J. Chem. Phys. 121, 7427 (2004); h ps://doi.o g/10.1063/1.1796253 121, 7427
© 2004 Ame ican Ins i u e o Physics.
Oxygen acancies on (110) om i s
p inciples calcula ions
Ci e as: J. Chem. Phys. 121, 7427 (2004); h ps://doi.o g/10.1063/1.1796253
Submi ed: 09 June 2004 . Accep ed: 29 July 2004 . Published Online: 01 Oc obe 2004
J. O iedo, M. A. San Miguel, and J. F. Sanz
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Oxygen acancies on TiO2„110… om i s p inciples calcula ions
J. O iedo, M. A. San Miguel, and J. F. Sanz
Depa amen o de Quı
´mica Fı
´sica, Facul ad de Quı
´mica Uni e sidad de Se illa, c/P o eso Ga cı
´a Gonza
´lez
s/n, 41012 Se ille, Spain
共Recei ed 9 June 2004; accep ed 29 July 2004兲
We ha e ca ied ou a sys ema ic s udy o oxygen acancy o ma ion on he TiO2共110兲su ace by
means o plane-wa e pseudopo en ial densi y- unc ional heo y calcula ions. We ha e used models
wi h he mean numbe o acancies pe su ace uni cell being
␪
⫽0.25 and
␪
⫽0.5. The s udy
comp ises se e al kind o acancies wi hin he ou e mos laye s o he su ace. The use o a sui able
se o echnical pa ame e is o en essen ial in o de o ge accu a e esul s. We ind ha he p esence
o b idging acancies is ene ge ically a o ed in acco dance o expe imen al da a, al hough he
o ma ion o sub-b idging acancies migh be possible a mode a e empe a u es. Su p isingly, he
spin s a e o he acancy has li le in luence on he esul s. A omic displacemen s a e also analyzed
and ound o be s ongly dependen on he pa icula a angemen o acancies. © 2004 Ame ican
Ins i u e o Physics. 关DOI: 10.1063/1.1796253兴
I. INTRODUCTION
The e has been an inc easing e o in s udying ansi ion
me al oxides du ing he las decade because o hei b oad
echnological applicabili y.1Among hese sys ems, i anium
dioxide has been one o he mos in es iga ed compounds.
The e a e wo impo an s uc u es o TiO2, ana ase and
u ile.2
Ru ile has been ex ensi ely cha ac e ized bo h expe i-
men ally1–4 and heo e ically.1,5–19 Among he di e en low
index su aces, he 共110兲is he mos s able and also he mos
s udied.2This su ace is eadily and ep oducibly p epa ed so
i has been conside ed a p o o ype o an oxide su ace.1I s
s uc u e can be desc ibed as o med by neu al laye s. Each
laye is made o h ee planes o composi ion O-Ti2O2-O. Fo
ins ance, in Fig. 1 a h ee-laye TiO2共110兲slab is shown. In
he su ace, he e a e wo kinds o i anium a oms, i e old
and six old coo dina ed, o ming al e na ing ows. All oxy-
gen a oms a e h ee old coo dina ed as in he bulk excep
hose ones in he ou e mos plane called b idging a oms,
which lose one bond when he uppe laye is emo ed.
The b idging oxygen a oms ha e a ac ed an eno mous
in e es because hey can be easily emo ed by he mal an-
nealing leading o poin de ec s ha al e a e he chemical
p ope ies o he su ace.1,2 Scanning unneling mic oscopy
共STM兲measu emen s quan i y as 7%–10% he a io
acancies/oxygen a oms when annealing a 900 K.20–22
The as amoun o expe imen al da a has become a
challenge o heo is s who ha e ex ensi ely s udied he su -
ace. Howe e , he esul s a e o en in disc epancy, and a e
di icul o compa e because o he a ie y o me hods and
echnical condi ions. The pe iodic densi y unc ional heo y
共DFT兲me hod wi h pseudopo en ials and plane wa es ha e
been used success ully o model oxide su aces in ecen
yea s.5–10,12–15,17,18,23 We ha e made use o i h oughou his
s udy.
