density functional theory calculations on microscopic aspects of oxygen diffusion in ceria-based...

Density Functional Theory Calculationson Microscopic Aspects of OxygenDiffusion in Ceria-Based Materials

CHRISTINE FRAYRET, ANTOINE VILLESUZANNE,MICHEL POUCHARD, SAMIR MATAR

Institut de Chimie de la Matiere Condensee de Bordeaux, CNRS, UPR 9048, Universite Bordeaux 1,87, Avenue du Dr. Albert Schweitzer, 33608 Pessac Cedex, France

Received 6 October 2003; accepted 12 January 2004Published online 13 October 2004 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.20343

ABSTRACT: The critical role of the ionic conductivity properties of materials in thedevelopment of Intermediate Temperature Solid Oxide Fuel Cell (IT-SOFC) technologyhas been emphasized over the past decade. However, from a fundamental point ofview, little is known of the physicochemical parameters influencing ionic diffusion inthese conductors. We attempted, through a Density Functional Theory investigation, todevelop a new approach for an oxygen diffusion study in solids at the atomic scale.This methodology relies on the evaluation and comparison of steric, energetic, andchemical bonding factors along with the consideration of charge transfers when mixedvalences do exist. Microscopic aspects of oxygen diffusion in the ionic conductor ceria-based materials have been investigated by means of this procedure. It has beendemonstrated that steric and polarizability effects prevail over other ones for bothnonstoichiometric and doped ceria. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem101: 826–839, 2005

Key words: density functional theory; topological analysis; diffusion; oxygen ionconductor; ceria-based materials

Introduction

F uel cells are widely viewed as the technologyof the future for replacing or at least consti-

tuting an alternative to conventional internal com-

bustion turbine generators. They indeed offer sev-eral attractive advantages over these traditionalengines: higher efficiencies, silent technology, andsignificantly lower emission of pollutants, espe-cially when hydrogen is used as fuel where the onlybyproduct is water.

Among the different fuel cell kinds, Solid OxideFuel Cells (SOFCs) possess many advantageous fea-tures as compared with other setups including, in

Correspondence to: C. Frayret; e-mail: [email protected]

International Journal of Quantum Chemistry, Vol 101, 826–839 (2005)© 2004 Wiley Periodicals, Inc.

particular, the highest efficiency, no corrosive prob-lems, no hazardous liquid electrolyte handling, andpotential fuel flexibility by the integration of aninternal reformer. The present challenge is to de-velop practical and cost-effective SOFCs systems. Inparticular, for small SOFC stacks used for applica-tions of intermediate power, it is now currentlyaccepted that the operating temperature has to belowered to 500°C in order to reduce costs associatedwith interconnect and construction materials. Con-sidering that the operating temperature is directlyrelated to the ionic conduction properties of thesolid electrolyte, this condition can be fulfilled ei-ther by employing thinner electrolyte films orthrough the use of better ionic conductor materials.Yttria-Stabilized Zirconia (YSZ), which was the firstenvisaged solid electrolyte, indeed reaches the tar-get ionic conductivity at 700°C for a 10–15 micronsthickness [1]. It could thus be thought that thecharacteristic operating temperature might be low-ered by further decreasing the thickness of the film.Unfortunately, this corresponds to the present limitthat is acceptable both for ceramic fabricationroutes and for a tolerable gas permeability. Other-wise, cathode materials which are currently pureelectronic conductors (LaxSryMnO3-� perovskitematerials) should be replaced by mixed (ionic andelectronic) conductors in view of increasing setupperformance. This efficiency augmentation wouldarise from the replacement of triple-point junctionsby more advantageous double interfaces. As a con-sequence, the research effort, in order to concretizeIT-SOFCs emergence, mainly relies on the searchfor better ionic conductors. Among the potentialmaterials likely to replace YSZ, ceria-based onesseem to be highly promising owing to the ability ofceria to provide the same ionic conductivity com-ponent at 500°C as would be afforded by YSZ at700°C [1]. Moreover, ceria reducibility and elec-tronic conductivity components, which constitute ahindrance to its use at high temperatures, are suf-ficiently low at this intermediate temperature. Asconcerns cathode materials, compounds belongingto the K2NiF4 family such as La2NiO4�� are theprospective ones considering their potentialities aswell in terms of electronic and ionic conductivity asfor electro-catalytic features [2]. As a conclusion,better-suited compounds for IT-SOFC technologyhave been identified at the present time. Neverthe-less, the origin of ionic conductivity differences be-tween materials still remains poorly understood. Inparticular, the reason why doped ceria has a higherionic conductivity than YSZ is not yet fully known.

It has been argued, for instance, that differencesmight come from grain boundary transport prop-erties [3], but reasons should also arise from theintrinsic features difference between both materials.Otherwise, for doped ceria, some general rules havebeen established concerning the optimization of thedopants nature and content. Among others, we canquote the fact that ionic conductivity seems to behighly dependent on dopant ionic radius, the high-est conductivity being measured for Sm3� andGd3� ions, which have ionic radii close to that ofthe Ce4� ion [4]. In the same way, if it is nowcurrently accepted that one order of magnitude inionic conductivity is gained by passing from per-ovskites such as La0.6Sr0.4Fe0.8Co0.2O3-� (LSFC) toLa2NiO4�� materials [2], little is known about theorigin of such a difference.

From the fundamental point of view, it is never-theless of crucial relevance to investigate micro-scopic aspects of oxygen diffusion. Moreover, if wewant to conceive still better ionic conductors thanthe present known ones, we have to be aware of thechemical features intrinsically governing diffusion.

