Identifying Doping Strategies To Optimize the Oxide Ion Conductivity

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Identifying Doping Strategies To Optimize the Oxide Ion Conductivity in Ceria-Based Materials Christine Frayret,*,† Antoine Villesuzanne,‡ Michel Pouchard,‡ Fabrice Mauvy,‡ Jean-Marc Bassat,‡ and Jean-Claude Grenier‡ LRCS-CNRS, UniVersite´ de Picardie, 33, Rue Saint-Leu, 80039 Amiens Cedex, France, and ICMCB-CNRS, UniVersite´ Bordeaux 1, 87, AV. Dr. A. Schweitzer, 33608 Pessac Cedex, France ReceiVed: February 5, 2010; ReVised Manuscript ReceiVed: September 13, 2010

One of the key objectives in developing solid oxide fuel cells and oxygen membranes is the improvement of ionic conductivity in electrolyte materials. Optimization of such materials relies on the understanding of the oxygen-ion diffusion mechanisms at the atomic scale. Getting a clear physical-chemical picture is thus the prerequisite to make an educated guess of the best choices of dopant. We highlight in this work some of the most salient recent advances in point defects and oxide ion conductivity studies obtained from atomic-scale simulations performed in the framework of the density functional theory using a supercell approach. First principles calculations have been performed on ceria doped with three kinds of trivalent cations (La3+, Y3+, and Lu3+) to probe the incidence of both dopant size and distribution on relative phase stability and oxygen diffusion efficiency. Ionic relaxation patterns indicate that the crystal structure reorganization after introduction of defects (DCe′, VO..) involves both electrostatic and steric parameters. A clear dopant site selectivity is evidenced for Y- and Lu- doping cases, while much less selective situation of dopants positioning characterizes lanthanum-doped ceria. The study of oxygen mobility has been extended to all possible successive atomic jumps within the supercell, along the three main directions. The set of energy barriers to diffusion can be rationalized in terms of stabilizing and destabilizing Coulomb interactions, elastic energy loss, and steric factors in link with the match between dopant size and its coordination number. On the basis of maximum energy barriers to diffusion and site selectivity features, yttrium and lanthanum dopants appear to be the most appropriate choices provided that a low to medium dopant concentration is preserved. Lutetium doping is clearly less favorable. At a higher doping rate, the increase of probability of occurrence of a very unfavorable configuration in La-doped ceria should be detrimental on the migration viewpoint. 1. Introduction Solid electrolytes exhibiting high oxygen ion conductivity are of special interest for their application in electrochemical devices such as solid oxide fuel cells (SOFC), solid oxide electrolysis cell (SOEC), oxygen separation membranes, and methane gas conversion reactors.1-3 The present challenge of SOFC technology is to develop practical and cost-effective setups. In particular, for smaller SOFC stacks, which are intended to mean power applications (3-5 kW), it is now currently accepted that the operating temperature has to be lowered down to 600-700 °C. The research effort to concretize intermediate temperature (IT) SOFCs emergence mainly relies on the search for electrolyte materials with higher oxygen ion conductivity than the state of the art, yttria-stabilized zirconia (YSZ), which reaches the target ionic conductivity at 700 °C only for a 10-15 µm thickness dense membrane.4 Ceria-based electrolytes seem to be highly promising, since they exhibit 4-5 times higher ionic conductivities, at IT, as compared to zirconia.4,5 Although ceria-based electrolytes are slightly reduced at low oxygen partial pressures and thereby develop increasingly higher electronic conductivity, it has been shown that gadolinia-doped ceria (GDC)-based SOFCs can be operated at temperatures as low as 500 °C4 with power output and efficiency as high as * To whom correspondence should be addressed. Tel: +33 322827586. Fax: +33 322827590. E-mail: [email protected]. † Universite´ de Picardie. ‡ Universite´ Bordeaux 1.

YSZ at 700 °C.6,7 Ceria reducibility and electronic conductivity components, which constitute a hindrance to its use at high temperatures, are effectively low enough at IT. A common feature of this class of materials is that bulk conductivity exhibits a maximum as a function of the dopant concentration at about 8-10 mol %, depending on the dopant.8-11 The ensuing decrease in conductivity with increasing dopant concentration has been ascribed to ordering (or so-called clustering) phenomena, related to the association between point defects (LnCe′ and VO · · ), which are generated through doping (see section 3.1). Moreover, at IT, additional association enthalpy due to this clustering effect is believed to be added to the migration enthalpy, exclusively linked to atomic diffusion according to the conventional definition. A clear physical picture on these defect clustering aspects along with atomic jump diffusion generating macroscopic ionic conductivity (and their correlation) are required for the improvement of ceria-based materials. Beyond tremendous experimental work performed on doped ceria, theoretical studies have been oriented especially toward the comprehension at the atomic level of such association and migration effects, which are not easy to catch at such scale with the usual characterization techniques. In the past, atomistic simulations have been widely employed in this topic.9-12 Such kinds of calculations also have been used very recently by Wei et al.13 to examine exclusively association energy values (and not migration ones) for a wide range of dopants within a large

10.1021/jp101156f  2010 American Chemical Society Published on Web 10/18/2010

Oxide Ion Conductivity in Ceria-Based Materials number of atomic defect structures. From the association barriers derived, the preference for vacancy dopant association was evidenced for small dopant, whereas large dopant cations lead to the prevalence of a vacancy location near Ce4+ host cations. Previous works using density functional theory (DFT) by Parrinello et al.,14,15 Bogicevic et al.,16 and Eichler17 focused on relaxation patterns and ionic conductivity in cubic and tetragonal Y-doped zirconia and have demonstrated that DFT constitutes a pertinent tool to investigate relaxation patterns, defect clustering, and oxygen mobility in these doped matrices. These works were especially focused on very accurate lattice models likely to reproduce experimental data on structure and ionic conductivity. Interesting findings were outlined, including in particular the positioning of dopants in the lattice and the role of vacancy-vacancy interactions in these materials. Yoshida et al.18 studied singly and doubly doped ceria with a content of dopants of 12.5% in the cation ratio, within DFT. A clear dependence of energy barriers to diffusion upon local structure was evidenced, but ionic relaxation phenomena, which are well-known to play a crucial role in systems with dopants, were not accounted for. More recently, Andersson et al.19 also investigated association effects and ionic conductivity in doped ceria from DFT calculations for several trivalent dopants (with atomic numbers comprised between 57 and 68). According to this work, a single value of migration enthalpy (amounting to 0.46 eV) was proposed for the local arrangement corresponding to the absence of trivalent ions in the vicinity of the vacancy, whatever the doping case. They also claim that thermodynamically stable defects always correspond to the configuration of NN sitting of the trivalent dopants (whatever its nature) and that the highest association energy is related to such an atomic distribution as well. The migration energy was evaluated when dopants were in the vicinity of the vacancy. A partition of the total energy into electronic and elastic parts has been extracted from the difference between unrelaxed and relaxed energy of the systems. They concluded that the minimum association energy should be reached when the best balance between repulsive elastic and attractive electronic parts is obtained. These results therefore complete previous interpretations of Brook and Kilner20 and Kim21 concerning lattice elastic strain minimization (due to size mismatch between host and dopant cations), which was suggested to be related to association enthalpy. However, recent experiments performed by Omar et al.22 on the Ce0.90D0.10O2-δ series (where D3+ ) Lu3+, Yb3+, Er3+, Y3+, Dy3+, Gd3+, Sm3+, or Nd3+) revealed that grain ionic conductivity differences for a given content of trivalent dopants are not a function solely of the elastic strain. Therefore, clear understanding of the microscopic factors that precisely determine the extent of migration barrier is still lacking. This implies separate analyzation of initial states (IS) and saddle point states (SPS) of the diffusion step in conjunction with atomic scale peculiarities. During the ion hopping event, an oxygen jumps from its site to a vacancy located in NN from it (vacancy hopping mechanism). The energy of the system at the initial state corresponds to the energy of the lattice containing point defects before the atomic jump, while the energy at the saddle point state is related to the positioning of the oxygen atom, which generates the highest constraint along the diffusion path. At such a position, the constraint on the oxygen is generated by the two nearest cations. The energy barrier to migration is expressed as ∆E ) |ESPS - EIS|. We complete here our previous theoretical works performed on both solid electrolyte (doped ceria)23,24 and cathode materials (La2NiO4+δ),24,25 to gain insight into factors governing ionic

