An alternative representation of impedance spectra of ceramics

Pergamon

Materials Research Bulletin 35 (2000) 727–740

An alternative representation of impedance spectra of ceramics Joa˜o C.C. Abrantesa, Joa˜o A. Labrinchab, Jorge R. Fradeb,* a

ESTG, Instituto Polite´cnico de Viana do Castelo, Ap. 574, 4900 Viana do Castelo, Portugal Ceramics and Glass Engineering Dep. (UIMC), University of Aveiro, 3810 Aveiro, Portugal

b

(Communicated A.R. West) Received 2 July 1999; accepted 2 July 1999

Abstract

Suitable approximations are derived to extract the relevant parameters of the bulk, grain boundary, and electrode contributions of impedance spectra from Z⬘ vs. Z⬙/f plots. Corrections are proposed for cases where the difference between bulk and grain boundary relaxation frequencies and/or the difference between grain boundary and electrode relaxation frequencies is relatively small. In these cases, the Nyquist plot (Z⬙ vs. Z⬘) shows overlapped arcs, but inspection of alternative Z⬘ vs. Z⬙/f representations may reveal contributions with relatively small differences in relaxation frequencies. These plots and the corresponding derivative dZ⬘/d(Z⬙/f) may be useful to interpret cases showing a significant distribution of grain boundary relaxation frequencies. © 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Ceramics; C. Impedance spectroscopy; D. Microstructures; D. Electrical properties

1. Introduction Impedance spectroscopy is often used to separate the bulk, grain boundary, and electrode processes of polycrystalline ceramic materials with ionic or mixed conduction [1]. This method has been widely used to assess if resistive grain boundaries may spoil the conduc-

* Corresponding author. Fax: ⫹351-34-425-300. E-mail address: [email protected] (J.R. Frade). 0025-5408/00/$ – see front matter © 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 5 - 5 4 0 8 ( 0 0 ) 0 0 2 6 9 - 5

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tivity of solid electrolytes and has been used to study other materials with useful grain boundary properties [2]. The most commonly assumed equivalent circuit for these types of materials consists of a series association of RC components. The real and imaginary parts of the impedance (Z* ⫽ Z⬘ – j Z⬙) obtained for this ideal equivalent circuit are Z⬘ ⫽ RB/[1 ⫹ (␻/␻B)2] ⫹ Rgb/[1 ⫹ (␻/␻gb)2] ⫹ Rel/[1 ⫹ (␻/␻el)2]

(1)

Z⬙ ⫽ RB(␻/␻B)/[1 ⫹ (␻/␻B)2] ⫹ Rgb(␻/␻gb)/[1 ⫹ (␻/␻gb)2] ⫹ R el共␻/␻ el兲/关1 ⫹ 共␻/␻ el兲 2兴

(2)

where subscripts B, gb, and el denote the bulk, grain boundaries, and electrode, and ␻i ⫽ 2␲fi ⫽ 1/(RiCi)

(3)

is the angular frequency of term i. The high frequency component usually corresponds to the true bulk properties, as measured for single crystals [2], the intermediate frequency term is ascribed to resistive grain boundaries, and the low frequency term corresponds to electrode processes or processes occurring at the material/electrode interface. The bulk relaxation frequency fB is often at least two orders of magnitude higher than the grain boundary relaxation frequency fgb, and the electrode relaxation frequency fel is also much smaller than fgb. In these conditions, the Nyquist plot should show nearly separate semicircles for the bulk, grain boundary, and electrode terms. Otherwise, one may need alternative representations such as those based on the modulus, admittance, and permittivity [3]. For example, a representation of the modulus is suitable for materials with similar bulk and grain boundary capacitances [4]. In this case, the Nyquist plot may conceal a minor bulk resistance, and the usual fitting codes [5] may be insufficient. In addition, the grain boundary behaviour of polycrystalline materials often deviates from a simple brick layer approximation [6,7], especially for materials with broad grain size distributions, irregular grain shapes, etc. Fleig and Maier [7] used a finite difference analysis to compute the impedance spectra for a variety of microstructural features, and to show that the current can make detours across large grains, thus lowering the overall grain boundary resistance across the sample. The grain boundary arc of an impedance spectrum may thus be affected by the bulk properties, yielding significant differences relative to the brick layer model, incorrect estimates of the activation energy, and possibly even an additional shoulder on the high frequency side of the grain boundary arc. The grain boundary arc of a typical impedance spectrum is often depressed, which might be at least partly due to microstructural features [7], or a distribution of relaxation times. This work proposes representations of Z⬘ vs. Z⬙/f to interpret impedance spectra. Simple relations are derived on assuming sufficient differences between the relaxation frequencies (fB ⬎⬎ fgb ⬎⬎ fel), and corrections can be used for cases when the relevant arcs overlap.

