Chemical Physics Letters 499 (2010) 126–130

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Chemical Physics Letters

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Hybrid density functional theory description of oxygen vacancies in the CeO2 (1 1 0)and (1 0 0) surfaces

Michael Nolan ⇑Tyndall National Institute, University College Cork, Lee Maltings, Cork, Ireland

a r t i c l e i n f o

Article history:Received 21 April 2010In final form 3 September 2010Available online 9 September 2010

0009-2614/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.cplett.2010.09.016

⇑ Fax: +353 21 4270 271.E-mail address: michael.nolan@tyndall.ie

a b s t r a c t

In ceria the formation of oxygen vacancies plays a key role. Density functional theory, corrected for on-site Coulomb interactions (DFT + U) provides a reasonable description of oxygen vacancies, but has issueswith the U dependence. We present a hybrid HSE06 study of oxygen vacancies in the (1 1 0) and (1 0 0)ceria surfaces. We find (i) the oxygen vacancy formation energy is larger with hybrid DFT compared withDFT + U, (ii) localised Ce3+ ions form and (iii) the position of the Ce3+ gap state is in good agreement withexperiment. Our results provide important information for assessing the reliability of the DFT + Uapproach.

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1. Introduction

Oxygen vacancies in CeO2 are key to its applications [1–3]. Uponformation of a neutral oxygen vacancy, the two electrons leftbehind reside on Ce ions near the vacancy site, reducing them toCe3+ [4–7]. This finding was established using density functionaltheory (DFT) corrected for on-site Coulomb interactions, DFT + U,with the PW91 exchange–correlation functional and U = 5 eV(hereafter denoted PW91 + U) a number of years ago [4–7] and fur-ther confirmed by hybrid DFT calculations on the (1 1 1) surface[8]. While DFT + U has proven to be a useful, pragmatic approachto modelling ceria, as evidenced by a number of publications in re-cent years [4–17], there remains the issue that the U parameter,applied to the Ce 4f states, is empirical and the interesting materialproperties depend on U. Our work has used U = 5 eV and the PW91approximate exchange–correlation functional, which was based onrecovering a consistent description of the position of the Ce3+ peakin the density of states of reduced ceria [4,5,18]. Other work hasused U = 4.5 eV with the generalized gradient approximation tothe exchange–correlation functional. While U has been fitted tothe description of a particular property, it has also been derivedfrom first principles [19].

However, properties of ceria depend strongly on U, as demon-strated by da Silva et al. [20], Huang and Fabris [9] and Castletonet al. [6]; one particular finding is that the position of the Ce3+ statein the band gap depends on U – if U is small, this state is found atthe bottom of the empty Ce 4f manifold, moderate values of U (4–5 eV) place this state in the middle of the band gap, while largervalues of U (>7 eV) actually place the Ce3+ states in the valenceband, which is inconsistent with experiment. A further issue is that

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even with the + U correction, the gap between the valance bandand the empty Ce 4f states remains underestimated. This meansthat no single value of U can be used to describe all properties ofceria to the same accuracy. Hence there is a need to use an ap-proach that can provide a consistent description of all properties,with less empiricism involved.

Hybrid DFT, in the shape of the HSE06 functional [21], has pro-ven to be a very good approach for metal oxides, with a number ofpapers providing the first accurate theoretical description of someimportant metal oxides [22–26]. For ceria, Hay et al. applied ascreened exchange hybrid DFT functional in a Gaussian basis [27]to provide a proper description of both Ce2O3 and oxidised CeO2,as noted by Beste et al. [28]. Subsequently, da Silva et al. appliedhybrid DFT (HSE06 screened exchange) in a plane wave basis tobulk ceria, highlighting the more consistent results obtained fromthis approach. Recently, Ganduglia-Pirovano et al. used HSE06 toshow that the most stable oxygen vacancy site in the (1 1 1) ceriasurface is a subsurface oxygen site [8] and that the most stable dis-tribution of Ce3+ depends sensitively on the structural distortionsin the surface upon reduction. Very recently Migani et al. [29] ap-plied PW91 + U (U = 4 eV) and HSE06 to oxygen vacancy formationon ceria nanoparticles. They found that oxygen vacancy formationis more favourable on nanoparticles compared to extended sur-faces. Hybrid DFT calculations, while expensive and not easily ap-plied to large model system, do provide very good results for metaloxides and can also be used to assess the performance of other DFTapproaches, such as LDA/GGA and DFT + U.

In this Letter we therefore present the results of a hybrid DFTstudy of a neutral oxygen vacancy in the (1 1 0) and (1 0 0) surfacesof ceria, providing important information on formation of oxygenvacancy defects in these surfaces, which are more reactive to-wards, e.g. CO oxidation and are thus of great interest in the catal-ysis community. In the latter application, these surfaces can be

M. Nolan / Chemical Physics Letters 499 (2010) 126–130 127

easily exposed by formation of nanorods or nanocubes and the im-pact of the exposed surface plane determined. In addition, thepresent results also provide a useful information for assessingthe impact of the value of U in DFT + U calculations [4,5,13].

