Application of Density Functional Theory to the Modeling of theMixed Ionic and Electronic Conductor La2NiO4+δ: Lattice

Relaxation, Oxygen Mobility, and Energetics of Frenkel Defects

Christine Frayret,* Antoine Villesuzanne, and Michel Pouchard

Institut de Chimie de la Matie`re Condense´e de Bordeaux, CNRS, 87,AVenue du Dr. Albert Schweitzer, 33608 Pessac Cedex, France

ReceiVed January 27, 2005. ReVised Manuscript ReceiVed October 20, 2005

The uptake of oxygen ions into the La2NiO4 host matrix was investigated by electronic structurecalculations within density functional theory (DFT). For La2NiO4.125, considered in a tetragonal simulationsupercell, a tilting phenomenon occurs, alternating one NiO2 plane on two along the [100] and [010]directions of the parentI4/mmmlattice. Beyond structural considerations, DFT calculations are used inconjunction with a topological analysis of the electron density to determine, at the atomic scale, theintrinsic features governing ionic conductivity for a direct interstitial mechanism within the (001) plane.Lattice relaxation and oxygen polarizability seem to be the key factors governing direct diffusionmechanisms in this compound. Charge-transfer phenomena were ruled out as a relevant parameter foroxygen mobility. The interstitial oxygen is identified as a formal O2- species both at its usualcrystallographic site and at the highly constrained saddle point state. Oxygen vacancies at the equatorialsites are found to be energetically more favorable than at the apical sites for the formation of Frenkel-type defects.

Introduction

During the past decade and with the discovery of mixedionic-electronic conduction in La2NiO4+δ,1-5 K2NiF4-typeoxides have gained a renewed interest as possible ion-conducting ceramic materials. There remains, however,debate as to the exact crystal structure of La2NiO4+δ (0.04< δ < 0.18) for which both orthorhombic (Fmmmor Bmab)and tetragonal (F4/mmm) unit cells have been proposed, aswell as biphasic mixtures, thus, involving different kinds ofcooperative tilting of the NiO6 octahedra.6-16 The knowledgeof structural details is essential for understanding the

mechanisms involved in ionic transport. Besides, the valenceof the interstitial oxygen as well as the nature of its possibleassociation with the neighboring apical oxygen atoms is alsostill controversial and difficult to characterize from experi-ment: several entities such as O2-, O- (i.e., holes in the pbands of oxygen), (O2)2-, (O2)- or (O3)5- have beenproposed.7,16-18

The stoichiometric nickelate La2NiO4 belongs to theRuddlesden-Popper series, of general formula An+1MnO3n+1

(n ) 1, 2, 3, ...). In such a structure,n AMO3 perovskitelayers alternate with AO rocksalt-type layers along theccrystallographic direction (Figure 1a). For ions such as Cu2+

(3d9) or Ni3+ (3d7; e.g., La2CuO4, LaSrNiO4), the Jahn-Teller effect leads to an elongation of the MO6 octahedraalong thec direction. This tetragonal distortion may befurther enhanced by the Lewis acid character of rare-earthcations A in the M-Oapical-A dumbbell entity, which inducesa shortening of the Oapical-A bond and an elongation of theopposite M-Oapical bond. A more schematic representationof the A2MO4 structure, based on the shortest bonds, consistsof a succession of MO2 planes and rocksalt-type A2O2 doublelayers (Figure 1b).

* To whom correspondence should be addressed. E-mail: frayret@icmcb-bordeaux.cnrs.fr. Tel.:+33 540002657. Fax:+33 540002761.(1) Kharton, V. V.; Viskup, A. P.; Naumovich, E. N.; Marques, F. M. B.

J. Mater. Chem.1999, 9, 2623.(2) Skinner, S. J.; Kilner, J. A.Solid State Ionics2000, 135, 709.(3) Kharton, V. V.; Viskup, A. P.; Kovalesky, A. V.; Naumovich, E. N.;

Marques, F. M. B.Solid State Ionics2001, 143, 337.(4) Boehm, E.A2MO4+δ nickelates, new cathode materials for intermediate

temperature Solid Oxide Fuel Cells. Ph.D. Thesis, University ofSciences and Technologies, Bordeaux, 2002.

(5) Boehm, E.; Bassat, J.-M.; Steil, M. C.; Dordor, P.; Mauvy, F.; Grenier,J.-C.Solid State Sci.2003, 5, 973.

(6) Demourgues, A.; Weill, F.; Darriet, B.; Wattiaux, A.; Grenier, J.-C.;Gravereau, P.; Pouchard, M.J. Solid State Chem.1993, 106, 317.