Survey of the Experimental andTheoretical Backgrounds

Although it can be easily understood, throughthese latter considerations, that there exists a needfor knowledge of the microscopic aspects of bulkdiffusion within these superionic oxide conductors,current experimental works essentially afford amacroscopic evaluation of the diffusion phenom-ena. Indeed, the information gained is frequently adiffusion coefficient, which cannot be regarded as amicroscopic feature of ionic transport, owing to anobservation over a long distance and to its collec-tive diffusion character, obeying a particle fluxequation (Fick laws). Moreover, for most currentmeans of investigation such as SIMS or electricalconductivity relaxation (except impedance spec-troscopy), very often applied to ceramics, the ob-served diffusion coefficients constitute a globalevaluation of the transport phenomena, in that allthe components (bulk, grain boundary, and dislo-cation core) are gathered in a single response of thematerial. This last point prevents extracting thebulk diffusion component directly related to thematerial’s intrinsic physicochemical features,whereas the other contributions are much more theresult of its microstructure. Accordingly, even iftraditional experiments performed on ceramics are

MICROSCOPIC ASPECTS OF OXYGEN DIFFUSION

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 827

very useful and justified from the technologicalviewpoint, they do not provide information con-cerning the fundamentals of diffusion at the micro-scopic scale.

Experimental methods more suited to the evalu-ation of the microscopic aspects of diffusion areprovided by nuclear probes [5–8] such as incoher-ent and coherent Quasielastic Neutron Scattering(QENS) [5, 6, 9] and NMR [5, 10]. In order to con-stitute a probe of the bulk diffusion alone, theyshould be devoted to the study of single crystalsrather than ceramics. These means of investigationallow one to obtain the atomic jump frequency, �,which constitutes the microscopic feature of thediffusion phenomenon [11]:

� � v*.exp��H/kT�

� v.exp��S/k�.exp��H/kT� (1)

where �* represents the Vineyard’s prefactor, � isthe effective frequency associated with vibration ofthe atom in the direction of the saddle point posi-tion, k is Boltzmann’s constant, and �H and �S arethe variations in enthalpy and entropy associatedwith atomic displacements towards the saddlepoint state.

Nevertheless, these kinds of studies are morescarce than other ones and mainly concern othertypes of materials than oxygen ion conductors,even if we can quote the NMR works of Fuda et al.[12, 13] dealing with yttria-doped ceria and those ofKim and Grey concerning BIMEVOX materials [14].

Theoretical studies undertaken for studying dif-fusion can be considered as providing at best apartial insight into microscopic aspects of this phe-nomenon. In this area, most investigations are per-formed with atomistic calculations [15] andthrough molecular dynamics (MD) methods. Thelatter essentially provides information on diffusioncoefficients (Einstein relation, implying an averageon all times and a summation on all ions) so that,through their use, one comes around again to thesame problem encountered in most current experi-mental works. Nevertheless, we have to mentionthat MD may help to define diffusion pathways andcan be used for gaining insight into the microscopicstructure, through the examination of pair correla-tion functions. This possibility notably allows forevaluating spatial environment effects as per-formed by Hayashi et al. [16] for Ce1-xDxO2 – 0.5xdoped ceria (x � 0.02–0.3; D � La, Gd, Y). Theyindeed discovered that D-O nearest-neighbor dis-

tances are almost independent of the dopant con-tent but exhibit a strong dependence on dopantionic radius. However, the role of properties suchas polarizability, charge transfer, and chemicalbonding cannot be extracted from these calcula-tions. Atomistic calculations are based on a classicalBorn model description of the lattice [17] and arenow widely used for studying a large range ofmaterials. Previous works have included, in partic-ular, investigations on both ceria [18–20] andLa2NiO4�� [18]. Atomistic calculations applied toceria-based materials have been mainly concernedwith defect clustering [19, 20] and ionic conductiv-ity dependence on dopants ionic radius [21] phe-nomena. However, these approaches mainly focuson the problem of energetics (defects formation andassociation energies along with diffusion activationones) but are not able to analyze the influence ofother physicochemical features on diffusion micro-scopic aspects. As concerns ab initio investigationsapplied to diffusion study, electronic structure cal-culations have dealt with Li� transport in LixCoO2-type materials [22], diffusion on Ag, Cu, and Ptsurfaces [23], or vacancy self-diffusion in bcc tung-sten [24]. As far as oxygen diffusion is concerned,no detailed investigation exists to date. Previousworks focused on peculiar aspects such as the mod-ification of electron density at critical points due tooxygen in Ag, Au, and Cu [25] or the role of exci-ton-vacancy binding on activation energy in �-SiO2[26].

In the face of this lack of complete informationon diffusion microscopic aspects, we developed amethodology based on first-principles DensityFunctional Theory (DFT) calculation with the aimof extending the knowledge of this phenomenon atthe chemical bonding scale. This procedure will bedevoted to the determination of the pertinent pa-rameters likely to account for both the diffusionphenomenon within a given solid and the differ-ences of properties such as ionic conductivity be-tween materials.

Calculation Methodology

As previously stressed, our investigations focusexclusively on bulk diffusion, which is directlylinked to the intrinsic physicochemical properties ofthe material. Studying bulk diffusion from firstprinciples can be achieved by comparing the phys-icochemical properties of two states characterizingthe system, at the microscopic scale, during diffu-

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828 VOL. 101, NO. 6

sion: the initial state (IS) and the saddle point state(SPS). The initial state corresponds to the systembefore diffusion, presenting a certain number ofdefects such as vacancies or interstitial atoms, sincebulk diffusion in materials is caused by the exis-tence of these point defects. The saddle point statecorresponds to the system in which the position ofthe diffusing atom is associated with the energeticmaximum along the diffusion pathway (transitionstate). The energy difference between both statesprovides the microscopic activation energy for thediffusion, which corresponds to the �H value in thepreviously described expression of the jump fre-quency (1). Diffusion phenomena in a given ionicconductor oxide can involve several diffusion path-ways, among which the predominant one is thatwhich is characterized by the lowest energy barrier,which is then called the activation energy of theatomic transport process.