2. Computational methods

We use a slab model of the ceria (1 1 0) and (1 0 0) surfaces anda plane wave basis set to describe the valence electronic wavefunctions with the VASP code [30]. The cut-off for the kinetic en-ergy is 396 eV. For the core–valence interaction we apply Blöchl’sprojector augmented wave (PAW) approach [31]. For Ce, we use12 valence electrons, and for O a [He] core. We use the screenedexchange Hyed-Scuseria-Ernzerhof (HSE06) hybrid DFT ex-change–correlation functional [21], with a screening length of0.2. k-Point sampling is performed using the Monkhorst–Packscheme, with a (2 � 2 � 1) sampling grid. For bulk fluorite struc-tured CeO2, we compute a lattice constant of 5.397 Å, in goodagreement with the experimental lattice constant of 5.411 Å.

The (1 1 0) surface is type I (in the Tasker classification [32])with neutral CeO2 planes along the slab and no dipole moment ispresent upon cleaving. However, the (1 0 0) surface has a dipoleupon cleaving and, in common with our previous work [4,5], wehave displaced half the surface terminating oxygen atoms fromone side of the slab to the other side of the slab [33]. For the(1 1 0) surface, a (2 � 2) expansion of the surface cell is used anda (4 � 2) expansion is used with the (1 0 0) surface. Formation ofone oxygen vacancy gives an overall oxygen vacancy concentrationof 1.79% in the (1 1 0) surface and 1.56% in the (1 0 0) surface; witheight oxygen atoms in the outermost layer of each surface super-cell, this translates into surface vacancy concentrations of 12.5%on both surfaces, consistent with vacancy concentrations in otherstudies [7,13]. The slab models are seven CeO2 layers (11.45 Å)thick for (1 1 0) and nine atomic layers (10.72 Å) thick for (1 0 0)with a 15 Å vacuum gap and the bottom two layers were fixed dur-ing the relaxations. All calculations are spin polarised with norestrictions on the overall spin.

The formation energy of an oxygen vacancy is given by

Evac ð110Þ surface ¼ ½EðCeO1:964Þ þ Eð1=2O2Þ� � EðCeO2Þ ð1ÞEvac ð100Þ surface ¼ ½EðCeO1:969Þ þ Eð1=2O2Þ� � EðCeO2Þ ð2Þ

Throughout, a positive energy signifies that there is a cost for for-mation of an oxygen vacancy.

3. Results

In Figure 1 we show a top view of the stoichiometric (1 1 0) and(1 0 0) ceria surfaces used in this work. Figure 2 shows the moststable oxygen vacancy structures in the (1 1 0) and (1 0 0) surfaces.For the (1 1 0) surface, there are two competing structures, which

Figure 1. Atomic structure of the stoichiometric ceria (1 1 0) [left panel] and (1 0 0) [indicating the edges of the supercells. Ce is white and O is read, with this colour schem

are characterized by the final position of the surface oxygen atomnearest the vacancy site. These sites were also found with DFT + U(PBE exchange–correlation functional, U = 5 eV, denoted PBE + U)by Scanlon et al. [13] and were termed the ‘simple’ oxygen vacancy(see Supplementary information) and the ‘split’ oxygen vacancy. Inthe ‘simple’ oxygen vacancy, the surface oxygen atom nearest thevacancy site remains at its lattice site, while in the ‘split’ vacancyconfiguration, this oxygen atom is displaced to bridge (henceforth‘bridging oxygen’) the two surface Ce ions nearest the vacancy site.In Ref. [13] the bridging oxygen vacancy was 0.32 eV more stable.In the present work, we find oxygen vacancy formation energies of2.31 and 2.00 eV for the simple and bridging oxygen vacancy struc-tures, respectively. These are ca. 0.40 eV higher than found withPBE + U [13] and the relative stability of the two configurations isthe same irrespective of the DFT approach used.