(7) Demourgues, A.; Weill, F.; Darriet, B.; Wattiaux, A.; Grenier, J.-C.;Gravereau, P.; Pouchard, M.J. Solid State Chem.1993, 106, 330.

(8) Demourgues, A.; Weill, F.; Grenier, J.-C.; Wattiaux, A.; Pouchard,M. Physica C1992, 192, 425.

(9) Hiroi, Z.; Obata, T.; Takano, M.; Bando, Y.; Takeda, Y.; Yamamoto,O. Phys. ReV. B 1990, 41, 11665.

(10) Tranquada, J. M.; Kong, Y.; Lorenzo, J. E.; Buttrey, D. J.; Rice, D.E.; Sachan, V.Phys. ReV. B 1994, 50, 6340.

(11) Tranquada, J. M. InNeutron Scattering in Layered Copper-OxideSuperconductors; Furrer, A., Ed.; Kluwer Academic Publishing:Dordrecht, 1998; pp 225-260.

(12) Tranquada, J. M.; Lorenzo, J. E.; Buttrey, D. J.; Sachan, V.Phys.ReV. B 1995, 52, 3581.

(13) Paulus, W.; Cousson, A.; Dhalenne, G.; Berthon, J.; Revcolevschi,A.; Hosoya, S.; Treutmann, W.; Heger, G.; Le Toquin, R.Solid StateSci.2002, 4, 565.

(14) Skinner, S. J.Solid State Sci.2003, 5, 419.(15) Jorgensen, J. D.; Dabrowski, B.; Pei, S.; Richards, D. R.; Hinks, D.

G. Phys. ReV. B 1989, 40, 2187.(16) Rodrı´guez-Carvajal, J.; Fernandez-Diaz, M. T.; Martinez, J. L.J.

Phys.: Condens. Matter1991, 3, 3215.(17) Buttrey, D. J.; Ganguly, P.; Honig, J. M.; Rao, C. N. R.; Schartman,

R. R.; Subbanna, G. N.J. Solid State Chem.1988, 74, 233.(18) Kuiper, P.; Rice, D. E.; Buttrey, D. J.; Lin, H. J.; Chen, C. T.Physica

B 1995, 208, 271.

6538 Chem. Mater.2005,17, 6538-6544

10.1021/cm050195f CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 11/23/2005

As for the perovskite structure, the Goldschmidt tolerancefactor19 can be used to account for the effects of the sizemismatch between anions and cations in the A2MO4 structure.It is written as

where rA, rM, and rO are the ionic radii of the An+, Mm+,and O2- ions, respectively.t ) 1 corresponds to an idealradii match (undistorted tetragonal structure). For La2NiO4,t ) 0.89,19 indicating that the NiO2 plane is under pressureas a result of the stretched La2O2 layer. The strain may bereleased by either a tilting of the NiO6 octahedra in thestoichiometric compound, which leads to an orthorhombicBmab structure, or an oxygen intercalation in the La2O2

double layers, resulting in overstoichiometric phasesLa2NiO4+δ. In the latter case, the release of strain in the NiO2

plane arises both from the partial oxidation of Ni2+ into Ni3+

and from a tilting of NiO6 octahedra. Oxygen can beintercalated either electrochemically at room temperature inan alkali aqueous electrolyte switching La2NiO4 as anodeor chemically by using oxidizing agents such as OCl-.17,20-23

The crystallographic site occupied by interstitial oxygen ionsin La2O2 layers corresponds to a double tetrahedron formedby four apical oxygen ions and four lanthanum ions. La2NiO4

can incorporate additional oxygen up toδ ) 0.25.6-8

Beyond δ ) 0.17, some ordering phenomena wereevidenced by Demourgues et al.6-8 from neutron diffractionand transmission electron microscopy. In the case ofLa2NiO4.25, ordering occurs both in the La2O2 layer and alongthe [111] direction of the parent tetragonal structure.7,8 Thiseffect leads to a 2× 2 × 1 superstructure corresponding toa monoclinic oxygen-ordered phase of space groupC2.7,8 Inthe intermediate range of oxygen incorporation (0.04< δ< 0.18), considerable debate surrounds the crystal structureadopted by La2NiO4+δ at both room and elevated tempera-tures. The electron diffraction work of Hiroi et al.9 performedfor 0.04< δ < 0.20 suggests a three-dimensional orderingof the interstitial oxygen ions. Hiroi et al. proposed thatordered phases occur whenδ equals to 1/2n, wheren is aninteger. Contrary to the conclusions of Hiroi et al., Tranquadaet al.10-12 found that the interstitial oxygen ions order one-dimensionally and periodically along thec axis. Severalstructural studies undertaken by these authors on a series ofLa2NiO4+δ single crystals forδ in the range 0.05-0.1110-12