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2. Solutions for fB >> fgb >> fel In these conditions, the high frequency range of the impedance spectra (for f ⬎⬎ fgb) only shows the bulk component, and Eqs. (1) and (2) reduce to Z⬘ ⬇ RB/[1 ⫹ (f/fB)2]

(1a)

Z⬙ ⬇ RB(f/fB)/[1 ⫹ (f/fB)2]

(2a)

with fB ⫽ ␻B/2␲. These approximate solutions can be combined to obtain the following: Z⬘ ⬇ fB(Z⬙/f)

(4)

and the slope of Z⬘ vs. Z⬙/f thus yields the bulk relaxation frequency. A similar method can be derived for the frequency range where the grain boundary contribution predominates (fB ⬎⬎ f ⬎⬎ fel), in which case Z⬘ ⬇ RB ⫹ Rgb/[1⫹ (f/fgb)2]

(1b)

On combining these approximate solutions one thus obtains Z⬘ ⬇ RB ⫹ (Z⬙/f)fgb

(5)

and the corresponding slope yields the grain boundary relaxation frequency fgb (in the intermediate frequency range). The intercept corresponds to the bulk resistance RB. On extending this method to the low frequency range (f ⬍⬍ fgb), one may extract parameters of the electrode processes, the corresponding solution being: Z⬘ ⬇ RB ⫹ Rgb ⫹ (Z⬙/f)fel,

(6)

The slope is identical to the electrode relaxation frequency, and the interception yields the sum of the bulk and grain boundary resistances. Eqs. (4)–(6) thus show that different parts of a generic plot can be used to extract all the relevant parameters.

3. Corrections for overlapped contributions 3.1. Grain boundary effects on the bulk contribution The grain boundary term may interfere with the bulk arc when the difference between the bulk and grain boundary relaxation frequencies is smaller than two orders of magnitude. The electrode processes are less likely to affect the bulk contribution. A combination of the bulk and grain boundary terms of Eqs. (1) and (2) thus yields Z⬘ ⫽ (Z⬙/f)fB䡠F(f)

(7)

where F(f) ⫽ {1 ⫹ r21[1 ⫹ (f/fB)2]/[1 ⫹ (f/fgb)2]}/{1 ⫹ (r21/f21)[1 ⫹ (f/fB)2]/[1 ⫹ (f/fgb)2]} (8)

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is a correction factor, with r21 ⫽ Rgb/RB; f21 ⫽ fgb/fB

(9)

Typical values of F(f) are easily evaluated for very high frequencies (f ⬎⬎ fB), and/or at the peak frequency (f ⫽ fB), as follows: F(⬁) ⫽ [1 ⫹ r21(f21)2]/(1 ⫹ r21f21)

(8a)

F(fB) ⫽ [1 ⫹ (1 ⫹ 2r21)(f21)2]/[1 ⫹ (2r21 ⫹ f21)f21]

(8b)