J. Phys. Chem. C, Vol. 114, No. 44, 2010 19063 conduction at the chemical bonding scale. These earlier works were based on a conjunction of DFT calculations and topological analysis of the electron density (“atoms in molecules” theory approach).26-28 The first studies performed on doped ceria (Ce0.75D0.25O1.875)23,24 have highlighted the high polarizability of the oxygen at SPS and the occurrence of more or less volume and shape evolutions of cations located near the diffusing oxygen, while diffusion is not assisted by noticeable charge transfers. In the present work, we will focus exclusively on the energetic aspects gained from DFT by trying to identify a rationalization of the vacancy hopping mechanism in doped ceria. Up to now, little work has been performed to correlate such features of atomic jumps with Coulomb interactions, elastic energy, dopant coordination, and lattice relaxation of the local defect structures. An important objective in this study is to bridge the gap between the standard diffusion barriers evaluation and the fundamental understanding of physical chemical parameters acting on their relative extent. The migration energy calculated using DFT can be directly connected to the probability that an oxygen ion jumps from its lattice site into an adjacent vacancy. There is a range of pathways for an atom to move. In contrast to the approach used by Andersson et al.,19 we account for a much broader range of diffusion barriers in the vicinity of a diffusing oxygen ion and oxygen vacancy. Instead of limiting diffusion pathways study to the six possible jumps of the oxygen in direct vicinity of the vacancy (Figure 1a), we have considered complete paths made of several successive jumps. Usually, activation energy for migration is indeed issued from these six single-jump events, based on the observation that for a barrier model in which the mobility of the vacancies is reduced to the neighborhood of the dopant ions, results are in agreement with experiment for significantly doped samples.29 However, these single jumps constitute only the first step of migration along a definite direction, and we will see that such restriction of diffusion paths is clearly insufficient (see section 3.2.3). Therefore, we have undertaken the thorough study of the four successive atomic jumps along a, b, and c directions within a 2 × 2 × 2 supercell since the material is likely to have macroscopic transport along privileged directions (depending on the crystallites orientation with respect to the electric field of the SOFC setup) instead of random walk. Moreover, in comparison with the work of Andersson et al.,19 we did not restrict the initial distributions to dopants both sitting next to the oxygen vacancy whatever the dopant type for the analysis of diffusion energy. We studied different configurations to determine which prevails initially (before diffusion) according to the dopant kind, and we clearly identify dramatic changes as a function of dopant size evolution. The aim of this study is first to analyze relaxation patterns and the respective incidence of stabilizing electrostatic interactions or steric effects on the preferential location of dopants with respect to the oxygen vacancy. This first step is also aimed at identifying the highest probable configuration according to the dopant nature, since it will in turn play the major role for the vacancy migration process. We then probed the energy profiles for oxygen ion migration along possible diffusion paths for all initial dopant distributions. A rationalization of both IS and SPS energies is proposed to provide a deep insight into the origin of the energy barrier extent. The present analysis provides useful information for the understanding of the vacancy diffusion mechanism, which is studied here for the first time through successive jump events using DFT. This work elucidates the origin and the influence

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Figure 1. (a) Six possible oxygen jumps toward an adjacent vacancy in the fluorite CeO2 lattice. (b) Fluorite structure of ceria and fractions of the 2 × 2 × 2 supercell showing “1D”, “2D”, and “0D” distributions (c, d, and e, respectively) of dopants. The oxygen vacancy is depicted as a small cube. Red, Ce4+ ions; green, O2- ions; and purple, D3+ ions. Dotted lines represent defect cluster interactions.

that dopant type has on the ionic conductivity of ceria-based solid oxide electrolyte materials. 2. Computational Methods The calculations were performed using the Vienna ab initio simulation package (VASP),30-33 which employs DFT and expands the electronic structure using a plane-wave basis set. Electron-ion interactions were described using the projectoraugmented wave (PAW) method,33,34 with plane waves up to the energy cutoff at 400 eV. We adopted the PW91 GGA exchange-correlation functional proposed by Perdew and Wang.35 The host matrix was modeled within the valence band model, which considers 4f electrons of Ce as a part of the valence band. The Ce (5s,5p,6s,5d,4f) and O (2s,2p) electrons were thus treated as valence states, while the remaining electrons were kept frozen as inner core states. To obtain a better insight into the real, locally disordered doped CeO2, the supercell method was applied to simulate low content doping effects: a 2 × 2 × 2 cubic supercell containing 95 atoms (2 dopant, 30 cerium, and 63 oxygen atoms) was used; the unique vacancy at the oxygen site is located at the center of the cell. For such very low dopant content (x ) 1/16), the k-point sampling was restricted to a single gamma (Γ) point: (0,0,0). Considering that structural relaxation only affects the local surrounding of the defects in the 2 × 2 × 2 supercell and that GGA tends to overestimate the lattice parameters of oxides, the Kohn-Sham equations were solved self-consistently using the lattice parameter fixed at the experimental values22,36-39 for doped ceria (a ) 5.42275 Å, 5.40825 Å, and 5.39845 Å for La-, Y-, and Lu-doped ceria, respectively) to avoid any underestimation of the energy barriers to diffusion. Therefore, only internal atomic positions in the cell were allowed to relax. All ions were initially set at the regular positions in the cubic fluorite crystal. The optimization of the atomic coordinates is performed via a conjugate-gradient algorithm based on the total energy and the Hellmann-Feynman forces on the atoms. The atomic coordinates were relaxed until the energy change between two ionic steps was smaller than 10-4 eV per supercell.

For pure CeO2 (space group Fm3jm, Z ) 4), the fcc fluorite structure can be described as a simple cubic array of anions in which half of the cubic sites are occupied alternatively, as schematically shown in Figure 1b. The dopants were selected within an isovalent series with clearly distinct sizes: lanthanum (La3+), yttrium (Y3+), and lutetium (Lu3+), for which ionic radii in the cubic environment are rLa3+ ) 1.160 Å, rY3+ ) 1.019 Å, and rLu3+ ) 0.977 Å (in comparison, rCe4+ ) 0.970 Å).40 Different starting configurations (dopant distribution) were studied. Among the large number of possibilities of defect ordering, three cases were selected as follows: (i) The 1D configuration (where D stands for dopant in the vicinity of the vacancy) (Figure 1c): It corresponds to a single-spaced distribution along [001], with the two dopants distant of a, in which the initial vacancy sits in a nearest neighbor (NN) position with respect to one dopant [(a × 31/2)/4] and in a next nearest neighbor (NNN) position with respect to the other one [(a × 111/2)/4]. (ii) A distribution with alternatively nonaligned short [(a × 21/2)/2] and large [(a × 101/2)/2] distances for which two configurations have been selected: the 2D configuration (Figure 1d), with the vacancy in the NN position [(a × 31/2)/4] with respect to both dopants and the 0D configuration (Figure 1e) in which the vacancy is located in the NNN position [(a × 111/2)/4] from the two dopants. 3. Results and Discussion 3.1. Relaxation Patterns, Dopant Site Selectivity, and Dopant Defect Association. Incorporation of trivalent dopants into CeO2 at the Ce4+ sites with the corresponding creation of oxygen vacancies is crucial to the ionic conductivity of these oxides since the number of vacancies thermally generated is insufficient to promote a required range of ionic transport. This doping process can be represented by the following defect reaction:

