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Grain size-dependent electrical conductivity of polycrystalline cerium oxide
Solid State Ionics 139 Ž2001. 255–265 www.elsevier.comrlocaterssi
Grain size-dependent electrical conductivity of polycrystalline cerium oxide I. Experiments A. Tschope ¨ ) , E. Sommer 1, R. Birringer FB Physik, Gebaude Germany ¨ 43, UniÕersitat ¨ des Saarlandes, 66041 Saarbrucken, ¨ Received 27 April 2000; received in revised form 17 November 2000; accepted 18 December 2000
Abstract The electrical conductivity of polycrystalline cerium oxide was investigated in the nanometer and micrometer size range. Nanocrystalline samples of different grain size were prepared by uniaxial hot-pressing of nanocrystalline powder at various temperatures and pressures. Additional annealing at high temperatures was employed in order to obtain microcrystalline samples. An equivalent-circuit analysis of ac-impedance spectra based on the brick-layer model was performed and the apparent bulk conductivity determined. The effect of a variation in temperature or oxygen partial pressure revealed the rather different nature of the electrical transport properties in the nano- and microcrystalline materials. Nanocrystalline cerium oxide exhibited electronic conductivity under conditions at which microcrystalline samples showed impurity-controlled ionic conductivity. The electronic conductivity of nanocrystalline samples was larger than the intrinsic electronic conductivity of pure single crystalline cerium oxide and was increasing with decreasing grain size. The experimental results were analyzed according to the defect chemistry of cerium oxide and consequences of a space charge effect on the partial electronic and ionic conductivity in polycrystalline cerium oxide will be discussed. q 2001 Elsevier Science B.V. All rights reserved. PACS: 77.84.B; 73.61.T; 66.30.D; 61.72.M; 84.37; 61.72.J Keywords: Cerium oxide; Conductivity; Grain size; Nanocrystalline materials
1. Introduction Ionic conductivity of polycrystalline solid oxide electrolytes depends on the chemical composition of the material, but to a large degree also on various
) Corresponding author. Tel.: q49-681-302-5187; fax: q49681-302-5222. E-mail address: [email protected] ŽA. Tschope ¨ .. 1 Present address: Max-Planck-Institut fur ¨ Metallforschung, 70174 Stuttgart, Germany.
microstructural parameters, e.g. porosity w1,2x, and the thermal treatment during processing of a particular specimen w3x. At lower temperatures, a grain boundary effect has been observed in polycrystalline materials, which is characterized by a reduction in dc ionic conductivity with decreasing grain size, i.e. with an increasing number of grain boundaries per unit length w4x. Various studies have shown that the resistance per grain boundary increases upon segregation of impurities such as Si or Ca which form thin blocking layers within the grain boundary network w5–8x. It has also been observed that this effect can
0167-2738r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 3 8 Ž 0 1 . 0 0 6 7 8 - 6
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A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
be counteracted by doping with elements that initiate the precipitation of silicate compounds as particulates or by quenching a high-temperature state at which impurities are dissolved in the host lattice rather than segregated w3x. A third way of reducing the layer concentration of segregated impurities is the use of very small grain sizes with correspondingly large grain boundary area per unit volume, since a fixed total amount of impurities will be spread over a larger interfacial area w9x. Although the number of grain boundaries per unit length increases, the net effect on dc conductivity is smaller than for coarse-grained microstructures w10x. It can be anticipated that a particularly high fraction of grain boundaries, as present in nanocrystalline materials, should have additional consequences on charge transport properties. For instance, it has been speculated that grain boundaries may provide fast diffusion pathways for ionic defects resulting in enhanced ionic conductivity in finegrained materials w11x. In tracer diffusion studies on nanocrystalline monoclinic zirconium oxide, oxygen diffusivity in grain boundaries was found to be increased by 3–4 orders of magnitude as compared to bulk zirconia w12x. According to the Nernst–Einstein equation, a large coefficient of oxygen self-diffusion indicates enhanced ionic conductivity in the grain boundaries of monoclinic zirconia as compared to the bulk phase. However, a correlation between high grain boundary density and increased ionic conductivity is not found in general, as demonstrated by the example of cerium oxide. Cerium oxide is a well-known fast oxygen ion conductor w13x. It has been frequently studied as a model system for the class of metal oxide electrolytes that exhibit the fluorite crystal structure. In contrast to zirconia, in which the cubic structure has to be stabilized by the addition of dopants, cerium oxide does not require such stabilization and therefore allows for investigations on defect chemistry at low doping concentrations at which the law of mass action may be applied on the various defect equilibria. Cerium oxide would also be attractive for solid oxide electrolyte applications since its ionic conductivity is even higher than that of zirconia at given doping concentration and temperature. However, the propensity for reduction to oxygen deficient CeO 2y x at high temperatures and low oxygen partial pres-
sures leads to an electronic partial conductivity, which reduces the ionic transference number w14,15x. Therefore, cerium oxide may also be regarded as a model system for mixed ionicrelectronic conductors ŽMIEC. depending on the degree of reduction. In recent studies on nanocrystalline cerium oxide, predominantly electronic conductivity was observed under conditions at which microcrystalline cerium oxide exhibits ionic conductivity w16–19x. In other words, a high grain boundary density in this material results in a loss of ionic conductivity. This effect is assumed to be correlated with a depletion of acceptor dopants in the bulk as a consequence of their segregation at grain boundaries w20x. The electronic conductivity of nanocrystalline cerium oxide was increased as compared to undoped single crystalline cerium oxide, which is explained by a donor effect of grain boundaries. Such a donor effect may originate from an enhanced reduction at the grain boundaries as compared to the bulk. In fact, a reduced activation energy for the electronic conductivity as compared to the single crystal value is found and attributed to a decrease in the enthalpy of reduction of nanocrystalline CeO 2y x w17,21x. However, the interpretation of experimental results so far was based on the defect thermodynamics of the cerium oxide volume phase. The impact of space charge layers along grain boundaries on the defect chemistry has not yet been taken into account. In space charge layers, the concentrations of point defects vary from their bulk values, depending on the sign and magnitude of their charge w22–25x. The enrichment or depletion of mobile charge carriers has significant consequences on the transport properties perpendicular or parallel to grain boundaries in fine-grained materials w26,27x. Taking into account the opposite sign of the two mobile charge carriers in mixed conductive cerium oxide, i.e. oxygen vacancies Žq. and electrons Žy., space charge segregation is expected to result in an increase of one and decrease of the other partial conductivity. It is the purpose of this study to investigate the grain size-dependent electrical conductivity of cerium oxide and to compare the experimental results with a theoretical model which is based on the single crystal defect chemistry and space charge theory. In the first paper Žpaper I., the experimental investigation of electrical conductivity of cerium oxide in the
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micro- and nanometer regime is presented. Among the samples of this study, a series of Gd-doped specimens was investigated. Gadolinium was chosen because of its almost perfect match of ionic radius in the host lattice w28x. Therefore, segregation of Gd in cerium oxide should be driven mostly by Coulomb interaction rather than ion size mismatch. Bulk depletion of gadolinium in nanocrystalline CeO 2 due to segregation would therefore indicate the presence and sign of a space charge potential. The experimental results on temperature and oxygen partial pressure dependence will be analyzed in the context of the single crystal defect chemistry of cerium oxide. In a following paper Žpaper II. w29x, a theoretical model based on space charge theory is presented, which explicitly takes into account space charge segregation of acceptor ions and of the two mobile charge carriers in mixed conductive cerium oxide. The present model is an extension of an earlier model by Maier w27x in so far that it includes the treatment of doubly charged species, such as fully ionized oxygen vacancies, together with singly charged electronic carriers. A brick-layer type geometry is employed to calculate the grain size dependence of ionic and electronic partial conductivity.