In the pursuit of correctness, calculations mustinclude ionic relaxation effects, which are particu-larly relevant when point defects are introduced—especially for ionic lattices and still more at thehighly constrained SPS—allowing for displacing at-oms towards their minimal energy positions. Thisaspect is treated by means of a pseudopotentialcalculation, which is performed using the VASPpackage [27]. Ground state energies provided bythis code constitute a first estimate of activationenergies. On the other hand, the relaxed atomicpositions provided by the VASP calculation areused as input in an all-electron code, the Wien2kpackage [28], in order to obtain more accurate en-ergy values and to perform a topological analysis ofthe charge density. The topological analysis imple-mented in the Wien2k code is based on the Bader“Atoms in Molecules” (AIM) theory [29] in whichinteratomic surfaces are defined as linking all thepoints of zero flux in the gradient vector field of theelectron density (i.e., “zero flux surfaces,” definedaccording to the equation ��(rs)n(rs) � 0 for everypoint rs on the surface S(rs) and where n(rs) repre-sents the unit vector normal to the surface at rs). Theexistence of these surfaces or separatives, represent-ing the boundaries between atoms, thus allows par-titioning the physical space into nonoverlappingregions. The latter contain, in general, only oneatom. Bader atomic surfaces, separating the so-called “atomic basins” of neighboring atoms, canthus be considered as atomic envelopes. The shapeevolution of these atomic envelopes along the dif-fusion path can be used for getting a qualitativetrend of atoms’ polarizability and especially that of

the diffusing atom. On the other hand, we candeduce from the atomic envelopes the volume as-sociated with each atom. Both atomic volumes andshapes evolution during diffusion have to be con-sidered in connection with interatomic distances inorder to estimate steric effects. In the same way, theintegration of the electron density within atomicvolumes provides the atomic charges, which can beused for evaluating charge transfers. On the otherhand, density of states (DOS) plots can be consid-ered to complement this topological analysis inview of gaining insight into chemical bonding fea-tures of the studied material.

The overall of this procedure has to be under-taken both for the initial state and for the saddlepoint one, so as to evaluate �H and compare thephysicochemical properties characterizing eachof these states. Otherwise, in the case of severalpathway possibilities, these calculations have todeal with the different saddle point states associ-ated with each diffusion path. The correspondingenergy barriers should then be estimated in orderto determine the most favorable diffusion path-way. In summary, our methodology consists ofdetermining four kinds of parameters likely toinfluence, on a microscopic scale, ionic diffusion:the nature of the chemical bonding, the existenceand extent of charge transfers, along with thesteric, polarizability, and energetic factors. Theaim of this research work is thus to evaluate andcompare the magnitude of these different param-eters in view of establishing a hierarchy of rele-vance and to investigate the possible connectionsbetween them.

The calculations presented here deal with ceria-based materials, which possess the fluorite struc-ture, with cerium and oxygen atoms occupyingcubic and tetrahedral sites, respectively. We haveperformed these investigations on both nonstoi-chiometric ceria and doped ceria. Ceria exhibits awide range of nonstoichiometry at high tempera-ture and low oxygen pressure [30]. When CeO2 isreduced to CeO2-x, defects in the form of Ce3� onnormal cationic sites are generated and electricallycompensated by oxygen vacancies (VO

.. ), which ispotentially interesting from the diffusivity view-point. High ionic conductivity is also reachedthrough the substitution of lower valence cationssuch as Gd3�, Y3�, or Ca2� for the Ce4� ones, themissing charge being balanced by the formation ofoxygen vacancies. Ceria, like most oxides crystal-lizing in the fluorite structure, is indeed character-ized by its ability to accommodate a large fraction



of lower valence cations, resulting in oxygen defi-ciencies as high as 20–25% or even more in somecases.

Our calculations began by studying oxygen-de-ficient ceria, taking first the case of a unit cell inwhich one oxygen site was left vacant (Fig. 1).This corresponds to Ce4O7, which is equivalent tothe formula CeO2-x with x, the deviation from

stoichiometry, equal to 0.25. In this structure,bulk diffusion takes place through oxygen jumpsfrom normal anionic sites in the lattice towardsvacant ones. Accordingly, this lattice offers a sin-gle diffusion pathway (Fig. 2) and the saddlepoint state is clearly identified as an oxygen ionlocated between two cerium ions (“dumbbell”configuration).

The second calculation concerning nonstoichio-metric ceria corresponds to a double cell in which asingle vacancy is present at the center of the cell(Fig. 3). The corresponding formula, Ce8O15, whichis equivalent to CeO1.875, is thus associated with areduction rate which is divided by two as com-pared with the previous case. Ce8O15 also corre-sponds to an oxygen deficiency closer to the exper-imentally defined one for the maximum of ionicconductivity (i.e., Ce1-xMxO2-x/2, x � 0.1 for triva-lent cations doped lattices).

