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Electrical properties of YSZ and CaSZ single crystals
Solid State Ionics 231 (2013) 37–42
Contents lists available at SciVerse ScienceDirect
Solid State Ionics journal homepage: www.elsevier.com/locate/ssi
Electrical properties of YSZ and CaSZ single crystals Saulius Kazlauskas ⁎, Algimantas Kežionis, Tomas Šalkus, Antanas Feliksas Orliukas Vilnius University, Saulėtekio al. 9/3, LT-10222 Vilnius, Lithuania
a r t i c l e
i n f o
Article history: Received 13 July 2012 Received in revised form 25 September 2012 Accepted 22 October 2012 Available online 25 November 2012 Keywords: AC conductivity Charge carrier relaxation Distribution of relaxation times INSPEC A6630
a b s t r a c t An impedance and electric conductivity of 10 mol%Y2O3–90 mol%ZrO2 and 15 mol%CaO–85 mol%ZrO2 single crystals were investigated in frequency range from 10 Hz to 3 GHz at temperatures from room temperature to 970 K by impedance spectroscopy methods. Investigations in the high frequency range were performed by the measurement of S-parameter matrix for a 2-port network of a coaxial waveguide and a sample, inserted in the gap of a central conductor. At the low frequencies 2- and 4-electrode methods were used for accurate measurements of the impedance and electrical conductivity. The quality of the experimental data was verified using Kramers–Kronig transforms. The obtained frequency dispersion of the electrical properties related to charge carrier relaxation in the electrical field was examined in terms of distribution of relaxation times (DRT). DRT function was obtained by numerically solving the Fredholm integral equation of first kind with newly developed Matlab application. Analysis of the experimental data has revealed slight spreading of the relaxation time distribution with temperature decrease for both single crystals. The temperature dependencies of the relaxation time corresponding to the DRT function peak showed Arrhenius behavior with activation energy close to that of electrical conductivity. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Y- and Ca-stabilized zirconia (YSZ and CaSZ) are well known as typical oxygen vacancy conductors. Currently, zirconia based ceramic materials are used in a number of applications and are widely investigated. The practical application of zirconia based single crystals is much more limited and there is a limited number of studies about electrical properties of the single crystals. We mention only a few works, where the electrical properties of YSZ and CaSZ single crystals were measured. The dielectric properties of yttria-stabilized cubic zirconia 14 mol%Y2O3–86 mol%ZrO2 over a wide ranges of frequencies (from 100 to 10 10 Hz) and temperature (from 25 to 200 °C) were reported in [1]. The value of the real part of the complex dielectric permittivity ε ' ≈ 30 was found, which decreases with frequency in the microwave region. Authors of [2–4] carried out the electrical conductivity measurements of 18 mol%Y2O3–82 mol%ZrO2 and 15 mol%CaO–85 mol%ZrO2 single crystals in the frequency range from 20 Hz to 1 MHz and in the temperature interval from 450 to 1200 K. According to the authors, the shape of bulk semicircle in the complex impedance plot is almost perfect for both single crystals, which indicates narrow distribution of relaxation times. Only slightly distorted impedance arcs for YSZ single crystals of various stoichiometries are presented in [5–7]. However, most of authors performed measurements in a very limited frequency or temperature
⁎ Corresponding author. Tel.: +370 5 2366064. E-mail address: [email protected] (S. Kazlauskas). 0167-2738/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ssi.2012.10.020
ranges and thus restricted opportunities to observe DRT changes with temperature. The AC response of ionic conductors is usually described by the use of the empirical frequency domain functions associated with certain properties of unknown time domain function (for example, see [8]). However, we applied another approach and determined the DRT function directly from the frequency domain data [9–11]. In this paper we present results of the investigation of electrical properties of single crystals YSZ and CaSZ by impedance spectroscopy in the broad frequency range from 10 Hz to 3 GHz, which allowed us to observe the frequency dispersion in the wide temperature interval (from ~ 500 K to 900 K). We also present the quantitative description of frequency dispersion of electrical properties attributed to charge carrier relaxation in the electrical field in terms of DRT. The aim of this study was to determine whether the DRT function depends on the temperature. 2. Experimental In the present study 10 mol%Y2O3–90 mol%ZrO2 and 15 mol%CaO– 85 mol%ZrO2 single crystals grown by the skull melting method [2–4] were used. For the measurements of complex specific ionic conductivity ðσ˜ Þ and complex specific impedance (˜z) two different impedance spectroscopy techniques were used. Investigations in a low frequency range (from 10 Hz to 2 MHz) were performed applying a 6 mV voltage to the sample and observing the current response [12]. 2- and 4-electrode methods were used in the above mentioned frequency range, which provided experimental data with 40 points per frequency decade.