2. Experimental Nanocrystalline cerium oxide powder was synthesized by thermal decomposition of CeŽIII.-acetate ŽAldrich, 99.9% or Alfa 99.99%. at 3008C for 5 h. By doping with Gd ŽHeraeus 99.9%., a mixed aqueous solution of the desired molar ratio was frozen in liquid nitrogen, and the solvent was extracted by freeze-drying to produce a mixed acetate precursor. Complete decomposition of the acetate and crystallization of cerium oxide in the fluorite structure was verified by thermogravimetric analysis and X-ray diffraction, respectively. Disk-shaped samples of 8 mm diameter and F 1 mm thickness were prepared by uniaxial hot pressing in vacuum Ž1.5 P 10y3 Pa. at various temperatures and pressures. Some of these samples were further annealed in air at 14008C in order to initiate grain growth into the micrometer size range. The geometric density of the desiccated specimen was measured by the Archimedes method using air
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and diethylphtalate as independent media. Warren– Averbach analysis of X-ray diffraction profiles was employed for characterization of the grain size in the nanometer range w30x. This analysis involved the Fourier transformation of the peak profiles of both the sample and microcrystalline reference spectra, deconvolution of instrumental line broadening and correction for strain contribution to diffraction peak broadening. The result of this analysis, the areaweighted mean column length ² L:area , is identical to the stereologic quantity of mean line intercept length and allows for the calculation of the grain boundary area per unit mass irrespective of the grain shape w31x: A gb s
2
r ² L:area
,
Ž 1.
where r denotes the density of cerium oxide Ž7.216 grcm3 .. Further details on the characterization of nanostructured materials by this method can be found in Ref. w32x. Samples that were annealed at high temperatures were polished and thermally etched at 13008C in order to determine the grain size by scanning electron microscopy. Platinum electrodes were applied to both sides of the specimens by magnetron sputtering. The ac-impedance in the frequency range v s 62.8 Hz–3.14 MHz was measured using a Quadtech 7400 analyzer. The temperature was varied between 3508C and 8008C and oxygen partial pressures in the range 10y4 –1 were established using gas mixtures of oxygen and argon. A least-square fit of a model equivalent circuit was obtained for each measurement using the following impedance function: Z Ž v . s Z1 Ž v . q Z 2 Ž v . s
R1 1 q Ž i vt 1 .
g1
q
R2 1 q Ž i vt 2 .
g2
Ž 2.
with the time constants t i s R i Ci . Eq. Ž2. describes the impedance of two ZARC—Cole distributed elements Z1Ž v . and Z2 Ž v . in series w33x. The two components will be assigned to various elements of the microstructure in the later discussion. For exponents g i s 1, the quantities R i and Ci can be associ-
A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
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Fig. 1. Nyquist plot of the complex impedance of microcrystalline cerium oxide, measured at 3508C and pO 2 r p 0 s10y4 and the result of a least-square fit. The numbers at the graph indicate the frequency logŽ v ..
Fig. 2. Nyquist plot of the complex impedance of nanocrystalline cerium oxide, measured at 4608C and pO 2 r p 0 s10y4 and the result of a least-square fit. The numbers at the graph indicate the frequency logŽ v ..
ated with an ideal resistor and capacitor in parallel, respectively.
values for the fit-parameters are summarized in Table 1. A nanocrystalline sample was prepared by uniaxial hot-pressing in vacuum at 6458C and 700 MPa for 90 min. Considerable reduction of the sample was evident from a dark coloration. Therefore, reoxidation by an annealing treatment for 15 h at 5508C in air was performed. The sample exhibited a density of 92.1% of theoretical and ² L:area s 26 nm. A typical impedance spectrum of this sample, measured at 4608C and pO 2rp 0 s 10y4 is shown in Fig. 2, together with the least-square fit of the equivalent circuit. The resulting values for the various fitparameters are also given in Table 1. Impedance spectra of polycrystalline ion conductors are commonly analyzed using the brick-layer model w34x. With regard to this model, we associate the high-frequency semi-circle, characterized by R 1 ,
3. Results A microcrystalline cerium oxide sample was prepared by uniaxial hot-pressing at 6558C and 330 MPa for 100 min followed by annealing in air at 14008C for 7 h. The grain size of the sample was 10 mm and a density of 97.1% of theoretical was determined. The impedance spectrum, measured at 3508C and pO 2rp 0 s 10y4 Ž p 0 s 1 atm s 1.013 P 10 5 Pa. is shown in Fig. 1. A small semi-circle was found in the high frequency range and was well discernible from the very large impedances at low frequencies. A least-square fit of the equivalent circuit impedance function Ž2. is also shown in Fig. 1, and the resulting
Table 1 Results of a least square fit of measured impedance spectra with the impedance function given by Eq. Ž2. for microcrystalline ŽFig. 1. and nanocrystalline ŽFig. 2. cerium oxide Sample Microcrystalline Nanocrystalline
R1 w V x
C1 wFx 4
7.4 P 10 3.6 P 10 5
g1 y11
1.9 P 10 2.3 P 10y11
0.95 0.95
R2 w V x 1.33 P 10 3.9 P 10 5
C2 wFx 8
g2 y9
5.0 P 10 2.4 P 10y10
0.92 0.69
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Table 2 Parameter of uniaxial hot-pressing and resulting density and grain size for various samples
Fig. 3. Bulk conductivity of micro- and nanocrystalline cerium oxide at 5008C as a function of oxygen partial pressure.
C1 , and g 1 , with the bulk impedance of the cerium oxide crystallites. This preliminary assignment, in which the contribution of grain boundaries parallel to the direction of current is neglected, as well as the origin of the low frequency arc will be further discussed below and in paper II w29x. From the resistance R 1 and the thickness d and cross-section A of the sample, the bulk conductivity s bulk s Ž drA. was calculated. A very distinct difference Ry1 1
Fig. 4. Temperature dependence of bulk conductivity of microand nanocrystalline cerium oxide at pO 2 r p 0 s10y4 .
Temperature w8Cx
Pressure wMPax
Density w% of theoreticalx
² L:area wnmx
700 720 740 780 800
780 740 600 600 600
91.8 92.6 94.8 96.5 97.2
24.7 26.4 33.7 39.3 44.0
between the micro- and nanocrystalline sample was observed in the oxygen partial pressure dependence of s bulk ŽFig. 3.. Whereas the bulk conductivity of the microcrystalline sample was virtually independent of pO 2 in the measured range, the conductivity of the nanocrystalline sample increased with decreasing oxygen partial pressure. The partial pressure dependence apparently followed a power-law with an exponent close to y1r6. It is also important to notice that the absolute value of electrical conductivity was 1–2 orders of magnitude lower for the nanocrystalline sample as compared to the microcrystalline sample. This implies that the partial ionic conductivity must be reduced by at least 1–2 orders of magnitude in the nanocrystalline sample.
Fig. 5. Bulk conductivity of 0.2 at.% Gd-doped cerium oxide at 5008C and pO 2 r p 0 s1 as function of grain boundary area per unit mass. The data point a A gb s 0 m2 rg corresponds to the electronic conductivity of undoped single crystalline cerium oxide.
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The temperature dependence of s bulk also exhibited different behavior for the two samples ŽFig. 4.. The activation energies, determined from the Arrhenius plot were 1.7 and 0.71 eV for nano- and microcrystalline cerium oxide, respectively. Various samples of 0.2 at.% Gd-doped cerium oxide were prepared as described above using the high-purity ceriumŽIII.acetate precursor. The conditions of uniaxial hot-pressing and the results of grain size and density characterization are summarized in Table 2. In Fig. 5, the bulk conductivity of these samples was plotted against the specific grain boundary area, which was calculated using Eq. Ž1.. The conductivity was found to increase with increasing grain boundary area, or in other words, with decreasing grain size.
4. Discussion The impedance spectrum of microcrystalline cerium oxide ŽFig. 1. exhibited a distinct semi-circle at high frequencies and a second and very large arc is observed at lower frequencies. The measured impedance spectrum is typical for polycrystalline ion conductors in which the ionic current is partially blocked by grain boundaries. The brick-layer model is commonly used to find a correlation between the components of an equivalent circuit and physical quantities characteristic of the material w34x. The high frequency semi-circle is assigned to the bulk impedance of the cerium oxide crystallites, based on the capacitance of C1 s 1.9 P 10y1 1 F, which corresponds to a reasonable value of e s 26 for the dielectric constant of cerium oxide w35x. In contrast, the value of C2 s 5.0 P 10y9 F reflects the rather high capacity of the thin grain boundary layers perpendicular to the electric field. A contribution to the high-frequency circle by grain boundaries parallel to the electric field can be neglected due to their large resistance with respect to the bulk phase. With the values of g 1 s 0.95 and g 2 s 0.92, the depression of both semi-circles was small. Variation of the oxygen partial pressure between 10y4 and 1 atm was found to have no effect on the bulk conductivity at 5008C ŽFig. 3., implying that impurity-controlled ionic conductivity was present. This means that the oxygen vacancy concentration was fixed by charge compen-
sation of acceptor cations, such as La3q, which is a typical impurity in cerium oxide. Assuming only trivalent acceptors at a concentration of w AXCe x, the oxygen vacancy concentration is given by: 1
w AXCe x . Ž 3. 2 With the mobility of oxygen vacancies m V O¨ and electron charge e 0 , the resulting extrinsic ionic conductivity: VO¨ s
s ionic Ž T . s 2e 0 m V O¨ Ž T . VO¨
Ž 4.
is independent of oxygen partial pressure. To a first approximation, the mobility of oxygen vacancies is the only temperature-dependent factor in Eq. Ž4.. Hence, the activation energy of ionic conductivity is identical to the activation energy of the vacancy mobility. From the temperature dependence of bulk conductivity ŽFig. 4., an activation energy of 0.71 eV was obtained, which is in agreement with values reported in literature w35–37x. Further analysis aiming at a quantitative expression for the absolute value of oxygen vacancy mobility requires a more detailed treatment of ionic charge transport in acceptor-doped cerium oxide. A number of studies clearly demonstrate, that a significant elastic and electrostatic interaction between acceptor ions AXCe and oxygen vacancies VO¨ causes the formation of associated pairs Ž AXCe VO¨ .P w35–37x. The association reaction: AXCe q VO¨ | Ž AXCeVO¨ .