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2CeO2

Ln2O3 98 2LnCe + VO · · + 3OOx

where, in Kro¨ger-Vink notation, LnCe′ signifies a dopant substitutional and VO · · an oxygen vacancy. Doping with Ln2O3 replaces host cations Ce4+ by Ln3+ cations, and for every two Ln3+, one oxygen vacancy needs to be created to satisfy charge neutrality. Having an effective charge of opposite signs, oxygen vacancies and trivalent dopants can attract each other. In this work, the configurations 2D and 1D correspond, respectively, to the trimer D-vacancy-D ({2DM′ - VO · · }x entities) and to the dimer D-vacancy ({DM′ - VO · · }• pairs) association defect entities (dotted lines in Figure 1c,d), whereas the 0D configuration (Figure 1e) does not involve any defect clustering phenomenon. The displacement patterns for the three dopant distributions are reported in Figures 2-4 respectively, whereas average shifts of NN oxygen, cerium, and dopant atoms are displayed in Table 1. For all dopant distributions (except 2La, see below), the six oxygen atoms NN to the vacancy move toward it (Figure 5), with displacement up to 0.32 Å (for the case of 0La). This is the result of a stabilizing electrostatic interaction due to the suppression of the strong repulsion between the NN anions. Conversely, an outward displacement from the vacancy along 〈111〉 directions is found for both dopants or host cations NN to the vacancy (Figures 2-4) because of the loss in stabilizing attraction between the anion and the cation. On the other hand, large dopants such as lanthanum ions tend to induce an outward displacement of neighbor oxygen atoms along 〈111〉 directions, whereas small dopants like lutetium generate a reverse relaxation pattern. The extent of displacement distances is strongly dependent upon the dopant distribution (and its correlated symmetry) as detailed hereafter. In the 0D configuration (Figure 2), the NNN location of dopants provides the most symmetrical environment near the vacancy (Td), among the three envisaged ones (Td, C3V, and C2V, corresponding, respectively, to 0D, 1D, and 2D distributions). In the three doping cases, cerium atoms are repelled by 0.15 Å from the vacancy. Oxygen ions NN to the vacancy move along 〈100〉 directions toward it within a nearly D4h symmetry related to a small oxygen octahedron flattening, due to the chosen [001] axis for the dopant distribution. Corresponding displacements range from 0.18 to 0.21 Å for 0Y and are comprised between 0.16 and 0.18 Å for 0Lu. For the 0La configuration, oxygen ions in the vicinity of LaNNN are strongly repelled, inducing a less symmetrical relaxation pattern. The large flattening along [001] (∆z ) 0.32 Å) of the O6 octahedron, which enhances the average ONN displacement as compared with that of Lu or Y systems (Table 1), is due to the strong long-range lattice compression induced by outsized lanthanum ions. For the 2D configuration (Figure 3), the relaxation pattern is clearly affected by the NN location of dopants with respect to the vacancy, with different trends as a function of dopant size. More mobile Y and Lu dopant ions are strongly repelled from the vacancy, as compared to the cerium atoms (0.16-0.18 Å vs 0.12-0.10 Å), leading to a C2V symmetry. In the 2La configuration, the two oversized lanthanum atoms are not mobile enough to move significantly along 〈111〉 (d ) 0.11 Å) as compared with cerium atoms (d ) 0.18 Å). NN oxygen atoms displacement toward the vacancy is also dependent upon the dopant radius: Larger displacements are observed when smaller dopants are introduced. The two NN La3+ ions even clearly prevent one NN oxygen atom from any displacement toward

Figure 2. Relaxation pattern for 0D distributions: (a) 0La, (b) 0Y, and (c) 0Lu. Direction and magnitude of atomic displacements are indicated.

the vacancy, emphasizing their large steric effects and the resulting lack of lattice softness. For the 1D distribution (Figure 4), the ionic relaxation is no longer symmetrical due to the different kinds of cation located in the vicinity of the vacancy. No La2 dumbbell prevents oxygen from its displacement toward the vacancy in the 1La case. Again, small dopants are more repelled by the vacancy than cerium, and the La3+ displacement is significantly lower due to its size. The relaxation patterns study has clearly highlighted both the electrostatic interactions due to charged point defects and the steric effects related to dopant size, with peculiarities in both direction and extent of displacements evidenced according to the relative defect distribution in the lattice. Beyond the consideration of such resulting structural features, energetics aspects characterizing initial distribution (before any atomic jump) shall also be studied to account for dopant site selectivity, related to prevalent cations to vacancy distributions. Such information is crucial since it will provide the picture of the

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predominant configurations, which will therefore have much greater statistical weights in the diffusion activation barriers during atomic jump processes. Relative values of energy of initial distribution before relaxation (with respect to the most stable one), displayed in Figure 6a as ∆E0, provide some information concerning the Coulomb interaction between the oxygen vacancy and the trivalent cations pair. For each doping case, the expected electrostatically driven ordering is observed before relaxation with an increasing stability from 0D to 2D distributions. Whereas electrostatically driven ordering is evidenced, whatever the doping case for nonrelaxed initial distribution, this situation is likely to be changed along with ionic relaxation. For Lu- and Y-doped ceria, such increasing stability by passing from 0D to 2D dopant distribution is preserved for relaxed initial distributions (Figure 6b): prevalent configurations correspond to the trimer ({2DCe′ - VO · · }x entities) and to a lesser extent to the dimer ({DCe′ - VO · · }• pairs), while distributions devoid of

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TABLE 1: Ce1-xLnxO2-x/2 (x ) 1/16): Average Displacements (in Å) of Dopant, Cerium, and Oxygen Atoms NN to the Vacancy at the Initial Distribution, with Respect to Ideal Fluorite Positions in the Configurations 2D, 1D, and 0D configuration

〈dD NN〉

〈dCe NN〉

〈dO NN〉

2La 2Y 2Lu 1La 1Y 1Lu 0La 0Y 0Lu

0.11 0.16 0.18 0.10 0.17 0.16

0.18 0.12 0.10 0.17 0.14 0.13 0.15 0.15 0.15

0.13 0.19 0.22 0.19 0.19 0.20 0.23 0.19 0.17

defect cluster (0D) should have very low occurrence in the material due to its low stability. Such clustering effects in


Figure 5. Directions of displacement for the six oxygen atoms NN to the oxygen vacancy.

Y-doped ceria are consistent with 17O MAS NMR peaks indicating prevailing oxygen atoms with both 3Ce + 1Y neighbors and 2Ce + 2Y neighbors as compared to oxygen with 4Ce neighbors.41,42 Apart from this qualitative identical trend in Lu- and Y-doped ceria, we should mention that the energy difference between 2D and 1D configurations (or between 1D and 0D ones) is higher for decreasing dopant size: the gain of stabilizing energy by passing from 1D to 2D and from 0D to 1D is indeed very high for Lu-doped ceria (0.30 and 0.44 eV, respectively) and approximately half of the magnitude for Y-doped ceria (0.17 and 0.24 eV, respectively). Conversely to previous cases, lanthanum doping clearly induces a huge strain counteracting and even overcoming the strong electrostatic interaction between dopant and vacancy. This feature results in a reverse trend for La-doped ceria, with the most stable relaxed state found for the 0La configuration. As a whole, the influence of stabilizing electrostatic interaction is yet only slightly inferior in magnitude to the stabilization afforded by steric effects, because of the very small difference in total energy after relaxation between the three considered configurations. Therefore, dopant distributions without defect clustering (0D) are only slightly privileged (0La is more stable than 2La and 1La by only 0.10-0.12 eV). Thus, configurations 2D and 1D may occur as well for this doping case. This global analysis therefore clearly highlights a dopant site selectivity depending upon dopant nature. Rationalization of such trends can be made through the following analysis. For small dopants, the {2DCe′ - VO · · }x defects association type is more favorable from the steric viewpoint insofar as it stabilizes the two cations in small sites ([7]-coordination). Therefore, in the case of lutetium doping, both electrostatic and steric effects favor a NN location of the dopant with respect to the oxygen vacancy. Conversely, for lanthanum doping, the marked preference for larger sites located in the NNN position with respect to the oxygen vacancy, giving them 8-fold coordination, is clearly in competition with Coulomb interactions. Having detailed both energetic trends for the nonrelaxed and relaxed initial distribution before diffusion, it is important to consider then the evolution of relaxation energy (Figure 6c), which evaluates the global amount of energy change occurring by minimizing forces on the atoms and which is related to the global optimization of the following effects: (i) the Coulomb (long-range electrostatic) interactions (ii) the Born type (short-range repulsive) interactions (iii) the steric effects connected to the match between coordination number and ionic radius. Indeed, the various values of relaxation energy explain the evolution of the differentiation in stability of the three considered distributions. As a whole, large values of relaxation energy