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S. Kazlauskas et al. / Solid State Ionics 231 (2013) 37–42
Table 1 The baseline data of numerical experiment. No. of curve
M
σ
Added noise, %
RMS error, %
Regularization parameter λ
Comments
1 2
−7.602 −7.602
0.33 0.33
– 0.01
– 0.35
– 0.15
3
−7.602
0.33
1
2.67
3.5
4
−7.602
0.33
1.6
2.22
6
5
−7.602
0.33
3.16
3.84
10
Initial DRT Backwards calculated DRT Backwards calculated DRT Backwards calculated DRT Backwards calculated DRT
Investigations in the high frequency range (from 0.3 MHz to 3 GHz) were based on the measurement of S-parameter matrix for a 2-port network made of a coaxial waveguide and a sample, inserted in the gap of a central conductor [13]. The amplitude of AC signal was 200 mV and 64 points per frequency decade were used. Samples of different sizes were required for distinct spectroscopy techniques. Measurements in low frequency range were performed using cylindrical samples of 1.5 mm in length and up to 4.5 mm in diameter (2-electrode method) and rectangular samples 8 × 1.5× 1.5 mm (4-electrode method). For the high frequency technique cylindrical samples of the same length and up to 2.5 mm in diameter were used. The electrodes of the samples were prepared using Pt paste (GWENT Electronic Materials LTD), fired at 1073 K, except for voltage electrodes in case of the 4-electrode method, where platinum wires turned around the sample were used. All measurements were carried out in air, the temperature was controlled by computer connected to K-type thermocouple and DC power supply. As will be shown later, the accuracy of DRT function estimation from experimental data is extremely sensitive to measurement errors. They were reduced in a variety of ways. The reliability of high frequency measurements was established by performing calibration measurement procedure as described in [13]. The determination of scattering matrix of the system during the calibration measurement procedure allowed elimination of the influence of parasitic capacity and inductance arising due to transmission line properties. The effect of non-homogeneous electric field at high frequencies was compensated in a similar way as using the model of dynamic capacitor in [14]. The obtained impedance spectra were also evaluated by means of Kramers–Kronig analysis. The calculations were performed by solving equations [15]:
zDC ¼
∞ ″ 2 z ðxÞ dx; ∫ π 0 x
ð1Þ
Fig. 2. Temperature dependencies of DC conductivity (1, 2), imaginary impedance peak frequency (3, 4) and relaxation time at DRT function peak (5, 6). Solid lines are linear fit of corresponding dependencies.
∞ ′ 2ω z ðxÞ−z′ ðωÞ ″ z ðωÞ ¼ − dx; ∫ π x2 −ω2
ð2Þ
0
which were modified to handle data sampled in the logarithmic frequency scale. In our notation ω and x are angular frequencies, z′ and z″ — real and imaginary parts of impedance, respectively, zDC — DC resistance. 3. Method of simulation In this paper we describe the method for finding DRT function, which allows us to represent the dispersive frequency response of YSZ and CaSZ ionic conductors. It was supposed that impedance spectrum z˜ðωÞ ¼ z′ ðωÞ−iz″ ðωÞ can be represented as superposition of the elements with individual relaxation times. In this model the relationship between distribution function and impedance spectrum of conducting system is specified by an equation in a similar way as in [16,17]: ∞
γ ðτÞ dτ; 1 þ iωτ 0
z˜ðωÞ ¼ z∞ þ zDC ∫
ð3Þ
where γ(τ) — DRT function, τ — relaxation time, and i — imaginary unit. z˜ðωÞ can be also considered as impedance of the circuit, where high frequency limit impedance (z∞) and distributed parallel RC = τ combinations are connected in series. Use of logarithmic time and frequency axes is more convenient taking into account available experimental data. Required expression of Eq. (3) can be obtained after substitution of variables: lgðτÞ→þ∞
z˜ðωÞ ¼ z∞ þ zDC þ∞
∫ lgðτÞ→−∞
¼ z∞ þ ∫ f ðτÞ⋅ −∞
τ⋅γðτ Þ⋅
1−iωτ dlnðτ Þ 1 þ ω2 τ 2
1−iωτ dlnðτÞ; 1 þ ω2 τ 2
ð4Þ
where f(τ) = zDC ⋅ γ(τ) ⋅ τ is the DRT function to be found. Analogous formulation of Eq. (4) for dielectric system can be found in [10]. The real and the imaginary parts of Eq. (4) can be expressed as: þ∞
′
z ðωÞ ¼ z∞ þ ∫ f ðτÞ⋅ −∞
Fig. 1. Initial and calculated backward DRT functions with different noise added, % (1 — initial, 2 — 0.01%, 3 — 1%, 4 — 1.6%, 5 — 3.16%).
″
þ∞
z ðωÞ ¼ ∫ f ðτ Þ⋅ −∞
1 dlnðτÞ; 1 þ ω2 τ 2
ωτ dlnðτ Þ: 1 þ ω2 τ 2
ð5aÞ
ð5bÞ
S. Kazlauskas et al. / Solid State Ionics 231 (2013) 37–42
39
Fig. 3. Complex plane plots of specific impedance. Solid points indicate decade frequencies; triangles represent zDC calculated using the Kramers–Kronig relation.
These equations for the DRT function f(τ) are known as Fredholm integral equations of the first kind. Finding f(τ) from Eqs. (5a) and (5b) is known as ill-posed problem, which can be solved by using the regularization methods. To solve this problem Matlab application has been developed, which uses regularization tool package [18,19]. In our application Eqs. (5a) and (5b) are sampled in the logarithmic frequency and time scale with equidistant steps: hω ¼ Δlgω;
hτ ¼ Δlgτ:
ð6Þ
So one can write: 8 N X > 1 2 2 −2 > > f k 1 þ ω1 τ k hτ ¼ 1; > > ′ > > z ω1 −z∞ k¼1 > > > > > ⋮ > > N > X > 2 2 −2 > 1 > f k 1 þ ωN τ k hτ ¼ 1; > > < z′ ω −z ∞ k¼1
N
N > > 1 X 2 2 −2 > > f k ω1 τ k 1 þ ω1 τ k hτ ¼ 1; > > > ″ > z ω1 k¼1 > > > > > ⋮ > > > N > 1 X > 2 2 −2 > > f k ωN τ k 1 þ ωN τ k hτ ¼ 1: : z″ ðω Þ N
solving Eqs. (7) and (8) revealed that in our case Tikhonov regularization method and function of regularization tool package tikhonov give the best matching of experimental and calculated results. However, this regularization method requires to set regularization parameter λ manually. The numerical experiment was performed to verify the effectiveness of this method. It was supposed to be Gaussian distributed relaxation times in the logarithmic time scale:
f ðτÞ ¼
! 1 ðlgτ−MÞ2 pffiffiffiffiffiffi exp − ; 2σ 2 σ 2π
ð9Þ
ð7Þ
k¼1
where τ1 ÷ τN, ω1 ÷ ωN — sampled relaxation time and frequency, respectively. If DC specific resistivity zDC is known, this system of 2N equations can be supplemented with one more equation: N X 1 f k hτ ¼ 1: z′ DC −z∞ k¼1
Fig. 5. Ratio of obtained by Kramers–Kronig transformation and measured imaginary part of impedance.
