Diffuseness as a Useful Parameter for Relaxor Ceramics

journal

1 Am. Cerom. SOC ,73I l O j 3122-25 (19901

Diffuseness as a Useful Parameter for Relaxor Ceramics Steven M, Pilgrim,* Audrey EmSutherland,* and Stephen R, Winzer* Martin Marietta Laboratories, Baltimore, Maryland, 21227

The low-field electrical behavior of relaxor ferroelectric ceramics has been qualitatively described by many parameters, including diffuseness (6), which characterizes the breadth of the transition peak and is roughly related to the standard deviation of the peak. Three major types of equations have been used to calculate 6: Gaussian distribution (originated by Smolenskii and Rolov et d.), power law (used by many authors), and variable power law (Uchino et al). In this work, the three calculational techniques are compared analytically and illustrated using dielectric data from ceramic 0.88Pb(Mg1,3Nbzi3)03. 0.12PbTi03. All three methods of analysis result in very linear plots; however, the calculated values of S are significantly different. A method of limiting the temperature range to that corresponding to 2/3 of the maximum relative dielectric constant is suggested in order to decrease the deviations. This limitation is shown to give consistent reproducible diffuseness values without extraneous contributions arising from the relaxor nature of the material or from differing temperature ranges of measurement, thus allowing 6 to be used as an effective comparative parameter for relaxor ceramics. [Key words: relaxation, diffusion, niobates, ferroelectrics, dielectric properties.] I.

the relative dielectric permittivity, T, the temperature at which K,,, occurs, and 6, the Gaussian diffuseness. Since determination of 6, involves calculation of natural logarithms, Smolenskii expanded Eq. (1)as a power series in order to ease calculation as shown in Eq. (2).

Truncating the expansion afterj = 1 results in Eq. (3),which is the commonly used power-law relation (e.g., Ref. 5):

(3) As Smolenskii noted, the trancated expansion is valid only for 6 >> T - T,. In addition to power laws, the variable power law was developed by Uchino et al. to describe behavior in selected ceramic systems.6 The variable power law (Eq. (4)) differs from Eq. (3) in that the exponent y has been introduced as an empirical variable and not as a pure square and in that the denominator is usually presented as a Curie-Weiss type constant.

Introduction

F

ERROELECTRICS with diffuse phase transitions (FDPT) were first mentioned in the literature in the early 1950s.’~~ These are typified by an absence of a sharp transition from the paraelectric to the ferroelectric phase. A subcategory of the FDPT, the relaxors, show dispersion of the maximum relative dielectric permittivity as a function of frequency in addition to a broad transition region. For relaxors as well as other FDPT the width of the transition region is of considerable importance for practical applications. Many authors have characterized the transition width by such parameters as full width at half-height, half-width at half-height, width at threequarters height, etc.; however, the concept of diffuseness (6) is most frequently employed. Smolenskii and Rolov introduced the concept of a Gaussian distribution of Curie temperatures from which the diffuseness parameter (6) could be ~ a l c u l a t e d The . ~ ~ ~diffuseness is roughly analogous to the standard deviation of the Gaussian distribution. This view has formed the basis for some descriptive models applicable to relaxors as well as the FDPT materials. Following Smolenskii, the relative dielectric permittivity ( K ) as a function of temperature approximates Gaussian behavior resulting in Kmox

where

-

- Ke(T-Tc)2/2S:

(1)

is the maximum relative dielectric permittivity,

K,,,~~

As shown in Eqs. (l), (3), and (4), the diffuseness values obtained by the various methods will be subscripted throughout this work for clarity. The possible magnitude of the variations between 6, and 6, can be illustrated by solving Eqs. (1) and (3) as functions of (T - Tc)/6to give Kmux

K -

= exp

[:(TS,TcI’] -

Figure 1 shows the permittivity ratio as a function of (T - Tc)/6.This graphically demonstrates the significant deviation between the two methods and reinforces the need to limit the temperature range as proposed by Smolenskii. An alternative way to assess the magnitude of variations between 6, and 6, can be found by solving Eqs. (1)and (3) as functions of K,,,/K: 6p42(%

- 1) = T

- T,

K

A plot of these equations is shown in Fig. 2. Although deviations between 6, and 6, are still present, Eq. (6) suggests an alternative method for calculating 6, within a desired error of 6,. In practice, limitation by K,,,,/K is much easier to achieve than limitation by temperature range. In order to quantify the required limits, i.e., refine Smolenskii’s requirement that T - T, 20000 IH

>

H

3123 I

t

IIH

r

[ I

W

a

. Y

u

x

H [ I

;

c u

10000 -

W

J

W

H

n W

w

H

I4

0

I

0

2

d ' , , . . . . lO [ I

I

. " . " 0

150

100

50

(T-Tc)/S

J

TEMPERATURE ('C)

Fig. 1. Permittivity ratio K".*/K as a function of (2' - T,)/S showing difference between Gaussian and power-law cases (from Eq. ( 5 ) ) .

Fig. 3. Low-field relative dielectric permittivity for a typical ceramic sample of 0.88Pb(Mg,,,Nbz,3)03.0.12PbTiO3.

the Gaussian case is less than some arbitrary limit (in this case 10%):

ondary reason is the frequent, questionable use of S in relation to ceramics.

0.96, < 6, < 1.16,

(7) The resulting conditions are given in Eq. (8) after substituting from Eqs. (5) and (6):

Since K,,, is by definition larger than K , the only physically meaningful limit is K,~~.K < 1.5. Therefore, if the power-law equations are to be used to approximate the Gaussian diffuseness within lo%, the temperature range must be limited to that corresponding to K equal to two-thirds of K , ~ , . The actual effects of these analytic conditions will be illustrated using dielectric data from a representative relaxor ceramic0.88Pb(Mg1/3Nb2/3)03. O.12PbTiOs. Although single-crystal data would form a more pure illustration of the computational effects, ceramic data were selected for two major reasons. The primary reason is the unavailability of a large number (>50) of raw datapoints on a single crystal (N.B., data derived from published plots are insufficiently precise); the sec-

w 0

x

95% theoretical), phase pure perovskite (by X-ray diffraction) materials. Pellets were polished to 1000 grit and electroded with sputtered Au followed by air-dried Ag to provide robust electrodes for electrical characterization. Capacitance and dielectric dissipation were measured on a computer-controlled system based on a Hewlett-Packard 4274A LCR* meter and a Delta Designs 2300 Thermal Chambert at four discrete frequencies (1, 10, 100, 1000 kHz) from 150" to -50°C while temperature was decreased at 4"C/min. Capacitance and the sample dimensions were used to calculate the relative dielectric permittivity ( K ) . The resulting dielectric data for a 0.88PMN -0.12PT sample are shown in Fig. 3 and Table I. The material shows relaxor behavior with a shift in the temperature of maximum K (Tc)of 9.5"C between 1 and 1000 kHz. The K curves contain 225 points at each frequency of which approximately 160 (i.e., those above T,) are used in the analysis. The K data were analyzed to determine the differences in diffuseness (6) calculated by the three methods as functions of T,, frequency, and temperature of maximum measurement. The methods used to calculate diffuseness were the Gaussian case (8J, the power-law case (ap), and the variable-power case (6"). For each method 6 was also calculated for a reduced temperature range, these are denoted by Sgl, 6pl,and 6,,, respectively. All 6 values were calculated by the method of linear least squares from these data. Tables I1 to IV contain the pertinent values