Eq. (8a) show that the grain boundary term may affect the slope even at very high frequencies. For example on assuming r21 ⫽ Rgb/RB ⫽ 1, and f21 ⫽ fgb/fB ⫽ 0.1, one still needs correction factors F(⬁) ⫽ 0.92, and F(fB) ⫽ 0.85. When the difference between the relaxation frequencies increases to two orders of magnitude, the correction factors converge to unity (F(⬁) ⫽ 0.99, and F(fB) ⫽ 0.98), and Eq. (4) is nearly true. The values obtained for Rgb/RB ⫽ 10 and fgb/fB ⫽ 0.1, (F(⬁) ⫽ 0.55, and F(fB) ⫽ 0.40) show that corrections are more likely to be required with increasing grain boundary to bulk resistance ratio. 3.2. Bulk effects on the grain boundary term The term (␻/␻B)2 decreases rapidly with decreasing frequency, and, in this case, one may assume 1 ⫹ (␻/␻B)2 ⬇ 1 even for (␻/␻B) ⫽ 0.1. In these conditions, one obtains: Z⬘ ⫽ RB(1 – f21) ⫹ (Z⬙/f)fgb

(10)

Eq. (10) shows that the intercept obtained on plotting Z⬘ vs. (Z⬙/f) in the grain boundary range is also dependent on the grain boundary to bulk relaxation frequency ratio f21 ⫽ fgb/fB; this must be taken into account to obtain a correct estimate of the bulk resistance. However, the slope is still identical to the grain boundary relaxation frequency. 3.3. Electrode effects on the grain boundary term The solutions that might apply to this situation are easily obtained by extending the previous methods; this yields Z⬘ ⫽ RB(1 – f21) ⫹ (Z⬙/f)fgbF(f)

(11)

where (1 – f21) is a correction for the estimate of the bulk resistance (obtained from the intercept), and F(f) is the correction factor for a correct estimate of the grain boundary relaxation frequency, as follows: F(f) ⫽ {1 ⫹ r32[1 ⫹ (f/fgb)2]/[1 ⫹ (f/fel)2]}/{1 ⫹ (r32/f32)[1 ⫹ (f/fgb)2]/[1 ⫹ (f/fel)2]} (12) with r32 ⫽ Rel/Rgb; f32 ⫽ fel/fgb

(13)

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Eqs. (11) and (12) show that the deviations from Eq. (5) increase with increasing relaxation frequency ratio (f32 ⫽ fel/fgb) or increasing resistance ratio r32 ⫽ Rel/Rgb. For example, on assuming f32 ⫽ 0.1, and r32 ⫽ 1 one obtains F(fgb) ⫽ 0.85, and this drops to F(fgb) ⫽ 0.40 when Rel/Rgb ⫽ 10 with an identical relaxation frequency ratio. 3.4. Grain boundary effects on the electrode contribution The method proposed in section 3.2 was extended for this case, and the relevant solution reduces to Z⬘ ⬇ RB ⫹ Rel(1 – f32) ⫹ fel(Z⬙/f)

(14)

Therefore, the slope is still close to the electrode relaxation frequency, and the frequency ratio f32 ⫽ fel/fgb can be used to obtain a correct estimate of the grain boundary resistance extracted from the intercept. 4. Simulated study cases Fig. 1a shows the impedance spectrum computed for RB ⫽ 102 ⍀, fB ⫽ 105 Hz, Rgb ⫽ 10 ⍀, fB ⫽ 104 Hz, Rel ⫽ 102⍀, fB ⫽ 102 Hz, and Fig. 1b shows the corresponding plot of Z⬘ vs. (Z⬙/f). The Nyquist plot (Z⬙ vs. Z⬘) is not suitable to distinguish a relatively small bulk resistance from the main grain boundary contribution but the alternative Z⬘ vs. (Z⬙/f) plot shows the expected changes in slope. The slopes in the grain boundary range and in the electrode range are very close to the original grain boundary and electrode relaxation frequencies respectively, as predicted by Eqs. (5) and (6). The limiting value of the slope in the high frequency range (54 kHz) is significantly lower than the bulk relaxation frequency, but the slope predicted by Eqs. (7) and (8a) is also 3

