. The I±V characteristics for barium titanate have been explained by the space charge limited currents (SCLC) theory [14] for polycrystalline insula-tors. This states that strongly defective surfaces of the crystalline grains provide a source of deep trap states in insulators where the trapped charge carriers are excited by the applied voltage. Therefore, Ba_0:65Sr_0:35TiO₃with larger grains has more defects and consequently yield larger leakage current, as shown in Fig. 1. The SCLC theory suggests that below an onset voltage, an ohmic current ¯ows. At higher voltages, the current transfers to a rapidly increasing behavior of the form of I ÿ Va_where

a is in the range of 1.2 to 2.0 [14]. It is seen in Fig. 1, that the current density is linearly dependent on the applied ®eld. The d.c. resistivities are then calculated and summarized in Table I. Resistivity decreases with the increase of grain size due to the increase of defects.

Considering the frequency response of an electrically tunable capacitor, an equivalent circuit model containing a d.c. resistor and a voltage-variable capacitor in parallel combination can be applied adequately. Both the d.c.

resistance as well as the capacitance are d.c. ®eld dependent under d.c. biasing.

R…V† ˆ V=I…V† …1†

C…V† ˆ C₀‰e0_{V† ÿ je}00_{V†Š …2†}

where C0is the geometrical capacitance of the capacitor

and the relative permittivity e…V† ˆ e0_{V† ÿ je}00_{V† has}

its real and imaginary parts which are both d.c. ®eld dependent. The current±voltage characteristics I…V† for Ba_0:65Sr_0:35TiO₃ with various grain sizes exhibit ohmic behavior, as illustrated in Fig. 1. Assuming that the onset of I±V behavior-transformation (SCLC theory) is not reached by the d.c. ®eld applied when a.c. measurements are being carried out, the resistance of these samples is then constant with ®eld. The real part of the relative permittivity can be calculated by the measured capaci-tance. As for the measured dielectric loss tand, it contains two terms, one for the imaginary permittivity, the other for loss due to the leakage current. Therefore at some speci®c measuring frequency o, the voltage variable loss tangent can be expressed as:

tan d…V† ˆe_e00_0…V†…V†‡_oRe0…V†C1

0 …3†

The ®rst term on the right hand side of Equation 3 is the intrinsic loss of the dielectrics, and the second term is the

Figure 1 Leakage current density as a function of d.c. ®eld for Ba0:65Sr0:35TiO3with various grain sizes.

T A B L E I Dielectric constant, dielectric loss, and resistivity for MgO and MnO2doped Ba0:65Sr0:35TiO3with various average grain sizes* Specimen (grain size (mm)) Dielectric constant Dielectric loss d.c. resistivity (GO cm)

10 kHz 4 MHz 10 kHz 4 kHz L1(0.87) 3783 3251 0.0115 0.1241 2477 L2(1.72) 5269 4801 0.0132 0.1200 1376 H1(2.16) 6959 5557 0.0288 0.2063 907.2 H4(3.74) 9808 7915 0.0134 0.2992 597.5 H8(5.43) 9947 8233 0.0168 0.3733 444.7

conduction loss. With the resistivity and dielectric constant listed in Table I, the conduction losses for the Ba_0:65Sr_0:35TiO₃with various grain sizes are estimated to be less than 10ÿ7_{for all frequency and ®eld ranges in this}

study. Hence, the conduction loss term can be neglected so that the imaginary permittivity e00_{is approximately the}

product of dielectric constant and tangent loss.