Finally, the doped ceria that we investigatedcorresponds to the insertion of two dopants re-placing two cerium atoms in the double cell, i.e.,a Ce6D2O15 formula, which is equivalent toCe0.75D0.25O1.875. The dopants were selected withthe aim of examining steric effects and thus cor-respond to two atoms having clearly differentfeatures in this respect: lanthanum (La3�) and

FIGURE 2. CeO1.875 tetragonal unit cell (a a 2a), with an oxygen vacancy at the center. [Color figure can beviewed in the online issue, which is available at www.interscience.wiley.com.]

FIGURE 1. Cubic unit cell for CeO1.75, with an oxygenvacancy at the center. [Color figure can be viewed inthe online issue, which is available at www.interscience.wiley.com.]

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830 VOL. 101, NO. 6

lutetium (Lu3�), for which ionic radii in cubicenvironment are: rLa3� � 1.160 Å, and rLu3� �

0.977 Å [31].

Computational Details

PSEUDOPOTENTIAL CALCULATIONS

The first step in our calculations, which wasintended to obtain relaxed atomic coordinates, wasachieved by making use of the Plane-Wave pseu-dopotential method as implemented in the VASPcode [27]. The generalized gradient approximation(GGA) from Perdew and Wang (PW91) [32] wasused for the exchange-correlation potential. Coreelectrons were represented by projector augment-ed-wave (PAW) pseudopotentials [33], whereas va-lence electron wavefunctions were expanded as aplane-wave basis set, taking a 400 eV cutoff energy.K-point grids of 6 6 6 and 4 4 2 sizewere used for the Brillouin zone sampling in thesingle (CeO1.75) and double (CeO1.875 andCe0.75D0.25O1.875) unit cell cases, respectively. Theatomic coordinates were relaxed until the energychange between two ionic steps was smaller than10-4 eV.

The Kohn-Sham equations were solved self-con-sistently using the lattice parameter fixed at theexperimental value for ceria. This initiative still re-mains correct for CeO1.75 owing to the fact that thelattice parameter is measured to be 5.411 Å even upto a 20% chemical reduction of the bulk [34]. Theeffect of temperature has been simulated by ex-panding the lattice parameter: 5.411 Å at 298 K and5.458 Å at 1073 K. For doped ceria, we undertookcalculations with the lattice parameter a � 5.458 Åonly, which is a realistic value for theCe0.75La0.25O1.875 lattice parameter at room temper-ature [35]. For Ce0.75Lu0.25O1.875, at the same tem-perature, the lattice parameter is expected to belower, owing to the smaller ionic radius of Lu3�.However, it was necessary, for estimating themethod potentiality to render an account of stericeffects, to keep the lattice parameter constant, sothat we also performed all our calculations onCe0.75Lu0.25O1.875 with a � 5.458 Å. Otherwise, wewould have to deconvolute the resultant effect ofboth differences in local steric hindrance (at thechemical bonding scale) and in “macroscopic”steric constraint (lattice parameters) generated bydopants.

ALL-ELECTRON CALCULATIONS

The all-electrons calculations were carried out inthe framework of the Augmented Plane Wave pluslocal orbitals (APW�lo) method [36], as imple-mented in the Wien2k package [28]. Exchange andcorrelation effects were treated within the general-ized gradient approximation (GGA) from Perdew,Burke, and Ernzerhof (PBE96) [37]. The muffin-tinspheres radii were fixed to 1.7 bohr for oxygen andto 1.9 bohr for cerium, lanthanum, and lutetiumatoms. Both core and valence states were calculatedself-consistently. Our calculations integrated the ce-rium and lutetium 4f states as a part of the valenceband. For core states, the relativistic effect wastaken into account in the scalar approximation butthe spin-orbit coupling was neglected owing to thelow 4f population. The basis sets were determined bya plane wave cut-off of RMT KMAX � 7. In theBrillouin zone, 250 k-points and 125 k-points sam-plings were used, respectively, for the single (CeO1.75)and double (CeO1.875 and Ce0.75D0.25O1.875) cells.

Recent works [38] suggest that cerium 4f elec-trons could be localized on cerium atoms close tooxygen vacancies (i.e., treated as Ce core electrons).Test calculations showed that the conclusions of thepresent article are, however, insensitive to whether

FIGURE 3. Passage from IS to SPS in CeO1.75. [Colorfigure can be viewed in the online issue, which is avail-able at www.interscience.wiley.com.]



or not 4f electrons are treated as valence electrons.Activation energies may be slightly affected; thiswill be the subject of a separate study.

Results and Discussion

FIRST CASE OF NONSTOICHIOMETRICCERIA: CeO1.75

In the case of the nonstoichiometric ceria CeO1.75,the passage of an oxygen from IS to SPS is repre-sented in Figure 2. Oxygen diffusion indeed takesplace along the straight line connecting the twonearest-neighbor anionic sites. This displacement isaccompanied by structural changes in terms ofspace groups. In particular, we notice that there aretwo kinds of cerium atoms in the saddle pointposition—two forming the “dumbbell” configura-tion with the diffusing oxygen (Ce1) and the twoothers (Ce2) being farther—whereas the initial stateis characterized by a single kind of cerium atom.

