Surface Science 576 (2005) 217–229

www.elsevier.com/locate/susc

Density functional theory studies of thestructure and electronic structure of pure and

defective low index surfaces of ceria

Michael Nolan a, Sonja Grigoleit a, Dean C. Sayle b,Stephen C. Parker c, Graeme W. Watson a,*

a Department of Chemistry, Trinity College, University of Dublin, Dublin 2, Irelandb Department of Environmental and Ordanance Systems, RMCS, Cranfield University, Shrivenham, Swindon SN6 8LA, United Kingdom

c Department of Chemistry, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom

Received 4 October 2004; accepted for publication 22 December 2004

Available online 31 December 2004

Abstract

We present periodic density functional theory (DFT) calculations of bulk ceria and its low index surfaces (111),

(110) and (100). We find that the surface energies increase in the order (111) > (110) > (100), while the magnitude

of the surface relaxations follows the inverse order. The electronic properties of the bulk and surfaces are analysed

by means of the electronic density of states and the electron density. We demonstrate that the bonding in pure ceria

is partially covalent and analysis of the resulting electronic states confirms the presence of localised Ce4f states above

the Fermi level. The surface atoms show only a small change in the charge distribution in comparison to the bulk and

from the DOS the main differences are due to the changes in the oxygen 2p and cerium 5 d states. Investigation of the

atomic and electronic structure of an oxygen vacancy on the (100) surface shows the problems DFT can have with

the description of strongly localised systems, wrongly predicting electron delocalisation over all of the cerium atoms

in the simulation cell. We demonstrate an improvement in the description of the strongly correlated cerium 4f states

in partially reduced ceria by applying the DFT+U methodology, which leads to the appearance of a new gap state

between the valence band and the empty Ce4f band. Analysis of the partial charge density shows that these states

are localised on the CeIII ions neighbouring the oxygen vacancy. In terms of classical defect chemistry, the vacancy

is bound by two neighbouring CeIII ions, which have been reduced from CeIV, i.e. V��Oþ 2Ce0Ce. The remaining Ce ions

are in the CeIV oxidation state. The localisation of Ce4f electrons modifies the predicted structure of the defective

surface.

� 2004 Published by Elsevier B.V.

0039-6028/$ - see front matter � 2004 Published by Elsevier B.V.

doi:10.1016/j.susc.2004.12.016

* Corresponding author. Tel.: +353 1 608 1357; fax: +353 1 671 2826.

E-mail address: watsong@tcd.ie (G.W. Watson).

218 M. Nolan et al. / Surface Science 576 (2005) 217–229

1. Introduction

The technological importance of cerium oxide,

CeO2, particularly its important role in automobile

three-way catalytic converters and in solid oxidefuel cells [1], has seen it become the focus of many

experimental and computational investigations.

Since the surface properties of ceria determine

the catalytic activity of this material, much atten-

tion has been given to the study of the low index

surfaces of ceria.

Ceria is an insulating, non-magnetic rare-earth

oxide. It has a cubic fluorite structure with fourcerium and eight oxygen atoms per unit cell and

an experimentally determined lattice parameter

of 5.411 A [2]. Upon partial reduction of ceria,

oxygen vacancies are formed [1]. In this reaction

the oxidation state of Ce changes reversibly from

Ce(IV) to Ce(III), the reaction being

OxO þ 2CexCe ! V��

Oþ 2Ce0Ce þ 1

2O2 ð1Þ

using Kroger–Vink notation, where OxO is a neutral

oxygen in an oxygen lattice site, CexCe is a neutral

cerium atom in a cerium site, V��O is a doubly pos-

itively charged vacancy in an oxygen site and

Ce0Ce is a single negatively charged cerium atom

(+3 oxidation state) in a cerium site. In the Ce(III)

state, a previously unoccupied 4f state is occupied,

giving the electronic configuration Ce4f1 [3]. It is

expected upon partial reduction, that the two cer-

ium atoms neighbouring the vacancy will be re-

duced to Ce(III). The reverse reaction where

Ce(III) changes to Ce(IV) is carried out through oxi-

dation. This redox process allows ceria to store or

release oxygen, depending on the oxygen partialpressure; the oxygen storage/release capability of

ceria being central to technological applications.

Reduction of ceria and oxygen ion migration is

energetically favourable in the low index surfaces

compared to bulk [4,5] and it is thus important

to obtain an understanding of the nature of pure

and reduced ceria surfaces.