A bias equation on the basis of Devonshire's theory to correlate the relative permittivity e0 _{and applying d.c.}

®eld E is proposed phenomenologically [2]: e0_{E† ˆ} e0…0†

…1 ‡ AE2_†1=3 …4†

where e0_{0† is the initial permittivity under zero d.c.}

biasing. The ®eld coef®cient A represents the strength of the d.c. ®eld effect. The value of A equals ae03 _{where a}

is the anharmonic coef®cient. As the anharmonic interaction becomes stronger, a larger value of A yields a larger value of ®eld coef®cient and thus the d.c. ®eld is more effective to reduce the permittivity. Johnson claimed that some deviations due to poly-crystallinity can be absorbed in the ®eld coef®cient A. Therefore Equation 4 can be applied with validity to the polycrystalline barium strontium titanate system. As for the imaginary part of dielectric permittivity, Johnson proposed another bias equation based on a RC series circuit:

Figure 2 Relative permittivity e00_{as a function of d.c. biasing ®eld for Ba}

0:65Sr0:35TiO3with various grain sizes (a) L1, (b) H1 and (c) H8. Curves are ®tted with the bias equation and the modi®ed bias equation. Note that the modi®ed equation gives better ®ts than the original one.

Recently, Outzourhit et al. [3] investigated the d.c. ®eld dependence of the dielectric constant of Ba0:1Sr0:9TiO3 ceramics prepared by the wet method

and the solid state method. They found that good ®tting of this bias equation to the experimental results can only be made separately in low and high ®eld ranges. For the BST system in this study, it is found that the suppression of the d.c. ®eld effect by the grain size is remarkable [10]. The validity of the bias equation was questioned when there is a transition broadening. In Outzourhit's work, the bias equation did not ®t the experimental data, presumably due to the diffuseness of the transition of Ba_0:1Sr_0:9TiO₃with grain sizes of about 1 mm and 10 mm. Attempts to employ the bias equation for the dielectric characteristics of Ba0:65Sr0:35TiO3in this study (shown in

Fig. 1 of Liou and Chiou [10]), reveal some controversies such as: (a) the bias equation does not predict well the dielectric behavior under d.c. biasing though the best ®tted ®eld coef®cient A is chosen, (b) if the ®eld coef®cient is calculated by Equation 4, it is no longer a constant with ®eld, and (c) the ®eld coef®cient is frequency dependent. Since the bias equation was explained adequately for single crystal or ceramics with high crystallization [2], it is argued that the controversies were attributed to the various microstruc-tures of the polycrystalline ceramics, especially the grain size. It was reported previously that there are various distributions of Curie temperatures occurring in the Ba_0:65Sr_0:35TiO₃ with various grain sizes [10]. This implies that each grain has its own initial permittivity e0_{0† which is different from each other. Thus, the}

dielectric behavior under d.c. ®eld changes profoundly due to the grain size effect. To realize the d.c effect by a distribution of e0_{0† in polycrystallites, the denominator}

of the bias equation is modi®ed by introducing a bias exponent b:

e0_{E† ˆ} e0…0†

…1 ‡ BE2_†1ÿb3 …6†

The bias exponent b ranges from 0 to 1 and represents the order of suppression of the d.c. ®eld effect. The related ®eld coef®cient B is supposed to be ®eld independent and represents the strength of the d.c. ®eld effect. The ®ttings of bias equation and modi®ed bias equation for samples L1, H1, and H4 at 10 kHz, 1 MHz and 4 MHz are shown in Fig. 2a to c for comparison. For the ®tting of the modi®ed bias equation, a two-parameter least square method was employed. The initial values of the two parameters, B and b, were calculated by solving two equations

constructed by two experimental data, e…E1† and

e…E2†. The optimum b was obtained by the least squares

method with the calculated B. The optimum B was then obtained by the least squares method with the optimum b. As shown in Fig 2a to c, the magnitude predicted by the bias equation in all samples is too high in the low ®eld region and too low in the high ®eld region. This is why Outzourhit et al. ®tted their results with the bias equation separately in high ®eld and low ®eld regions. Meanwhile, less deviation of the modi®ed bias equation

( 5 0.6%) is observed at d.c. ®elds up to 5000 V cmÿ1_.

