Lower valent cation-doped CeO2 materials have attracted remarkable research interest for the electrolyte application in solid oxide fuel cells operating in the intermediate temperature range (500–700°C). At these temperatures, the oxygen-ion conductivity of gadolinium-doped ceria is about an order of magnitude higher than that of yttria-stabilized zirconia. The oxygen-ion diffusion in the cubic fluorite structure of CeO2 is dependent on several factors such as charge valence and size of dopant cation, doping amount, etc. In the literature, several conductivity trends have been reported as a function of these parameters and are explained by the atomistic computational models. This chapter describes the highlights of the various activities that have been done in this regard to provide insights into the mechanisms underlying the oxygen-ion conduction process in acceptor-doped ceria.
- bulk ionic conductivity
- doped ceria
- local defect complexes
- defect chemistry
- solid oxide fuel cells
Cerium oxide is the most thoroughly investigated material for the application in solid oxide fuel cells (SOFCs) working in the intermediate-temperature range (500–700°C) [1, 2]. This is because of its pronounced catalytic properties, substantial oxygen-ion conductivity (on doping with lower valent metal oxides) and good phase and thermal stability in the given temperature range. The high catalytic activity in ceria is associated with the presence of Ce4+/Ce3+ redox couple which is a key to improve the electrochemical properties of the electrodes. Although this is one area where most of the papers related to ceria have been produced, in this chapter, we restrict our discussion to its usefulness as an electrolyte in SOFCs .
Ceria exhibits a cubic fluorite crystal structure that possesses a relatively large interionic open space facilitating fast oxygen-ion diffusion inside the lattice (shown in Figure 1). This open structure is stable from room temperature to its melting point (~2400°C) and allows to accommodate high levels of point defect disorder . Several properties such as ionic conduction, thermoelectric effect, mixed ionic-electronic conductivity, etc., can be altered and tailored by controlling the concentration of these zero-dimensional defects . As a result, the defect engineering in ceria plays a vital role in improving its properties for the electrolyte application.
Oxygen-ion conduction in ceria is mediated through a vacancy diffusion mechanism. In oxidizing atmospheres, the cubic fluorite structure of pure CeO2 does not have any oxygen vacancy defects. As a result, pure CeO2 itself is a poor oxygen-ion conductor. The oxygen vacancy defects are introduced into the CeO2 structure by partially substituting Ce4+ with acceptor cations inside the lattice. This can be expressed through the following defect reaction in Kröger-Vink notation.
Thus, the impurity center and oxygen vacancy are the dominant extrinsic ionic defects in trivalent metal oxide doped ceria. Nearly, all oxygen anions vibrate in the region of their tetrahedral sites and under a sufficient driving force, jump from their respective site to the adjacent vacant oxygen-ion site in (100) direction. Though the other conduction pathways are also present, they have a substantially large jump barrier . As a result, the jump distance of oxygen-ion is half the length of the unit cell. During migration from one tetrahedral site to another, oxygen-ion is required to pass through an edge formed by cations shared by the involved tetrahedrons (shown in Figure 2).
For a pure oxygen-ion conductor, oxygen-ion conductivity (σ) can be expressed by,
where No is the concentration of oxygen sites, is the fraction of oxygen-ion vacancy and μi is the mobility of oxygen-ions with charge denoted as qi. Several studies have demonstrated that at high temperatures (>700°C) and under the reducing environments (present on the anode side of SOFC device), Ce4+ present in ceria tends to reduce to Ce3+ leading to the formation of oxygen vacancy and two additional electrons as a charge compensating defect.
These electrons are typically localized in the periphery of Ce3+, leading to the formation of small polarons (). These polarons not only break the translation symmetry of crystal but also induce a local lattice distortion around the site. The migration of polaron results in electronic conduction which contributes a portion of the measured total conductivity in bulk ceria. However, for the solid oxide electrolyte, the electronic conductivity should be kept as low as possible not only to minimize the leakage current which if present decreases the overall cell efficiency, but also to avoid the warping and delamination of electrolyte layer because of the chemical expansion . As a result, the use of doped ceria is restricted to conditions (less than ~700°C, > 10−18 atm) where the electronic conduction is substantially lower than the ionic conduction and thus can be neglected .