The calculations undertaken on CeO1.75 werestarted by comparing results obtained with andwithout ionic relaxation for the purpose of evaluat-ing the impact of this effect. As expected, ionicrelaxation has the consequence of lowering both theinitial state energy and the saddle point state one.The involved energy decrease amount is around 0.3eV (Table I), which is nonnegligible. Even if wefound this value to be of nearly the same extent forboth states in this special case, which could indicatethat there is no incidence of ionic relaxation onactivation energies by “compensation of systematicerrors,” it might not always be the case. Further-more, the crystal structure in the vicinity of a defectdramatically changes, which in turn largely modi-

fies steric, polarizability, and even charge transfereffects. By ionic relaxation, in the presence of avacancy at the center of the unit cell, cerium atomstend to move towards the corners of the cell (i.e.,along the direction 111�) in order to minimizeelectrostatic interactions. Table I gives the values ofatomic displacements. In the SPS, the cerium dis-placement associated with relaxation effects takesplace along the direction 011�. As would beguessed, relaxation is much more important in thecase of the SPS, where there are more constraints.Both analyses highlight the necessity to perform inour calculations this first step, allowing us to renderan account of ionic relaxation effects. Consideringthe activation energy values provided in this workfor two temperatures (T � 298 K; T � 1073 K) byexpanding the lattice parameter, as previouslystressed, we obtain as expected a decrease in acti-vation energies with rising temperatures (Table I).This trend comes from the increasing facility ofperforming ionic relaxation for ions in the largercell, lowering further the steric hindrance at thesaddle point state. Otherwise, comparison of VASPand Wien2k results (after having introduced therelaxed positions) indicates the good agreement be-tween both codes. Together, these considerations,along with the correct order of magnitude obtainedfor activation energy values, which has been estab-lished for the most current oxygen deficiency x �

0.125 (as stated in the following section), validatesour choices of both computational method andstructural model. Accordingly, we can now focusour study on the influence of the parameters likelyto have an effect on diffusion phenomenon from themicroscopic viewpoint.

TABLE I ______________________________________________________________________________________________CeO1.75: activation energies (Ea), maximum cerium displacements due to the oxygen vacancy presence (�uCe),Ce-O distances in the dumbbell configuration of the saddle point state (dCe-O), and energy gain due to theionic relaxation (�Erelax), for the unit cell parameters at 298 K and 1073 K

a (Å) Ea (eV)

�uCe (Å)

dCe-O (Å)

�Erelax (eV)

IS SPS IS SPS

5.411 1.08a 0.07a 0.14a 2.057a �0.26a �0.34a

1.12b

5.458 0.92a 0.08a 0.14a 2.071a �0.33a �0.29a

0.96b

a Calculated with VASP.b Calculated with Wien2k.

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832 VOL. 101, NO. 6

By plotting the density of states for the initialstate, we first noticed that these are characterizedby a valence band of preponderant oxygen 2p char-acter, and a conduction band of predominant ce-rium 4f (not shown). This result is in accordancewith previous work [39] and underlines the highionic character of the material. The same trendsoccur for the saddle point state. However, Figure 4shows that, for the conduction band, Ce1-projectedDOS are now shifted towards higher energies ascompared with Ce2 ones, due to destabilizingshorter Ce1-O distances. By considering in moredetail the zone located at the limit of the conductionband, we notice that areas below EF associated with

Ce1 and Ce2 are different in magnitude, whichcould be an indication of charge transfers betweencerium atoms. By integrating these two areas,charge transfers of about 0.13 electron from Ce1 toCe2 are found. Ce1, which is near the saddle pointposition, tends to lose the same amount of charge asis gained by Ce2, which is situated farther from thisconstrained configuration.

This first study on CeO1.75 has thus allowed usboth to validate our methodology and to suspectthe existence of charge transfers facilitating the dif-fusion phenomenon. These preliminary resultshave thus proven the feasibility of such a study.Nevertheless, we must underline the fact that thesmall size characterizing this first cell leads to aquite constrained system, which does not leaveenough room for performing a sufficient ionic re-laxation. The next step will thus correspond to thestudy of a larger cell where space conditions arefulfilled to allow for sufficient ionic relaxation. As aconsequence, we are now interested with “morerealistic” materials also characterized by oxygendeficiency closer to the one of interest for techno-logical applications in SOFCs.

SECOND CASE OF NONSTOICHIOMETRICCERIA: CeO1.875

In the case of CeO1.875, the two cerium atomslabeled Ce1 are located near the diffusing oxygen,whereas the two cerium atoms labeled Ce2 are char-acterized by the same values along the c axis but arefarther from this position (Fig. 3). Table II showsthat ionic relaxations are indeed facilitated for thedouble cell as compared with the single one.

Considering first activation energy values, TableII shows that lower values are obtained as com-

FIGURE 4. Density of states for the conduction bandof CeO1.75, in the saddle point state, calculated withthe APW�lo method (Wien2k code). [Color figure canbe viewed in the online issue, which is available atwww.interscience.wiley.com.]

TABLE II ______________________________________________________________________________________________CeO1.875: activation energies (Ea), maximum cerium displacements due to the oxygen vacancy presence(�uCe), Ce-O distances in the dumbbell configuration of the saddle point state (dCe-O), and energy gain due tothe ionic relaxation (�Erelax), for the unit cell parameters at 298 K and 1073 K

a (Å) Ea (eV)

�uCe (Å)

dCe-O (Å)

�Erelax (eV)

IS SPS IS SPS

5.411 0.78a 0.11a 0.20a 2.113a �0.69a �0.73a

0.82b

5.458 0.74a 0.12a 0.19a 2.119a �0.83a �0.70a

0.70b

a Calculated with VASP.b Calculated with Wien2k.



pared with the ones characterizing CeO1.75 (TableI), owing to the higher part of ionic relaxation in thedouble cell. The activation energy characterizingoxygen jumps in CeO1.875 is indeed around 0.8 eV atroom temperature and 0.7 eV at 1073 K, whereas itwas nearly 1.0 eV and 0.9 eV in the case of CeO1.75.However, we checked our calculations to obtain theright order of magnitude for the activation energiesvalues. Even if a direct comparison between ourinvestigations performed on nonstoichiometric ce-ria and the NMR work of Fuda et al. [13] on dopedceria is somewhat difficult, we can say that ourvalues seem effectively to be of the right order ofmagnitude, since an activation energy of nearly 0.6eV was derived from their work. The quite satisfy-

ing agreement between this value and the onesobtained in our calculation further validates thepresent approach.