indicate that a cooperative tilting of NiO6 octahedra takesplace at low temperature (T < 290 K) as a consequence ofinterstitial ions. This tilting effect occurs periodically alongthe c direction and leads to an orthorhombic lattice (spacegroup Bmab) in which the La3+ coordination becomes[1+7+1]. According to this model, local distortions generate(001) antiphase boundaries, which correspond to an inversionfor the direction of octahedra tilting. A so-called “stagen”structure corresponds to a succession ofn La2O2 layers ofthe same tilting direction terminated by antiphase boundaries.For 0.05< δ < 0.11, stage 4 to 2 ordering was observed byTranquada et al.10 on cooling below 290 K. On the otherhand, Paulus et al.13 found no evidence for lower symmetrythan tetragonal (F4/mmm) for excess oxygen content below0.18, whereas Skinner14 reported an orthorhombicFmmmphase forδ ) 0.17, in agreement with the early results ofJorgensen et al.15 on La2NiO4.18. Otherwise, according toRodrıguez-Carvajal et al.,16 the 0.04< δ < 0.13 system canbe described as a mixture ofF4/mmm-type overstoichiometricstructures and an almost stoichiometric [δ ∼ 0.00(2)]Bmab-type phase.

We report here on density functional theory (DFT)calculations of the lattice relaxation around interstitial oxygenions in La2NiO4.125. Furthermore, the intrinsic microscopicfeatures governing ionic conductivity at the atomic scale wereinvestigated from a topological analysis of the calculatedelectron density. We focused our study on the directinterstitial diffusion mechanism within the La2O2.25 layer ofLa2NiO4+δ. We also identified the nature of the interstitialoxygen through the evaluation of its atomic charge. Finally,the energetics of Frenkel pair formation as intrinsic latticedisorder was examined.

Computational Details

Our calculation procedure relies on the combined use oftwo ab initio codes at the level of DFT and employing planewave basis sets. The first step consists of determining therelaxed atomic positions by using a pseudopotential DFTcode as implemented in the VASP package,24-26 which wasused with success in earlier works by Eichler27 for the study

(19) Singh, K. K.; Ganguly, P.; Goodenough, J. B.J. Solid State Chem.1984, 52, 254.

(20) Wattiaux, A.; Fourne`s, L.; Demourgues, A.; Bernaben, N.; Grenier,J.-C.; Pouchard, M.Solid State Commun.1991, 77, 489.

(21) Bezdicka, P.; Wattiaux, A.; Grenier, J.-C.; Pouchard, M.; Hagenmuller,P. Z. Anorg. Allg. Chem.1993, 619, 7.

(22) Grenier, J.-C.; Wattiaux, A.; Doumerc, J.-P.; Dordor, P.; Fourne`s, L.;Chaminade, J.-P.; Pouchard, M.; Hagenmuller, P.J. Solid State Chem.1992, 96, 20.

(23) Demourgues, A.; Wattiaux, A.; Grenier, J.-C.; Pouchard, M.; Soubey-roux, J.-L.; Dance, J.-M.; Hagenmuller, P.J. Solid State Chem.1993,105, 458.

Figure 1. Two schematic representations of the tetragonal A2MO4 structurebased (a) on MO6 perovskite-type octahedra and (b) on square-planar MO4

entities, corresponding to a succession of MO2 planes alternating with A2O2

double layers of NaCl type.

t )rA + rO

x2(rM + rO)(1)

Application of DFT to the Modeling of La2NiO4+δ Chem. Mater., Vol. 17, No. 26, 20056539

of oxygen diffusion in zirconia. We used the projectoraugmented wave method28,29 for the treatment of electron-ion core interactions and a plane-wave cutoff of 400 eV.The generalized gradient approximation (GGA) from Perdewand Wang (PW91)30 was used for the exchange-correlationpotential. As a starting point, we used the cell parametersand atomic positions of La2NiO4.14 as determined by Pauluset al.13 The cell parameters were kept fixed throughout thestudy. The simulation cell is represented schematically inFigure 2. It corresponds to lattice parameters (ax2, ax2, c)with respect to the parameters of theF4/mmm tetragonalstructure, in which an interstitial oxygen is introduced (i.e.,La16Ni8O33 ≡ La2NiO4.125, Z ) 8). The single interstitialoxygen is initially at position (1/2 1/4 1/4) in the supercell andleads to a La2O2.25 layer. The second La2O2 rocksalt-typelayer, centered atz ≈ 3/4, does not possess any interstitialoxygen. Singlek-point calculations were performed usingVASP for k ) Γ and (1/4 1/4 1/4). As a result of the large unitcell used here, no significant differences were found exceptfor some apical Ni-O distances, with a 7% difference whichdoes not affect the conclusions of this work. The atomiccoordinates were relaxed until the energy change betweentwo ionic steps was smaller than 10-4 eV.