One o he main aims o he p esen wo k is o ge a
deepe insigh in o he ene ge ics o oxygen acancy o ma-
ion and also in o he s uc u e o he elaxed de ec i e su -
aces. In addi ion, we ha e es ed he in luence o some im-
po an echnical pa ame e s on he esul s. We shall show
ha he se o condi ions used in a calcula ion may be c ucial
in o de o ge meaning ul esul s. Despi e he ac ha c e-
a ion o b idging acancies has al eady been s udied wi h
di e en me hods and models,6–14 he e is li le in o ma ion
abou o he ype o acancies. In o de o in es iga e se e al
kind o acancies we ha e emo ed di e se oxygen a oms,
which a e named in Fig. 1. As a consequence o his sys em-
a ic s udy we shall es ablish some ules ha could assis in
he p ocess o choosing models o ep esen de ec i e su -
aces.
The es o he pape is o ganized as ollows. In he
second sec ion we desc ibe some echnical de ails abou he
me hodology applied. In he hi d sec ion we discuss ou
esul s. We s a wi h some p elimina y calcula ions on he
bulk and he s oichiome ic su ace and ollow wi h an analy-
sis o how he choice o a su ace model and a se o pa am-
e e s a ec he ene ge ics o o ma ion o de ec i e su aces.
Finally, we discuss he s uc u e o elaxed su aces. In he
ou h sec ion we es ablish he conclusions om ou wo k.
II. TECHNICAL METHODS
In he p esen wo k we use he implemen a ion o DFT
wi h pseudopo en ials and plane wa es.24 We use he gene -
alized g adien app oxima ion 共GGA兲 o he exchange-
co ela ion ene gy, which seems o desc ibe be e he o ma-
ion and c ea ion o bonds on su aces.25 We use he GGA
due o Pe dew e al.26,27
The calcula ions we e pe o med wi h he VASP 共Vienna
ab ini io simula ion package兲code.28–31 This is a pa icula ly
e icien and s able code, which sea ches o he sel -
consis en g ound s a e using esidual minimiza ion and
cha ge mixing. As in all pseudopo en ial plane-wa e codes,
he o ces on he a oms a e au oma ically calcula ed, and
JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 15 15 OCTOBER 2004
74270021-9606/2004/121(15)/7427/7/$22.00 © 2004 Ame ican Ins i u e o Physics
elaxa ion o all a oms in he sys em o hei equilib ium
posi ion is also au oma ic.
In pseudopo en ial calcula ions, we mus decide which
s a es a e ea ed as co e s a es and which as alence s a es.
Fo oxygen, he co e consis s only o 1ss a es whe eas o
Ti, up o and including he 3pshells a e ozen and he
e e ence s a e o pseudopo en ial gene a ion is s1d3.We
used he s anda d ul aso pseudopo en ials included in he
VASP package.32
III. RESULTS
A. P elimina y es s: bulk and s oichiome ic su aces
In o de o analyze some echnical de ails, a se o p e-
limina y calcula ions ha e been pe o med. Fi s , we made
ene gy con e gence es s o he TiO2bulk wi h ene gy cu -
o s up o 495 eV. Fo k-poin sampling, we used he
Monkho s -Pack scheme33 wi h se s o se e al sizes. We con-
cluded ha an ene gy cu o o 396 eV and a 4⫻4⫻4kpoin
se including he ⌫poin a e enough o ge ene gy con e -
gence wi hin 0.002 eV/a om. The op imized la ice pa am-
e e s a e a⫽4.616Å, c⫽2.974Å, and u⫽0.304, which
compa es well wi h he expe imen al ones 关a⫽4.594Å, c
⫽2.959Å, and u⫽0.305 共Re . 34兲兴 and p e ious heo e ical
wo k.5The calcula ed aand c alues ha e been used o he
共110兲su ace h oughou he p esen wo k.