J. Phys. Chem. C, Vol. 114, No. 44, 2010 19067 ranging from -1.59 to -3.17 eV reflect the huge amount of energy gained upon atomic reorganization. The relaxation energy for the 0D distribution, corresponding to the isolated defects (VO · · and DCe′ far from each other) describes how the lattice is initially constrained by these separated defects and then reorganizes around them. By comparing relaxation energies in Lu- (-2.03 eV), Y- (-2.11 eV), and La-doped ceria (-3.17 eV), it is straightforward to see that the lanthanum doping modifies much more strongly the pristine lattice. From the comparison of the various kinds of relative defect positioning (2D, 1D, and 0D), it must be underlined that Erelax does not evolve a lot for Lu- and Y-doped ceria, amounting to about -2 eV for the three dopant distributions. We can remark, however, a resultant gain in stability of 0.13 and 0.07 eV afforded by differences in lattice relaxation when going from 0D to 2D situation for Lu- and Y-doped ceria, respectively. On the reverse, La-doped ceria exhibits a very broad distribution of relaxation energies, which progressively increases by passing from the most strained situation, 2La, to the 0La configuration: -1.59, -2.28, and -3.17 eV for 2La, 1La, and 0La, respectively. Such large differences indicate that a considerably higher amount of lattice reorganization is possible for lanthanum doping when La3+ ions are away from the vacancy position, in agreement with the gradual raise in ONN displacement from 2La to 0La distributions (Table 1). In fact, all Erelax values (even whatever the doping case) can be mainly related to ONN shift toward the vacancy since among the different distributions, a constant balancing between dopants and host cations displacement evolution is evidenced (average values of dopants + host cations displacement ∼0.14-0.15 Å). Concerning distinctions appearing for 〈dONN〉 in the 2D distribution, observed trends have to be correlated with polarizing effect of cations. When such a trimer defect clustering is involved (2D), large La3+ ions in the vicinity of oxygen atoms do not polarize oxygen ions in their neighborhood, while small Lu3+ ions polarize a lot of these anions. Beyond the evidence of the quasi absence of selective dopants positioning for La doping and the clear dopant site selectivity for Y- and Lu-doped ceria, origins of the distinction of preferred positioning as a function of dopant nature have been identified and quantified. All of these results therefore nicely complete the first conclusions provided by atomistic calculations concerning the preferential location of oxygen vacancies according to the size of the dopant.12,13 3.2. Energetics of Oxygen Ion Migration. In ceria, the SPS position lies at halfway from an initial state position for the oxygen and its final state position (corresponding initially to a vacancy in the IS) along the rectilinear pathway. Four successive atomic jumps labeled (1), (2), (3), and (4) along [h00], [0k0], and [00l] directions have been considered for the study of oxygen ion migration. As a whole, 12 ISs and 12 SPSs per distribution (0D, 1D, and 2D) were involved for each doping case (with a total of 216 calculations, Figure 7a-c). For each doping and dopant configuration, the values of energy at the IS and SPS along with the resultant energy barriers to diffusion are analyzed in the next parts to identify the correlation between these features and atomic scale peculiarities. 3.2.1. ISs. By plotting the energy of all initial states after relaxation, EIS, as a function of decreasing energy barrier to migration, ∆E, one can particularly probe the spreading in energy of IS (∆EIS ) |Emost stable IS - E less stable IS|) and identify the different energy ranges that are involved. The smallest energy spreading of IS is found for lanthanum doping, with

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Figure 6. Ce1-xLnxO2-x/2 (x ) 1/16); relative energies (in eV) with respect to the most stable configuration: (a) before relaxation (∆E0) and (b) after relaxation (∆E1) and (c) relaxation energies |Erelax| (in eV) for the initial distribution before diffusion in the dopant configurations 2D, 1D, and 0D.


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Figure 7. Description of the successive diffusion steps along [h00] and [0k0] within the (ab) plane and along [00l] within the (ac) plane for the 2D (a), 1D (b), and 0D (c) distribution.

∆EIS around 0.38 eV (Figure 8a). For Lu substitution, it is twice as large as the La-doped case (0.75 eV, Figure 8b). Yttrium doping provides a moderate dispersion in energy values for IS, with ∆EIS ∼ 0.44 eV (Figure 8c). Therefore, the energy difference between “fundamental” (most stable) and less stable IS increases in the order: La, Y, Lu. Rationalization of the different energy ranges shall now be provided on the basis of stabilizing (VO · · /DCe′) and destabilizing (DCe′/DCe′) Coulomb interactions (along with their extent according to involved

interatomic distances) and matching between dopant ionic radius and its coordination. For lanthanum doping, the fundamental IS arises for the atomic jumps (3) and (4) of [h00] 2D paths. In such a configuration, which can be labeled as [Ce4]NN[La2]NNN with respect to the vacancy environment (analogous to a 0D configuration), both La3+ cations are kept at a (a × 111/2)/4 distance with respect to the vacancy, thus preserving their 8-fold coordination. The fundamental state for Lu doping has a very low energy as

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Figure 8. Distribution of absolute energy values of the doped system (-E in eV) for initial EIS and saddle point states, ESPS for the whole dopant distributions (2D, 1D, and 0D) as a function of decreasing energy barrier to migration, ∆E. (a) La doping, (b) Lu doping, and (c) Y doping.

Oxide Ion Conductivity in Ceria-Based Materials compared to others. The strongly stabilizing Coulomb component related to the [Lu2Ce2]NN configuration is responsible in part for these very low energy states, with the shortest lutetium vacancy distances that might be involved: (a × 31/2)/4. On the other hand, a strong stabilization is afforded by the positioning of small Lu3+ ions in the vicinity of the vacancy, giving them favorable [7]-coordination. Moreover, the outward displacements of lutetium ions tend to minimize the repulsive interaction between dopants (LuCe′/LuCe′). In yttria-doped ceria, the fundamental state also corresponds to the [Y2Ce2]NN distribution, in which Coulomb effects are stabilizing and for which the Y cation lies in a somewhat more favorable [7]-coordination site for this kind of doping, while YCe′/YCe′ repulsion effects are still slightly diminished by outward displacement of yttrium ions. Very unstable ISs are found in La-doped ceria for [h00] 0D (3) and (4) and [0k0] 0D (3) jumps. For these distributions, both La3+ cations are located at (a × 191/2)/4 with respect to the vacancy. No dopant is therefore characterized by the unfavorable [7]-coordination, but the stabilizing Coulomb energy is rather low. For lutetium doping, many unstable ISs are found for the [Ce4]NN[Lu2]NNN configuration, due to both low stabilizing Coulomb interactions and a mismatch between the Lu3+ ionic radius and their [8]-coordination number. A completely similar trend is observed for Y doping, with higher energy states associated with the [Ce4]NN[Y2]NNN distribution. The ISs of intermediate energy correspond, for the whole of doping cases, to the [Ce3D]NN[D]NNN configuration, all involving dopant-vacancy distances equal to (a × 31/2)/4 and (a × 111/2)/4 but with a distinct shift in energy from the fundamental state: an increase of 0.15-0.25, 0.19, and 0.28-0.43 eV for La-, Y-, and Lu-doped ceria, respectively. This analysis has proven that all IS energies can be rationalized through the accurate consideration of both Coulomb interactions and the match between dopant ionic radius and its coordination. 3.2.2. SPSs. Figure 8a-c shows the energy of SPS (ESPS) as a function of the decreasing energy barrier to diffusion, ∆E. During oxygen migration, a further oxygen vacancy, VO · · , and one interstitial oxygen atom, Oi′′, are involved according to the anionic Frenkel defect formation: OOx + Vix f VO · · + Oi′′. Similarly to IS, the relative energy of SPS can be rationalized by considering: (i) short-range Born type repulsions between Oi′′ and CeCex (or DCe′) (ii) VO · · /DCe′ attractive Coulomb interaction (iii) Oi′′/DCe′ repulsive Coulomb interaction (iv) the match between the dopant ionic radius and the coordination shell. The analysis of SPS energy trends as a function of atomic distribution in the lattice should indicate which of these components prevail(s) in each case. Concerning the first contribution (i), we should outline that SPSs are by nature characterized by strong elastic energy losses (EEL) corresponding to large Born type interactions due to the passage of the oxygen within a very constrained area in between two cations located at short distance from it. The SPS is characterized by two vacancies [VO · · (i) and VO · · (f)], corresponding, respectively, to the initial (i) and the final (f) position of the vacancy before and after the jump, respectively. A good approximation of such short-range repulsive energies resulting from the overlapping of the closed shells of the three neighboring atoms (Pauli exclusion principle) should be given by the difference between the total energy of the lattice at this SPS state, ESPS, and the (Ei + Ef)/2 value, as detailed hereafter. In both initial and final states, the VO · · /DCe′ attractive Coulomb