ð8Þ
The solutions of Eqs. (7) and (8) must satisfy inequality fk > 0. Regularization tool package used for the solution of ill-posed problem implements several regularization methods. The tests on
where M and σ are the mean and standard deviation of lgτ, respectively. During this experiment impedance versus frequency was calculated (it was supposed that z∞ = 0) using Eqs. (5a) and (5b). Measuring errors were simulated by adding Gaussian white noise to the real and imaginary parts of impedance. Finally DRT function has been calculated backwards using Matlab application and choosing regularization parameter λ. The choice of λ was based on minimal RMS (Root Mean Square) error of impedance, simulated using DRT function obtained from noisy data. The baseline data and results of our numerical experiment are presented in Table 1 and Fig. 1. The minimum RMS error and very good correspondence of initial and simulated DRT was found for λ ≅ 0.15, when 0.01% noise was added (Fig. 1 curves 1, 2). In more noisy cases the minimum of RMS error was found at its higher values and higher λ. Corresponding DRT functions are presented in Fig. 1 (curves 3, 4, and 5). Good agreement of initial and calculated results is observed. 4. Results and discussion Temperature dependencies of DC conductivity (σDC) of single crystals YSZ and CaSZ are presented in Fig. 2. σDC measurements in the temperature interval 450–700 K were performed using the 4-electrode method. The σDC at temperatures higher than 600 K was extracted from the low frequency region of the impedance data by plotting the measured impedance data on the complex z˜ plane Table 2 Activation energies of σDC (ΔEσ), νp (ΔEν) and τp (ΔEτ).
Fig. 4. Complex plane plots of specific impedance. Solid points indicate decade frequencies; triangles represent zDC calculated using the Kramers–Kronig relation.
YSZ CaSZ
ΔEσ
ΔEν
ΔEτ
1.19 1.40
1.19 1.43
1.18 1.40
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S. Kazlauskas et al. / Solid State Ionics 231 (2013) 37–42
Fig. 6. Frequency dependencies of real part of normalized impedance. Solid lines are normalized impedance frequency dependencies calculated backwards from obtained DRT functions.
Fig. 8. Frequency dependencies of real part of normalized impedance. Solid lines are normalized impedance frequency dependencies calculated backwards from obtained DRT functions.
(z″ = f(z′)) and extrapolating the data to the intersection with the real axis (Figs. 3 and 4). Both 2-electrode and 4-electrode methods showed nearly identical values of conductivity. Solid lines in Fig. 2 are least square linear fits of experimental results and represent Arrhenius behavior. The triangles in Figs. 3 and 4 indicate zDC obtained by the Kramers–Kronig transformation (Eq. (1)) of the measured z″(ω) data. The comparison of reconstructed (using Eq. (2)) and measured z″(ω) data (Fig. 5) showed acceptable quality of the experiment, except for significant deviation at low frequencies, which are related to the limited frequency range of experimental data. Activation energies of conductivity of YSZ and CaSZ single crystals are given in Table 2. These values are in sufficient agreement with the results reported by other authors (for example, 1.22 eV for YSZ [6] and 1.32 eV for CaSZ [2]). The data for DC conductivity slightly varied: the obtained value 9.1⋅10 −3 S/m compared to 9.7⋅10 −4 S/m [6] at 670 K for YSZ and 1.1⋅10 −4 S/m compared to 2.1⋅10 −4 S/m [2] at 690 K for CaSZ. Though, at a closer look, the experimental conductivity data in Fig. 2 shows a gradual change in slope towards lower activation energies at high temperatures for both single crystals. For
example, the YSZ single crystal exhibits change of activation energy from 1.23 eV (460–510 K) to 1.05 eV (870–920 K). This effect has been observed in many studies of solid oxide electrolytes [4,20,21]. Frequency dependencies of real and imaginary parts of impedance for both single crystals are shown in Figs. 6–9. The y axis of these figures represents normalized specific impedance z =zDC . The DC resistivity decreases with increasing temperature and leads to decrease in the average charge carrier relaxation time. So, the impedance dispersion region and the position of the imaginary impedance peak (z″p ) shift to higher frequencies with the increase of the temperature (Figs. 6–9). These figures also show a slight but regular increase of normalized impedance . ″ peak value zp with temperature. Thus at lower temperatures semizDC . . ″ ′ ¼f z circles of normalized impedance z are slightly more
Fig. 7. Frequency dependencies of imaginary part of normalized impedance. Solid lines are normalized impedance frequency dependencies calculated backwards from obtained DRT functions.