In order to facilitate the study of surface struc-ture, thin films of ceria have been grown on a

number of support materials including alumina,

yttrium stabilised zirconia, platinum and palla-

dium [6–8]. Experimentally, the (111) surface is

observed to be stable and undergoes little relaxa-

tion [9]. The (110) and (100) surfaces have been

observed to undergo surface relaxations [9,10]. It

has been determined through STM [10], AFM

[11], ion scattering spectroscopy [12] and low en-

ergy electron diffraction [13] that the (111) surfaceis oxygen terminated. The (110) surface has been

studied by Norenberg and Briggs using STM and

electron diffraction [14], demonstrating that it is

terminated with a stoichiometric layer. The least

stable, and most studied, surface is (100). Cleaving

this surface gives a dipole moment perpendicular

to the surface and therefore requires a reconstruc-

tion since dipolar surfaces are unstable [15]. Gen-erally removal of 50% of the terminating oxygen

species is observed [4,10], which Hermann demon-

strated using angular resolved mass spectroscopy

of recoiled ions [12]. Norenberg and Harding [10]

have presented an STM study of the pure and par-

tially reduced (100) surface of ceria. These authors

have found that surface relaxations take place in

order to reduce the surface energy. While cationtermination of the (100) surface is possible, the

anion terminated (100) surface is found to have

the lowest surface energy [4,10] and is taken as

the observed termination. Evidence for the appear-

ance of naturally occurring oxygen defects was de-

rived from the nature of the bright spots in the

STM image coupled to atomistic simulations.

Despite the fundamental importance of thesespecies to automotive catalysis, an understanding

of how oxygen vacancy defects modify the proper-

ties of ceria is still lacking. It is necessary to devel-

op our understanding of the oxygen storage

mechanism in cerium dioxide, in order to develop

more efficient catalysts. While the electronic struc-

ture of pure bulk ceria has been well studied, with

much debate regarding the exact nature of the elec-tronic structure, it is only in recent years that the

electronic structure of reduced ceria has been stud-

ied [3,9]. The resulting features in the UPS spec-

trum are dependent on the occupation of the

cerium 4f states [16]. In the work of Henderson

et al. [3], it was demonstrated that upon reduction

of the ceria (111) surface, a new occupied Ce4f

state appears in the gap between the valence bandand the previously unoccupied Ce4f states, 1.2 eV

above the valence band. Mullins et al. have ob-

served the formation of this same peak for reduced

M. Nolan et al. / Surface Science 576 (2005) 217–229 219

ceria (111) thin films grown on an Ru(0001) sup-

port [13]. This peak is due to formation of Ce3+

and the intensity of the XPS peak is related to

the amount of Ce3+ present. The Ce5d spectrum

is also modified upon reduction, with a new peakappearing at a binding energy of 903.8–904.0 eV

[3,10]. These observations are characteristic of

the presence of Ce3+ species.

The defective (111) surface has also been stud-

ied by Norenberg and Briggs with STM [17]. No

significant lateral relaxation of the surface was ob-

served and oxygen termination of this surface was

found. These authors claim that oxygen vacancydefects initially form in triangular clusters and

upon further annealing form line defects, indicat-

ing that the clustering of the oxygen vacancy de-

fects on ceria surfaces is energetically favourable.

Namai et al. [11] have also observed these triangu-

lar and linear oxygen defects in the partially re-

duced CeO2(111) surface. These authors have

also concluded that an oxygen defect density ofgreater than 1 · 1013 cm�2 (approximately 1% of

top layer O2�) is necessary for vacancy clustering

to occur. Oxygen vacancy clustering is also pre-

dicted from atomistic simulation to be energeti-

cally more favourable than isolated surface

oxygen vacancies [4,11]. There have been limited

studies on the reduced (110) and (100) surfaces,

although Mullins et al. have studied the electronicstructure of reduced CeO2(110) with UPS and

have observed the appearance of the Ce4f peak

at 2.0 eV above the valence band [16].

Many studies have applied atomistic simulation

methods, using interatomic potentials, abbreviated

hereafter as IP, in order to gain an insight into

bulk and surface properties of ceria. Interatomic

potentials are parameterised analytical functionsused to describe the interactions between ions in

a material. The parameters in the functions are

generally chosen to reproduce experimental data.

A number of interatomic potentials have been

developed for the calculation of the structure of

bulk ceria and the surface energies and relaxed

structures of ceria surfaces [4,5,18–27]. In compar-

ison, the number of studies of bulk and surfaces ofceria using ab initio methods is smaller. In CeO2,

the formal oxidation state of cerium is +4, how-

ever, two different approaches have been devel-

oped in order to treat the ground state electronic

structure. In the first, cerium is seen as tetravalent

with an unoccupied 4f-band (4f0) and a completely

filled O2p-band [28]. The second considers the

ground state of ceria to be a mixture of two Ceconfigurations, 4f0 and 4f1 with a filled O2p va-

lence band for the former and a partially filled

O2p-valence-band in the latter [29]. In this model,

cerium is no longer strictly tetravalent.