J. Phys. Chem. C, Vol. 114, No. 44, 2010 19071 interaction might occur if the SPS involves D3+ ions. Such joined Coulomb interactions of IS and FS are gathered in the SPS energy. The |ESPS - (Ei + Ef)/2| difference should thus roughly cancel these electrostatic interactions. Otherwise, each initial [related to VO · · (i)] or final [related to VO · · (f)] state is devoid of the strong repulsion specifically associated to the SPS but is, however, characterized by a certain degree of Born type repulsion of similar extent in both cases. In the expression (Ei + Ef)/2, they appear twice but are averaged. Such short-range contributions, existing in all states (including SPS) and of smaller extent than EEL (which is exclusively due to the high constraint at SPS and not present in other states), should thus be canceled by the difference between the ESPS and the (Ei + Ef)/2 term. Therefore, the difference between ESPS and this (Ei + Ef)/2 quantity provides some quantitative information on the main EEL at SPS related to the jump. EEL values are provided in the tables of the Supporting Information. For the second parameter (ii), we should note that there are now four VO · · /DCe′ interactions to consider (instead of two for the IS), since there are now two vacancies involved during diffusion (instead of one for the IS). The discussion on SPS energy rationalization is partitioned according to the three doping cases, and selected identical pathways will be studied as a function of the dopant nature. 3.2.2.1. La Doping. A huge dispersion in energy characterizes SPS for La doping, with ∆ESPS ) |Emost stable SPS - Eless stable SPS| as high as 1.17 eV (Figure 8a). The very prohibitive energy by passing in between two large lanthanum ions at the SPS in the [h00] 2D (1) jump is responsible for this feature. The diffusing oxygen ion is too much compressed at La-Oi-La SPS, therefore, generating an important short-range Born type repulsion between Oi′′ and the two DCe′: EEL is very high (∼1.5 eV). Indeed, as a whole, the mean oxygen-cerium distance at the SPS is approximately 2.10 Å. By taking into account the La3+ ionic radius at the SPS configurationsthat is, in the [6 + 1]-coordinationswhich is nearly of 1.10 Å,35 there is thus a 0.18 Å difference between Ce4+ (ionic radius ) 0.92 Å in [7]-coordination) and La3+ ions. This implies that the La-Oi distance at such SPS should reach 2.10 + 0.18 ) 2.28 Å, whereas it is only 2.13 Å. In our previous work, using a topological analysis of the electron density (AIM approach), we indeed noticed the strong compression of the mobile atom along the metal-metal direction but also, and surprisingly, the joined volume expansion within the plane containing the diffusion path.23,24 We see here that huge short-range Born type interactions (Oi′′/DCe′ repulsive contribution) along with the two La3+ ions lying in unfavorable [7]-coordination clearly overcome the gain in Coulomb energy due to the occurrence of four short La-vacancy distances. By avoiding taking into account the exception of this very destabilized SPS distribution, the energy spreading of SPS considering the state immediately below the most unstable one as new unstable state, “∆ESPS′”, decreases below 0.65 eV. Lower but nevertheless elevated energy values of SPS are found, which are ascribable to the involvement of one La3+ ion only in the SPS configuration. In [h00] 1D (1), the La-Oi distance at SPS is notably longer than previously (2.21 Å) but still constrained as compared to the sum of ionic radii (2.28 Å). In the same way, the Ce-Oi distance after relaxation (2.03 Å) also reveals a constrained state. The EEL value amounts to 0.945 eV, lower than that of La-Oi-La SPS but still elevated. This trend can be generalized since EEL values for the different La-Oi-Ce SPSs vary from 0.77 to 0.96 eV (average, 0.82 eV). Once more,

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the lack of softness of the La-doped ceria lattice in the vicinity of dopants, despite the unit cell parameter enhancement, is evidenced. Eleven stable values of SPS all corresponding to the most stable Ce-Oi-Ce configuration are found, for which the electron cloud of mobile oxygen is significantly less deformed. The EEL for the various Ce-Oi-Ce SPSs indeed ranges from 0.33 to 0.65 eV with an average value of 0.49 eV. For the [h00] 2D (2) jump, the vacancy is globally farther from dopants than initially [with now two La-vacancy distances of (a × 31/2)/4 and two other ones of (a × 111/2)/4], indicating a loss of Coulomb energy as compared with IS. The Ce-Oi distance at SPS is 2.13 Å, which is slightly higher than the mean value of 2.10-2.11 Å, because of the perturbation of the La3+ ions in the La2Ce2 tetrahedron. This prevents high Born repulsion between oxygen and cerium electron clouds (EEL ) 0.42 eV) and tends to stabilize this SPS. However, two La3+ ions are positioned in unfavorable [7]-coordination. For the [h00] 2D (3) path, Coulomb energy is on the reverse gained as compared with IS, since another vacancy is introduced at the same distance with respect to the dopants [four (a × 111/2)/4 La-vacancy distances instead of two (a × 111/2)/4 distances for the IS]. By comparison with the [h00] 2D (2) path, Coulomb interactions are, however, less stabilizing. Nevertheless, lower repulsion between Oi′′ and DCe′ as compared with this jump is involved (due to longer Oi′′ - DCe′ distances) along with the absence of La3+ ions in unfavorable [7]-coordination. These favorable effects counterbalance the slight increase in EEL, 0.47 eV (Ce-Oi distance at SPS is now 2.11 Å), and the less favorable Coulomb interactions, therefore stabilizing as a whole by 0.04 eV this jump as compared with the [h00] 2D (2) one. By proceeding further with the whole of stable states, it is straightforward to notice that these 11 distributions (all devoid of very strong Born type repulsion) can be classified into three main groups according to the lanthanum-vacancy distances, EEL ranges, and La3+ coordination: (i) two distances of (a × 31/2)/4 and two others of (a × 111/2)/ 4, along with two La3+ ions in [7]-coordination, with EELs varying in the range 0.33-0.42 eV (ii) one distance of (a × 31/2)/4 and three others of (a × 111/2)/ 4, along with one La3+ ion in [7]-coordination, with EELs varying in the range 0.40-0.44 eV (iii) four distances of (a × 111/2)/4, without any La3+ ion in [7]-coordination, with EELs varying in the range 0.42-0.44 eV. Stabilizing VO · · /LaCe′ Coulomb interactions increase progressively by going from (iii) to (i), and EEL values also slightly decrease in average in the same sense, while the stability regarding the match between the La3+ ionic radius and the coordination follows a reverse trend. The very low dispersion in energy (-833.29 to -833.37 eV) for such different contributions underlines once more the balance effects between dopant coordination and VO · · /DCe′ Coulomb interactions in La-doped ceria (to which EEL evolution is added at the SPS). Among these 11 very stable states, two belong to the 0La system {[00l] (1) and [00l] (4)}, with EEL values equal to 0.42 and 0.44 eV, respectively. Because we have proven previously that the 0La is the most stable and probable configuration for IS, other 0La SPS should also be considered. The 10 other states related to this configuration are characterized by energies higher by 0.2-0.4 eV, which is at least related to the increase in EEL values (all roughly augmented by 0.16-0.23 eV). Within such 0La distribution, spreading of SPS energy values is limited to a very low amount: 0.42 eV.