At the (1 0 0) surface, we have also found more than one possi-ble solution. The solutions differ in the distribution of the reducedCe3+ ions, which we shall return to below. The computed vacancyformation energy for the most stable vacancy structure is2.60 eV, compared to 2.27 eV for the PW91 + U results of Ref. [5].A second minimum energy structure for an oxygen vacancy in thissurface, which lies 0.20 eV higher in energy than this structure andis shown in the Supporting information. Hybrid DFT thereforetends to show higher vacancy formation energies compared to pre-vious DFT + U results using PW91 + U [5] and PBE + U [13], with thedifference between the two methods being between 0.30 and0.40 eV. We find further support for this result in the literature.For the (1 1 1) surface [8] the oxygen vacancy formation energywith hybrid DFT is larger by 0.80 eV compared with DFT + U(PBE + U, U = 4.5 eV) [8]. In [8], it was also found that there are anumber of oxygen vacancy structures that lie close in energy tothe most stable structure, differentiated by the position of the re-duced Ce ions. For nanoparticles [29], it was found that theHSE06 oxygen vacancy formation energies were ca. 0.4 eV largerthan the PW91 + U formation energies. The most favourable oxy-gen vacancy site is on the (1 0 0)-like facet on the nanoparticleand the difference between DFT + U and HSE06 in [29] is consistentwith our results on the extended (1 0 0) surface.

In terms of structure, the split vacancy structure on the (1 1 0)surface shows strong distortions around the vacancy site. Thebridging oxygen atom has Ce–O distances of 2.05 and 2.31 Å tothe nearest surface Ce ions and is pushed out of the surface layerby 0.43 Å. These surface Ce ions are pushed away from their initiallattice sites with Ce–O distances to the remaining neighbouringsurface oxygens of 2.16/2.20 Å for one of the Ce ions and 2.32/2.35 Å for the other Ce ion. The Ce–Ce distance for these Ce ionsis 4.15 Å, compared to 3.87 Å in the defect-free surface. Distortionsin the structure are present into the second and third subsurfacelayers, similar to the PW91 + U and PBE + U results [5,13].

In the (1 0 0) surface, the removal of a twofold coordinated ter-minal surface oxygen atom displaces the neighbouring Ce ions by

right panel] surfaces. The view is a top view of each surface with the black linese used throughout.

Figure 2. Atomic structure of (a) a surface oxygen vacancy in the (1 1 0) ceria surface in the most stable split vacancy configuration and (b) a surface oxygen vacancy in the(1 0 0) ceria surface. In both surfaces, the vacancy site is indicated with a ‘V’ and in part (a) the bridging oxygen discussed in the text is indicated by ‘Obr’. The right hand partof (b) shows the surface tilted to better indicate the site of the vacancy.

128 M. Nolan / Chemical Physics Letters 499 (2010) 126–130

0.30 Å away from the vacancy site. The Ce–O distances to the near-est surface oxygen atoms are 2.25 and 2.24 Å, so that these oxygenatoms are also displaced away from the vacancy site. This shortensthe distance of these oxygen atoms to the next Ce ions from 2.19 Åin the pure surface to 2.03 Å in the defective surface. The Ce–Cedistance between the Ce ions nearest the vacancy site is elongatedto 4.40 Å from 3.80 Å in the pure surface. The hybrid DFT geometryis similar to that from PW91 + U [5].

In Figure 3 we show the excess spin density for the most stableoxygen vacancy in both surfaces; this information is also shown for

Figure 3. Excess spin density for (a) surface oxygen vacancy in the split vacancy configusurface. The spin density isosurfaces are at 0.02 electrons/Å3.

the other structures in Supplementary information. The excessspin density is the difference between the spin up and spin downelectron density and allows the location of the electrons left behindupon oxygen vacancy formation to be visualised. For the (1 1 0)surface with a split vacancy structure we see two Ce ions carryingspin density. This is an indicator that these Ce ions are in their re-duced Ce3+ state [4,8,13,17–19]. Interestingly, only one of the Ce3+

ions is an immediate neighbour of the bridging oxygen at the sur-face, while the second Ce3+ ion is found in the next subsurfacelayer; the spin density for the simple vacancy structure has two

ration in the (1 1 0) ceria surface and (b) surface oxygen vacancy in the (1 0 0) ceria

M. Nolan / Chemical Physics Letters 499 (2010) 126–130 129

reduced surface Ce ions neighbouring the vacancy site (see Sup-porting information). A similar result was found with PBE + U[13] and for Ti-, Zr- and Hf-doped CeO2 (where PBE + U, U = 5 eV,was used) [34]. We find that the Ce–Ce distance between this pairof reduced Ce ions is shorter than between the two surface Ce ionscoordinated to the bridging oxygen atom, giving a simple origin forthis Ce3+ distribution. The surface Ce3+ has the longer Ce–O dis-tance to the bridging oxygen; consistent with the larger ionic ra-dius of a Ce3+ ion.

For the (1 0 0) surface, the two Ce ions neighbouring the va-cancy site are reduced to Ce3+, again similar to the PW91 + U result[4,5]. In the higher energy solution, one reduced Ce ion is on thenext nearest neighbour Ce ion to the vacancy site, see Supportinginformation. Thus, in all ceria surfaces, there is more than one sta-ble distribution of Ce3+ ions in the defective surface and these min-ima can be quite close in energy as indicated for the (1 1 1) surfaceand in other studies [8,13,29].