P
Ž 5.
can be taken into account by applying the corresponding law of mass-action: P
GA0 Ž AXCe VO¨ . s K T s 4 exp y AŽ . VO¨ w AXCe x k BT
ž /
Ž 6.
with the standard Gibbs free energy of association GA0 s HA0 y TSA0 w35x. As a consequence of association, not all oxygen vacancies that are generated for charge compensation do necessarily contribute to ionic conductivity and the effective concentration of mobile oxygen vacancies is dependent on temperature. A general expression for the temperature dependence of ionic conductivity of cerium oxide can be written as:
s ionic Ž T . s VO¨ Ž T .
A T
ž
exp y
Hm k BT
/
Ž 7.
A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
with an activation energy of vacancy mobility Hm s 0.61 eV and the pre-exponential A s 4 P 10 5 KS cmy1 w35x. At low temperatures, almost complete association can be anticipated and the activation energy, as determined from the temperature dependence of conductivity is the sum of 1 2HA and Hm . At high temperatures, the measured activation energy represents only the mobility term, since the concentration of associated pairs can be neglected. The analysis of our data with a result of 0.71 eV indicated that association should not be neglected. Wang et al. w35x suggest to fit the experimental data of the temperature-dependent conductivity with the enthalpy of association and the total acceptor concentration as fitting parameters Ž SA is assumed to be negligible.. This was performed on our data and the line in Fig. 4 through the data points of the microcrystalline sample is the result of such a fit with an enthalpy of association HA s 0.31 eV and a total acceptor concentration of 1340 ppm. This concentration seemed reasonable as the chemical analysis of the ceriumŽIII.acetate ŽAldrich, 99.9%. provided by the manufacturer, yielded 2000 ppm maximum total metallic impurities. In the further discussion, the association of oxygen vacancies will not be explicitly taken into account. Alternatively, an expression for the temperature-dependent electrical conductivity, as given in Eq. Ž4., can be obtained by considering all, i.e. dissociated and associated oxygen vacancies as mobile charge carriers with a reduced effective mobility, in which the average residence time in an associated state is included. The total oxygen vacancy concentration is controlled by charge compensation of the acceptor impurities through Eq. Ž3. so that the effective mobility can be evaluated from the measured electrical conductivities using Eq. Ž4.. The result:
m V O¨ Ž T . s
m0V O¨ T
ž
exp y
0.71 eV k BT
/
Ž 8.
with m0V O¨ s 1.5 P 10 2 cm2 KVy1 sy1 is a reasonable approximation, since the temperature dependence of conductivity in Fig. 4 follows almost a straight line. However, it should be noticed that this effective mobility is also dependent on the total acceptor concentration since the association enthalpy tends to increase with the acceptor concentration in the dilute limit w35x.
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As a test for consistency, we calculate the residual electronic partial conductivity of cerium oxide:
sel Ž T . s e 0 mel Ž T . n Ž T .
Ž 9.
for the given acceptor concentration. The concentration of electrons and oxygen vacancies are related through the external reductionroxidation equilibrium with the gas phase: VO¨ n2 pO1r2 2 OOx
ž
s K R Ž T . s K R0 exp y
HR k BT
/
.
Ž 10 .
With the equilibrium constant K R0 s 3.73 P 10 6 atmy1 r2 , H R s 4.67 eV w14x and the estimated acceptor concentration given above, the concentration of electronic charge carriers can be calculated. By using the electron mobility w38x:
me Ž T . s
me0 T
ž
exp y
0.4 eV k BT
/
cm2 Vs
Ž 11 .
with me0 s 3.9 P 10 2 cm2 KVy1 sy1 , a partial electronic conductivity at 5008C of 3.8 P 10y1 0 S cmy1 was obtained.2 This value was much lower than the measured conductivity of 2.5 P 10y5 S cmy1 indicating that the assumption of a predominant ionic conductivity is consistent with the defect chemistry of cerium oxide. It should be noticed that cerium oxide is a small-polaron electron conductor w38x with an electron mean free path of an atomic distance. This may justify the assumption of an electron mobility in the cerium oxide bulk phase, which is hardly affected by the microstructure but dependent on temperature only. As shown in this discussion, the present experimental results on the electrical conductivity of microcrystalline cerium oxide are consistent with the defect chemistry of single crystalline cerium oxide. In the following section, the results obtained for nanocrystalline cerium oxide of the same chemical composition will be discussed. The impedance spectrum of the nanocrystalline cerium oxide sample exhibited two overlapping semi-circles Žsee Fig. 2..
2
Calculation of the electrical conductivity from molar concentrations of charge carriers requires a value for the number of CeO 2 formula units per unit volume. This quantity was obtained from the density and molar mass of cerium oxide as 2.525P10 22 cmy3 .
A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
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Again, the circle at high frequencies was attributed to the bulk impedance based on the value of the capacitance C1. As a first major difference with respect to the microcrystalline sample, the resistance of the second semi-circle R 2 was comparable in size to R 1 of the bulk semi-circle and the depression of the low-frequency circle was also much larger. Obviously, the grain boundary effect in nanocrystalline cerium oxide was much smaller. Taking into account the high density of grain boundaries in nanocrystalline samples, the specific resistance per grain boundary was very low, similar to reported values in literature w17x. More important was the observation of a clear dependence of the ‘bulk’ conductivity on the oxygen partial pressure, which was close to a y1r6 power law ŽFig. 3.. Such a power-law dependence is expected if the charge neutrality equation is reduced to the Brouwer approximation: 2 VO¨ s n
Ž 12 .
and the concentrations of both electrons and oxygen vacancies are controlled by the external equilibrium with the gas phase, Eq. Ž10.. Although the concentrations of both defects are of the same order of magnitude, as evident from Eq. Ž12., electronic conductivity prevails over ionic conductivity due to a higher mobility of the electrons Žcompare Eqs. Ž8. and Ž11... An activation energy of 1.7 eV was obtained by analyzing the temperature dependence of electrical conductivity ŽFig. 4.. Two factors in Eq. Ž9. for the electronic conductivity are temperature-dependent, the electron mobility and the electron density. By inserting Eqs. Ž10. – Ž12. into Eq. Ž9., an apparent activation energy for single crystalline cerium oxide of: Eapp ,sx s Em e q
1 3
H R s 1.98 eV
Ž 13 .
is expected. The measured value of 1.7 eV for nanocrystalline cerium oxide was lower, which could be due to a reduced hopping energy andror a reduced H R . The absolute value of the overall electrical conductivity of nanocrystalline cerium oxide was found to be increased by a factor of 3 if compared to the electronic conductivity of 1 P 10y7 S cmy1 for undoped single crystalline cerium oxide. The latter
was calculated using Eqs. Ž9. – Ž12., with w AXCe x s 0, T s 5008C and pO 2rp 0 s 1. The enhanced electronic conductivity of nanocrystalline cerium oxide is consistent with the lower apparent activation energy. On the one hand, a reduced hopping energy would translate into a higher electron mobility. On the other hand, a reduced enthalpy of the external reductionroxidation equilibrium would result in a larger concentration of electronic charge carriers. The present experimental data do not allow for a separation of these effects. However, previous studies reported activation energies even below 1.0 eV, which could obviously not be explained by a reduction of the hopping energy alone w16–19x. Hence, a significant decrease of the enthalpy of redox equilibrium seems to play the more important role. The increased electronic conductivity in nanocrystalline cerium oxide has also been reported in literature. If compared with the intrinsic value of undoped cerium oxide, the electronic conductivity of the nanocrystalline sample in Ref. w17x was enhanced by a factor of 21 at T s 5008C. The analysis of the present results indicates, that bulk conductivity of the nanocrystalline cerium oxide sample is controlled by the external equilibrium and dominated by the electronic contribution. It is two orders of magnitude lower than the ionic conductivity measured for the polycrystalline sample, but larger than the electronic conductivity of pure single crystalline cerium oxide. Comparing the results on the electrical conductivity of microcrystalline and nanocrystalline cerium oxide, the most significant aspect is the change from impurity-controlled ionic to electronic conductivity controlled by the external equilibrium. It is important to notice that this transition was not due to a change in the overall chemical composition, since care was taken that only specimens from the same batch of powder were compared. The loss of acceptor-controlled ionic conductivity upon reduction in grain size can be readily explained by an enhanced segregation of acceptor impurities to the grain boundaries in the nanocrystalline material. While maintaining the overall chemical composition, segregation of acceptor cations would decrease the bulk concentration as the grain boundary area per unit mass increases, i.e. the grain size decreases w20x. This effect of acceptor segregation will now be elucidated in more detail.