A number of studies of ceria have been con-

cerned with elucidating the role of the Ce4f elec-

trons in the electronic structure of CeO2. In early

SCF band calculations of bulk ceria Koellinget al. [30] concluded that some covalent bonding

is present, so that ceria is not completely ionic.

Fujimori also concluded that partial occupancy

of the Ce4f states is present [29], corresponding

to the second model above. However, Wuilloud

et al. [28] and Wachter et al. [31] have concluded

that the cerium 4f states in CeO2 are fully unoccu-

pied and localised, corresponding to the firstmodel above.

In their study of the electronic properties of

bulk ceria, with Hartree–Fock theory, Hill and

Catlow (who use a minimal basis set on cerium

and oxygen) [32] and Gennard et al. (who use a

more extended basis set) [33] have neglected com-

pletely the Ce4f basis functions, under the assump-

tion that doing so does not affect the bulkproperties of ceria, since the Ce4f orbitals are as-

sumed to be unoccupied. These studies found that

the bulk properties of ceria can be well described

even without the Ce4f electrons, indicating the

validity of the first model. Recent density func-

tional theory calculations of bulk CeO2 and

Ce2O3 were presented by Skorodumova et al.,

[34] in the framework of the full-potential linearmuffin-tin orbital (FP-LMTO) method. The best

agreement with experiment for CeO2 was obtained

by treating the cerium 4f-functions as part of the

valence region. However, in studying fully reduced

ceria, Ce2O3, the same authors found that in order

for the Ce4f electrons to be correctly localised,

they had to be treated as core states. Treating

the 4f electrons as valence electrons, resulted inan incorrect partially filled f-band at the Fermi

level. Choosing the f electrons to be core or

valence depending on the problem at hand is

220 M. Nolan et al. / Surface Science 576 (2005) 217–229

clearly not a satisfactory way of understanding the

electronic structure of ceria.

In addition to studying the bulk properties of

ceria, Gennard et al. [33] also studied the (111)

and (110) surfaces using Hartree–Fock. Recently,Skorodumova et al. [35] have studied the surface

energies and structures of the (111), (110) and

(100) surfaces using density functional theory.

Both of the ab initio studies are in agreement

with atomistic simulations regarding the relative

stability of the surfaces, (111) > (110) > (100),

although DFT predicts smaller surface energies

than atomistic simulations and Hartree–Fock [35].In this paper we present periodic density func-

tional theory (DFT) calculations of bulk and the

three low index surfaces of pure ceria. We also

consider reduction of the (100) surface through

formation of oxygen vacancies on the (100) sur-

face. We analyse the structural and electronic

properties by means of the density of states,

charges and the charge density. We demonstratethe presence of unoccupied Ce4f electronic states

above the Fermi level. Reduction of the (100) sur-

face leads to occupied Ce4f states which are found

to be delocalised over all of the cerium atoms with-

in the simulation cell using GGA-DFT. In order to

correctly describe the localisation of these elec-

trons, we use the DFT+U [36] methodology to

correct this failing of DFT for the (100) surfaceof ceria. In addition we also examine how the

DFT+U approach affects the resulting atomic

structure of the defective surface.

2. Computational methods

The DFT calculations were performed using theVienna ab initio simulation package VASP [37–39]

which utilises a plane-wave basis set for the

description of the valence electrons. In the present

study, we have employed the projector-augmented

wave (PAW) method [40,41] to accurately repro-

duce the effect of the core electrons on the valence

electrons, with [He] and [Xe] cores for oxygen and

cerium. For the exchange-correlation functionalthe generalized gradient approximation (GGA)

of Perdew and Wang (PW91) [42] was used. The

ions were relaxed towards equilibrium using the

quasi-Newton method until the forces were less

than 0.01 eV/A (applying a tighter convergence

criterion of 0.005 eV/A in the forces had no signif-

icant effect on the distances). For the oxygen defect

vacancy structures, fully spin polarised calcula-tions were carried out and the calculations were

also performed within the DFT+U methodology

[36], using a value of U = 5 eV. This is in reason-

able agreement with earlier DFT+U studies of

Ce metal [43], where a value of 6.1 eV was used.