Fig. 9. Frequency dependencies of imaginary part of normalized impedance. Solid lines are normalized impedance frequency dependencies calculated backwards from obtained DRT functions.
zDC
zDC
depressed compared to the ones at the higher temperatures. That indicates the narrowing of DRT function distribution with increasing temperature. This phenomenon is confirmed by observing the form of normalized DRT function at different temperatures (Figs. 10 and 11).
S. Kazlauskas et al. / Solid State Ionics 231 (2013) 37–42
41
Table 3 RMS error and DRT function peak values. YSZ
Fig. 10. Normalized DRT functions at different temperatures.
The calculated DRT function is very sensitive to measurement errors. Measurement of z˜ðf Þ in full frequency range 10 Hz–3 GHz was carried out in two or three stages with the help of two sets of measurement equipment using different samples. The sample shaping, temperature measurement and other inaccuracies accompanied by transition from one method to another lead to variation of measured
Fig. 11. Normalized DRT functions at different temperatures.
CaSZ
Temperature, K
RMS error, %
lgτ at DRT peak, s
Peak value of DRT
Temperature, K
RMS error, %
lgτ at DRT peak, s
Peak value of DRT
520 560 700 750 800 850 900
1.73 0.67 0.91 0.92 0.83 0.58 0.60
−4.66 −5.47 −7.52 −8.04 −8.51 −8.88 −9.21
1.13 1.14 1.16 1.19 1.16 1.28 1.31
600 650 700 820 850 900 910
3.3 0.79 0.49 5.5 2.14 1.22 0.84
−4.04 −4.93 −5.76 −7.10 −7.38 −7.78 −7.86
1.06 1.10 1.16 1.28 1.24 1.30 1.36
specific impedance data by several percent. These errors lead to a critical increase of RMS error as DRT function is calculated from z˜ðf Þ measured by distinct methods. In Figs. 10 and 11 only those DRT functions which are calculated from z˜ðf Þ data measured by single method are shown. Therefore DRT functions in the regions about 650 K for YSZ and 750 K for CaSZ are not shown. At first sight the obtained DRT function is close to Gaussian with some low intensity peaking far from the mean lgτ. For comparison, the Gaussian curve and DRT function of CaSZ single crystal at 600 K are presented in Fig. 12. The accuracy is not good enough to define the temperature behavior of DRT function directly from the peak width. However, a slight narrowing of DRT function with increasing temperature can be linked to the increase of the DRT peak value at high temperatures. In our opinion, the distribution of the relaxation times of charge carriers may be related with irregular dopant distribution in investigated zirconia-based single crystals. These irregularities may result in microregions (their size is of the order of the Debye length) with distinct local dopant concentration, conductivity and charge carrier relaxation time. Debye length increases proportionally to the square root of the temperature and therefore increasing temperature leads to the expansion of the microregions. This leads to more uniform average local dopant concentration in individual microregions and results in narrowing of DRT function. RMS error of backwards calculated impedance and peak values of normalized DRT function at different temperatures for both single crystals are presented in Table 3. Backwards calculated frequency dependencies of the normalized impedance are shown as solid lines in Figs. 6–9. Temperature dependencies of z″ peak frequency (νp) and relaxation time at DRT function peak (τp) for both single crystals are presented in Fig. 2. Solid lines in this figure are least square linear fits of corresponding dependencies. These dependencies show Arrhenius behavior with the activation energies close to that of σDC. The activation energies of these parameters are presented in Table 2. 5. Conclusions The electrical properties of the single crystal YSZ and CaSZ in the broad frequency and temperature intervals were investigated. The observed frequency dispersion was attributed to the charge carrier relaxation in the electrical field, and examined in terms of distribution of relaxation times. It showed DRT function shape in the logarithmic time scale close to the Gaussian. The slight narrowing of DRT function with increasing temperature was also observed. The performed studies have revealed that temperature behavior of DC conductivity, z″ peak frequency and relaxation time at DRT function peak can be described by Arrhenius relationship with the same activation energy. References
Fig. 12. DRT function for CaSZ at 600 K and Gaussian distribution.
[1] M.T. Lanagan, J.K. Yamamoto, A. Bhalla, S.G. Sankar, Mater. Lett. 7 (1989) 437–440. [2] A.F. Orliukas, P. Bohac, K. Sasaki, L.J. Gauckler, J. Eur. Ceram. Soc. 12 (1993) 87–96.
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[3] A.F. Orliukas, P. Bohac, K. Sasaki, L.J. Gauckler, in: F.W. Poulsen, J.J. Bentzen, T. Jacobsen, E. Skou, M.J.L. Östergård (Eds.), Proceedings of the 14th. Risø International Symposium on Materials Science: High temperature Electrochemical Behaviour of fast Ion and Mixed Conductors, Risø National Laboratory, Dennmark, 1993, pp. 369–374. [4] A.F. Orliukas, P. Bohac, K. Sasaki, L.J. Gauckler, Solid State Ionics 72 (1994) 35–38. [5] P.S. Manning, J.D. Sirman, R.A. De Souza, J.A. Kilner, Solid State Ionics 100 (1997) 1–10. [6] A. Pimenov, J. Ullrich, P. Lunkenheimer, A. Loidl, C.H. Rüscher, Solid State Ionics 109 (1998) 111–118. [7] S. Komine, F. Munakata, J. Mater. Sci. 40 (2005) 3887–3890. [8] J.R. Dygas, Solid State Ionics 176 (2005) 2065–2078. [9] D.L. Misell, R.J. Sheppard, J. Phys. D: Appl. Phys. 6 (1973) 379–389. [10] J. Macutkevic, J. Banys, A. Matulis, Nonlinear Anal. Model. Control 9 (2004) 75–88. [11] H. Schichlein, A.C. Muller, M. Voigts, et al., J. Appl. Electrochem. 32 (2002) 875–882.