To check the accuracy of the calculated energy barriersand to perform a topological analysis of the electron densityusing Bader’s method (see below), the as-obtained equilib-rium geometry is then used as input for an all-electron, full-potential DFT calculation, using the augmented plane wave+ local orbitals (APW+lo) method,31,32 as implemented in

the WIEN2k package.33 The GGA from Perdew et al.34 wasused for the exchange-correlation potential, and muffin-tinsphere radii were fixed to 1.7, 1.8, and 1.9 au for oxygen,nickel, and lanthanum atoms, respectively. A 38k-pointsampling was used for the irreducible wedge of the Brillouinzone.

Concerning the study of atomic transport, this procedureis undertaken for the two states characterizing the systemduring diffusion at the microscopic scale: the saddle pointstate (SPS), linked to the atomic position for the energymaximum along the oxygen diffusion pathway, and the initialstate (IS), corresponding to the oxygen on a lattice site. Theenergy difference between these two states provides anestimation of the diffusion activation barrier, which can betentatively used for discriminating several diffusion path-ways. The lowest activation barrier among all diffusionpathways can then be compared to the experimental activa-tion enthalpy. Besides the research of energy barrier values,these calculations are aimed at identifying the relevant factorswhich govern, at the atomic scale, the oxygen ion conductiv-ity. These factors may be of four kinds: the structural factor(lattice type and chemical bonding), the steric factor, the atompolarizability, and the existence of charge transfers, whichmay assist diffusion especially when mixed valences arepresent in the compound. This calculation procedure provedefficient for the study of oxygen diffusion in ceria-basedmaterials.40,41

Bader’s “Atoms in Molecules” (AIM) theory35-39 definesinteratomic surfaces as linking all points of zero flux in thegradient vector field of the electron densityF(r ). Accordingto this definition, the physical space can be partitioned intonon-overlapping regions which contain, in general, one atomonly. The boundaries between these so-called atomic basinscan thus be considered as atomic envelopes. The AIM theoryprovides thus a nonambiguous way to divide the electrondensity into atomic contributions. Although atomic electronpopulations from the AIM analysis remain model-dependent,their evolution with structural parameters is more reliablethan if based on Mulliken-type analysis. The evolution ofthe shape of the atomic envelopes along the diffusion pathis related to atom polarizability, depicts the ability of thediffusing atom electron cloud to deform, and gives informa-

(24) Kresse, G.; Hafner, J.Phys. ReV. B 1993, 47, RC558.(25) Kresse, G.; Furthmu¨ller, J. Comput. Mater. Sci.1996, 6, 15 (http://

cms.mpi.univie.ac.at/vasp).(26) Kresse, G.; Furthmu¨ller, J. Phys. ReV. B 1996, 54, 11169.(27) Eichler, A.Phys. ReV. B 2001, 64, 174103.(28) Kresse, G.; Joubert, J.Phys. ReV. B 1999, 59, 1758.(29) Blochl, P. E.Phys. ReV. B 1994, 50, 17953.(30) Perdew, J. P.; Wang, Y.Phys. ReV. B 1992, 45, 13244.

(31) Sjostedt, E.; Nordstro¨m, L.; Singh, D. J.Solid State Commun.2000,114, 15.

(32) Madsen, G. K. H.; Blaha, P.; Schwarz, K.; Sjo¨stedt, E.; Nordstro¨m,L. Phys. ReV. B 2001, 64, 195134.

(33) Blaha, P.; Schwarz, K.; Madsen, G.; Kvasnicka, D.; Luitz, J.WIEN2k;An Augmented Plane Wave+ Local Orbitals Program for CalculatingCrystal Properties; Techn. Universita¨t Wien: Vienna, 2001 (ISBN3-9501031-1-2; http://www.wien2k.at/).

(34) Perdew, J. P.; Burke, K.; Ernzerhof, M.Phys. ReV. Lett. 1996, 77,3865.