Oxygen-ion diffusion in doped ceria is a thermally activated process. The temperature dependent behavior of the oxygen-ion conductivity can be expressed through Arrhenius relationship given by,
where σo is a pre-exponential factor and EA is the activation energy of oxygen-ion conduction. The value of EA is determined from the gradient of the linear-fit of the σ data plotted between logσT versus 1/T. The detailed form of Eq. (4) is given by,
where a is the jump distance of oxygen-ion, is an appropriate lattice vibration frequency, and ΔSm and ΔHm are the change in entropy and enthalpy, respectively, during oxygen-ion diffusion. On comparing Eqs. (4) and (5), it is apparent that the change in enthalpy during oxygen-ion diffusion (ΔHm) is, in fact, the activation energy (EA) shown in Eq. (4). Thus, an explicit relationship between the conductivity and temperature is provided in Eq. (5).
It is noteworthy to mention that even though the total conductivity of polycrystalline samples is influenced by the conductivity inside the grains and grain boundaries, in this chapter, we only refer to the grain (bulk) conductivity which occurs by oxygen-ion jump through the regular lattice and is not affected by the microstructural characteristics of the sample. Thus, the bulk ionic conduction is the inherent property of the doped material . On the other hand, the conduction in grain boundaries is dependent on several factors such as impurity segregation, space-charge region, etc. . Both the bulk and grain boundary conductivity can be estimated by acquiring an impedance spectrum over a frequency range (typically 10 MHz to 0.01 Hz) on the polycrystalline sample . As grain boundaries show higher capacitance value (in the range from 10−8 to 10−11 F) compared to the bulk region (~ 10−12 F), two distinct arcs are typically observed associated with these phenomena . On fitting the impedance data using an analog equivalent electric circuit, the resistance offered by bulk and grain boundary regions is estimated which can be then be used in determining the corresponding conductivities.
The bulk conductivity has a significant dependence on the concentration of oxygen vacancy which is controlled by the doping amount according to in A2O3-doped CeO2. It can be observed from Eq. (5) that the conductivity would likely to follow a parabolic behavior as a function of oxygen vacancy concentration and reaches its maximum value when half of the oxygen sublattice is unoccupied. From several investigations on fluorite-structured oxides (including ceria), it has been found that even though conductivity exhibits maxima when plotted against the dopant concentration, the maximum in conductivity appears at significantly lower dopant fractions (less than 0.4) [11, 12, 13]. Figure 3 shows the bulk conductivity of Smx/2Nd x/2Ce1-xO2-x/2 as a function of doping amount (x) at various temperatures . At 550°C, the maximum in conductivity is found at x = 0.15. On lowering the temperature, the doping fraction leading to the maximum in conductivity (xmax) shifts toward the lower value. Thus, the conductivity data obtained at low temperatures results in low xmax, while the high-temperature data will possess a high value of xmax . Further, the xmax value is also dependent on the charge valence and size of acceptor dopant used to generate oxygen vacancies in ceria. This is attributed to the formation of local defect complexes which are discussed in the later section.
Besides the doping amount, the physical properties of dopant cations also influence the bulk conductivity of ceria. Numerous investigations have been performed to understand these correlations in doped ceria systems [1, 15, 16, 17, 18, 19]. Figure 4 depicts the dependence of ionic conductivity in A0.10Ce0.90O1.95 on the ionic size of the trivalent dopant cation (A3+) at 600°C. While the work performed by Omar et al.  showed a continuous increase in the bulk conductivity with the increase in ionic size, others [17, 20, 21, 22] reported a conductivity maximum around Gd3+ and Sm3+. It is interesting to note that the maximum in conductivity observed around these cations is independent of dopant concentration . In the literature, several empirical, semi-empirical and atomistic models have been reported to clarify this and elucidate the effect of doping amount and the dopant-type on the conductivity. This chapter reviews some of these models to provide insights into the mechanisms underlying the oxygen-ion conduction process in doped ceria.
2. Formation of local defect complexes
Whenever charged point defects () are incorporated into the crystal structure, to maintain electroneutrality, defects possessing opposite charges must be created. These oppositely charged defects strongly interact with each other because of electrostatic attraction. Further, the elastic interactions also exist to relax the local stresses surrounding these defects. For example, an oversized substitutional impurity cation may attract an open space of vacancy. This leads to the formation of local defect complexes such as and . These defect complexes trap the oxygen vacancies and effectively reduce the number of mobile oxygen-ions. In most literature, it is commonly accepted that the primary reason for the significant decrease in ionic conductivity of doped ceria at higher dopant content is essentially because of the formation of local defect complexes.