The topological analysis undertaken for CeO1.875allowed us to note a relevant result which corre-sponds to the high capability to deformation of theoxygen atom by passing from the initial state to thesaddle point one (Fig. 5a). Effectively, diffusingoxygen is characterized at the saddle point positionby a parallelepipedic shape and has thus beenhighly expanded towards the direction perpendic-ular to the cerium–cerium axis, and compressedalong it. This corresponds to a very short Ce-Obond length in the dumbbell configuration (TableII). In this respect, we underline the fact that theoxygen ion is located here in a 2-coordinated envi-ronment, which is likely to facilitate oxygen ionpolarizability since the coordinance number is quitelow. This first feature, proving the ease of diffusionprovided by the ability to achieve deformation ofthe oxygen electron cloud, is also complemented bythe fact that the associated atomic volume increasesby 10.3% by passing from the initial state to thesaddle point one. This constitutes indeed anotherindication of the facility offered by the polarizabil-ity of oxygen for performing its jump towards thevacant site. Furthermore, Table III shows thatatomic volumes evolution of the nearest neighborcerium atoms (Ce1) also tends to facilitate the oxy-gen diffusion in CeO1.875 since these ones are char-acterized by a volume decrease. The Ce2 atoms,farther from the saddle point position, undergo onthe reverse a volume augmentation.

This topological analysis also supplied us withatom charge evolutions between the initial stateand the saddle point one (Table IV). These valuesseem to indicate that the amount of charge lost bythe association (diffusing O � 2 Ce1) is nearly

FIGURE 5. Atomic envelopes as defined in Bader’s“Atoms in Molecules” theory [29], calculated by theAPW�lo method (WIEN2k code) and rendered withGEOMVIEW [Phillips, M.; Munzner, T.; Levy, S. GEOM-VIEW, http://www.geomview.org]. Top (a): Initial (left)and saddle point (right) states for CeO1.875. Bottom (band c): Saddle point states for Ce0.75La0.25O1.875 (b)and Ce0.75Lu0.25O1.875 (c) cases in “2D” configuration.[Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com.]

TABLE III _____________________________________CeO1.875: atomic volumes (V) provided by thetopological analysis and evolution in % of theseatomic volumes between the IS and the SPS(Evolution � VIS � VSPS/VIS) for the unit cellparameter at 1073 K, calculated using the APW�lomethod (Wien2k code)

Species VIS (Å3) VSPS (Å3) Evolution (%)

O 14.10 15.55 �10.3Ce1 19.17 17.89 �6.7Ce2 19.17 20.40 �6.4

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834 VOL. 101, NO. 6

equivalent to the quantity gained by (2 Ce2). Thisobservation leads us to the conclusion that Ce-Ceand Ce-O charge transfers should help the diffu-sion, even if the amounts involved are very weak.

Another calculation on CeO1.875 was performedwith an “artificially charged lattice” in order tosupport the hypothesis on charge transfers occur-ring due to mixed valences. The removal of twoelectrons per cell, together with a backgroundcharge application, was carried out in order to getthe formal charge �4�e� for cerium atoms. Thecharge evolutions obtained from this calculation areshown in Table V. These indicate that cerium atomsnear the saddle point position cannot lose furtherelectrons, due to their charge �4�e�, and that theamount of charge lost by the diffusing atom is nowcompletely counterbalanced by the charge gain ofCe2, which is highly reducible. These values of Ce1charge evolution, which are nearly one order ofmagnitude lower than the corresponding ones inthe uncharged lattice, effectively show that chargetransfers come from the mixed valence nature ofcerium ions.

This result tends to demonstrate that lower val-ues of charge transfers obtained from the DOS in-tegration in the conduction band in CeO1.875 ascompared with the results obtained in the CeO1.75

case (0.09 electron for CeO1.875 versus 0.13 forCeO1.75) are much more ascribable to the fact thatcerium atoms near the diffusing oxygen are lessable to be oxidized in the CeO1.875 material (owingto a mean value of the cerium formal charge:�3.75�e� for CeO1.875 versus �3.5�e� for CeO1.75)rather than to the higher Ce-O distance at the sad-dle point state in the case of CeO1.875, which re-duces the local constraint in this dumbbell config-uration (Tables I and II). Comparison of chargetransfer values obtained by DOS integration in theconduction band (Table IV) with those obtained byintegration in atomic volumes, involving all elec-trons, for the uncharged CeO1.875 lattice tends todemonstrate that reverse charge transfers—fromdistant cerium atoms towards nearby saddle pointposition ones, involving deeper states of valenceband—should occur to partially compensate con-duction band charge donation from Ce1 to Ce2 iontypes. We may conclude from this analysis thatcharge transfers are not a simple phenomenon andthat, moreover, the charge transfers of outer elec-trons during atomic transport are not very signifi-cant insofar as the comparison of charges evolutionobtained from the DOS integration in the conduc-tion band and the integration within Bader atomicvolumes leads us to the conjecture that they arelargely counterbalanced by reverse charge dona-tions involving states of valence band.