A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
A major driving force for segregation of acceptor cations is the size mismatch between the impurity and host ions which is more easily accommodated in the open atomic arrangement of grain boundaries. However, enrichment of aliovalent cations such as acceptors in the space charge layer along grain boundaries due to Coulomb interaction is a second possible mechanism for segregation. The latter effect of space charge segregation should dominate if the acceptor ions exhibit only a small size mismatch with respect to the Ce 4q host, which is the case for Gd 3q. In other words, if the loss of ionic conductivity is also observed in Gd-doped nanocrystalline cerium oxide, this effect could not be explained by a segregation due to size mismatch. However, the segregation of gadolinium would indicate the presence and sign of a space charge potential. A set of 0.2 at.% Gd-doped samples with different grain sizes were prepared and the bulk conductivity was determined. A microcrystalline sample from that powder batch with a grain size of 1 mm exhibited a bulk conductivity of 1 P 10y4 S cmy1 at 5008C as expected for this acceptor concentration. The bulk conductivity of the nanocrystalline samples as function of grain boundary area per unit mass is shown in Fig. 5. The calculated conductivity of acceptor-free single crystalline cerium oxide Ž A gb s 0 m2rg. is also included. Very similar to the measurements discussed above, the bulk conductivity of the Gd-doped nanocrystalline cerium oxide was two orders of magnitude lower than that of the microcrystalline sample and bulk conductivity increased with increasing grain boundary area per unit mass, i.e. with decreasing grain size. The loss of ionic conductivity in the nanocrystalline sample could again be explained by the segregation of Gd acceptor ions. However, the size mismatch between Gd 3q and Ce 4q is too small to provide sufficient driving force for such an extended segregation. As discussed above, segregation of negatively charged Gd acceptor ions provides strong evidence for the presence of a space charge layer at grain boundaries. Because space charge layers inevitably influence the concentration of all charged point defects, in particular the mobile charge carriers, the important question arises, whether other experimental results such as the enhancement of electronic conductivity or the reduced activation energy of electronic conductivity could also be ex-
263
plained by the space charge effect. Accumulation of Gd in the space charge layer implies a positive excess charge at the grain boundaries. The same space charge potential results in the accumulation of electrons and the depletion of oxygen vacancies in the space charge layer. Upon reduction of the grain size, the space charge effect on the charge carrier concentrations would result in a decreased ionic and increased electronic partial conductivity, which is in fact the central observation of the present study. This effect was also considered by Lohwasser et al. w39x in the discussion on the electrical conductivity of cerium oxide thin films. An important point to notice is that a grain size of 20 nm is the same order of magnitude as the screening length l, which characterizes the width of space charge layers. Therefore, a contribution to the ‘bulk’ conductivity by grain boundaries parallel to the current must not be neglected. In fact, the overlap of space charge layers may result in a flat band situation, in which the conductivity is homogeneous over the whole crosssection of the crystallites. With regards to this discussion, it seems appropriate to develop a model for the effect of space charge layers on the grain size dependence of mixed ionicrelectronic conductivity in polycrystalline cerium oxide. Such a model will be presented in paper II w29x. So far, the discussion of the present measurements was limited to the ‘bulk’ conductivity, which was calculated from the resistance R 1 of the highfrequency semi-circle in the impedance spectra. We have also determined the apparent activation energies associated with the grain boundary effect, using R 2 . For the microcrystalline sample, the activation energy was found to be 1.9 eV, which is a little larger than the value of 1.7 eV reported in literature w5x, yet, this value was much larger than that of the bulk ionic conductivity Ž0.71 eV. with the consequence that the grain boundary effect becomes negligible at temperatures above 7008C. One possible reason for such a large difference in activation energies of bulk and grain boundary conductivity is the presence of a continuous network of impurity-rich grain boundaries. The grain boundary effect was much smaller in the nanocrystalline samples than in the microcrystalline specimens, indicating that the impurities were diluted by spreading over a much larger grain boundary area per unit volume. We also
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A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
determined the activation energy of the apparent grain boundary conductance as 1.7 eV, which was identical to the value of the ‘bulk’ conductivity. Hence, scattering of electrons upon crossing the grain boundaries can be ruled out as origin of the second semi-circle. This second arc at low frequencies could also originate from a depletion of charge carriers in space charge layers along grain boundaries. However, this effect requires that the grain size is sufficiently large as compared to the screening length l, which is not the case as shown in Ref. w29x. We assume that a constriction effect due to the residual porosity w1x or inhomogeneous contacts at the electrodes w40x of the nanocrystalline specimen is the origin of this second semi-circle.
5. Conclusions The effect of grain size on the electrical conductivity of polycrystalline cerium oxide was investigated. Nanocrystalline specimens exhibited electronic conductivity under conditions at which microcrystalline samples showed impurity-controlled ionic conductivity. The electronic conductivity was larger than the intrinsic electronic conductivity of pure single crystalline cerium oxide and was increasing with decreasing grain size. The loss of impuritycontrolled ionic conductivity was also observed in Gd-doped samples, which is indicative of a significant space charge effect related to a positive grain boundary excess charge. As a consequence of the accumulationrdepletion of electronicrionic mobile charge carriers in the space charge layers, the partial electronicrionic conductivity increasedrdecreased with increasing grain boundary area per unit mass, i.e. with decreasing grain size. In a following paper, an analytical model for the conductivity of a polycrystalline mixed ionicrelectronic conductor will be presented, which explicitly takes into account the effect of space charge layers and grain size on the partial conductivities.
Acknowledgements The authors acknowledge technical assistance by A. Kohm-Witt and financial support by the German ¨
National Science Foundation ŽDeutsche Forschungsgemeinschaft, Sonderforschungsbereich 277 ‘Grenzflachenbestimmte Materialien’.. ¨
References w1x A.D. Brailsford, D.K. Hohnke, Solid State Ionics 11 Ž1983. 133. w2x M. Kleitz, L. Dessemond, M.C. Steil, Solid State Ionics 75 Ž1995. 107. w3x C.S. Chen, M.M.R. Boutz, B.A. Boukamp, A.J.A. Winnubst, K.J. de Vries, A.J. Burggraaf, Mater. Sci. Eng., A 168 Ž1993. 231. w4x S.J. Hong, K. Mehta, A.V. Virkar, J. Electrochem. Soc. 145 Ž1998. 638. w5x R. Gerhardt, A.S. Nowick, J. Am. Ceram. Soc. 69 Ž1986. 641. w6x R. Gerhardt, A.S. Nowick, M.E. Mochel, I. Dumler, J. Am. Ceram. Soc. 69 Ž1986. 647. w7x D.Y. Wang, A.S. Nowick, J. Solid State Chem. 35 Ž1980. 325. w8x M.J. Verkerk, B.J. Middelhuis, A.J. Burggraaf, Solid State Ionics 6 Ž1982. 159. w9x M. Aoki, Y.-M. Chiang, I. Kosacki, L.J.-R. Lee, H.L. Tuller, Y. Liu, J. Am. Ceram. Soc. 79 Ž1996. 1169. w10x J.H. Hwang, D.S. McLachlan, T.O. Mason, J. Electroceram. 3 Ž1999. 7. w11x D. Vaßen, D. Stover, L.G.J. de Haart, M. Cappadonia, Br. ¨ Ceram. Proc. 56 Ž1996. 35. w12x U. Brossmann, R. Wurschum, U. Sodervall, H.-E. Schaefer, ¨ ¨ J. Appl. Phys. 85 Ž1999. 7646. w13x H.L. Tuller, A.S. Nowick, J. Electrochem. Soc. 122 Ž1975. 255. w14x H.L. Tuller, A.S. Nowick, J. Electrochem. Soc. 126 Ž1979. 209. w15x H.L. Tuller, in: T.O. Sørensen ŽEd.., Nonstoichiometric Oxides, Academic Press, New York, 1981, 271. w16x A. Tschope, ¨ J.Y. Ying, H.L. Tuller, Sens. Actuators, B 31 Ž1992. 111. w17x Y.-M. Chiang, E.B. Lavik, I. Kosacki, H.L. Tuller, J.Y. Ying, Appl. Phys. Lett. 69 Ž1996. 185. w18x J.H. Hwang, T.O. Mason, Z. Phys. Chem. 207 Ž1998. 21. w19x I. Kosacki, T. Suzuki, H.U. Anderson, in: E.D. Wachsman, M.-L. Liu, J.R. Akridge, N. Yamazoe ŽEds.., Solid State Ionic Devices, ECS Proceedings, vol. 99-13, Electrochemical Society, Pennington, 1999, p. 190. w20x Y.-M. Chiang, E.B. Lavik, D.A. Blom, Nanostruct. Mater. 9 Ž1997. 633. w21x Y.-M. Chiang, E.B. Lavik, I. Kosacki, H.L. Tuller, J.Y. Ying, J. Electroceram. 1 Ž1997. 7. w22x J. Frenkel, Kinetic Theory of Liquids, Oxford Univ. Press, New York, 1946. w23x K. Lehovec, J. Chem. Phys. 21 Ž1953. 1123.