105(1 ⫹ 10 ⫻ 0.01)/(1 ⫹ 10 ⫻ 0.1) ⫽ 55 ⫻ 103 Hz The intercept obtained by extrapolation of the grain boundary branch in Fig. 1b is 86 ⍀, which is significantly lower than the bulk resistance. However, Eq. (11) predicts an intercept 102(1 – 104/105) ⫽ 90 ⍀, which nearly accounts for the difference. The intercept obtained on extrapolating the electrode branch (1.09 k⍀) is identical to the value predicted by Eq. (11) with a correction factor F(fgb) given by Eq. (12). Fig. 2a and b show the results computed for a second study case when RB ⫽ 400 ⍀, fB ⫽ 5 10 Hz, Rgb ⫽ 400 ⍀, fB ⫽ 103 Hz, Rel ⫽ 400 ⍀, and fB ⫽ 102 Hz. This example shows significant overlapping of the grain boundary and electrode branches. The slope drawn from the electrode branch (103 Hz) is close to the electrode relaxation frequency, and the intercept (748 ⍀) is also close to the value predicted by Eq. (14); this predicts 400 ⫹ 400(1 – 102/103) ⫽ 760 ⍀. The slope of the grain boundary branch (880 Hz) also differs from the corresponding relaxation frequency, but is close to the value predicted by Eqs. (11) and (12) (851 Hz). The difference between the bulk resistance, and the intercept (397 ⍀) is only about 1% because the frequency ratio fgb/fB ⫽ 0.01 is small; this agrees with Eq. (11).

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Fig. 1. Impedance spectrum computed for RB ⫽ 102 ⍀, fB ⫽ 105 Hz, Rgb ⫽ 103 ⍀, fB ⫽ 104 Hz, Rel ⫽ 102 ⍀, fB ⫽ 102 Hz, and the corresponding representation of Z⬘ vs. Z⬙/f.

Figs. 3a– c simulate the effects of grain boundaries with a significant distribution of relaxation times. This study case corresponds to RB ⫽ 1 k⍀, fB ⫽ 106 Hz, Rel ⫽ 1 k⍀, fB ⫽ 102 Hz, and the following distribution of the grain boundary resistance: (Rgb/Nt)(dN/dln(fgb)) ⫽ 10exp{–[ln(fgb/104Hz)]2} k⍀

(15)

N being the number of grain boundaries and Nt the total number of grain boundaries across the sample. The grain boundary distribution yields a depressed arc in the Nyquist representation (Fig. 3a), and a variable slope in the grain boundary range of the alternative Z⬘ vs. (Z⬙/f) plot (Fig. 3b). This variable slope is shown in Fig. 3c and represents a variable equivalent relaxation frequency: feq ⫽ dZ⬘/d(Z⬙/f)

(16)

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Fig. 2. Impedance spectrum of RB ⫽ 400 ⍀, fB ⫽ 105 Hz, Rgb ⫽ 400 ⍀, fB ⫽ 103 Hz, Rel ⫽ 400 ⍀, and fB ⫽ 102 Hz, and the corresponding representation of Z⬘ vs. Z⬙/f.

5. Impedance spectra of SrTiO3 samples Two SrTiO3 samples were characterized by impedance spectroscopy and these spectra were used to demonstrate the applicability of the proposed representation. The samples, and the experimental methods were the same as described elsewhere [8]. One of those samples (Sample A) was dense and with relatively large grain size, and the second sample (Sample B) retained a significant porosity and its average grain size was small. Fig. 4a shows the impedance spectrum obtained for sample A at 450°C, and in air; Fig. 4b shows the corresponding Z⬘ vs. (Z⬙/f) plot. Both representations show the bulk and grain boundary terms. In the high frequency range the slope of Z⬘ vs. Z⬙/f (0.78 MHz) is close to

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Fig. 3. Impedance spectrum, and representations of Z⬘ vs. Z⬙/f, and log(feq) vs. Z⬘, where feq ⫽ dZ⬘/d(Z⬙/f), computed for RB ⫽ 1 k⍀, fB ⫽ 106 Hz, Rel ⫽ 1 k⍀, fB ⫽ 102Hz, and (Rgb/Nt)[dN/dln(fgb)] ⫽ 10exp{–[ln(fgb/ 104Hz)]2} k⍀.