[12] A. Kežionis, E. Kazakevičius, T. Šalkus, A.F. Orliukas, in: Programme and book of abstracts: Lithuanian–French–Ukrainian Workshop on Materials for Functional Elements of Solid State Ionics, Vilnius, 2008, p. 10. [13] A. Kežionis, E. Kazakevičius, T. Šalkus, A.F. Orliukas, Solid State Ionics 188 (2011) 110–113. [14] J. Banys, S. Lapinskas, S. Rudys, S. Greicius, R. Grigalaitis, Ferroelectrics 414 (2011) 64–69. [15] D.D. Macdonald, M. Urquidi-Macdonald, J. Electrochem. Soc. 132 (1985) 2316–2319. [16] R.M. Fuoss, J.G. Kirkwood, J. Am. Chem. Soc. 63 (1941) 329–341. [17] J.R. Macdonald, M.K. Brachman, Rev. Mod. Phys. 28 (1956) 393–422. [18] P.C. Hansen, Numer. Algorithms 46 (2007) 189–194. [19] A.N. Tikhonov, DAN SSSR 153 (1963) 49–52. [20] S.P.S. Badwal, J. Mater. Sci. (1984) 1767–1776. [21] K.L. Ngai, Philos. Mag. B 77 (1998) 187–195.
Contents lists available at SciVerse ScienceDirect
Solid State Ionics journal homepage: www.elsevier.com/locate/ssi
Electrical properties of YSZ and CaSZ single crystals Saulius Kazlauskas ⁎, Algimantas Kežionis, Tomas Šalkus, Antanas Feliksas Orliukas Vilnius University, Saulėtekio al. 9/3, LT-10222 Vilnius, Lithuania
a r t i c l e
i n f o
Article history: Received 13 July 2012 Received in revised form 25 September 2012 Accepted 22 October 2012 Available online 25 November 2012 Keywords: AC conductivity Charge carrier relaxation Distribution of relaxation times INSPEC A6630
a b s t r a c t An impedance and electric conductivity of 10 mol%Y2O3–90 mol%ZrO2 and 15 mol%CaO–85 mol%ZrO2 single crystals were investigated in frequency range from 10 Hz to 3 GHz at temperatures from room temperature to 970 K by impedance spectroscopy methods. Investigations in the high frequency range were performed by the measurement of S-parameter matrix for a 2-port network of a coaxial waveguide and a sample, inserted in the gap of a central conductor. At the low frequencies 2- and 4-electrode methods were used for accurate measurements of the impedance and electrical conductivity. The quality of the experimental data was verified using Kramers–Kronig transforms. The obtained frequency dispersion of the electrical properties related to charge carrier relaxation in the electrical field was examined in terms of distribution of relaxation times (DRT). DRT function was obtained by numerically solving the Fredholm integral equation of first kind with newly developed Matlab application. Analysis of the experimental data has revealed slight spreading of the relaxation time distribution with temperature decrease for both single crystals. The temperature dependencies of the relaxation time corresponding to the DRT function peak showed Arrhenius behavior with activation energy close to that of electrical conductivity. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Y- and Ca-stabilized zirconia (YSZ and CaSZ) are well known as typical oxygen vacancy conductors. Currently, zirconia based ceramic materials are used in a number of applications and are widely investigated. The practical application of zirconia based single crystals is much more limited and there is a limited number of studies about electrical properties of the single crystals. We mention only a few works, where the electrical properties of YSZ and CaSZ single crystals were measured. The dielectric properties of yttria-stabilized cubic zirconia 14 mol%Y2O3–86 mol%ZrO2 over a wide ranges of frequencies (from 100 to 10 10 Hz) and temperature (from 25 to 200 °C) were reported in [1]. The value of the real part of the complex dielectric permittivity ε ' ≈ 30 was found, which decreases with frequency in the microwave region. Authors of [2–4] carried out the electrical conductivity measurements of 18 mol%Y2O3–82 mol%ZrO2 and 15 mol%CaO–85 mol%ZrO2 single crystals in the frequency range from 20 Hz to 1 MHz and in the temperature interval from 450 to 1200 K. According to the authors, the shape of bulk semicircle in the complex impedance plot is almost perfect for both single crystals, which indicates narrow distribution of relaxation times. Only slightly distorted impedance arcs for YSZ single crystals of various stoichiometries are presented in [5–7]. However, most of authors performed measurements in a very limited frequency or temperature
⁎ Corresponding author. Tel.: +370 5 2366064. E-mail address: [email protected] (S. Kazlauskas). 0167-2738/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ssi.2012.10.020
ranges and thus restricted opportunities to observe DRT changes with temperature. The AC response of ionic conductors is usually described by the use of the empirical frequency domain functions associated with certain properties of unknown time domain function (for example, see [8]). However, we applied another approach and determined the DRT function directly from the frequency domain data [9–11]. In this paper we present results of the investigation of electrical properties of single crystals YSZ and CaSZ by impedance spectroscopy in the broad frequency range from 10 Hz to 3 GHz, which allowed us to observe the frequency dispersion in the wide temperature interval (from ~ 500 K to 900 K). We also present the quantitative description of frequency dispersion of electrical properties attributed to charge carrier relaxation in the electrical field in terms of DRT. The aim of this study was to determine whether the DRT function depends on the temperature. 2. Experimental In the present study 10 mol%Y2O3–90 mol%ZrO2 and 15 mol%CaO– 85 mol%ZrO2 single crystals grown by the skull melting method [2–4] were used. For the measurements of complex specific ionic conductivity ðσ˜ Þ and complex specific impedance (˜z) two different impedance spectroscopy techniques were used. Investigations in a low frequency range (from 10 Hz to 2 MHz) were performed applying a 6 mV voltage to the sample and observing the current response [12]. 2- and 4-electrode methods were used in the above mentioned frequency range, which provided experimental data with 40 points per frequency decade.