To ensure the convergence of the calculations,

we computed the total energy and the equilibrium

volume of the bulk CeO2 unit cell for plane-waveenergy cut-offs in the range of 300 eV and 600 eV

and on Monkhorst–Pack grids of (2 · 2 · 2),(4 · 4 · 4) and (6 · 6 · 6) k-points. The equilib-

rium structure was obtained by fitting a series of

volume-energy data to the Murnaghan equation

of state. The results show that the calculations

are well converged for a k-point grid of

(4 · 4 · 4) and an energy cut-off of 500 eV andare accurate to 0.01 A3 in the equilibrium volume

and to 0.02 eV in the total energy.

The surface calculations were performed using

the slab method [44], in which a finite number of

crystal layers in a three dimensional periodic cell

is used to generate two surfaces via the introduc-

tion of a vacuum gap perpendicular to the surface.

The vacuum gap must be large enough that inter-actions between the periodic images perpendicular

to the surface are minimised. The slab must also be

of sufficient thickness that the structure in the mid-

dle of the slab is sufficiently bulk-like. The surface

energies calculated in this work are converged to

0.01 J/m2 with respect to both the number of layers

in the slab and the vacuum thickness, where a 15 A

vacuum gap was used throughout. For the (111)surface a thickness of 10.5 A (4 atomic layers)

was sufficient. For the (110) and (100) surface

calculations a thickness of 11.5 A (7 atomic lay-

ers) and 18.6 A (15 atomic layers) was required.

The thicker slabs for the latter two surfaces are

necessary given their stronger relaxations in com-

parison to the (111) surface. For the surface calcu-

lations a (4 · 4 · 1) k-point grid was used, with thethird vector perpendicular to the surface, and the

energy cut-off of the converged bulk calculation

(500 eV).

Table 1

Lattice parameter and bulk modulus of ceria from the DFT and

IP calculations

Method Lattice parameter (A) Bulk modulus (GPa)

DFTa 5.470 172

LDA [34] 5.390 214.7

GGA-DFT [34] 5.480 187.7

LDA [35] 5.370 –

GGA-DFT [35] 5.470 –

HF [32] 5.385 357

HF [33] 5.546 221

IP [4] 5.411 263

IP [19] 5.411 268

IP [19] 5.411 289

Experiment 5.411 [2] 236 [48]

a Present work.

M. Nolan et al. / Surface Science 576 (2005) 217–229 221

3. Results and discussion

3.1. The low index surfaces of ceria

We have investigated the energies and proper-ties of the three low index surfaces (111), (110)

and (100), see Fig. 1. The (111) surface is classi-

fied as a type 2 surface [15] and consists of neutral

(anion–cation–anion) repeat units and thus has no

net dipole moment perpendicular to the surface.

The (110) surface is a type 1 surface and is

composed of stoichiometric layers and is therefore

also charge neutral. In contrast, the type 3 (100)surface consists of a sequence of charged (cation–

anion) planes, which results in a dipole perpendic-

ular to the surface vector. Since the surface energy

of dipolar surfaces is theoretically infinite we fol-

lowed the prescription of Ref. [45] and removed

the dipole moment by moving every second row

of oxygen atoms from one side of the slab to the

other side resulting in a surface termination witha 50% vacant oxygen layer. While some atomistic

studies have considered cation terminated (100)

surfaces [4,10,23], these studies and experimental

data [4,10,16] support anion termination of this

surface.

The lattice constant of bulk ceria obtained in

this work is 5.470 A, and is compared to the results

of previous studies in Table 1. This is within 1.09%of the experimental value of 5.411 A, indicating

that the DFT calculations give a reliable descrip-

tion of the structural properties of ceria. However,

Fig. 1. Relaxed structures of the (a) (111), (b) (110), and (c) (100) su

and the light spheres are cerium ions.

the calculated bulk modulus of ceria is notably

smaller than the experimentally determined value.

Previous HF [33] and DFT [34] calculations have

also underestimated the bulk modulus of ceria,

while interatomic potentials generally overestimate

this quantity [4,19,33].