Frayret et al. 3.2.2.2. Lu Doping. SPS energies for Lu-doped ceria are characterized by a very high spreading, with ∆ESPS around 0.80 eV (Figure 8b). The energy trends tend to follow those of initial states. Furthermore, some crossing between unstable IS and SPS does appear between -833.3 and -833.4 eV. EEL values are globally comprised between 0.47 and 0.70 eV. Contrary to La doping, the most stable SPS corresponds to the [h00] 2D (1) jump for which Lu3+ is twice involved, with Lu-Oi distances (2.05 Å) only slightly inferior to Ce-Oi corresponding ones (∼2.10 Å). It should be noticed that in the light of our previous study,23,24 the Lu3+ ion appears to be always smaller than Ce4+, in opposition to Shannon and Prewitt tables.40 The small size of the substituants further stabilizes the SPS configuration through favorable steric effects (diminution of steric hindrance). Indeed, despite a lower interatomic distance at the SPS as compared with Ce-Oi-Ce dumbbell configurations, the small ionic radius of Lu3+ allows us to reduce considerably the steric constraints as compared especially with the case where two Ce atoms would be separated from Oi with only 2.05 Å. The EEL value for this Lu-Oi-Lu SPS is 0.64 eV. This SPS is also characterized by a gain in Coulomb energy (four short Lu-vacancy distances) and strongly favorable positioning of the two Lu3+ ions in [7]-coordination. In [h00] 1D (1), the distances at the Lu-Oi-Ce SPS are dissymmetrical like for La doping, but a higher value (2.12 Å) characterizes the Lu-Oi distance as compared with the Ce-Oi one (2.04 Å), which corresponds to a reverse situation as compared with La-doped ceria, due to the inverted mobility features between La3+ and Lu3+ ions with respect to Ce4+ one. For this SPS, the EEL value is equal to 0.59 eV, which is much lower than that of the corresponding SPS for La doping (0.945 eV). The constraint is indeed the same for Ce-Oi but is doubly diminished for Lu-Oi by the increase in distance, on one hand, and the small size of the substituent, on the other hand. Moreover, the positioning of one Lu3+ ion in [7]-coordination is favorable. Similarly, the whole of the Lu-Oi-Ce SPS values for EEL lie between 0.53 and 0.64 eV (average, 0.56 eV), which are much lower than those encountered for La-doped ceria, highlighting the strong differences of Born repulsion between the two kinds of doping. For the [h00] 2D (2) step, the Ce-Oi distance at SPS is lower (2.08 Å) than the mean encountered value (2.10-2.11 Å). This implies that the mobile oxygen atom is roughly 0.2 Å less in diameter in the Ce-Oi-Ce dumbbell configuration as compared with the same jump in La-doped ceria. This effect tends to destabilize significantly this SPS (EEL ) 0.61 eV instead of 0.42 eV for the La-doped ceria). If electrostatic interactions play the same role here, unfavorable positioning of Lu3+ ions in [8]-coordination also enhances such destabilization. For the [h00] 2D (3) path, a higher EEL value is involved (0.70 eV) as compared with the [h00] 2D (2) jump along with less stabilizing Coulomb interactions [four (a × 111/2)/4 Lu-vacancy distances instead of two equal to (a × 31/2)/4 and two other ones of (a × 111/2)/4] and unfavorable positioning of Lu3+ ions in [8]-coordination, while they were in [7]-coordination during the other jump. Such a situation yields the most unstable SPS in Ludoped ceria since the corresponding energy is the highest one for this jump. As a whole, the EEL for Ce-Oi-Ce SPSs lies between 0.47 and 0.70 eV (average 0.62 eV) in Lu-doped ceria. Among all of the 2Lu SPSs (corresponding to the prevalent initial distribution), we observe a spreading in energy equal to 0.8 eV, whereas the dispersion in EEL is about 0.2 eV. 3.2.2.3. Y Doping. Similarly to IS, Y doping provides a moderate dispersion in energy values for the SPS, with ∆ESPS

Oxide Ion Conductivity in Ceria-Based Materials around 0.44 eV, which seems to be a simple energy shift from IS values (Figure 8c). Most stable SPSs are found between -836.7 and -836.8 eV. In the [h00] 2D (1) pathway, the Y-Oi distance at the YOi-Y SPS is 2.10 Å (like for Ce-Oi-Ce SPS), and the corresponding EEL value is 0.65 eV. Like for Lu3+ ions, the positioning of Y3+ ions in [6 + 1]-coordination is quite suited according to their ionic radius. Moreover, in this distribution, as already outlined, the four Y-vacancy distances correspond to (a × 31/2)/4, affording the most stable Coulomb contribution among the different possible cases. However, a slightly more stable distribution (by 0.06 eV) is found, for example, for the [0k0] 2D (1) path in which the two Y3+ ions are also in [7]- and [6 + 1]-coordinations but for which three Y-vacancy distances are (a × 31/2)/4 and one is (a × 111/2)/4. Here, the SPS configuration corresponds to Y-Oi-Ce, with Ce-Oi and Y-Oi distances of 2.03 and 2.16 Å, respectively, and EEL equals 0.595 eV. As a whole, EEL values for Y-Oi-Ce SPS lie between 0.59 and 0.61 eV with very low spreading (average, 0.60 eV). Other SPSs lying at the same fundamental energy also correspond to Y-Oi-Ce SPS: [0k0] 2D (4), [00l] 2D (1) and (4), and [00l] 0D (1) and (4). EEL values for the different (Ce-Oi-Ce) SPSs lie between 0.52 and 0.61 eV (average, 0.55 eV). From the detailed analysis of SPSs presented above, general trends concerning EEL values as a function of dopant nature can be drawn as follows: (i) For the unique D-Oi-D SPS configuration, the EEL trend plays a considerable role, and a decrease from the largest La dopant to the smallest Lu one is observed (from 1.50 to 0.64 eV, with an intermediate value of 0.75 eV for Y). (ii) The same evolution is observed for D-Oi-Ce SPS: EEL values range from 0.82 eV for La to 0.60 eV for Lu, with an intermediate value of 0.56 eV for Y. (iii) For the various Ce-Oi-Ce SPS, EEL values increase from La to Lu, showing a decrease in lattice softness with the lattice parameter (average values of 0.49, 0.55, and 0.62 eV for La, Y, and Lu, respectively). Having detailed the rationalization of the various SPS energies, it is important to identify now the conditions providing either high SPS or low SPS energy according to dopant choice: (i) Most stable SPSs for La-doped ceria correspond to a Ce-Oi-Ce distribution associated with the [h00] 2D (3) jump, while [h00] 2D (1) path, in which atoms distribution is La-Oi-La, corresponds to the highest SPS energy. (ii) The exactly reverse trend is found for lutetium doping with the highest and lowest energy corresponding, respectively, to SPS of pathways [h00] 2D (3), associated with Ce-Oi-Ce distribution, and [h00] 2D (1), associated with Lu-Oi-Lu distribution. (iii) Like for Lu-doped ceria, less stable SPSs of Y-doped ceria are found for the [h00] 2D (3) jump, while on the other hand, several paths give access to the most stable SPS: [0k0] 2D (1) and (4), [00l] 2D (1) and (4), and [00l] 0D (1) and (4), corresponding to a Y-Oi-Ce configuration. Such clear observations highlight the role played by EEL strength on the more or less stabilized SPS to which some other effects are added, as previously discussed. 3.2.3. Energy Barriers to Diffusion: Rationalization and Identification of the Best Doping Choice. Tables a-c of the Supporting Information display the various values of energy barrier to diffusion calculated as the difference between SPS