A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265 w24x K.L. Kliewer, J.S. Koehler, Phys. Rev. 140 Ž1965. 1226. w25x J. Jamnik, J. Maier, S. Pejovnik, Solid State Ionics 75 Ž1995. 51. w26x J. Maier, Ber. Bunsenges. Phys. Chem. 88 Ž1984. 1057. w27x J. Maier, Ber. Bunsenges. Phys. Chem. 90 Ž1986. 26. w28x J.A. Kilner, in: R. Metselaar, H.J.M. Heijligers, J. Schoonman ŽEds.., Solid State Chemistry, Elsevier, Amsterdam, 1983, p. 189. w29x A. Tschope, ¨ Solid State Ionics 139 Ž2001. 267. w30x B.E. Warren, B.L. Averbach, J. Appl. Phys. 21 Ž1950. 595. w31x R.T. DeHoff, in: G.F. Van der Voort ŽEd.., Applied Metallography, Van Nostrand, New York, 1986, p. 89. w32x C.E. Krill, R. Birringer, Philos. Mag. A 77 Ž1998. 621.
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w33x R.L. Hurt, J.R. MacDonald, Solid State Ionics 20 Ž1986. 111. w34x J.E. Bauerle, J. Phys. Chem. Solids 30 Ž1969. 2657. w35x D.Y. Wang, D.S. Park, J. Griffith, A.S. Nowick, Solid State Ionics 2 Ž1981. 95. w36x R. Gerhardt-Anderson, A.S. Nowick, Solid State Ionics 5 Ž1981. 547. w37x V. Butler, C.R.A. Catlow, B.E.F. Fender, J.H. Harding, Solid State Ionics 8 Ž1983. 109. w38x H.L. Tuller, A.S. Nowick, J. Phys. Chem. Solids 38 Ž1977. 859. w39x W. Lohwasser, J. Gerblinger, U. Lampe, H. Meixner, J. Appl. Phys. 75 Ž1994. 3991. w40x J. Fleig, J. Maier, J. Electroceram. 1 Ž1997. 73.
Grain size-dependent electrical conductivity of polycrystalline cerium oxide I. Experiments A. Tschope ¨ ) , E. Sommer 1, R. Birringer FB Physik, Gebaude Germany ¨ 43, UniÕersitat ¨ des Saarlandes, 66041 Saarbrucken, ¨ Received 27 April 2000; received in revised form 17 November 2000; accepted 18 December 2000
Abstract The electrical conductivity of polycrystalline cerium oxide was investigated in the nanometer and micrometer size range. Nanocrystalline samples of different grain size were prepared by uniaxial hot-pressing of nanocrystalline powder at various temperatures and pressures. Additional annealing at high temperatures was employed in order to obtain microcrystalline samples. An equivalent-circuit analysis of ac-impedance spectra based on the brick-layer model was performed and the apparent bulk conductivity determined. The effect of a variation in temperature or oxygen partial pressure revealed the rather different nature of the electrical transport properties in the nano- and microcrystalline materials. Nanocrystalline cerium oxide exhibited electronic conductivity under conditions at which microcrystalline samples showed impurity-controlled ionic conductivity. The electronic conductivity of nanocrystalline samples was larger than the intrinsic electronic conductivity of pure single crystalline cerium oxide and was increasing with decreasing grain size. The experimental results were analyzed according to the defect chemistry of cerium oxide and consequences of a space charge effect on the partial electronic and ionic conductivity in polycrystalline cerium oxide will be discussed. q 2001 Elsevier Science B.V. All rights reserved. PACS: 77.84.B; 73.61.T; 66.30.D; 61.72.M; 84.37; 61.72.J Keywords: Cerium oxide; Conductivity; Grain size; Nanocrystalline materials
1. Introduction Ionic conductivity of polycrystalline solid oxide electrolytes depends on the chemical composition of the material, but to a large degree also on various
) Corresponding author. Tel.: q49-681-302-5187; fax: q49681-302-5222. E-mail address: [email protected] ŽA. Tschope ¨ .. 1 Present address: Max-Planck-Institut fur ¨ Metallforschung, 70174 Stuttgart, Germany.
microstructural parameters, e.g. porosity w1,2x, and the thermal treatment during processing of a particular specimen w3x. At lower temperatures, a grain boundary effect has been observed in polycrystalline materials, which is characterized by a reduction in dc ionic conductivity with decreasing grain size, i.e. with an increasing number of grain boundaries per unit length w4x. Various studies have shown that the resistance per grain boundary increases upon segregation of impurities such as Si or Ca which form thin blocking layers within the grain boundary network w5–8x. It has also been observed that this effect can
0167-2738r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 3 8 Ž 0 1 . 0 0 6 7 8 - 6
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A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
be counteracted by doping with elements that initiate the precipitation of silicate compounds as particulates or by quenching a high-temperature state at which impurities are dissolved in the host lattice rather than segregated w3x. A third way of reducing the layer concentration of segregated impurities is the use of very small grain sizes with correspondingly large grain boundary area per unit volume, since a fixed total amount of impurities will be spread over a larger interfacial area w9x. Although the number of grain boundaries per unit length increases, the net effect on dc conductivity is smaller than for coarse-grained microstructures w10x. It can be anticipated that a particularly high fraction of grain boundaries, as present in nanocrystalline materials, should have additional consequences on charge transport properties. For instance, it has been speculated that grain boundaries may provide fast diffusion pathways for ionic defects resulting in enhanced ionic conductivity in finegrained materials w11x. In tracer diffusion studies on nanocrystalline monoclinic zirconium oxide, oxygen diffusivity in grain boundaries was found to be increased by 3–4 orders of magnitude as compared to bulk zirconia w12x. According to the Nernst–Einstein equation, a large coefficient of oxygen self-diffusion indicates enhanced ionic conductivity in the grain boundaries of monoclinic zirconia as compared to the bulk phase. However, a correlation between high grain boundary density and increased ionic conductivity is not found in general, as demonstrated by the example of cerium oxide. Cerium oxide is a well-known fast oxygen ion conductor w13x. It has been frequently studied as a model system for the class of metal oxide electrolytes that exhibit the fluorite crystal structure. In contrast to zirconia, in which the cubic structure has to be stabilized by the addition of dopants, cerium oxide does not require such stabilization and therefore allows for investigations on defect chemistry at low doping concentrations at which the law of mass action may be applied on the various defect equilibria. Cerium oxide would also be attractive for solid oxide electrolyte applications since its ionic conductivity is even higher than that of zirconia at given doping concentration and temperature. However, the propensity for reduction to oxygen deficient CeO 2y x at high temperatures and low oxygen partial pres-
sures leads to an electronic partial conductivity, which reduces the ionic transference number w14,15x. Therefore, cerium oxide may also be regarded as a model system for mixed ionicrelectronic conductors ŽMIEC. depending on the degree of reduction. In recent studies on nanocrystalline cerium oxide, predominantly electronic conductivity was observed under conditions at which microcrystalline cerium oxide exhibits ionic conductivity w16–19x. In other words, a high grain boundary density in this material results in a loss of ionic conductivity. This effect is assumed to be correlated with a depletion of acceptor dopants in the bulk as a consequence of their segregation at grain boundaries w20x. The electronic conductivity of nanocrystalline cerium oxide was increased as compared to undoped single crystalline cerium oxide, which is explained by a donor effect of grain boundaries. Such a donor effect may originate from an enhanced reduction at the grain boundaries as compared to the bulk. In fact, a reduced activation energy for the electronic conductivity as compared to the single crystal value is found and attributed to a decrease in the enthalpy of reduction of nanocrystalline CeO 2y x w17,21x. However, the interpretation of experimental results so far was based on the defect thermodynamics of the cerium oxide volume phase. The impact of space charge layers along grain boundaries on the defect chemistry has not yet been taken into account. In space charge layers, the concentrations of point defects vary from their bulk values, depending on the sign and magnitude of their charge w22–25x. The enrichment or depletion of mobile charge carriers has significant consequences on the transport properties perpendicular or parallel to grain boundaries in fine-grained materials w26,27x. Taking into account the opposite sign of the two mobile charge carriers in mixed conductive cerium oxide, i.e. oxygen vacancies Žq. and electrons Žy., space charge segregation is expected to result in an increase of one and decrease of the other partial conductivity. It is the purpose of this study to investigate the grain size-dependent electrical conductivity of cerium oxide and to compare the experimental results with a theoretical model which is based on the single crystal defect chemistry and space charge theory. In the first paper Žpaper I., the experimental investigation of electrical conductivity of cerium oxide in the
A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
micro- and nanometer regime is presented. Among the samples of this study, a series of Gd-doped specimens was investigated. Gadolinium was chosen because of its almost perfect match of ionic radius in the host lattice w28x. Therefore, segregation of Gd in cerium oxide should be driven mostly by Coulomb interaction rather than ion size mismatch. Bulk depletion of gadolinium in nanocrystalline CeO 2 due to segregation would therefore indicate the presence and sign of a space charge potential. The experimental results on temperature and oxygen partial pressure dependence will be analyzed in the context of the single crystal defect chemistry of cerium oxide. In a following paper Žpaper II. w29x, a theoretical model based on space charge theory is presented, which explicitly takes into account space charge segregation of acceptor ions and of the two mobile charge carriers in mixed conductive cerium oxide. The present model is an extension of an earlier model by Maier w27x in so far that it includes the treatment of doubly charged species, such as fully ionized oxygen vacancies, together with singly charged electronic carriers. A brick-layer type geometry is employed to calculate the grain size dependence of ionic and electronic partial conductivity.