(35) Bader, R. F. W.Atoms in Molecules: A Quantum Theory; ClarendenPress: Oxford, 1990.

(36) Bader, R. F. W.Chem. ReV. 1991, 91, 893.(37) Bader, R. F. W.J. Phys. Chem. A1998, 102, 7314.(38) Penda´s, A. M.; Costales, A.; Luan˜a, V. Phys. ReV. B 1997, 55, 4275.(39) Aray, Y.; Rodriguez, J.; Vega, D.Comput. Phys. Commun.2002, 143,

199.(40) Frayret, C.; Villesuzanne, A.; Pouchard, M.; Matar, S.Int. J. Quantum

Chem.2005, 101, 826.(41) Frayret, C.Application of Density Functional Theory (DFT) to the

modelling of oxygen ion diffusion within model solid electrolytes andmixed conductors. Ph.D. Thesis; University of Sciences and Technolo-gies, Bordeaux, 2004.

Figure 2. Schematic representation of the simulation cell (2a × 2a × clattice with respect to the unit cell of symmetryI4/mmm) for several planesalong thec axis:z ) 0 (NiO2 plane),z ≈ 0.15 (LaO layer),z ≈ 0.35 (LaOlayer), z ≈ 1/2 (NiO2 plane),z ≈ 0.65 (LaO layer), andz ≈ 0.85 (LaOlayer); Oeq, equatorial oxygen; Oap, apical oxygen.

6540 Chem. Mater., Vol. 17, No. 26, 2005 Frayret et al.

tion on the constraints at the SPS configuration. On the otherhand, the volume associated with each atom can be calculatedfrom its envelope. Finally, the integration of the electrondensity within atomic volumes provides the atomic charges,from which the degree of ionic character and, above all, thecharge transfers during diffusion can be estimated.

Results and Discussion

1. Lattice Relaxation. The atomic positions were firstrelaxed in La2NiO4.125with the interstitial oxygen (OI) in itsinitial position, that is, (1/2 1/4 1/4) in the supercell. Nosignificant displacements are obtained for the lanthanum andnickel ions. The main displacements involve the oxygen ionsfrom the four NiO6 octahedra first neighbors of OI (Figure3). In each of these octahedra, the apical oxygen (Oap) ionnext to OI is repelled by∼0.47 Å; the other Oap ion in theoctahedron is displaced by∼0.15 Å in the opposite direction.The equatorial oxygen (Oeq) ion sitting below OI rises by∼0.1 Å alongc, and the one at the opposite corner is lowered

by ∼0.3 Å (Figure 4). The two other Oeq ions are notaffected.

These combined displacements constitute a tilting effectof the octahedra along the [010]I4/mmmaxis and the [100]I4/mmm

axis of the next layer. The NiO6 octahedra are tilted by anangle of about 12° in the vicinity of the interstitial oxygen,that is, more than twice the tilt angle determined for thestoichiometricBmabstructure at room temperature, whichamounts to 5°. This suggests that the oxygen intercalationin La2NiO4 is a more efficient way to reduce the inherentlattice strain than the orthorhombic distortion. In the lowerpart of the NiO6 octahedra, the tilting is rather lesspronounced (â′ ∼ 4°), indicating that the octahedra areslightly distorted.

The tilting phenomenon obtained from our calculations isin agreement with the observations of Paulus et al.13 obtainedfor La2NiO4.14 using neutron diffraction on single crystals.The calculated atomic displacements of Oap ions induced bythe lattice relaxation are indeed in the same direction([100]I4/mmm and [010]I4/mmm) and of the same order ofmagnitude as the ones observed by Paulus et al. The presentwork thus constitutes the first theoretical evidence that theintercalation of interstitial oxygen actually causes a tiltingof octahedra, not a simple lattice distortion. The supplemen-tary information gained from calculations lies in the iden-tification of a displacement for four (two equatorial and twoapical) oxygen ions among the six of the octahedron; inparticular, no information on the displacements of Oeq atomscould be obtained from neutron diffraction.13 The crystallinestructure determined from our DFT calculations does notcorrespond to the tilting direction indicated by Tranquadaet al.10 for an antiphase boundaries model between layers ofBmabsymmetry in the compound La2NiO4+δ. Indeed, thetilting described by Tranquada et al.10 involves a rotation

Figure 3. La2NiO4.125(IS): oxygen atom displacement generated by ionicrelaxation for the apical [(a)z ∼ 0.15; (b)z ∼ 0.35] and equatorial [(a)z∼ 0; (b) z ∼ 1/2) oxygen atoms. The arrows show the displacement in theLa2O2.25 layer whereas the signs (+) and (-) indicate the direction ofdisplacement along thec axis.