J. Phys. Chem. C, Vol. 114, No. 44, 2010 19073 and IS energies. First of all, it is straightforward to see on the example of the 2Lu case that limiting the exploration of energy barriers to the six paths usually studied (Figure 5) is clearly insufficient. Indeed, for these six diffusion paths, energy barriers are comprised between 0.64 and 1.01 eV, whereas for the 12 successive paths, energy barriers are in the following range of energy: 0.27-0.98 eV. Therefore, such restriction does not allow probing of the whole energy range found for successive paths, thus demonstrating the usefulness of our approach. As a general comment on energy barriers to diffusion, the largest spreading of values is found for La-doped ceria, with ∆E ) |ESPS - EIS| in the range 0.33-1.50 eV. This situation originates from the combination of very dispersed SPS and slightly dispersed IS. The spreading is quite lower for Lu-doped ceria (∆E comprised between 0.27 and 0.98 eV)seven if energiesofbothISandSPSarethemselveslargelydispersedswhereas the smallest dispersion corresponds to Y doping (∆E ) 0.37-0.83 eV), with a large part of ∆E values of the order of 0.55-0.62 eV. First, it is crucial to identify the conditions leading either to the smallest or to the highest ∆E value in each doping case. For lanthanum doping, the smallest energy barrier (0.33 eV) is achievable by involving IS nearly 0.2 eV higher than the less stable one and SPS very close to the most stable one. Concerning the highest energy barrier (1.50 eV), the same IS is involved, but SPS corresponds to the most unstable one. In Lu-doped ceria, the smallest ∆E barrier (0.27 eV) is ascribable to the association between the most unstable IS and an SPS state lying approximately 0.4 eV higher than the most stable one. The same SPS is involved for the highest energy barrier to diffusion (0.98 eV), while IS corresponds then to the most stable one. Therefore, the situation is inverted between small and large dopants. For lutetium doping, the difference between the lowest and the highest energy barrier is exclusively linked to the passage from the [Ce4]NN[Lu2]NNN IS distribution to the [Lu2Ce2]NN one (the SPS being identical for both jumps). In La-doped ceria, the evolution between the lowest and the highest ∆E is only correlated to the passage from a stable Ce-Oi-Ce SPS distribution, characterized by a low EEL component, to the involvement of very strong EEL for the La-Oi-La SPS. From these two extreme cases, we may conclude that the highest values for energy barriers may arise from either a very stable IS or a strong EEL at SPS, according to dopant nature. Similarly, the lowest values of ∆E originate either from very unstable IS or low EEL at SPS. For yttria-doped ceria, the highest ∆E (0.83 eV) is observed when IS lies ∼0.2 eV higher than the most stable IS and SPS is the most unstable one, while the lowest ∆E (0.37 eV) is associated with both an IS nearly equivalent to the most stable one and an SPS very close to the most stable one. Beyond the global analysis and rationalization of the energetics of activation barriers to diffusion, it is worth considering also the maximal energy barrier (∆Emax) along a given direction of diffusion pathway ([h00], [0k0], and [00l]), since this will govern the ionic conductivity when an electric field occurs as in the SOFC technology (Figure 9a-c). The best doping choice should therefore be strongly oriented by the comparison of ∆Emax among the various substitutions. For Y-doped ceria in the 2D configuration, which is the first one to consider for this doping since it is the most stable one and must therefore be the dominant configuration in the material, ∆Emax is the lowest for [0k0] and [00l] paths (Figure 9c). For the 1D distribution, the path to privilege corresponds to [00l], whereas [h00] and [0k0] ones are favored in the 0D distribution.

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Frayret et al. distribution remains quite prohibitive (0.86 eV). Such a result is in accordance with the experimental study of grain ionic conductivity of Ce0.90D0.10O2-δ, which has demonstrated that Lu doping produces the lowest one in the series of trivalent dopants D3+ ) Lu3+, Yb3+, Er3+, Y3+, Dy3+, Gd3+, Sm3+, and Nd3+.22 As for Y doping, ∆Emax is significantly lower for the 0D distribution (0.66-0.75 eV), which has unfortunately a weak probability of occurrence because of both Coulombic and coordination effects. The La doping case presents specific features as compared with the previous ones, with in particular a completely prohibitive pathway (1.50 eV) along [h00]scorresponding to jump (1)sfor the 2D distribution. Even if the distribution 0La appears to be slightly more stable, the quasi nonpreference of La site occupancy for 2D or 0D and 1D distributions, due to compensating effects of coordination and Coulomb interactions, leads to the consideration of all configurations at the same level. To optimize ionic conductivity in La-doped ceria, it is necessary either to avoid the [h00] path for 2D configuration or to find a way to generate only 0D distribution in the material (Figure 9a). However, the possibility of avoiding the [h00] 2D (1) path is expected for very low dopant concentrations only. Conclusion

Figure 9. Representation of maximal energy barrier along the three diffusion paths ([h00], [0k0], and [00l]) as a function of dopant distribution (2D, 1D, and 0D) for (a) La doping, (b) Lu doping, and (c) Y doping.

Given that 2Y and 1Y configurations should existsespecially in the envisaged temperature rangesthe most suited path for ionic conductivity of the material is [00l], with maximum barriers to diffusion of 0.69 and 0.61 eV, respectively. Nevertheless, ∆Emax does not exhibit drastic changes of energy range as a whole and can be considered as quite isotropic among the different dopant distributions. Yet, it can be noticed that stabilization of Y-doped ceria in the 0D distributionsif achievedswould afford on average the best ionic conductivity properties if no direction of jump is imposed, with an energy barrier of only 0.60 eV for [h00] and [0k0] paths and of 0.69 eV along [00l]. Doping ceria with lutetium appears to be clearly unfavorable for ionic transport properties since the 2D configuration is largely prevalent over other ones and provides the highest values of maximum energy barrier (0.84, 0.86, and 0.98 eV) (Figure 9b). Even if a favorable path is found for [00l] direction in the 1D distribution (0.59 eV), this configuration is not expected to dominate and the corresponding value for the prevailing 2D

Quantum simulations using DFT have been used to derive rules concerning the optimal strategy of doping in ceria-based solid electrolytes at a low doping rate. In this work, three initial dopant distributions within the matrix were studied before diffusion (2D, 1D, and 0D), corresponding, respectively, to trimer (D-vacancy-D) and dimer (D-vacancy) entities for the two first ones and to a distribution devoid of defect clustering for the third one. Contrary to other studies, the study of diffusion energetics has been extended to the complete set of successive atomic jumps along the three directions ([h00], [0k0], and [00l]) instead of being restricted to the first step, as usually performed. One of the goals of this study was first to establish the quantitative defect ordering preferences according to the dopant type, giving access to the most prevalent initial distribution before diffusion in each case. The second most relevant objective was to develop a rationalization of both IS and SPS energy values on the one hand and energy barriers for diffusion on the other hand. The final aim was to identify the best doping choice according to the maximal energy barriers observed among the different diffusion steps for the most initial favored or predominant(s) distribution(s). From this work, the following general conclusions can be drawn. (i) The requirement of ionic relaxation consideration resulting from the introduction of point defects through doping has been outlined. The resultant structure (before diffusion) is clearly influenced by both Coulomb interactions and steric effects, with a strong dependence of displacement distances upon dopant type and relative positioning of dopants and vacancy. Energy relaxation evolution among the three distributions can be essentially connected to the extent of displacement of oxygen atoms NN to the vacancy, due to the constant balancing between dopants and host cations displacement distances. (ii) Electrostatically driven ordering is preserved for Lu- and Y-doped ceria only after relaxation. For large dopants (La), a marked preference for large sites of coordination higher than [7] is found, interfering with these Coulomb interactions and leading to a slightly more stable situation of the 0D configuration over other ones. Small dopants (Lu) exhibit on the reverse a clear prevalence of 2D configuration, more favorable from both