38
S. Kazlauskas et al. / Solid State Ionics 231 (2013) 37–42
Table 1 The baseline data of numerical experiment. No. of curve
M
σ
Added noise, %
RMS error, %
Regularization parameter λ
Comments
1 2
−7.602 −7.602
0.33 0.33
– 0.01
– 0.35
– 0.15
3
−7.602
0.33
1
2.67
3.5
4
−7.602
0.33
1.6
2.22
6
5
−7.602
0.33
3.16
3.84
10
Initial DRT Backwards calculated DRT Backwards calculated DRT Backwards calculated DRT Backwards calculated DRT
Investigations in the high frequency range (from 0.3 MHz to 3 GHz) were based on the measurement of S-parameter matrix for a 2-port network made of a coaxial waveguide and a sample, inserted in the gap of a central conductor [13]. The amplitude of AC signal was 200 mV and 64 points per frequency decade were used. Samples of different sizes were required for distinct spectroscopy techniques. Measurements in low frequency range were performed using cylindrical samples of 1.5 mm in length and up to 4.5 mm in diameter (2-electrode method) and rectangular samples 8 × 1.5× 1.5 mm (4-electrode method). For the high frequency technique cylindrical samples of the same length and up to 2.5 mm in diameter were used. The electrodes of the samples were prepared using Pt paste (GWENT Electronic Materials LTD), fired at 1073 K, except for voltage electrodes in case of the 4-electrode method, where platinum wires turned around the sample were used. All measurements were carried out in air, the temperature was controlled by computer connected to K-type thermocouple and DC power supply. As will be shown later, the accuracy of DRT function estimation from experimental data is extremely sensitive to measurement errors. They were reduced in a variety of ways. The reliability of high frequency measurements was established by performing calibration measurement procedure as described in [13]. The determination of scattering matrix of the system during the calibration measurement procedure allowed elimination of the influence of parasitic capacity and inductance arising due to transmission line properties. The effect of non-homogeneous electric field at high frequencies was compensated in a similar way as using the model of dynamic capacitor in [14]. The obtained impedance spectra were also evaluated by means of Kramers–Kronig analysis. The calculations were performed by solving equations [15]:
zDC ¼
∞ ″ 2 z ðxÞ dx; ∫ π 0 x
ð1Þ
Fig. 2. Temperature dependencies of DC conductivity (1, 2), imaginary impedance peak frequency (3, 4) and relaxation time at DRT function peak (5, 6). Solid lines are linear fit of corresponding dependencies.
∞ ′ 2ω z ðxÞ−z′ ðωÞ ″ z ðωÞ ¼ − dx; ∫ π x2 −ω2
ð2Þ
0
which were modified to handle data sampled in the logarithmic frequency scale. In our notation ω and x are angular frequencies, z′ and z″ — real and imaginary parts of impedance, respectively, zDC — DC resistance. 3. Method of simulation In this paper we describe the method for finding DRT function, which allows us to represent the dispersive frequency response of YSZ and CaSZ ionic conductors. It was supposed that impedance spectrum z˜ðωÞ ¼ z′ ðωÞ−iz″ ðωÞ can be represented as superposition of the elements with individual relaxation times. In this model the relationship between distribution function and impedance spectrum of conducting system is specified by an equation in a similar way as in [16,17]: ∞
γ ðτÞ dτ; 1 þ iωτ 0
z˜ðωÞ ¼ z∞ þ zDC ∫
ð3Þ
where γ(τ) — DRT function, τ — relaxation time, and i — imaginary unit. z˜ðωÞ can be also considered as impedance of the circuit, where high frequency limit impedance (z∞) and distributed parallel RC = τ combinations are connected in series. Use of logarithmic time and frequency axes is more convenient taking into account available experimental data. Required expression of Eq. (3) can be obtained after substitution of variables: lgðτÞ→þ∞
z˜ðωÞ ¼ z∞ þ zDC þ∞
∫ lgðτÞ→−∞
¼ z∞ þ ∫ f ðτÞ⋅ −∞
τ⋅γðτ Þ⋅
1−iωτ dlnðτ Þ 1 þ ω2 τ 2
1−iωτ dlnðτÞ; 1 þ ω2 τ 2
ð4Þ
where f(τ) = zDC ⋅ γ(τ) ⋅ τ is the DRT function to be found. Analogous formulation of Eq. (4) for dielectric system can be found in [10]. The real and the imaginary parts of Eq. (4) can be expressed as: þ∞
′
z ðωÞ ¼ z∞ þ ∫ f ðτÞ⋅ −∞
Fig. 1. Initial and calculated backward DRT functions with different noise added, % (1 — initial, 2 — 0.01%, 3 — 1%, 4 — 1.6%, 5 — 3.16%).