At this stage in our study, we have seen thatsteric along with polarizability effects and chargetransfers might play a role in oxygen ion diffusionin ceria. However, from the latter observation, wemay suspect that the first two cases should consti-

TABLE IV _____________________________________________________________________________________________CeO1.875: total electron populations (NBader) obtained from the integration within atomic volumes, andconduction band populations (nCB) obtained through the integration of partial density of states, for the unitcell parameters at 1073 K, calculated using the APW�lo method (Wien2k code)

Species

NBader (e�)

�NBadera (e�)

nCB (e�)

�nCBa (e�)IS SPS IS SPS

Ce1 55.792 55.759 �0.033 0.22 0.13 �0.09Ce2 55.792 55.856 �0.064 0.22 0.28 �0.09O 9.236 9.178 �0.058

a �X � XSPS � XIS.

TABLE V ______________________________________Atomic electron population (NBader) results obtainedfrom the topological analysis in charged CeO1.875

(Ce8O152�) for the unit cell parameter at 1073 K,

calculated using the APW�lo method (Wien2k code)

Species

NBader (e�)

�NBadera (e�)IS SPS

O 9.169 9.111 �0.058Ce1 55.651 55.642 �0.009Ce2 55.651 55.691 �0.040

a �X � XSPS � XIS.



tute the predominant factors. Doped ceria closer tomaterials likely to have a potential application inSOFCs were then investigated, which results arediscussed in the following section.

DOPED CERIA: Ce0.75D0.25O1.875

Doped ceria calculations correspond to the in-vestigations performed for Ce0.75La0.25O1.875 andCe0.75Lu0.25O1.875. For each of these two com-pounds, three dopant configurations were under-taken with the specific aim of studying steric ef-fects: the first one (“2D”) corresponds to the twodopants located in the nearest-neighbor position ofan SPS one, the second one (“1D”) to one dopantsituated in its nearest-neighbor position, whereasthe other dopant is farther, and the third case(“0D”) where the two dopants are both in a secondneighbor metal position from it (Fig. 6). The diffu-sion pathway studied here is the same as the onepreviously studied for CeO1.875 and implies thusthat SPS in these three cases should correspond,respectively, to D-O-D, D-O-Ce, and Ce-O-Cedumbbell configurations. Not all the SPS discussedin the following should belong to the “actual” dif-fusion pathway, especially when high energies arefound. Therefore, we use the term “energy barrier”rather than “activation energy.” The configurations“2D” and “1D” studied in this work correspond,respectively, to the trimer D-vacancy-D and dimer

D-vacancy association defects entities mentioned inthe literature as existing in doped ceria.

From energy barriers, shown in Table VI, it fol-lows, first, that there is a high barrier to diffusionwhen two lanthanum ions are situated near the sad-dle point position; this barrier progressively decreas-ing when fewer lanthanum ions are located in thevicinity of the diffusing oxygen. In the case of lute-tium doping, energy barriers are, in the reverse,nearly identical in all cases and are furthermore lowerthan the one characterizing the corresponding un-doped ceria ( 0.6 eV for Ce0.75Lu0.25O1.875 versus 0.7eV for CeO1.875). This last point underlines the highlyfavorable character of lutetium as a dopant from theionic conductivity viewpoint in such a structure.

The strong dependence of energy barriers on thedopant nature should now be ascribed to the mi-croscopic features. Considering first the interatomicdistances as supplied in Table VI, we note that, inall cases, D-O or Ce-O distances in the dumbbellconfiguration are very short. This confirms the abil-ity of the diffusing oxygen to undergo a deforma-tion of its electronic cloud during diffusion. Figure5b,c effectively shows the striking oxygen shapemodification at the SPS for both cases of doping inthe configuration “2D.” On the other hand, theBader surfaces represented in this figure indicateclearly that the strongest deformation (flattest oxy-gen shape) corresponds to the highest calculatedenergy barrier, i.e., the La-doped ceria in case “2D.”

FIGURE 6. Ce0.75D0.25O1.875 unit cell in “2D” (a), “1D” (b), and “0D” (c) cases. [Color figure can be viewed in theonline issue, which is available at www.interscience.wiley.com.]

FRAYRET ET AL.

836 VOL. 101, NO. 6

They therefore evidence the higher constraint expe-rienced by the oxygen at the SPS in the case of thelanthanum-doped ceria as compared with the situ-ation characterizing the lutetium one. These clearlydistinct steric features of both cases of doping arethe consequence of the difference in atomic volumebetween both dopants (Vmean SPS La 19.40 Å3;Vmean SPS Lu 13.65 Å3) for a dumbbell configura-tion in which the dopant-diffusing oxygen inter-atomic distance is quite similar in both cases (TableVI), which is undeniably unfavorable for the lan-thanum doping in front of the lutetium one. To thisrespect, we underline that Bader atomic volumesfollow the trend of ionic radii, which is a quitesatisfying result from the method validity view-point. D-O distances at the SPS are only slightlyhigher in the “2D” and “1D” cases of the La-dopedceria in comparison with the Lu-doped one, whichis more evidence of the higher local steric hindrancegenerated by lanthanum ions.

To pursue the steric investigations, we now con-sider the evolution of the atomic volumes provided inTable VI as a function of the defect configuration. Firstof all, we notice, as in the case of CeO1.875, the samegeneralized trend of a volume augmentation for thediffusing oxygen atom and a volume decrease of thetwo atoms surrounding it during the jump. There is,thus, like for nonstoichiometric ceria, a strong influ-ence of the spatial environment on the diffusion phe-nomenon. In other respects, some clear differences

between doped and undoped ceria can be identified.In particular, the volume enhancement for oxygen ismore pronounced for La-doped ceria as comparedwith CeO1.875 whereas it is less pronounced in thecase of Lu-doped ceria. This result constitutes a firstevidence of differences in microscopic features be-tween lanthanum and lutetium doped ceria from theatomic volume effects viewpoint. Furthermore, thelarger enhancements of oxygen volume for lantha-num doped ceria are correlated with the larger de-crease of dopant volumes: Table VI shows that afactor 2 is found for both oxygen and dopant volumeevolutions, between the La-doped and the Lu-dopedcases in the configurations “2D” and “1D.” On theother hand, the oxygen volume enhancement de-creases from the “2D” to the “0D” configurations inthe case of La-doped ceria, whereas it remainsroughly constant in the case of Lu-doped ceria. Theevolution from the “2D” case towards the “0D” one of�V/VIS O and -�V/VIS M indeed completely matchesthe evolution of energy barrier for both doping cases,as can be deduced from Figure 7, underlying thestrong influence of atomic volume effects on energybarriers.