Oxide Ion Conductivity in Ceria-Based Materials electrostatic effects and match between ionic radius and dopant coordination ([7]-coordination). Quantitative dopant site selectivity has been provided according to dopant kind (La, Y, Lu), through the evaluation of energy differences after relaxation among the three dopant to vacancy distributions. (iii) Rationalization of the different energy ranges of IS has been performed on the basis of the relative extent of stabilizing (VO · · /DCe′) and destabilizing (DCe′/DCe′) Coulomb interactions (linked to involved interatomic distances) and the match between dopant ionic radius and its coordination. Similarly, understanding of more or less high energy of SPS has been achieved by calling to the same contributions, to which some other effects occurring exclusively at this constrained state should be added, including especially short-range Born type repulsions between Oi′′ and CeCex (or DCe′), associated with the EEL value. Strong differentiation and even often reverse trends concerning stabilization or destabilization of the various IS or SPS have been evidenced for dopants characterized by clearly distinct atomic sizes (La and Lu). According to dopant nature, the highest values for energy barriers may arise from either a very stable IS (Lu) or a strong EEL at SPS (La), while the lowest values of ∆E originate from either very unstable IS (Lu) or low EEL at SPS (La). (iv) From the maximal energy barrier consideration, doping ceria with lutetium appears to be clearly unfavorable for ionic conductivity, even at low dopant content since the 2D configuration is largely prevalent over other ones and provides the highest values of maximum energy barrier (0.9-1.0 eV). Y doping is favorable because of the moderate barriers found for the prevalent 2D and 1D distributions, provided that one can orientate selectively path along [00l] direction. At intermediate dopant ratio, Y doping should be the most suited one due to the prevalent 2D configuration, allowing energy barriers of 0.7-0.8 eV. La-doped ceria also provides a quite attracting lattice, because of the moderate maximum values of barriers (0.7-0.8 eV) along the three directions for the 0La distribution. Fortunately, such a configuration is the most stable one. To take benefit from this effect, it is necessary to use small contents of La dopant to be devoid of associated defects, which generate limiting paths. Furthermore, quenching samples after high temperature treatments would favor more random dopant distributions within the lattice. The above-mentioned results suggest that limiting the examination of diffusion energy to the distribution associated with NN sitting of the dopants with respect to the vacancy is clearly insufficient. Similarly, the calculation of association energy (defined as the energy difference between the thermodynamically stable supercell and the less stable one) does not provide any information concerning energy barriers to diffusion evolution among the different distributions. Through this study, DFT calculations have proven to constitute a pertinent tool to extract selection criteria of materials likely to have applications in the field of ionic conductor materials. This methodology is a way to provide a completely new conception of conducting oxides synthesis, based upon a prospective way. From the fundamental standpoint, this study tends to clarify defect clustering issues in relation with oxygen ion conductivity. Instead of considering additional association enthalpies, as usually suggested for the interpretation of decrease in ionic conductivity at rising doping contents, it is proposed that a more appropriate treatment would be to consider the incidence of Coulomb interactions, repulsive elastic energy of Born type, and steric factors in link with dopant-adapted coordination number, on both IS, SPS, and resulting energy barriers.

J. Phys. Chem. C, Vol. 114, No. 44, 2010 19075 Finally, at a low doping rate, heat treatments should be used to orientate the selection of the most suited configurations according to results found concerning maximum energy barriers as a function of the distributions and the dopant kind. An increase of dopant concentrations within the host matrix modifies the ratio of dopant distributions in the lattice, for which relative initial thermodynamic stability resulting from Coulomb and steric contributions is clearly distinct according to dopant size. Further theoretical studies including a higher number of initial vacancies (introduced at larger extent of dopants within the supercell) could now be envisaged to map out the energy landscape and see how we can extend our single-vacancy approach of successive jumps and rationalizations to a higher content of dopants (made of more or less concerted multiple jumps). Acknowledgment. We acknowledge the computational facilities provided within the intensive numerical simulation facilities network “M3PEC” of the Bordeaux University (http:// www.m3pec.u-bordeaux1.fr), partly financed by the Conseil Re´gional d’Aquitaine and the French Ministry of Research and Technology. Supporting Information Available: Tables of data for Ce0.9375La0.0625O1.96875, Ce0.9375Lu0.0625O1.96875, and Ce0.9375Y0.0625O1.96875. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Kudo, T.; Fueki, K. Solid State Ionics; VCH: New York, 1990. (2) Steele, B. C. H. Mater. Sci. Eng., B 1992, 13, 79. (3) Minh, N. Q. J. Am. Ceram. Soc. 1993, 76, 563. (4) Steele, B. C. H. J. Mater. Sci. 2001, 36, 1053. (5) Etsell, T. H.; Flengas, S. N. Chem. ReV. 1970, 70, 339. (6) Riess, I. Solid State Ionics 1992, 52, 127. (7) Riess, I.; Go¨dickemeier, M.; Gauckler, L. J. Solid State Ionics 1996, 90, 91. (8) Dell, R. M.; Hooper, A. In Solid Electrolytes; Hagenmu¨ller, P., van Gool, W., Eds.; Academic Press: New York, 1987; p 291. (9) Takahashi, T.; Iwahara, H.; Nagai, Y. J. Appl. Electrochem. 1972, 2, 97. (10) Inaba, H.; Tagawa, H. Solid State Ionics 1996, 83, 1. (11) Kilner, J. A.; Steele, B. C. H. In Nonstoichiometric Oxides; Sorensen, O. T., Ed.; Academic Press: New York, 1981. (12) Minervini, L.; Zacate, M. O.; Grimes, R. W. Solid State Ionics 1999, 116, 339. (13) Wei, X.; Wei, P.; Lafei, C.; Bin, L. Solid State Ionics 2009, 180, 13. (14) Stapper, G.; Bernasconi, M.; Nicoloso, N.; Parrinello, M. Phys. ReV. B 1999, 59, 797. (15) Pietrucci, F.; Bernasconi, M.; Laio, A.; Parrinello, M. Phys. ReV. B 2008, 78, 094301. (16) Bogicevic, A.; Wolverton, C. Phys. ReV. B 2003, 67, 024106. (17) Eichler, A. Phys. ReV. B 2001, 64, 174103. (18) Yoshida, H.; Inagaki, T.; Miura, K.; Inaba, M.; Ogumi, Z. Solid State Ionics 2003, 160, 109. (19) Andersson, D. A.; Simak, S. I.; Skorodumova, N. V.; Abrikosov, I. A.; Johansson, B. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 3518. (20) Kilner, J. A.; Brook, R. J. Solid State Ionics 1982, 6, 237. (21) Kim, D.-J. J. Am. Ceram. Soc. 1989, 72, 1415. (22) Omar, S.; Wachsman, E. D.; Jones, J. L.; Nino, J. C. J. Am. Ceram. Soc. 2009, 92, 2674. (23) Frayret, C.; Villesuzanne, A.; Pouchard, M.; Matar, S. Int. J. Quantum Chem. 2005, 101, 826. (24) Frayret, C. Application of Density Functional Theory (DFT) to the Modelling of Oxygen Ion Diffusion within Model Solid Electrolytes and Mixed Conductors; Ph.D. thesis, University of Bordeaux 1, 2004; http:// tel.ccsd.cnrs.fr/docs/00/04/86/98/PDF/tel-00010824.pdf. (25) Frayret, C.; Villesuzanne, A.; Pouchard, M. Chem. Mater. 2005, 17, 6538. (26) Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Clarendon Press: Oxford, 1994. (27) Bader, R. F. W. Chem. ReV. 1991, 91, 893. (28) Bader, R. F. W. J. Phys. Chem. A 1998, 102, 7314.

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JP101156F

Identifying Doping Strategies To Optimize the Oxide Ion Conductivity

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