Figure 4. La2NiO4.125(IS): schematic representation of the tilting of NiO6

octahedra centered atz∼ 0 along the [010] direction; displacement distancesassociated with this lattice relaxation phenomenon.

Application of DFT to the Modeling of La2NiO4+δ Chem. Mater., Vol. 17, No. 26, 20056541

around the [110]I4/mmm axis for which the six atoms of theoctahedron undergo a displacement (Figure 5). The structuralmodel obtained through the lattice relaxation for La2NiO4.125

rather corresponds to the one described by Axe,43 becausefiles similar to the ones proposed for theP42/ncmtetragonalphase have been evidenced. The tilting of octahedra in thisphase indeed takes place according to the [110]P42/ncm and[11h0]P42/ncmdirections, which are equivalent to [100]I4/mmmand[010]I4/mmm (Figure 6). However, the alternate tilting ofoctahedra observed for La2NiO4.125only appears one file ontwo, whereas it concerns all files in the case of theP42/ncmphase described by Axe.43

The displacement of Oap ions in La2O2 and La2O2.25 layersinduces a widening of the interstitial site (Figure 2). Theeight adjacent tetrahedral sites forming a sheet of edge-sharing tetrahedra are, therefore, not equivalent: Figure 7bshows that the occupied site (1) is enlarged as mentioned,those of type (2) are compressed, and the sites labeled (3)are unchanged whereas those of type (4) are weakly enlargedand distorted as compared to the undistorted tetragonalstructure (Figure 7a). OI lies in a symmetrically dilated Oap

tetrahedron leading to an Oap-Oi distance of 2.61 Å. Thisdistance, close to the sum of ionic radii of O2- (rO2- [VI] (1.40Å) + rO2- [IV] (1.38 Å) ) 2.78 Å),42 rules out hypotheses ofan interstitial oxygen involved in peroxide or trioxide species,or being O-. Our topological analysis of the electron densityled to the same conclusion (see below), with OI beingunambiguously an O2- ion, even in the SPS of the diffusionprocess. Atomic volumes and total electron populationsdeduced from the topological analysis exhibit some peculiar

features. Oap ions are systematically more voluminous thanOeq ions, by about 7-10%. This may be related to the largerNi-Oap bond length. Otherwise, it is worth mentioning thatOeq and Oap ions have a nearly identical total electronpopulation. The difference in atomic volumes is, therefore,not correlated with a difference in electron population. Thus,Oap ions are expected to have a higher polarizability thanOeq ions; this effect may be relevant for atomic transport.

2. Direct Diffusion Mechanism within the La2O2.25

Layer. The diffusion pathway considered in this studycorresponds to direct atomic jumps of the interstitial oxygenbetween vacant adjacent interstitial sites within the La2O2.25

layer. A SPS related to this ionic transport process is locatedat (5/8, 3/8, 1/4) (Figure 8). Other diffusion paths andmechanisms involve coupled displacements of both inter-stitial and apical oxygen atoms; in the case of diffusion alongthe c direction, equatorial oxygen atoms participate too.44

Our study of these more complex mechanisms will bereported at a later time.

At the saddle point, all oxygen ions in the simulation cellare displaced. OI nearest-neighbor oxygen atoms move bynearly 0.8 Å, thus, enlarging considerably the passagewindow and reducing the steric constraints (dOap-OI ≈ 2.44Å after relaxation). The relaxation of the two nearest-

(42) Shannon, R. D.Acta Crystallogr., Sect. A1976, 32, 751.(43) Axe, J. D. Private communication cited in ref 10.

(44) Minervini, L.; Grimes, R. W.; Kilner, J. A.; Sickafus, K. E.J. Mater.Chem.2000, 10, 2349.

Figure 5. Top: tilting of the octahedra obtained by DFT calculations onLa2NiO4.125. Bottom: tilting of the octahedra in the model proposed byTranquada et al.10-12 (Bmabstructure+ antiphase boundaries: the tiltingtakes place around the [110]I4/mmmaxis).

Figure 6. Axe model43 for the displacement of apical oxygen atoms inLa2NiO4+δ. Dashed line:P42/ncm lattice.

Figure 7. (a) Edge-sharing tetrahedral sites for interstitial oxygen ions inthe (001) plane atz∼ 0.25 before ionic relaxation. (b) Various deformationsof the tetrahedral sites after relaxation; ionic relaxation generates four kindsof tetrahedral sites: (1) enlarged, (2) compressed, (3) unchanged, and (4)weakly enlarged and distorted.