2. Experimental Nanocrystalline cerium oxide powder was synthesized by thermal decomposition of CeŽIII.-acetate ŽAldrich, 99.9% or Alfa 99.99%. at 3008C for 5 h. By doping with Gd ŽHeraeus 99.9%., a mixed aqueous solution of the desired molar ratio was frozen in liquid nitrogen, and the solvent was extracted by freeze-drying to produce a mixed acetate precursor. Complete decomposition of the acetate and crystallization of cerium oxide in the fluorite structure was verified by thermogravimetric analysis and X-ray diffraction, respectively. Disk-shaped samples of 8 mm diameter and F 1 mm thickness were prepared by uniaxial hot pressing in vacuum Ž1.5 P 10y3 Pa. at various temperatures and pressures. Some of these samples were further annealed in air at 14008C in order to initiate grain growth into the micrometer size range. The geometric density of the desiccated specimen was measured by the Archimedes method using air
257
and diethylphtalate as independent media. Warren– Averbach analysis of X-ray diffraction profiles was employed for characterization of the grain size in the nanometer range w30x. This analysis involved the Fourier transformation of the peak profiles of both the sample and microcrystalline reference spectra, deconvolution of instrumental line broadening and correction for strain contribution to diffraction peak broadening. The result of this analysis, the areaweighted mean column length ² L:area , is identical to the stereologic quantity of mean line intercept length and allows for the calculation of the grain boundary area per unit mass irrespective of the grain shape w31x: A gb s
2
r ² L:area
,
Ž 1.
where r denotes the density of cerium oxide Ž7.216 grcm3 .. Further details on the characterization of nanostructured materials by this method can be found in Ref. w32x. Samples that were annealed at high temperatures were polished and thermally etched at 13008C in order to determine the grain size by scanning electron microscopy. Platinum electrodes were applied to both sides of the specimens by magnetron sputtering. The ac-impedance in the frequency range v s 62.8 Hz–3.14 MHz was measured using a Quadtech 7400 analyzer. The temperature was varied between 3508C and 8008C and oxygen partial pressures in the range 10y4 –1 were established using gas mixtures of oxygen and argon. A least-square fit of a model equivalent circuit was obtained for each measurement using the following impedance function: Z Ž v . s Z1 Ž v . q Z 2 Ž v . s
R1 1 q Ž i vt 1 .
g1
q
R2 1 q Ž i vt 2 .
g2
Ž 2.
with the time constants t i s R i Ci . Eq. Ž2. describes the impedance of two ZARC—Cole distributed elements Z1Ž v . and Z2 Ž v . in series w33x. The two components will be assigned to various elements of the microstructure in the later discussion. For exponents g i s 1, the quantities R i and Ci can be associ-
A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
258
Fig. 1. Nyquist plot of the complex impedance of microcrystalline cerium oxide, measured at 3508C and pO 2 r p 0 s10y4 and the result of a least-square fit. The numbers at the graph indicate the frequency logŽ v ..
Fig. 2. Nyquist plot of the complex impedance of nanocrystalline cerium oxide, measured at 4608C and pO 2 r p 0 s10y4 and the result of a least-square fit. The numbers at the graph indicate the frequency logŽ v ..
ated with an ideal resistor and capacitor in parallel, respectively.
values for the fit-parameters are summarized in Table 1. A nanocrystalline sample was prepared by uniaxial hot-pressing in vacuum at 6458C and 700 MPa for 90 min. Considerable reduction of the sample was evident from a dark coloration. Therefore, reoxidation by an annealing treatment for 15 h at 5508C in air was performed. The sample exhibited a density of 92.1% of theoretical and ² L:area s 26 nm. A typical impedance spectrum of this sample, measured at 4608C and pO 2rp 0 s 10y4 is shown in Fig. 2, together with the least-square fit of the equivalent circuit. The resulting values for the various fitparameters are also given in Table 1. Impedance spectra of polycrystalline ion conductors are commonly analyzed using the brick-layer model w34x. With regard to this model, we associate the high-frequency semi-circle, characterized by R 1 ,
3. Results A microcrystalline cerium oxide sample was prepared by uniaxial hot-pressing at 6558C and 330 MPa for 100 min followed by annealing in air at 14008C for 7 h. The grain size of the sample was 10 mm and a density of 97.1% of theoretical was determined. The impedance spectrum, measured at 3508C and pO 2rp 0 s 10y4 Ž p 0 s 1 atm s 1.013 P 10 5 Pa. is shown in Fig. 1. A small semi-circle was found in the high frequency range and was well discernible from the very large impedances at low frequencies. A least-square fit of the equivalent circuit impedance function Ž2. is also shown in Fig. 1, and the resulting
Table 1 Results of a least square fit of measured impedance spectra with the impedance function given by Eq. Ž2. for microcrystalline ŽFig. 1. and nanocrystalline ŽFig. 2. cerium oxide Sample Microcrystalline Nanocrystalline
R1 w V x
C1 wFx 4
7.4 P 10 3.6 P 10 5
g1 y11
1.9 P 10 2.3 P 10y11
0.95 0.95
R2 w V x 1.33 P 10 3.9 P 10 5
C2 wFx 8
g2 y9
5.0 P 10 2.4 P 10y10
0.92 0.69
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259
Table 2 Parameter of uniaxial hot-pressing and resulting density and grain size for various samples
Fig. 3. Bulk conductivity of micro- and nanocrystalline cerium oxide at 5008C as a function of oxygen partial pressure.
C1 , and g 1 , with the bulk impedance of the cerium oxide crystallites. This preliminary assignment, in which the contribution of grain boundaries parallel to the direction of current is neglected, as well as the origin of the low frequency arc will be further discussed below and in paper II w29x. From the resistance R 1 and the thickness d and cross-section A of the sample, the bulk conductivity s bulk s Ž drA. was calculated. A very distinct difference Ry1 1
Fig. 4. Temperature dependence of bulk conductivity of microand nanocrystalline cerium oxide at pO 2 r p 0 s10y4 .
Temperature w8Cx
Pressure wMPax
Density w% of theoreticalx
² L:area wnmx
700 720 740 780 800
780 740 600 600 600
91.8 92.6 94.8 96.5 97.2
24.7 26.4 33.7 39.3 44.0
between the micro- and nanocrystalline sample was observed in the oxygen partial pressure dependence of s bulk ŽFig. 3.. Whereas the bulk conductivity of the microcrystalline sample was virtually independent of pO 2 in the measured range, the conductivity of the nanocrystalline sample increased with decreasing oxygen partial pressure. The partial pressure dependence apparently followed a power-law with an exponent close to y1r6. It is also important to notice that the absolute value of electrical conductivity was 1–2 orders of magnitude lower for the nanocrystalline sample as compared to the microcrystalline sample. This implies that the partial ionic conductivity must be reduced by at least 1–2 orders of magnitude in the nanocrystalline sample.
Fig. 5. Bulk conductivity of 0.2 at.% Gd-doped cerium oxide at 5008C and pO 2 r p 0 s1 as function of grain boundary area per unit mass. The data point a A gb s 0 m2 rg corresponds to the electronic conductivity of undoped single crystalline cerium oxide.