Figure 5 depicts a case where defect is trapped in a tetrahedral site of the fluorite structure surrounded by oppositely charged dopant cations and neutral host cations. There are several experimental studies which support the formation of local defect structures. Extended X-ray absorption fine structures (EXAFS) have been extensively utilized to study the local structures around the cations and anions in lower-valent cation-doped ceria [34, 35, 36]. In case of GdxCe1-xO2-x/2, the addition of Gd3+ leads to a decrease in the interatomic distance Gd-O which is attributed to the formation of the defect complexes with two Gd3+ and one oxygen vacancy . The formation of local defect structure is also evident in the nuclear magnetic resonance study where the average coordination number is estimated for cations in Y3+- and Sc3+-doped CeO2 . The tendency of oxygen vacancies to preferentially associate with Sc3+ was found to be stronger than with Y3+ in these systems.
The binding energy associated with the local defect structures is mainly dependent on the electrostatic attraction of the defects caused by the effective charges in the lattice. Further, it also includes terms due to the relaxation of the lattice around the defect which depends on the polarizability and the size of dopant cation. Several groups have found a difference in the magnitudes of EA when measured in the high- and low-temperature regimes [14, 39]. Typically, the activation energy value estimated in a low-temperature regime is higher than that in the high-temperature region. In a few investigations, a kink is noticed around 350–600°C in the Arrhenius plot of the measured ionic conductivity . Nevertheless, there are several other studies where instead of a sharp, a gradual change in slope with temperature is reported [8, 40]. It has been argued that in a low-temperature regime, most of the oxygen vacancies are bound to various traps to form local defect complexes . However, at higher temperatures, the thermal vibrations of the local defect structures become dominant. At a certain temperature, the thermal energy overcomes the binding energy of these local defect complexes, and the majority of oxygen vacancy defects are set free to migrate from one site to another. Thus, the oxygen vacancies require only migration (ΔHm) to cross the energy barrier at higher temperatures. As a result, in the low-temperature region, the total EA is composed of ΔHm and association enthalpy (ΔHa), whereas in the high-temperature region, the measured EA is only because of migration enthalpy of oxygen-ions.
In the literature, several methods have been used to determine the association enthalpy. Omar et al.  have taken a difference in the magnitudes of EA obtained in the high- and low-temperature regimes and reported the association enthalpy in Smx/2Ndx/2Ce1-xO2-x/2 in the range of 0.02–0.05 eV . Nowick et al.  obtained the association enthalpy by estimating the migration enthalpy in pure CeO2 to be 0.67 eV. Stephen and Kilner  have taken the minimum value of EA measured in the low-temperature regime in NdxCe1-xO2-x/2 as a migration enthalpy. The association enthalpy initially decreases and then increases after reaching a minimum value at x = 0.03 (shown in Figure 6) . The increase in association enthalpy is attributed to the development of deep vacancy association. A similar trend in association enthalpy is also observed for other dopant cations such as Y3+, Gd3+ and Sm3+ and so on [11, 14, 44].
3. Elastic lattice strain
The amount of local defect structures increases with the incorporation of oxygen vacancies inside the lattice. For achieving high oxygen-ion conductivity, the association of defects needs to be minimized. Kilner and Brook  stated that the oxygen-ion conductivity in doped CeO2 could be enhanced if the elastic strain present in the lattice is lowered. Kim  partially replaced acceptor cations in fluorite-structured MO2 (where M is a tetravalent cation) and studied the change in lattice parameter of the host oxides. By performing multiple regression analysis, an empirical model was developed that can predict the elastic strain in doped ceria systems for any given dopant cation. This model takes into consideration the size mismatch and difference in charge valence between the dopant and host cations and the dopant concentration. Further, Kim  proposed the concept of critical ionic radius (rc) which was described as the ionic radius of a cation which on doping leads to a negligible distortion in the host fluorite lattice. For the trivalent dopant cation and host ceria, the value of rc was estimated to be 1.038 Å. Based on Kilner and Brook’s prediction, the cation with the ionic radius of rc is the optimal dopant that can lead to the maximum oxygen-ion conductivity in CeO2 . As a result, the oxygen-ion conductivity in doped CeO2 depend upon the () value, where rd is the ionic radius of dopant. The lower the value for a particular dopant, the higher the ionic conductivity for that system is expected to be. Kim  argued that the highest ionic conductivity is observed around Gd3+ (see Figure 4) as rc value is close to the ionic radius of Gd3+ (= 1.053 Å).
In a similar study, Hong and Virkar  calculated the rc value to be 1.024 Å using a simple analytical model that can also predict the elastic lattice strain present in doped CeO2. In this model, the oxygen vacancy is treated as one of the chemical species whose size remains constant irrespective of the size and concentration of dopant cation.