With the ®ttings to the dielectric characteristics shown in Fig. 1 of [10], the bias exponents b and ®eld coef®cients B for Ba0:65Sr0:35TiO3 with various grain

sizes at different frequencies are shown in Fig. 3a and b, respectively. No apparent frequency dependence of b and B suggest that these two parameters are material parameters. The increase of b and the decrease of B with the decrease of grain size is clearly seen. Larger b and smaller B make the denominator in the modi®ed bias equation (Equation 6) closer to one, thus the reduction of dielectric constant by the d.c. ®eld is suppressed. This suppression is presumably due to the broad distribution of Curie temperatures in polycrystalline materials.

Figure 3 (a) Bias exponent b and (b) ®eld coef®cient B of permittivity e0 _{as a function of frequency for Ba}

0:65Sr0:35TiO3 with various grain sizes.

Equation 5 is modi®ed in a similar way as: e00_{E† ˆ} e00…0†

…1 ‡ BE2_†2…1ÿb†3

…7† Fig. 4 shows the ®ttings with the bias equation (Equation 5) and the modi®ed bias equation (Equation 7). The modi®ed equation gives a better ®t, as in the case of e0_{E†. As reported previously, dopants contribute to the}

split of dipolar relaxation, and the losses at frequencies below 100 kHz are higher than those predicted by the approximated power law [10]. It is noted in Fig. 4 that the behavior of the dielectric loss e00 _{at 10 kHz is also in}

agreement with the prediction of the modi®ed bias equation. As shown in Fig. 5, the bias exponent b

increases and the ®eld coef®cient B decreases with the increase of grain size. Similar trends of e0_{E† and e}00_E†

are observed, b ranges from 0.27 to 0.70 for e0_{and from}

0.29 to 0.70 for e00_{. Although the bias ®eld coef®cients}

for both e0_{and e}00_{are also comparable, more ¯uctuations}

of b and B for e00_{at different frequencies are observed in}

Fig. 5. The calculation of the imaginary permittivity e00_is

based on Equation 3 neglecting the second term as the leakage current is very small. There are differences between the b and B in Fig. 3 and those in Fig. 5. It is believed that ®tting data with Equation 6 (Fig. 3) gives more accurate b and B values, since the contribution of conduction loss and dopant-related relaxation to e00 _are

neglected in Equation 7 (Fig. 5).

Using the modi®ed bias equation proposed in this

Figure 4 Relative permittivity e00_{as a function of d.c. biasing ®eld for Ba}

0:65Sr0:35TiO3with various grain sizes (a) L1, (b) H1 and (c) H8. Curves are ®tted with the bias equation and the modi®ed bias equation. Note that the modi®ed equation gives better ®ts than the original one.

study, the ®ttings of Outzourhit's results are shown in Fig. 6. The bias exponent b is 0.42 for Ba_0:1Sr_0:9TiO₃ prepared by the wet method and is 0.28 for that prepared by the solid state method (ss method). The grain sizes of Outzourhit's samples are about 1 mm (wet method) and 10 mm (ss method), respectively. Outzourhit also showed that samples prepared by the wet method had broader Curie peaks than those prepared by the solid state method. The larger value of bias exponent is explained by the broader transition distribution in the sample prepared by the wet method.

One may argue that an equation with two ®tting parameters always gives better ®ts than one with only one ®tting parameter. This is true with regard to the mathematics. The bias exponent b varies, the trend of modi®ed bias equation intrinsically, i.e., different b values determine different function forms of Equations 6 and 7. Together with the frequency independence of b, as shown in Figs 3a and 5a, these features suggest that b is a material parameter. Perhaps the bias coef®cient B plays a tuning role during the ®tting procedure. Nevertheless, once these two parameters are

Figure 6 Cure ®tting with the modi®ed bias equation for the permittivity e0_{. Ba}

0:1Sr0:9TiO3prepared by wet and solid state methods (data obtained from Outzourhit et al. [15]).