260
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The temperature dependence of s bulk also exhibited different behavior for the two samples ŽFig. 4.. The activation energies, determined from the Arrhenius plot were 1.7 and 0.71 eV for nano- and microcrystalline cerium oxide, respectively. Various samples of 0.2 at.% Gd-doped cerium oxide were prepared as described above using the high-purity ceriumŽIII.acetate precursor. The conditions of uniaxial hot-pressing and the results of grain size and density characterization are summarized in Table 2. In Fig. 5, the bulk conductivity of these samples was plotted against the specific grain boundary area, which was calculated using Eq. Ž1.. The conductivity was found to increase with increasing grain boundary area, or in other words, with decreasing grain size.
4. Discussion The impedance spectrum of microcrystalline cerium oxide ŽFig. 1. exhibited a distinct semi-circle at high frequencies and a second and very large arc is observed at lower frequencies. The measured impedance spectrum is typical for polycrystalline ion conductors in which the ionic current is partially blocked by grain boundaries. The brick-layer model is commonly used to find a correlation between the components of an equivalent circuit and physical quantities characteristic of the material w34x. The high frequency semi-circle is assigned to the bulk impedance of the cerium oxide crystallites, based on the capacitance of C1 s 1.9 P 10y1 1 F, which corresponds to a reasonable value of e s 26 for the dielectric constant of cerium oxide w35x. In contrast, the value of C2 s 5.0 P 10y9 F reflects the rather high capacity of the thin grain boundary layers perpendicular to the electric field. A contribution to the high-frequency circle by grain boundaries parallel to the electric field can be neglected due to their large resistance with respect to the bulk phase. With the values of g 1 s 0.95 and g 2 s 0.92, the depression of both semi-circles was small. Variation of the oxygen partial pressure between 10y4 and 1 atm was found to have no effect on the bulk conductivity at 5008C ŽFig. 3., implying that impurity-controlled ionic conductivity was present. This means that the oxygen vacancy concentration was fixed by charge compen-
sation of acceptor cations, such as La3q, which is a typical impurity in cerium oxide. Assuming only trivalent acceptors at a concentration of w AXCe x, the oxygen vacancy concentration is given by: 1
w AXCe x . Ž 3. 2 With the mobility of oxygen vacancies m V O¨ and electron charge e 0 , the resulting extrinsic ionic conductivity: VO¨ s
s ionic Ž T . s 2e 0 m V O¨ Ž T . VO¨
Ž 4.
is independent of oxygen partial pressure. To a first approximation, the mobility of oxygen vacancies is the only temperature-dependent factor in Eq. Ž4.. Hence, the activation energy of ionic conductivity is identical to the activation energy of the vacancy mobility. From the temperature dependence of bulk conductivity ŽFig. 4., an activation energy of 0.71 eV was obtained, which is in agreement with values reported in literature w35–37x. Further analysis aiming at a quantitative expression for the absolute value of oxygen vacancy mobility requires a more detailed treatment of ionic charge transport in acceptor-doped cerium oxide. A number of studies clearly demonstrate, that a significant elastic and electrostatic interaction between acceptor ions AXCe and oxygen vacancies VO¨ causes the formation of associated pairs Ž AXCe VO¨ .P w35–37x. The association reaction: AXCe q VO¨ | Ž AXCeVO¨ .
P
Ž 5.
can be taken into account by applying the corresponding law of mass-action: P
GA0 Ž AXCe VO¨ . s K T s 4 exp y AŽ . VO¨ w AXCe x k BT
ž /
Ž 6.
with the standard Gibbs free energy of association GA0 s HA0 y TSA0 w35x. As a consequence of association, not all oxygen vacancies that are generated for charge compensation do necessarily contribute to ionic conductivity and the effective concentration of mobile oxygen vacancies is dependent on temperature. A general expression for the temperature dependence of ionic conductivity of cerium oxide can be written as:
s ionic Ž T . s VO¨ Ž T .
A T
ž
exp y
Hm k BT
/
Ž 7.
A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
with an activation energy of vacancy mobility Hm s 0.61 eV and the pre-exponential A s 4 P 10 5 KS cmy1 w35x. At low temperatures, almost complete association can be anticipated and the activation energy, as determined from the temperature dependence of conductivity is the sum of 1 2HA and Hm . At high temperatures, the measured activation energy represents only the mobility term, since the concentration of associated pairs can be neglected. The analysis of our data with a result of 0.71 eV indicated that association should not be neglected. Wang et al. w35x suggest to fit the experimental data of the temperature-dependent conductivity with the enthalpy of association and the total acceptor concentration as fitting parameters Ž SA is assumed to be negligible.. This was performed on our data and the line in Fig. 4 through the data points of the microcrystalline sample is the result of such a fit with an enthalpy of association HA s 0.31 eV and a total acceptor concentration of 1340 ppm. This concentration seemed reasonable as the chemical analysis of the ceriumŽIII.acetate ŽAldrich, 99.9%. provided by the manufacturer, yielded 2000 ppm maximum total metallic impurities. In the further discussion, the association of oxygen vacancies will not be explicitly taken into account. Alternatively, an expression for the temperature-dependent electrical conductivity, as given in Eq. Ž4., can be obtained by considering all, i.e. dissociated and associated oxygen vacancies as mobile charge carriers with a reduced effective mobility, in which the average residence time in an associated state is included. The total oxygen vacancy concentration is controlled by charge compensation of the acceptor impurities through Eq. Ž3. so that the effective mobility can be evaluated from the measured electrical conductivities using Eq. Ž4.. The result:
m V O¨ Ž T . s
m0V O¨ T
ž
exp y
0.71 eV k BT
/
Ž 8.
with m0V O¨ s 1.5 P 10 2 cm2 KVy1 sy1 is a reasonable approximation, since the temperature dependence of conductivity in Fig. 4 follows almost a straight line. However, it should be noticed that this effective mobility is also dependent on the total acceptor concentration since the association enthalpy tends to increase with the acceptor concentration in the dilute limit w35x.
261
As a test for consistency, we calculate the residual electronic partial conductivity of cerium oxide:
sel Ž T . s e 0 mel Ž T . n Ž T .
Ž 9.
for the given acceptor concentration. The concentration of electrons and oxygen vacancies are related through the external reductionroxidation equilibrium with the gas phase: VO¨ n2 pO1r2 2 OOx
ž
s K R Ž T . s K R0 exp y
HR k BT
/
.
Ž 10 .
With the equilibrium constant K R0 s 3.73 P 10 6 atmy1 r2 , H R s 4.67 eV w14x and the estimated acceptor concentration given above, the concentration of electronic charge carriers can be calculated. By using the electron mobility w38x:
me Ž T . s
me0 T
ž
exp y
0.4 eV k BT
/
cm2 Vs
Ž 11 .
with me0 s 3.9 P 10 2 cm2 KVy1 sy1 , a partial electronic conductivity at 5008C of 3.8 P 10y1 0 S cmy1 was obtained.2 This value was much lower than the measured conductivity of 2.5 P 10y5 S cmy1 indicating that the assumption of a predominant ionic conductivity is consistent with the defect chemistry of cerium oxide. It should be noticed that cerium oxide is a small-polaron electron conductor w38x with an electron mean free path of an atomic distance. This may justify the assumption of an electron mobility in the cerium oxide bulk phase, which is hardly affected by the microstructure but dependent on temperature only. As shown in this discussion, the present experimental results on the electrical conductivity of microcrystalline cerium oxide are consistent with the defect chemistry of single crystalline cerium oxide. In the following section, the results obtained for nanocrystalline cerium oxide of the same chemical composition will be discussed. The impedance spectrum of the nanocrystalline cerium oxide sample exhibited two overlapping semi-circles Žsee Fig. 2..
2
Calculation of the electrical conductivity from molar concentrations of charge carriers requires a value for the number of CeO 2 formula units per unit volume. This quantity was obtained from the density and molar mass of cerium oxide as 2.525P10 22 cmy3 .