To examine the validity of minimum strain’s hypothesis, Omar et al.  have carried out the experimental studies using a novel co-doping strategy in ceria. A co-dopant pair of Lu3+ and Nd3+ were chosen and added in a ratio such that the weighted average dopant ionic radius of co-dopants matches the magnitude of rc. Using this approach, it is expected that, on average, the positive elastic strain because of larger dopant cation (i.e., Nd3+) is compensated by the negative elastic strain caused by the smaller dopant cation (i.e., Lu3+). This, as a result, prevents any distortion in fluorite lattice that is usually present in singly doped ceria systems. On calculating the lattice parameter, almost negligible elastic lattice strain was seen in these systems (see Figure 7). However, the ionic conductivity in Lu3+ and Nd3+ co-doped CeO2 system was found to be lower than that of Gd3+-doped CeO2 . Thus, even though the co-doped CeO2 system exhibits minimal elastic strain, its oxygen-ion conductivity remains inferior to that of Gd3+-doped CeO2.
On plotting the activation energy for oxygen-ion diffusion in trivalent cation-doped CeO2, it has been found that the minimum value for activation energy occurs for Sm3+ which possesses a much larger ionic radius than the rc value . Thus, both co-doping results and the obtained activation energy values do not support the hypothesis given by Kilner and Brook  that the highest oxygen-ion conductivity is obtained in the system having a negligible elastic lattice strain. However, it is noteworthy to mention that all the rc values reported in the literature are derived from the empirical relationships which are based on the lattice expansion data acquired at room temperature. As the oxygen-ion conductivity of doped CeO2 materials is usually measured between 400°C and 700°C, there is a possibility that the rc value increases with the increasing temperature. Omar et al.  tested this hypothesis by measuring the lattice distortion for various doped CeO2 systems (with 10 mol.% dopant concentration) at higher temperatures. The rc value was then determined, where the lattice parameter mismatch between doped and pure CeO2 becomes zero (see Figure 8). It was noticed that the rc value decreases with the increase in temperature and does not follow the conductivity trend. The obtained results clearly indicate that the oxygen-ion conductivity is not solely dependent on elastic strain, and therefore, a structure-conductivity relationship based on the critical ionic radius concept is insufficient to describe the conduction behavior in doped ceria.
4. Dopant cation-oxygen vacancy interaction energetic
In the past, multiple atomistic computer simulation techniques have been used to study the interaction energetic involved between the cations and oxygen vacancies in acceptor cation-doped CeO2. Butler et al.  reported that the ionic radius of dopant has a major influence on the stability of defect complexes. Minervini et al.  studied the binding energy of an oxygen vacancy with dopant cation using the energy minimization techniques. It was found that the oxygen vacancies prefer to reside in the first neighboring sites of small dopant cations and in the second neighboring sites of large dopant cations. For Gd3+ dopant cation, the oxygen vacancy shows similar preference to reside in first and second neighboring sites. Moreover, the binding energy was also estimated to be lowest in Gd3+-doped CeO2. Although these theoretical studies show a close match with the experimental results, they are based on empirical atomistic models.
Andersson et al.  performed the quantum mechanical calculations within the density functional theory (DFT) formalism in trivalent cation-doped CeO2. Both electrostatic and elastic interaction energies between the dopant cation and oxygen vacancy located in nearest neighbor (NN) site and in the next to nearest neighbor (NNN) site (of dopant cation) were predicted using the ab-initio calculations. It was found that for Pm3+ dopant cation, the total interaction energy values are similar for both the configurations (shown in Figure 9). As a result, the Pm3+-doped CeO2 system contains the maximum number of oxygen sites with equi-interaction energy inside the host lattice. There will be no site preference for oxygen vacancies which leads to minimal deep traps inside the doped ceria lattice. Thus, Pm3+-doped CeO2 was predicted to exhibit higher oxygen-ion conductivity than any other acceptor-doped ceria materials.