In Table 2 the surface energies of the unrelaxed

and relaxed surfaces are shown. It is found that therelative stability of the surfaces decreases in the

order (111) > (110) > (100) and the relaxation

energy increases in the order (111) < (110) <

(100). Physically that is what one would ex-

pect—the defective type 3 (100) surface is unstable

and thus undergoes significant rearrangements,

which results in a greater energy gain due to the

relaxation. The relative magnitudes of the surface

rfaces of ceria. In these figures, the dark spheres are oxygen ions

Table 2

Unrelaxed and relaxed surface energies of the low index

surfaces of ceria from ab initio and IP calculations

Method DFTa LDA

[35]

GGA

[35]

HF

[33]

IP-MD

[23]

IP

[4]

IP

[19]

IP

[35]

(111)

Eunrelaxed 0.69 1.06 0.69 1.34 1.44 1.70 1.63 1.65

Erelaxed 0.68 1.04 0.68 1.31 1.12 1.54 1.35 1.05

(110)

Eunrelaxed 1.26 1.55 1.25 2.61 3.37 3.59 – 3.47

Erelaxed 1.01 1.35 1.05 2.11 2.07 2.45 2.10 1.19

(100)

Eunrelaxed 2.05 – 2.06 – 6.31 6.46 – 6.23

Erelaxed 1.41 – 1.41 – 2.41 3.25 – 3.11

a Present work.

Fig. 2. Ionic displacements for the low index surfaces of ceria.

(a): (111) surface, (b): (110) surface and (c): (100) surface. The

X-axis is the depth into the structure from the surface layer,

while the Y-axis is the ionic relaxation in the vertical direction.

222 M. Nolan et al. / Surface Science 576 (2005) 217–229

energies are in agreement with experimental obser-

vations that the (110) and (100) surfaces are less

stable than the (111) surface. The difference in sur-

face energy between the (110) and (100) surfaces

is smaller than the difference between these sur-

faces and the (111) surface.

While the ordering of the surface energies ob-

tained in the present work is the same as thatfound in earlier studies, the magnitudes of the sur-

face energies are significantly smaller than the val-

ues obtained using atomistic simulation [4,17,21].

The surface energies calculated with HF for the

(111) and (110) surfaces in [33] lie within the

range of values obtained in the atomistic simula-

tion studies. In [33] an a posteriori correlation cor-

rection was added using the functional of Perdewand Wang and resulted in an increase in the sur-

face energies so that they lie slightly above the

atomistic simulation results. The present results

are similar to the results from Skorodumova

et al., also obtained within DFT [34]. A possible

explanation for the comparatively smaller DFT

surface energies could be due to the self-consistent

inclusion of electronic correlation effects in theDFT formalism.

Fig. 2 shows the ionic displacements perpendic-

ular to the surfaces after relaxation. Positive values

indicate that the ions relax out of the surface,

whereas negative values indicate a displacement

into the bulk. As the (111) surface is the most clo-

sely packed and stable surface, the relaxations

(Fig. 2a) are rather small. The overall effect is an

inward shift of the surface layer. The atomic dis-

placements in this surface are smaller than those

using HF [33] and atomistic simulation [4,33].

The Ce–O distances around the surface layer con-

tract, but by no more than 0.01 A (0.5%).

M. Nolan et al. / Surface Science 576 (2005) 217–229 223

For the relaxation of the (110) surface (Fig. 2b)

one can observe rumpling in the surface layer sim-

ilar to that seen for the (100) surfaces of rock salt

structured oxides [46]; while both ions are shifted

inwards, the inward displacement of the ceriumions is notably larger than the displacement of

the oxygen ions. This is due to the charge on the

cerium atom causing it to be pulled into the bulk.

The (110) surface is the only surface which under-

goes lateral displacement. The oxygen ions are

shifted towards the nearest cerium ion in the same

layer, so that the extension of the bond-length due

to the vertical relaxation is compensated. The re-sult is that the Ce–O distances in the surface layer

are 2.33 A (contraction of 1.7%), while in the sec-

ond layer, a contraction of 2.50% in the Ce–O dis-

tance is observed; the bulk Ce–O distances are

2.37 A. The displacements computed in the present

work are in agreement with HF results [33], and

smaller than those computed with the IP methods

[23,33,35].The (100) surface undergoes the strongest

relaxations, Fig. 2c. The surface oxygen ions are

only two coordinate resulting in a large shift in-

Fig. 3. Ce (left) and O (right) PDOS for bulk CeO2 (top) and the (10

valence band and the smearing width is set to 0.07 eV.

wards and a contraction of 7.60% in the Ce–O dis-

tance directly below the surface vacancy. Each of

the other oxygen layers is split up—one half of

the oxygen ions are shifted out towards the sur-

face, while the other half (directly below thesurface oxygen) relaxes into the bulk. Moving in-

wards, we observe that contraction of the Ce–O

distances is less strong compared to that for the

surface layer; a contraction of 5.50% is found in

the second layer with the distances in the middle

of the slab close to bulk values.