″
þ∞
z ðωÞ ¼ ∫ f ðτ Þ⋅ −∞
1 dlnðτÞ; 1 þ ω2 τ 2
ωτ dlnðτ Þ: 1 þ ω2 τ 2
ð5aÞ
ð5bÞ
S. Kazlauskas et al. / Solid State Ionics 231 (2013) 37–42
39
Fig. 3. Complex plane plots of specific impedance. Solid points indicate decade frequencies; triangles represent zDC calculated using the Kramers–Kronig relation.
These equations for the DRT function f(τ) are known as Fredholm integral equations of the first kind. Finding f(τ) from Eqs. (5a) and (5b) is known as ill-posed problem, which can be solved by using the regularization methods. To solve this problem Matlab application has been developed, which uses regularization tool package [18,19]. In our application Eqs. (5a) and (5b) are sampled in the logarithmic frequency and time scale with equidistant steps: hω ¼ Δlgω;
hτ ¼ Δlgτ:
ð6Þ
So one can write: 8 N X > 1 2 2 −2 > > f k 1 þ ω1 τ k hτ ¼ 1; > > ′ > > z ω1 −z∞ k¼1 > > > > > ⋮ > > N > X > 2 2 −2 > 1 > f k 1 þ ωN τ k hτ ¼ 1; > > < z′ ω −z ∞ k¼1
N
N > > 1 X 2 2 −2 > > f k ω1 τ k 1 þ ω1 τ k hτ ¼ 1; > > > ″ > z ω1 k¼1 > > > > > ⋮ > > > N > 1 X > 2 2 −2 > > f k ωN τ k 1 þ ωN τ k hτ ¼ 1: : z″ ðω Þ N
solving Eqs. (7) and (8) revealed that in our case Tikhonov regularization method and function of regularization tool package tikhonov give the best matching of experimental and calculated results. However, this regularization method requires to set regularization parameter λ manually. The numerical experiment was performed to verify the effectiveness of this method. It was supposed to be Gaussian distributed relaxation times in the logarithmic time scale:
f ðτÞ ¼
! 1 ðlgτ−MÞ2 pffiffiffiffiffiffi exp − ; 2σ 2 σ 2π
ð9Þ
ð7Þ
k¼1
where τ1 ÷ τN, ω1 ÷ ωN — sampled relaxation time and frequency, respectively. If DC specific resistivity zDC is known, this system of 2N equations can be supplemented with one more equation: N X 1 f k hτ ¼ 1: z′ DC −z∞ k¼1
Fig. 5. Ratio of obtained by Kramers–Kronig transformation and measured imaginary part of impedance.
ð8Þ
The solutions of Eqs. (7) and (8) must satisfy inequality fk > 0. Regularization tool package used for the solution of ill-posed problem implements several regularization methods. The tests on
where M and σ are the mean and standard deviation of lgτ, respectively. During this experiment impedance versus frequency was calculated (it was supposed that z∞ = 0) using Eqs. (5a) and (5b). Measuring errors were simulated by adding Gaussian white noise to the real and imaginary parts of impedance. Finally DRT function has been calculated backwards using Matlab application and choosing regularization parameter λ. The choice of λ was based on minimal RMS (Root Mean Square) error of impedance, simulated using DRT function obtained from noisy data. The baseline data and results of our numerical experiment are presented in Table 1 and Fig. 1. The minimum RMS error and very good correspondence of initial and simulated DRT was found for λ ≅ 0.15, when 0.01% noise was added (Fig. 1 curves 1, 2). In more noisy cases the minimum of RMS error was found at its higher values and higher λ. Corresponding DRT functions are presented in Fig. 1 (curves 3, 4, and 5). Good agreement of initial and calculated results is observed. 4. Results and discussion Temperature dependencies of DC conductivity (σDC) of single crystals YSZ and CaSZ are presented in Fig. 2. σDC measurements in the temperature interval 450–700 K were performed using the 4-electrode method. The σDC at temperatures higher than 600 K was extracted from the low frequency region of the impedance data by plotting the measured impedance data on the complex z˜ plane Table 2 Activation energies of σDC (ΔEσ), νp (ΔEν) and τp (ΔEτ).
Fig. 4. Complex plane plots of specific impedance. Solid points indicate decade frequencies; triangles represent zDC calculated using the Kramers–Kronig relation.
YSZ CaSZ
ΔEσ
ΔEν
ΔEτ
1.19 1.40
1.19 1.43
1.18 1.40
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S. Kazlauskas et al. / Solid State Ionics 231 (2013) 37–42
Fig. 6. Frequency dependencies of real part of normalized impedance. Solid lines are normalized impedance frequency dependencies calculated backwards from obtained DRT functions.
Fig. 8. Frequency dependencies of real part of normalized impedance. Solid lines are normalized impedance frequency dependencies calculated backwards from obtained DRT functions.