A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
262
Again, the circle at high frequencies was attributed to the bulk impedance based on the value of the capacitance C1. As a first major difference with respect to the microcrystalline sample, the resistance of the second semi-circle R 2 was comparable in size to R 1 of the bulk semi-circle and the depression of the low-frequency circle was also much larger. Obviously, the grain boundary effect in nanocrystalline cerium oxide was much smaller. Taking into account the high density of grain boundaries in nanocrystalline samples, the specific resistance per grain boundary was very low, similar to reported values in literature w17x. More important was the observation of a clear dependence of the ‘bulk’ conductivity on the oxygen partial pressure, which was close to a y1r6 power law ŽFig. 3.. Such a power-law dependence is expected if the charge neutrality equation is reduced to the Brouwer approximation: 2 VO¨ s n
Ž 12 .
and the concentrations of both electrons and oxygen vacancies are controlled by the external equilibrium with the gas phase, Eq. Ž10.. Although the concentrations of both defects are of the same order of magnitude, as evident from Eq. Ž12., electronic conductivity prevails over ionic conductivity due to a higher mobility of the electrons Žcompare Eqs. Ž8. and Ž11... An activation energy of 1.7 eV was obtained by analyzing the temperature dependence of electrical conductivity ŽFig. 4.. Two factors in Eq. Ž9. for the electronic conductivity are temperature-dependent, the electron mobility and the electron density. By inserting Eqs. Ž10. – Ž12. into Eq. Ž9., an apparent activation energy for single crystalline cerium oxide of: Eapp ,sx s Em e q
1 3
H R s 1.98 eV
Ž 13 .
is expected. The measured value of 1.7 eV for nanocrystalline cerium oxide was lower, which could be due to a reduced hopping energy andror a reduced H R . The absolute value of the overall electrical conductivity of nanocrystalline cerium oxide was found to be increased by a factor of 3 if compared to the electronic conductivity of 1 P 10y7 S cmy1 for undoped single crystalline cerium oxide. The latter
was calculated using Eqs. Ž9. – Ž12., with w AXCe x s 0, T s 5008C and pO 2rp 0 s 1. The enhanced electronic conductivity of nanocrystalline cerium oxide is consistent with the lower apparent activation energy. On the one hand, a reduced hopping energy would translate into a higher electron mobility. On the other hand, a reduced enthalpy of the external reductionroxidation equilibrium would result in a larger concentration of electronic charge carriers. The present experimental data do not allow for a separation of these effects. However, previous studies reported activation energies even below 1.0 eV, which could obviously not be explained by a reduction of the hopping energy alone w16–19x. Hence, a significant decrease of the enthalpy of redox equilibrium seems to play the more important role. The increased electronic conductivity in nanocrystalline cerium oxide has also been reported in literature. If compared with the intrinsic value of undoped cerium oxide, the electronic conductivity of the nanocrystalline sample in Ref. w17x was enhanced by a factor of 21 at T s 5008C. The analysis of the present results indicates, that bulk conductivity of the nanocrystalline cerium oxide sample is controlled by the external equilibrium and dominated by the electronic contribution. It is two orders of magnitude lower than the ionic conductivity measured for the polycrystalline sample, but larger than the electronic conductivity of pure single crystalline cerium oxide. Comparing the results on the electrical conductivity of microcrystalline and nanocrystalline cerium oxide, the most significant aspect is the change from impurity-controlled ionic to electronic conductivity controlled by the external equilibrium. It is important to notice that this transition was not due to a change in the overall chemical composition, since care was taken that only specimens from the same batch of powder were compared. The loss of acceptor-controlled ionic conductivity upon reduction in grain size can be readily explained by an enhanced segregation of acceptor impurities to the grain boundaries in the nanocrystalline material. While maintaining the overall chemical composition, segregation of acceptor cations would decrease the bulk concentration as the grain boundary area per unit mass increases, i.e. the grain size decreases w20x. This effect of acceptor segregation will now be elucidated in more detail.
A. Tschope ¨ et al.r Solid State Ionics 139 (2001) 255–265
A major driving force for segregation of acceptor cations is the size mismatch between the impurity and host ions which is more easily accommodated in the open atomic arrangement of grain boundaries. However, enrichment of aliovalent cations such as acceptors in the space charge layer along grain boundaries due to Coulomb interaction is a second possible mechanism for segregation. The latter effect of space charge segregation should dominate if the acceptor ions exhibit only a small size mismatch with respect to the Ce 4q host, which is the case for Gd 3q. In other words, if the loss of ionic conductivity is also observed in Gd-doped nanocrystalline cerium oxide, this effect could not be explained by a segregation due to size mismatch. However, the segregation of gadolinium would indicate the presence and sign of a space charge potential. A set of 0.2 at.% Gd-doped samples with different grain sizes were prepared and the bulk conductivity was determined. A microcrystalline sample from that powder batch with a grain size of 1 mm exhibited a bulk conductivity of 1 P 10y4 S cmy1 at 5008C as expected for this acceptor concentration. The bulk conductivity of the nanocrystalline samples as function of grain boundary area per unit mass is shown in Fig. 5. The calculated conductivity of acceptor-free single crystalline cerium oxide Ž A gb s 0 m2rg. is also included. Very similar to the measurements discussed above, the bulk conductivity of the Gd-doped nanocrystalline cerium oxide was two orders of magnitude lower than that of the microcrystalline sample and bulk conductivity increased with increasing grain boundary area per unit mass, i.e. with decreasing grain size. The loss of ionic conductivity in the nanocrystalline sample could again be explained by the segregation of Gd acceptor ions. However, the size mismatch between Gd 3q and Ce 4q is too small to provide sufficient driving force for such an extended segregation. As discussed above, segregation of negatively charged Gd acceptor ions provides strong evidence for the presence of a space charge layer at grain boundaries. Because space charge layers inevitably influence the concentration of all charged point defects, in particular the mobile charge carriers, the important question arises, whether other experimental results such as the enhancement of electronic conductivity or the reduced activation energy of electronic conductivity could also be ex-
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plained by the space charge effect. Accumulation of Gd in the space charge layer implies a positive excess charge at the grain boundaries. The same space charge potential results in the accumulation of electrons and the depletion of oxygen vacancies in the space charge layer. Upon reduction of the grain size, the space charge effect on the charge carrier concentrations would result in a decreased ionic and increased electronic partial conductivity, which is in fact the central observation of the present study. This effect was also considered by Lohwasser et al. w39x in the discussion on the electrical conductivity of cerium oxide thin films. An important point to notice is that a grain size of 20 nm is the same order of magnitude as the screening length l, which characterizes the width of space charge layers. Therefore, a contribution to the ‘bulk’ conductivity by grain boundaries parallel to the current must not be neglected. In fact, the overlap of space charge layers may result in a flat band situation, in which the conductivity is homogeneous over the whole crosssection of the crystallites. With regards to this discussion, it seems appropriate to develop a model for the effect of space charge layers on the grain size dependence of mixed ionicrelectronic conductivity in polycrystalline cerium oxide. Such a model will be presented in paper II w29x. So far, the discussion of the present measurements was limited to the ‘bulk’ conductivity, which was calculated from the resistance R 1 of the highfrequency semi-circle in the impedance spectra. We have also determined the apparent activation energies associated with the grain boundary effect, using R 2 . For the microcrystalline sample, the activation energy was found to be 1.9 eV, which is a little larger than the value of 1.7 eV reported in literature w5x, yet, this value was much larger than that of the bulk ionic conductivity Ž0.71 eV. with the consequence that the grain boundary effect becomes negligible at temperatures above 7008C. One possible reason for such a large difference in activation energies of bulk and grain boundary conductivity is the presence of a continuous network of impurity-rich grain boundaries. The grain boundary effect was much smaller in the nanocrystalline samples than in the microcrystalline specimens, indicating that the impurities were diluted by spreading over a much larger grain boundary area per unit volume. We also
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determined the activation energy of the apparent grain boundary conductance as 1.7 eV, which was identical to the value of the ‘bulk’ conductivity. Hence, scattering of electrons upon crossing the grain boundaries can be ruled out as origin of the second semi-circle. This second arc at low frequencies could also originate from a depletion of charge carriers in space charge layers along grain boundaries. However, this effect requires that the grain size is sufficiently large as compared to the screening length l, which is not the case as shown in Ref. w29x. We assume that a constriction effect due to the residual porosity w1x or inhomogeneous contacts at the electrodes w40x of the nanocrystalline specimen is the origin of this second semi-circle.
5. Conclusions The effect of grain size on the electrical conductivity of polycrystalline cerium oxide was investigated. Nanocrystalline specimens exhibited electronic conductivity under conditions at which microcrystalline samples showed impurity-controlled ionic conductivity. The electronic conductivity was larger than the intrinsic electronic conductivity of pure single crystalline cerium oxide and was increasing with decreasing grain size. The loss of impuritycontrolled ionic conductivity was also observed in Gd-doped samples, which is indicative of a significant space charge effect related to a positive grain boundary excess charge. As a consequence of the accumulationrdepletion of electronicrionic mobile charge carriers in the space charge layers, the partial electronicrionic conductivity increasedrdecreased with increasing grain boundary area per unit mass, i.e. with decreasing grain size. In a following paper, an analytical model for the conductivity of a polycrystalline mixed ionicrelectronic conductor will be presented, which explicitly takes into account the effect of space charge layers and grain size on the partial conductivities.
Acknowledgements The authors acknowledge technical assistance by A. Kohm-Witt and financial support by the German ¨
National Science Foundation ŽDeutsche Forschungsgemeinschaft, Sonderforschungsbereich 277 ‘Grenzflachenbestimmte Materialien’.. ¨
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