In the following we analyse the electronic prop-

erties of the surfaces. The projected density ofstates (PDOS) of the surface layers are displayed

in Fig. 3, along with the PDOS for the cerium

and oxygen atoms of the bulk. For the projection

onto spherical harmonics, we used Wigner–Seitz

radii of 1.5 A for Ce and O. For the (100) surface

we have plotted the atom and angular momentum

projected DOS of the surface layer together with

the total and projected DOS curve for bulk CeO2

(see Fig. 3). The low lying cerium 5s band found

at an energy of �33 eV is not included in the

plot. The band found between �20 eV and

0) surface (bottom). The zero of energy is set to the top of the

224 M. Nolan et al. / Surface Science 576 (2005) 217–229

�10 eV below the valence band arises from interac-

tions between cerium 5p and oxygen 2s states. The

valence band is composed of predominantly oxy-

gen 2p states with a small admixture of Ce5d and

4f. Above the valence band we find a narrow unoc-cupied band derived from Ce4f. Finally, the con-

duction band is made up of Ce5d states. Note

also the presence of O2p states in the Ce5p/O2s

bands, presumably an artefact that is due to the

Wigner–Seitz radii used in the projection. The

O2p contribution to the unoccupied Ce4f band

on the other hand is probably real andmay indicate

that ceria is not fully ionic. This is also indicated bythe presence of a Ce4f contribution to the valence

band DOS. The band-gaps of 1 eV between the va-

lence band and the Ce4f band and 5 eV between

the valence band the empty Ce5d band show the

usual underestimation of band gaps obtained from

DFT calculations; the experimental band gaps

taken from XPS spectra [28] are 3 eV and 6 eV.

A comparison of the projected DOS of the sur-face layer and the DOS of the bulk shows that the

main changes occur in the valence states, those

being predominantly oxygen 2p. The change in

the DOS for the (111) and (110) surfaces com-

pared to bulk are very small with the largest

changes seen for the (100) surface, which also

undergoes the strongest relaxation. Due to the

Fig. 4. Partial charge densities for the (100) surface. (a) Semi-core sta

0 to 0.16 electrons/A3.

low coordination number of 2 there is a destabili-

sation of the 2p states in the surface oxygen atoms,

which results in a shift of the 2p projected DOS to

higher energies, Fig. 3.

From the projection onto spherical harmonics,the ionic charges can be obtained in the different

layers of each of the surface slabs. This procedure

does not allow the absolute charge on the ions to

be determined and therefore we discuss qualita-

tively how the ionic charge is modified in the sur-

face compared to bulk. The charges in the bulk

are +2.72 on cerium and �1.49 on oxygen, sug-

gesting that ceria is not completely ionic. We findthat the charges of the ions in the different layers

in the slab differ slightly from the values in the

bulk. It is only in the surface layer that any notable

change in the charge of the cerium and oxygen ions

is observed. The most stable surface, (111), shows

the smallest change from bulk, while the other two

surfaces display a relatively larger change in the

surface layer charges compared to bulk ceria.For the (100) surface, we find a charge of +2.61

on surface cerium and �1.34 on surface oxygen.

In succeeding layers, the charges are very similar

to the bulk charges.

We further investigate the electronic properties

using the electron density distribution. Fig. 4

shows the partial electron density of the semi-core,

tes, (b) valence states, (c) localised Ce4f states. The scale is from

Fig. 5. DFT+U optimised structure of CeO2 with two oxygen

vacancies. (a) A view of the structure, where the large spheres

denote the reduced Ce3+ ions. In (b) we show an expansion of

the simulation cell in plan view, in which the large spheres

indicate the positions of the reduced Ce3+ ions.

M. Nolan et al. / Surface Science 576 (2005) 217–229 225

valence and Ce4f states for the (100) surface. The

charge density plots confirm that only the surface

layer displays any (small) change in the distribu-

tion of the electron density. We note that the

charge distributions in the surface layers are in linewith the contraction of the Ce–O distances in the

surface layers already discussed. The partial

charge density in the energy range of the Ce4f

states for the (100) surface demonstrates clearly

that the Ce4f states above the Fermi level are in-

deed localised on the cerium atoms. Note also

the asymmetric density for the atoms in the surface

layer, compared to the more spherical distributionfor the atoms in the middle of the slab. This is due

to the asymmetric electric field present at the sur-

face. The distortion of the 4f states at the surface

for the (111) and (110) surfaces is much weaker

compared to the (100) surface.

Before we consider defects in the (100) surface,

we briefly demonstrate that the DFT+U approach

has little effect on the properties of the pure (100)surface. We find that there is no modification to

the surface relaxations and that the charge density

distributions and ionic charges are unmodified.

The only effect of U that is of note is that the en-

ergy gap between the top of the valence band

and the empty Ce4f band is increased by 0.67 eV

over the GGA value.