(z″ = f(z′)) and extrapolating the data to the intersection with the real axis (Figs. 3 and 4). Both 2-electrode and 4-electrode methods showed nearly identical values of conductivity. Solid lines in Fig. 2 are least square linear fits of experimental results and represent Arrhenius behavior. The triangles in Figs. 3 and 4 indicate zDC obtained by the Kramers–Kronig transformation (Eq. (1)) of the measured z″(ω) data. The comparison of reconstructed (using Eq. (2)) and measured z″(ω) data (Fig. 5) showed acceptable quality of the experiment, except for significant deviation at low frequencies, which are related to the limited frequency range of experimental data. Activation energies of conductivity of YSZ and CaSZ single crystals are given in Table 2. These values are in sufficient agreement with the results reported by other authors (for example, 1.22 eV for YSZ [6] and 1.32 eV for CaSZ [2]). The data for DC conductivity slightly varied: the obtained value 9.1⋅10 −3 S/m compared to 9.7⋅10 −4 S/m [6] at 670 K for YSZ and 1.1⋅10 −4 S/m compared to 2.1⋅10 −4 S/m [2] at 690 K for CaSZ. Though, at a closer look, the experimental conductivity data in Fig. 2 shows a gradual change in slope towards lower activation energies at high temperatures for both single crystals. For
example, the YSZ single crystal exhibits change of activation energy from 1.23 eV (460–510 K) to 1.05 eV (870–920 K). This effect has been observed in many studies of solid oxide electrolytes [4,20,21]. Frequency dependencies of real and imaginary parts of impedance for both single crystals are shown in Figs. 6–9. The y axis of these figures represents normalized specific impedance z =zDC . The DC resistivity decreases with increasing temperature and leads to decrease in the average charge carrier relaxation time. So, the impedance dispersion region and the position of the imaginary impedance peak (z″p ) shift to higher frequencies with the increase of the temperature (Figs. 6–9). These figures also show a slight but regular increase of normalized impedance . ″ peak value zp with temperature. Thus at lower temperatures semizDC . . ″ ′ ¼f z circles of normalized impedance z are slightly more
Fig. 7. Frequency dependencies of imaginary part of normalized impedance. Solid lines are normalized impedance frequency dependencies calculated backwards from obtained DRT functions.
Fig. 9. Frequency dependencies of imaginary part of normalized impedance. Solid lines are normalized impedance frequency dependencies calculated backwards from obtained DRT functions.
zDC
zDC
depressed compared to the ones at the higher temperatures. That indicates the narrowing of DRT function distribution with increasing temperature. This phenomenon is confirmed by observing the form of normalized DRT function at different temperatures (Figs. 10 and 11).
S. Kazlauskas et al. / Solid State Ionics 231 (2013) 37–42
41
Table 3 RMS error and DRT function peak values. YSZ
Fig. 10. Normalized DRT functions at different temperatures.
The calculated DRT function is very sensitive to measurement errors. Measurement of z˜ðf Þ in full frequency range 10 Hz–3 GHz was carried out in two or three stages with the help of two sets of measurement equipment using different samples. The sample shaping, temperature measurement and other inaccuracies accompanied by transition from one method to another lead to variation of measured
Fig. 11. Normalized DRT functions at different temperatures.
CaSZ
Temperature, K
RMS error, %
lgτ at DRT peak, s
Peak value of DRT
Temperature, K
RMS error, %
lgτ at DRT peak, s
Peak value of DRT
520 560 700 750 800 850 900
1.73 0.67 0.91 0.92 0.83 0.58 0.60
−4.66 −5.47 −7.52 −8.04 −8.51 −8.88 −9.21
1.13 1.14 1.16 1.19 1.16 1.28 1.31
600 650 700 820 850 900 910
3.3 0.79 0.49 5.5 2.14 1.22 0.84
−4.04 −4.93 −5.76 −7.10 −7.38 −7.78 −7.86
1.06 1.10 1.16 1.28 1.24 1.30 1.36
specific impedance data by several percent. These errors lead to a critical increase of RMS error as DRT function is calculated from z˜ðf Þ measured by distinct methods. In Figs. 10 and 11 only those DRT functions which are calculated from z˜ðf Þ data measured by single method are shown. Therefore DRT functions in the regions about 650 K for YSZ and 750 K for CaSZ are not shown. At first sight the obtained DRT function is close to Gaussian with some low intensity peaking far from the mean lgτ. For comparison, the Gaussian curve and DRT function of CaSZ single crystal at 600 K are presented in Fig. 12. The accuracy is not good enough to define the temperature behavior of DRT function directly from the peak width. However, a slight narrowing of DRT function with increasing temperature can be linked to the increase of the DRT peak value at high temperatures. In our opinion, the distribution of the relaxation times of charge carriers may be related with irregular dopant distribution in investigated zirconia-based single crystals. These irregularities may result in microregions (their size is of the order of the Debye length) with distinct local dopant concentration, conductivity and charge carrier relaxation time. Debye length increases proportionally to the square root of the temperature and therefore increasing temperature leads to the expansion of the microregions. This leads to more uniform average local dopant concentration in individual microregions and results in narrowing of DRT function. RMS error of backwards calculated impedance and peak values of normalized DRT function at different temperatures for both single crystals are presented in Table 3. Backwards calculated frequency dependencies of the normalized impedance are shown as solid lines in Figs. 6–9. Temperature dependencies of z″ peak frequency (νp) and relaxation time at DRT function peak (τp) for both single crystals are presented in Fig. 2. Solid lines in this figure are least square linear fits of corresponding dependencies. These dependencies show Arrhenius behavior with the activation energies close to that of σDC. The activation energies of these parameters are presented in Table 2. 5. Conclusions The electrical properties of the single crystal YSZ and CaSZ in the broad frequency and temperature intervals were investigated. The observed frequency dispersion was attributed to the charge carrier relaxation in the electrical field, and examined in terms of distribution of relaxation times. It showed DRT function shape in the logarithmic time scale close to the Gaussian. The slight narrowing of DRT function with increasing temperature was also observed. The performed studies have revealed that temperature behavior of DC conductivity, z″ peak frequency and relaxation time at DRT function peak can be described by Arrhenius relationship with the same activation energy. References
Fig. 12. DRT function for CaSZ at 600 K and Gaussian distribution.
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