Overall, these trends tend to demonstrate thatdopant atomic volumes appear to drive the energybarrier values in the studied cases of doped ceria.The steric hindrance aspects should thus be consid-ered as the predominant factor explaining the

TABLE VI _____________________________________________________________________________________________Ce0.75D0.25O1.875: energy barriers (�E), oxygen-rare earth distances in the dumbbell configuration of the saddlepoint state (dM-O), and atomic volumes evolution (�V/VIS) from the IS to the SPS (Unit cell parameter a � 5.458Å), calculated using the APW�lo method (Wien2k code)

Case �E (eV) Species dM-O (Å) VIS (Å3) VSPS (Å3) Evolution (%)

2 Lu 0.58 O 16.07 17.31 �7.7Lu1 2.068 14.66 13.65 �6.9

2 La 1.97 O 14.34 16.69 �16.4La1 2.130 21.81 19.35 �11.3

1 Lu 0.53 O 15.35 16.41 �6.9Ce1 2.121 19.24 18.08 �6.0Lu1 2.071 14.43 13.65 �5.4

1 La 1.26 O 13.79 15.85 �14.9Ce1 2.036 18.61 17.59 �5.5La1 2.195 22.00 19.55 �11.1

0 Lu 0.63 O 14.75 15.90 �7.8Ce1 2.133 19.18 18.03 �6.0

0 La 0.81 O 13.62 15.17 �11.4Ce1 2.103 18.97 17.84 �6.0



strong dependence of energy barriers on the natureof the dopant ions.

Conclusions and Prospects

In this article, a new procedure was presentedfor gaining insight into the fundamentals of diffu-sion. This methodology was conceived to shedmore light on the until now ill-known microscopicaspects of diffusion in ionic conductor oxides.

The procedure was applied to ceria-based mate-rials as test and first investigation of the approach.Even if we could expect in both nonstoichiometricceria and doped ceria cases that charge transfers ofthe Ce-Ce or Ce-D type might help the diffusionphenomenon, due to the mixed valence nature ofcerium, it has emerged that these seem to be of verylow magnitude if we consider the net charge effect(topological analysis). Accordingly, this observa-tion has allowed us to rule out charge transfers asconstituting the prevailing factor for driving diffu-sion in ceria-based materials. It has been shown thatthe predominant factor influencing oxygen atomic

jumps in these materials should be primarily of asteric nature. On the other hand, the polarizabilityof the oxygen seems to be the other key parameterinfluencing diffusion at the atomic scale. Spectacu-lar atomic volume and shape evolutions clearlydemonstrate that diffusion in ceria-based materialsis far from a simple rigid ionic spheres description.We have then succeeded in assigning the diffusionphenomenon in ceria-based materials to some pe-culiar parameters directly related to the intrinsiccompound features.

This methodology has in particular clearly provento be a pertinent tool for studying the dopant ionicradius effect on the diffusion phenomenon. We there-fore now apply it to the “actual” comparison of lan-thanum-doped ceria and lutetium doped-ceria, takenthis time with its own lattice parameter. The determi-nation of the favorable jump for each configurationassociated with the above-described procedure willallow us to determine whether it is preferable to insertsmall dopants, which are favorable on the atomicscale from the steric viewpoint but unfavorable at themacroscopic one, since they induce a lattice parame-

FIGURE 7. (a) Atomic volume evolution of the diffusing oxygen between IS and SPS (�V/VIS O) in Ce0.75D0.25O1.875

(D � La, Lu) for the “2D,” “1D,” and “0D” cases. (b) Atomic volume evolution of the metal nearest-neighbor from thediffusing oxygen between IS and SPS (-�V/VIS M) in Ce0.75D0.25O1.875 (D � La, Lu) for the “2D,” “1D,” and “0D”cases. (c,d): Column representation of the energy barrier for the “2D,” “1D,” and “0D” cases in Ce0.75La0.25O1.875 andin Ce0.75Lu0.25O1.875, respectively. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

FRAYRET ET AL.

838 VOL. 101, NO. 6

ter decrease or large dopants, which are characterizedby reverse features.

Future work has to be applied also to other ionicconductor oxides in order to discover the origin ofionic conductivity differences between these mate-rials. Under way are calculations notably focusingon La2NiO4��. Otherwise, prospective worksshould also include the development of other pro-cedures in view of increasing the list of tools thatwe have at our disposal for studying the micro-scopic aspects of diffusion. In particular, the Vine-yard prefactor (Expression 1) can be evaluatedthrough a dynamical matrix calculation. Moreover,maps of electrostatic energy density and electronlocalization function (ELF) should provide furtherinsight into the role of chemical bonding in trans-port phenomena within oxygen ion conductors.

ACKNOWLEDGMENTS

Computational facilities supplied by the scien-tific pole ’M3PEC’ of the University Bordeaux 1 aregratefully acknowledged.

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