3.2. Oxygen vacancies in reduced ceria

Since the exchange of oxygen between ceria and

the atmosphere is of significant importance in the

application of ceria as an oxygen storage material,

we have studied oxygen vacancy defects in the

(100) surface. Upon reduction of ceria, CeIV ions

are formally reduced to CeIII, Eq. (1), and it is ex-pected that these ions will be localised around the

oxygen vacancy sites.

For the present study, we consider the elec-

tronic structure of the partially reduced (100)

surface. To enable investigation of the oxygen

vacancy a 2 · 2 expansion of the pure (100) sur-

face was created with two vacancies, one on each

of the surfaces, thus ensuring that the slab hasno net dipole moment, Fig. 5. The slab thickness is

10.94 A, 9 layers. The same plane wave cut-off

energy (500 eV) and relaxation criteria (conver-

gence in the forces to less than 0.01 eV/A) and a

2 · 2 · 1 k-point sampling mesh were applied.

In UPS studies [3,13] of the electronic structureof partially reduced ceria surfaces, a new state is

found in the energy gap between the top of the va-

lence band and the bottom of the conduction

band, which is suggested to be due to partial occu-

pation of the Ce4f states, leading to formation of

CeIII [3,13].

The electronic structure resulting from GGA-

DFT gives rise to a DOS with no gap state be-tween the valance band and the conduction band

226 M. Nolan et al. / Surface Science 576 (2005) 217–229

with the Ce4f states crossing the Fermi level. The

partial charge density isosurfaces are displayed in

Fig. 6(a) and (b), illustrating that the Ce4f elec-

trons are delocalised over all the cerium atoms in

the simulation cell. This behaviour is not that ex-pected for these electronic states and we conclude

from the lack of a gap state and the charge density

that GGA-DFT gives an incorrect description of

the electronic structure of reduced ceria. This find-

ing is consistent with the work of Skorodumova

et al. [34] who showed that standard DFT is un-

able to describe correctly the localisation of cerium

4f states in fully reduced bulk ceria, Ce2O3.

Fig. 6. (a) Isosurfaces of charge density for the defective (100) ceria s

(c) Front view for GGA, (d) side view for GGA. The isosurface cont

In other materials, similarly localised states are

poorly described with standard DFT e.g. known

insulators are found to be metallic with states

crossing the Fermi level or an electronic hole is

delocalised instead of being localised [47].To correctly describe the electronic structure of

the vacancy defect, we apply the DFT+U method-

ology [36], which is one approach to correct for the

inability of approximate DFT to properly describe

strongly localised systems [47]. Briefly, due to the

fact that the self interaction of an electron is not

correctly cancelled in DFT, there is an artificial

barrier to electronic localisation, since the self

urface. (a) Front view for U = 5 eV, (b) side view for U = 5 eV.

ours are 0.16 electrons/A3.

Fig. 7. Ce Partial density of states for partially reduced

CeO2(100) surface computed with DFT+U, where U is 5 eV.

The zero of energy is set to the top of the valence band and the

smearing width is 0.07 eV.

M. Nolan et al. / Surface Science 576 (2005) 217–229 227

interaction is biased towards delocalising elec-

tronic states. The introduction of the on-site Cou-

lomb interaction, U, helps to remove some of the

self-interaction error, reducing the artificial barrier

to localisation, forming localised electronic states,in agreement with experiment and contrary to

the GGA result. The actual value of U is usually

chosen to recover the correct value of an experi-

mentally measured parameter, e.g. the magnetic

moment or the band gap. For the present system,

this is difficult, since we are dealing with a defect

state. In the present work, we use the PW91

GGA and choose a value of U that leads to local-isation of two electrons in the Ce4f states for

each oxygen vacancy and an electronic structure

corresponding to that observed in experimental

UPS spectra [3,13]. Using values of U in the range

of 2–7 eV we have found that the degree of

delocalisation decreases with increasing U. For

U < 5 eV, significant delocalisation still persists,

while for U = 5 eV and greater, the results areessentially converged, in that localisation is found

to be essentially independent of the choice of U.

In this work, we use a value of U of 5 eV, which

is consistent with earlier work on cerium metal

[43].