6542 Chem. Mater., Vol. 17, No. 26, 2005 Frayret et al.

neighbor lanthanum atoms also contributes to this effect butto a less extent, with a displacement of 0.3 Å (dLa-OI ≈ 2.16Å after relaxation). The lanthanum ions of the La2O2.25 layermove quite significantly (up to 0.3 Å); displacements oflanthanum ions in the La2O2 layer do not exceed 0.08 Å.The alternate tilting of NiO6 octahedra described abovebecomes a double tilting: the Oap ions are now displacedalong the [110]I4/mmm and [11h0]I4/mmm directions (Figure 9).This remains a tilting phenomenon because the displacementsof Oap ions belonging to the La2O2 layer are inverted ascompared to the ones from the La2O2.25 layer.

The direct interstitial diffusion pathway within the La2O2.25

layer is characterized by a moderate calculated energy barrier,of the order of 1.2 eV. Beyond lattice softness, another factorexplaining this rather low value is the high polarizability (ordeformability) of the oxygen ions, which is evidenced bythe shape of the diffusing oxygen at the SPS (Figure 10).

The evolution of atomic volumes, derived from thetopological analysis of the electron density, reinforces theargument of strong oxygen deformability, as already em-phasized in ceria-based materials.40,41 Although highly

constrained at the saddle point configuration, the OI atomexhibits a 12.5% volume increase when passing from itsnormal crystallographic site to the saddle point site. A volumeincreasemeans that the electron cloud extends significantlyalong the direction of diffusion, whereas it is compressedperpendicular to it. Meanwhile, the volume of Oap andlanthanum atoms located in nearest-neighbor position withrespect to OI decreases by 4.5 and 2.5% on average,respectively. Beyond steric effects, it is of primary interestto check possible charge transfers during the diffusionprocess, in relation with the possible existence of O- species(or holes delocalized over several oxygen atoms) and M-Oor M-M′ charge transfers. First, one of the most interestingresults provided by the topologic analysis lies in the evidenceof the O2- character of OI at the IS, owing to the very closecharge values for lattice and interstitial oxygen atoms (Table1). Second, Table 1 shows that the charge evolution betweenthe IS and the SPS is quite weak: OI tends to lose only 0.1electron in the SPS. The total amount of charge lost byoxygen atoms (Oap and OI) at the SPS does not exceed 0.2electron for three atoms. The charge compensation seemsto occur via the overall lattice, and the mixed valencecharacter of nickel plays a minor role. A slight change inelectron population for nickel is observed only in next-nearest-neighbor positions (Table 1). Anyhow, charge trans-fers are very small during the direct interstitial transportprocess within the La2O2.25 layer and can be clearly excluded

Figure 8. Schematic representation of the interstitial oxygen displacementin the La2O2.25 layer (direct mechanism). The SPS position is situatedmidway from two vacant interstitial sites (crosses). Nearest neighbors ofthe interstitial oxygen atom are shown in projection.

Figure 9. Displacements of apical oxygen atoms induced by the ionicrelaxation at the SPS in the direct interstitial mechanism. IS and FS:positions of the interstitial oxygen ion in the IS and final state of the hoppingprocess, respectively.

Figure 10. La2NiO4.125 (SPS in the direct interstitial mechanism withinthe La2O2.25 layer): atomic envelopes obtained by the topological analysisof the calculated electron density (rendered with GEOMVIEW);45 mobileoxygen (blue), apical oxygen (purple), and lanthanum (red) atoms. The arrowindicates thec direction.

Table 1. La2NiO4.125: Electron Populations (NBader) Obtained fromthe Integration within Atomic Volumes at the IS and SPSs and

Their Differences for the Interstitial Oxygen Atom, ItsNearest-Neighbor Apical Oxygen, Lanthanum and Nickel (NN)

Atoms, and Its Next-Nearest-Neighbor (NNN) Nickel Atoms

atom NBader(IS) NBader(SPS) ∆N

OI 9.32 9.22 -0.10Oap 9.31 9.26 -0.05La 54.91 54.94 +0.03NiNNN 26.68 26.71 +0.03NiNN 26.67 26.67 0.0

Application of DFT to the Modeling of La2NiO4+δ Chem. Mater., Vol. 17, No. 26, 20056543

as a prominent factor for oxygen mobility. As in ceria-basedmaterials,40,41 our results suggest that steric factors as wellas lattice relaxation and oxygen polarizability play the majorroles for oxygen diffusion in La2NiO4+δ, at least as far asdirect interstitial mechanisms are concerned.