The electronic density of states projected (PDOS) onto the Ce 4fstates is shown in Figure 4. Here the strongest differences with theresults from the DFT + U calculations are evident, due to the largerband gap obtained with hybrid DFT. For both methods, theformation of an oxygen vacancy produces a new state in the bandgap, indicated as region II in the PDOS. With hybrid DFT, this stateis found 2.0 eV above the valence band edge for the (1 1 0) surfaceand 2.1 eV above the valence band edge for the (1 0 0) surface. Forthe (1 1 0) surface, this is in very good agreement with experiment

Figure 4. Ce 4f PDOS for (a) surface oxygen vacancy in the split vacancyconfiguration in the (1 1 0) surface and (b) surface oxygen vacancy in the (1 0 0)surface. The zero of energy is the top of the valence band. The regions marked I, IIand III are discussed in the text.

[35,36], while this information is not available for the (1 0 0) sur-face. The gap state is derived from occupation of a Ce 4f state, lead-ing to formation of Ce3+.

With standard DFT the Ce3+ states are present at the bottom ofthe otherwise unoccupied Ce 4f manifold, region III in the PDOS,and are delocalised over all Ce atoms in the surface [4]. WithPW91 + U and PBE + U, U = 5 eV, the localised occupied Ce 4f statessplit off from the unoccupied Ce 4f states and these states lie 1.00and 0.90 eV above the valence band edge for the (1 1 0) and (1 0 0)surfaces [4,5,13]. This offset will depend on the value of U used inthe DFT + U calculation, as will the overall O 2p (valence band) toCe 4f energy gap [6]. The present hybrid DFT results put the latterenergy gap at ca. 2.9 eV for the (1 1 0) surface and 3.0 eV for the(1 0 0) surface and the offset of the occupied Ce 4f states fromthe empty Ce 4f states at 0.9 eV for both surfaces. This is slightlyreduced compared to the original DFT + U results and indicatesthat U values in the range 4.5–5.0 eV (as widely used in studiesof ceria) are suitable for describing the position of the Ce3+ statesrelative to the unoccupied Ce 4f states.

However, due to the underestimation of the band gap, theDFT + U approach will be unable to provide consistent offsets forthe gap state from both the valence and conduction bands. Valuesof U smaller than 5 eV position the Ce 4f states nearer the empty Ce4f manifold, while values of U larger than 5 eV will position the Ce4f states towards the valence band. Although values of U largerthan 5 eV could reproduce the band gap, these will push the Ce3+

state into the valence band. Thus, obtaining an accurate band gapof ceria is crucial for the correct position of the Ce3+ states in theband gap upon oxygen vacancy formation. Future work on thereactivity of ceria surfaces will determine the importance of thediffering descriptions of the relative position of the Ce3+ statesfound by DFT + U and hybrid DFT.

4. Conclusions

To provide a benchmark for previous DFT + U studies of oxygenvacancy formation in ceria surfaces we have carried out a hybridDFT study, using the HSE06 functional, of oxygen vacancy forma-tion in the (1 1 0) and (1 0 0) surfaces of ceria. Upon comparisonwith previous DFT + U results we find

(i) The oxygen vacancy formation energy on the (1 1 0) and(1 0 0) ceria surfaces is larger with hybrid DFT than withPW91 + U and PBE + U (U = 5 eV), by between 0.3 and0.4 eV. A similar difference is found for nanoparticles [29].Although the difference between HSE06 and PBE + U(U = 4.5 eV) for the (1 1 1) surface is larger (0.8 eV), the qual-itative trend is the same.

(ii) Localised Ce3+ ions are found, confirming the DFT + U results.The exact position of the Ce3+ ions depends on the structuraldistortions upon vacancy formation, which was previouslydiscussed for the (1 1 1) surface by Ganduglia-Pirovanoet al. with hybrid DFT [8] and the (1 1 0) surface withDFT + U in Ref. [13]. Different solutions can be separatedby as little as 0.20 eV.

(iii) The position of the Ce3+ state in the band gap is improvedover DFT + U, with the offset to the valence band (ca. 2 eV)in very good agreement with the available experimentaldata for the (1 1 0) surface, 2 eV [35,36].

These hybrid DFT results thus confirm much of the findingsdeveloped from DFT + U studies and highlight issues with the latterapproach. We are now studying adsorption of molecules at thesesurfaces with the hybrid approach to benchmark the descriptionof adsorption energetics obtained from DFT + U.

130 M. Nolan / Chemical Physics Letters 499 (2010) 126–130

Acknowledgements

We acknowledge support from Science Foundation Irelandthrough the Starting Investigator Research Grant Program, project‘EMOIN’ Grant No. SFI 09/SIRG/I1620, and computing resourcesprovided by the SFI and Higher Education Authority Funded IrishCentre for High End Computing.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.cplett.2010.09.016.

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