The partial charge density from the DFT+U

electronic structure is presented in Fig. 6(c) and

(d) and demonstrates clearly the localisation ofthe Ce4f electrons on the surface cerium atoms

neighbouring the oxygen vacancy. There is no

extension of these states onto the subsurface atoms

below the oxygen vacancy. The fact that the sub-

surface Ce atoms nearest the oxygen vacancies

are reduced is expected due to Coulombic interac-

tions. Fig. 7 shows the partial cerium density of

states of the oxygen vacancy defect (100) surfacecalculated with DFT+U. Here, we observe the

appearance of a state in the previously empty

gap between the top of the valence band and the

unoccupied cerium 4f band. We note from the par-

tial charge density in Fig. 6(c) and (d) that cerium

4f electronic states are occupied in the gap state

PDOS peak and that the electrons are predomi-

nantly localised on the cerium atoms neighbouringthe oxygen vacancies; there is a small oxygen con-

tribution to the PDOS and the charge density in

the gap state.

The DFT+U approach results in an energy gap

of 0.90 eV from the top of the valence band to the

new Ce4f gap state and an essentially unchanged

band gap of 5 eV from the valence band to the

Ce5d conduction band. The band gap from the

valence band to the unoccupied Ce4f band is in-

creased to 2.1 eV compared to the pure surface.The opening up of this band gap is not unex-

pected, given that the DFT+U formalism is ap-

plied to the Ce4f states and one function of U is

to shift up the energy of unoccupied states, i.e.

the unoccupied Ce4f states. The formation energy

of an oxygen vacancy in ceria was calculated with

GGA and DFT+U, see Eq. (1). With DFT+U, the

calculated vacancy formation energy is 4.55 eV,which is a reduction of 0.83 eV from the calculated

GGA value of 5.38 eV.

Comparing the geometry of the defective sur-

face optimised with DFT+U to the geometry of

the pure (100) surface, we find for the defective

surface that the Ce–O distances neighbouring the

vacancy are shortened by 0.60%. Relaxation of

the defective geometry with GGA-DFT results ina geometry whereby the contraction of the Ce–O

distances is stronger compared to DFT+U. The

contraction of the Ce–O distance s neighbouring

the vacancy is 2.10% for the GGA-DFT geometry

compared to the contraction of 0.60% obtained in

the DFT+U geometry suggesting that GGA-DFT

overestimates contraction of the Ce–O distances

in the vacancy defect structure. The differing

228 M. Nolan et al. / Surface Science 576 (2005) 217–229

structures found for GGA-DFT and DFT+U are

consistent with the localisation of CeIII states and

demonstrate the importance of self-consistent

relaxations with DFT+U.

4. Conclusion

We have presented the results of DFT calcula-

tions of the low index surfaces of ceria. It has been

shown that the order of stability of the surfaces is

(111) > (110) > (100), while the extent of surface

relaxations is in the inverse order. The electronicproperties of the surfaces were studied through

the charges, the density of states and the charge

density. The investigation of ceria surfaces

confirms the presence of some covalent bonding be-

tween cerium and oxygen and the localised atomic-

like character of the unoccupied Ce4f electronic

states. Only the surface atoms show a significant

change in the charge distribution in comparisonto the bulk. The main changes occur in the oxygen

2p and cerium 5d states of the surface atoms.

Reduction of the (100) surface was studied with

the introduction of oxygen vacancies. It was found

that application of GGA-DFT to this structure re-

sulted in delocalisation of the electronic hole over

all of the cerium atoms within the simulation cell.

Applying the DFT+U formalism (U = 5 eV), wefind upon introducing oxygen vacancies:

(i) Only the Ce ions neighbouring the vacancy

are reduced from CeIV to CeIII, i.e. OxO þ

2CexCe ! V��Oþ 2Ce0Ce. The charge density

plots demonstrate that the electronic states

are localised on the two CeIII ions neighbour-

ing the oxygen vacancy site. The remainingCe ions remain in the unreduced CeIV state.

(ii) In the density of electronic states, a new peak,

due to the CeIII ions, appears in the previ-

ously empty band gap between the top of

the valence band and the unoccupied cerium

4f (CeIV) states.

(iii) In addition, use of the DFT+U formalism

results in different structures compared toGGA-DFT, consistent with the localised CeIII

species, indicating the importance of self-con-

sistent relaxation when applying DFT+U.

Thus, a modification of DFT to correct for the

self interaction error, such as the DFT+U method-

ology is necessary for the determination of the nat-ure of the electronic structure of defective sites in

CeO2.

Acknowledgements

We acknowledge support from Enterprise Ire-

land, grant number SC/2001/233, the Donors ofthe Petroleum Research Fund administered by

American Chemical Society and the EPSRC, grant

numbers GR/548431/01 and GR/548448/01. We

also wish to thank Dr. Peter Oliver at Rutherford

Appleton Laboratory for access to and assistance

with Hrothgar, a 16 node Beowulf cluster.

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