3. Energetics of Frenkel Defects.The defects consideredin this work are anion Frenkel pairs composed of aninterstitial oxygen and (i) a vacancy on the equatorial site,(ii) a vacancy on an apical site in the La2O2.25 layer, and(iii) a vacancy on the apical site in the La2O2 layer. Thesethree types of Frenkel pairs are associated with the followingreaction in Kroger-Vink notation: OO

× f VO‚‚ + Oi′′. Their

calculated formation energies are 1.59, 2.34, and 1.79 eV,respectively. Thus, the dominant Frenkel pair in the com-pound La2NiO4.125, from the viewpoint of formation energy,would be vacancies at the equatorial site. This trend is inagreement with atomistic calculations carried out by Min-ervini et al.44 The preferential localization of the oxygenvacancy in the basal plane contributes to a decrease in thestrain arising from the mismatch between the La2O2 layerand the NiO2 plane. A non-negligible energy difference wasfound between the formation of apical vacancies in thevicinity of the interstitial oxygen and those away from theLa2O2.25 layer. The formation of Frenkel defects involving avacancy at the apical site in the La2O2 layer is indeed favored.

The widening of interstitial sites and the predominanceof equatorial vacancies among the different intrinsic latticedisorders of Frenkel type may provide an element ofexplanation for the low activation energy for diffusion alongthec axis, as compared to the one in the (a, b) plane, foundexperimentally by Bassat et al.46 using secondary ion massspectrometry on La2NiO4+δ single crystals. The occurrenceof Frenkel pairs may be a prerequisite for efficient atomictransport in oxides with Ruddlesden-Popper type structure.

Conclusions

This work provides the first computational evidence thatthe intercalation of interstitial oxygen in La2NiO4 actuallycauses a tilting of NiO6 octahedra. For La2NiO4.125consideredin a tetragonal simulation supercell, this tilting phenomenon

occurs alternating one NiO2 plane on two along the directions[100] and [010] of the parent tetragonal lattice (i.e., rotationaround the axes [010]I4/mmmand [100]I4/mmm). The interstitialoxygen is formally O2- both at its usual crystallographic siteand at the highly constrained SPS. Owing to the tilt angleand to the valence of the interstitial species, oxygenintercalation is a more efficient way to partially reduce thesteric strain resulting from the lattice parameter mismatchbetween the La2O2

2+ and NiO22- layers, compared to the

orthorhombic distortion of the stoichiometric phase.The topological analysis of the electron density has proven

to be a pertinent tool for the study of features at the atomicscale in the ionic diffusion phenomena. In La2NiO4.125, areasonable value for the activation energy for the directinterstitial diffusion mechanism within the La2O2.25 layer wasobtained by DFT calculations. Lattice relaxation, atomicvolumes evolution, and high oxygen polarizability seem tobe the key factors governing direct diffusion mechanisms inthis type of compound. Charge-transfer phenomena play onlya minor role, if at all.

In agreement with previous theoretical studies, an oxygenvacancy on the equatorial site associated with an interstitialoxygen atom corresponds to prevalent Frenkel pairs. Workis in progress to investigate indirect diffusion mechanismsand the role of Frenkel defects, in relation with theexperimental observation of low activation energy fordiffusion along thec axis in La2NiO4+δ. Otherwise, the tiltingdirections evidenced in this study may depend on the initialstructural choice (tetragonal vs orthorhombic supercell).Different simulation cells might lead to other types ofcollective tilting.

Note Added after ASAP Publication.There was an errorin the Kroger-Vink notation in section 3 of Results andDiscussion in the version published ASAP November 23,2005; the corrected version was published ASAP November29, 2005.

Acknowledgment. The authors are grateful to J.-C. Grenier,J.-M. Bassat, and the SOFC group from ICMCB for usefuldiscussions. C. Domain and P. Stevens are also acknowledgedfor helpful discussions and for financial support from EDF.Computational facilities were provided by the Pole M3PEC,Bordeaux 1 University.

CM050195F

(45) Phillips, M.; Munzner, T.; Levy, S. GEOMVIEW. http://www.geomview.org; accessed March 2003.

(46) Bassat, J.-M.; Odier, P.; Villesuzanne, A.; Marin, C.; Pouchard, M.Solid State Ionics2004, 167, 341.

6544 Chem. Mater., Vol. 17, No. 26, 2005 Frayret et al.