Unfortunately, Pm is a radioactive element and cannot be used as a dopant in ceria. The best dopant should be having an effective atomic number around Pm3+ (61) with an ionic radius of 1.093 Å. According to Andersson et al. , a co-doping approach, with Sm3+ and Nd3+ as co-dopants, provides an experimental scenario for examining the validity of this hypothesis. Omar et al. [8, 42] have studied the influence of co-doping Sm3+ and Nd3+ on the oxygen-ion conductivity of CeO2. Sm3+ and Nd3+ were added in an equal ratio to obtain the effective atomic number of Pm3+, that is, 61. By doing so, similar total interactions between the dopant cations and oxygen vacancies sitting in NN and NNN sites (of dopant cations) are expected, which may lead to enhancement in the oxygen-ion conductivity. It was reported that for compositions containing 10 mol.% of dopant, Sm3+ and Nd3+-doped ceria exhibits 14% higher grain ionic conductivity than that of Gd0.10Ce0.90O2.95 at 550°C, in air (shown in Figure 3). The obtained high conductivity of co-doped samples validates the density functional theory prediction about Pm3+ to be the best dopant cation for achieving high oxygen-ion conductivity in CeO2.
5. Migration enthalpy in the local environment
Although several investigations have shown the formation of local defect structures in doped ceria systems, only a few models exist which describe the doping fraction at which they start appearing and directly influencing the conductivity. Also, in several experimental investigations, the observed trend in macroscopic migration enthalpy is attributed to the microscopic level association between dopant cation and oxygen vacancies without a thorough understanding of underlying atomistic level mechanisms. In earlier studies, the local defect structures are considered as one of the chemical species whose concentration is described using the equilibrium thermodynamics [1, 39]. However, this approach does not take into account the interactions between the local defect structures, especially at higher doping concentrations (>10 mol.% A2O3).
Multiple atomistic simulations studies have been performed to comprehend the relationships between the macroscopic conductivity and the migration in various local environments. In one of the earlier theoretical studies, Murray et al.  used Kinetic Monte Carlo (KMC) simulations to consider the migration enthalpy in the local environment to estimate the oxygen-ion conductivity as a function of doping level. It was reported that at higher doping concentration, oxygen vacancies tend to reside next to dopants, and thus do not contribute toward ionic conduction.
In a similar investigation, Nakayama et al.  used ab-initio density functional theory calculations to identify two key relationships that govern the oxygen-ion migration in rare-earth cation-doped ceria. First, the lowering of migration energy barrier by doping with a smaller trivalent cation would be accompanied by trapping of an oxygen vacancy at the nearest neighboring sites of the dopant. Second, doping with a larger trivalent cation increases the energy barrier but decreases the trapping effect of oxygen vacancies. Thus, the relative magnitude of these two effects is dependent on the size of dopant cation which in turn decides the magnitude of oxygen-ion conductivity.
In recent work by Koettgen et al. , the oxygen-ion conductivity was calculated as a function of the doping amount by combining ab-initio density functional theory and Kinetic Monte Carlo simulations. A model was developed that can estimate migration energies for all the possible jump configurations present in rare-earth-doped CeO2 system. Migration energies were analyzed for energy contributions that are either energetically symmetric for both migration directions, that is, forward and backward jumps or energetically asymmetric for the forward and backward directions. If the presence of dopant changes the migration energy identically in both the backward and forward directions, the energy contribution, in this case, is referred to as blocking. On the other hand, if the migration energy is different for forward and backward jumps, the energy contribution is referred to as trapping.
Figure 10 shows the schematic of trapping and blocking effects. The energy of the system as a function of a reaction coordinate for the corresponding configuration is also shown. It was demonstrated that both these effects have an impact on the observed ionic conductivity. While blocking effect determines the doping fraction at which the maximum in conductivity is observed, it is the trapping effect which limits the maximum ionic conductivity value. Thus, this study clarifies that the optimal doping concentration to achieve the maximum ionic conductivity cannot be predicted only by trapping effect (association between dopant cation and oxygen vacancies) which is commonly assumed in the literature.
6. Summary and conclusion
This chapter reviews some of the highlights of investigations performed on lower-valent cation-doped ceria materials which have been considered strong candidates for the electrolyte application in SOFCs operating at the intermediate-temperature range. Some of the basic characteristics of doped ceria relating to its high oxygen-ion conductivity are discussed. The maximum in conductivity observed by adding a large amount of lower-valent dopant cation is explained on the basis of formation of local defect structures. The extent of formation of these local defect structures cannot be lowered by just minimizing the elastic lattice distortion. It has been emphasized that the concept of critical ionic radius alone cannot explain the maximum oxygen-ion conductivity observed in Pm3+-doped CeO2 as found by the first-principles density functional theory calculations. Particular attention has been given to a more recent atomistic simulations study on rare-earth-doped ceria which calculates the migration energies for all the possible jump configurations that may present in rare-earth-doped CeO2. This study explains the importance of the shape of migration energy barrier and its impact on the ionic conductivity.
Author would like to acknowledge IIT Kanpur for supporting in writing this chapter.