Figure 7 Grain size dependence of variance of lattice distortion and the suppressing parameters a, b for Ba0:65Sr0:35TiO3 with various grain sizes.

properly chosen, the dielectric behavior can be predicted well with the initial value …e0_{0† or e}00_0††

known. The modi®ed bias equation is then useful for applications. Furthermore, from the same trends of the tilt parameter a [10], critical exponent g [10] and bias exponent b for Ba_0:65Sr_0:35TiO₃ with various grain sizes, the variance of b is also believed to be due to the ferroelectric to paraelectric transition broadening. These parameters are called ``suppression parameters'' to represent the suppression order of temperature depen-dence, frequency response and d.c. ®eld effect. Fig. 7 exhibits the grain size dependence of these parameters. In this ®gure the lattice distortion is obtained from the X-ray analysis at room temperature, g is obtained at 10 kHz, b is obtained at 10_{C and 10 kHz, and a is}

obtained at 10_{C. All these four parameters decrease as}

grain size increases.

4. Conclusions

1. A decrease of the dielectric constant for Ba_0:65Sr_0:35TiO₃ with the increase of d.c. biasing ®eld is observed. A modi®ed bias equation with a bias exponent b, is proposed to predict the dielectric behavior under d.c. biasing. It is found that the ®ts are better than the original bias equation for Ba0:65Sr0:35TiO3 with

various grain sizes. Once the parameters in the modi®ed bias equation are properly chosen, the dielectric behavior under the d.c. ®eld can be predicted accurately with the initial value …e0_{0† or e}00_{0†† known.}

2. Frequency independence of the bias exponent b suggests that it is a material parameter. The increase of b with the decrease of grain size indicates that the suppression of the d.c. ®eld effect occurs in polycrystal-line Ba_0:65Sr_0:35TiO₃.

3. The grain size dependence of suppression para-meters a, b and g obtained from the dielectric characteristics of frequency response, d.c. ®eld effect

and temperature dependence, show the same trend with the variance of lattice distortion for polycrystalline Ba_0:65Sr_0:35TiO₃.

Acknowledgments

This work is supported by the Chung-Shan Institute of Science and Technology and the National Science Council of Taiwan (contract numbers: NSC 86-2623-E-009-002 and NSC 86-2216-E-009-019).

References

1. H . D I A M A N D, J. Appl. Phys. 32(5) (1961) 909. 2. K . M . J O H N S O N, ibid. 33(9) (1962) 2826. 3. A . O U T Z O U R H I T,J . U . T R E F N Y,T. K I TO andB . YA R A R, J. Mater. Res. 10(6) (1995) 1411. 4. R . W. B A B B I T T,T. E . KO S C I CAandW. C . D R AC H, Microwave Journal June (1992) 63. 5. F. S E L M I,R . H U G H E S,V. K . VA R A D A NandV. V. VA R A D A N, Proc. SPIE 1916 (1993) 180.

6. J . W. L I O UandB . S . C H I O U, J. Am. Ceram. Soc. 80(12) (1997) 3093.

7. J . W. L I O UandB . S . C H I O U, Jpn. J. Appl. Phys. 36(7A) (1997) 4359.

8. F. B AT T LO, E . D U V E R G E R, J . C . J U L E S, J . C . N I E P C E,

B . J A N N OTandM . M A G L I O N E, Ferroelectrics 109 (1990) 113.

9. B . S . C H I O U,C . M . KO HandJ . G . D U H, J. Mater. Sci. 22 (1987)

3893.

10. J . W. L I O UandB . S . C H I O U, submitted to J. Mater. Sci.: Mater.

Electron.

11. G . V. L E W I S,C . R . A . CAT LOWandR . E . W. CA S S E LTO N,

J. Am. Ceram. Soc. 68(10) (1985) 555.

12. Y. C . C H E N,G . M . LO,C . R . S H I H,L . W U,M . H . C H E Nand

K . C . H U A N G, Jpn. J. Appl. Phys. 33 (1994) 1412.

13. H . T. L A N G H A M M E R, T. M UÈ L L E R, A . PO L I T Y, K . H . F E L G N E RandH . P. A B I C H T, Mater. Lett. 26 (1996) 205. 14. W. O S A K andK . T K AC Z. J. Phys. D: Appl. Phys. 22 (1989)

1746.

Received 23 August 1999 and accepted 17 May 2000