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Fig. 4. Impedance spectrum of a SrTiO3 sample with large grain size at 450°C, and the corresponding representations of Z⬘ vs. (Z⬙/f), and feq ⫽ dZ⬘/d(Z⬙/f) vs. Z⬘.

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the peak frequency of the bulk (0.8 MHz), and extrapolation of the second branch of Z⬘ vs. (Z⬙/f) yields an intercept at about 7.7 k⍀, which is also close to the value obtained by fitting the experimental data with a code developed by Boukamp[5]. The slope of the second branch in Fig. 4b is clearly much higher than the expected range shown in the Nyquist plot, probably because the range of values shown in Fig. 4b still shows the transition from the bulk to grain boundary branches. This is supported by the grain boundary plateau shown in Fig. 4c, which confirms that the grain boundary relaxation frequency is about 200 Hz. Fig. 4c also shows the bulk plateau. At higher temperatures the frequency range covered is insufficient to show the bulk contribution (Fig. 5a), and the corresponding plot of Z⬘ vs. (Z⬙/f) fails to show the relevant features of the bulk (Fig. 5b). However, a plot of equivalent frequency (Fig. 5c) shows the onset of the bulk contribution in the high frequency ranges and the onset of the electrode contribution in the low frequency range. The Nyquist plots obtained for sample B (Fig. 6a) show a single arc. This arc was ascribed to the resistive grain boundaries because the amplitude of the arc is orders of magnitude higher than the expected range of values of bulk resistance. Note that the plot of equivalent frequency (Fig. 6c) shows a sudden jump in the high frequency range, which corresponds to the transition from the grain boundary relaxation frequency to the bulk relaxation frequency. The decay in the low frequency range indicates the onset of the electrode contribution. Fig. 4a has already shown that the bulk relaxation frequency should be lower than 1 MHz at temperatures below 450°C. However, the Nyquist plot of Sample B fails to show the bulk contribution even at 400°C (Fig. 7a), probably due to the interference caused by the main grain boundary contribution. The bulk contribution is revealed on plotting the equivalent frequency (Fig. 7b) but this plot still fails to show the expected high frequency plateau.

6. Conclusions Plots of Z⬘ vs. (Z⬙/␻), or Z⬘ vs. (Z⬙/f), may be used to extract the resistance and the relaxation frequencies of bulk, grain boundary, and electrode contributions from impedance spectra. This new representation may be suitable even for cases when the difference between the bulk and grain boundary relaxation frequencies may not exceed one order of magnitude, thus yielding Nyquist plots with overlapped arcs. In this case, the parameters extracted from representations of Z⬘ vs. (Z⬙/f) must be corrected using suitable formulae. The representations of Z⬘ vs. (Z⬙/f) is mainly useful to characterize the grain boundary to electrode transition and to assess the electrode relaxation frequency. The derivative dZ⬘/d(Z⬙/f) may be an alternative representation for materials with a significant distribution of relaxation frequencies in the grain boundary range, and to reveal the onset of the bulk and/or electrode terms when the Nyquist plot is largely dominated by very resistive grain boundaries.

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Fig. 5. Impedance spectrum obtained for Sample A at 550°C, and the corresponding representations of Z⬘ vs. (Z⬙/f), and feq ⫽ dZ⬘/d(Z⬙/f) vs. Z⬘.

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Fig. 6. Impedance spectrum obtained for a SrTiO3 sample with small grain size (Sample B) at 500°C, and the corresponding representation of feq ⫽ dZ⬘/d(Z⬙/f) vs. Z⬘.

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Fig. 7. Impedance spectrum obtained for Sample B at 400°C, and the corresponding representation of feq ⫽ dZ⬘/d(Z⬙/f) vs. Z⬘.

Acknowledgments This work was sponsored by JNICT and Programme PRAXIS XXI, under Contract PRAXIS/3/3.1/MMA/1760/95, and by the European Commision under a network research contract 030-TMRX-CT97-0130.

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