In o de o do calcula ions on he u ile su ace we use
he adi ional app oach, ha is, o employ a cell ha in-
cludes a po ion o acuum, and eplica e i in he h ee di-
ec ions o space, so an a ay o slabs sepa a ed by acuum is
c ea ed. We mus ensu e ha he slab hickness and he
acuum wid h a e wide enough o ob ain meaning ul esul s.
We ha e checked ha using a acuum wid h o 5 Å absolu e
ene gy alues a e con e ged wi hin 0.001 eV/a om ha is
p ecise enough o he p esen wo k. The in luence o slab
hickness on he esul s is mo e c i ical.
The su ace ene gy 关de ined in he usual way 共Re s. 5
and 23兲兴 e sus he numbe o laye s is plo ed in Fig. 2 o
a共1⫻1兲model in he single s a e. The con e ged su ace
ene gy is abou 0.5 J/m2 ha is in ag eemen wi h p e ious
epo ed alues o calcula ions using GGA.5,14 We can see
ha he su ace ene gy does no con e ge un il a conside -
able numbe o laye s is used. The e is also an oscilla ing
odd-e en beha io ha has been no ed p e iously.5,15 The
su ace ene gy o slabs wi h an odd numbe o laye s is
highe because hey ha e an addi ional symme y plane a
he slab cen e 共i.e., in Fig. 1兲so he a omic displacemen s on
bo h sides o he slab a e mi o ed and end o ein o ce each
o he . As a consequence, he ene gy o he slab inc eases
and he elaxa ion ene gy dec eases. As we inc ease he num-
be o laye s, he in e ac ion be ween bo h sides o he slab
become weake and he su ace ene gy con e ge o a limi
alue, al hough he con e gence a e is as e o slabs wi h
an e en numbe o laye s. Fo compa ison pu poses we ha e
included in Fig. 2 a e age alues be ween odd and e en
numbe o laye s 共i.e., 3.5 laye s is a mean alue be ween 3
and 4 laye su ace ene gies兲. I is qui e e iden om hese
numbe s ha he a e age alues con e ge much as e and
all he alues om 4.5 laye s a e con e ged wi hin 0.05 J/m2.
We will show ha he same beha io is obse ed o he
acancy ene gy o ma ion.
B. Ene ge ics o acancy o ma ion
The main aim o his wo k is o s udy acancy o ma ion
ene gies o u ile su aces. When we c ea e a acancy on a
su ace we b eak he su ace symme y and, depending on
he acancy co e age, di e en a angemen s o acancies
a e possible. I is expec ed ha he in e ac ions among a-
cancies will be s onge when he acancy densi y is high.
We ha e made calcula ions o se e al acancy concen a-
ions and di e en a angemen s o acancies 共see Fig. 3兲.
We should keep in mind ha in expe imen al condi ions he
acancy concen a ion is usually low 关less han 10% 共Re .
20兲兴. The e o e, in o de o do sui able compa isons we
should use p ohibi i e la ge uni cells. In ou wo k we ha e
conside ed wo di e en acancy concen a ions
␪
⫽0.5 and
␪
⫽0.25, whe e
␪
is he mean numbe o acancies pe su -
ace uni cell. Fo
␪
⫽0.5 we ha e used wo a angemen s,
named 共1⫻2兲and 共2⫻1兲models. The uni cell we use in he
calcula ions is made o se e al p imi i e su ace uni cells:
he i s digi 共i.e., 1⫻2兲 ep esen s he numbe o p imi i e
su ace uni cells, o size c⫻
冑
2a, in he 共001兲di ec ion and
he second digi ep esen s he numbe o p imi i e uni cells
FIG. 1. Th ee laye slab o model a s oichiome ic TiO2共110兲su ace.
Se e al kind o acancies discussed in he ex a e labeled.
FIG. 2. Su ace ene gy s numbe o laye s in he slab o a 共1⫻1兲model in
he single s a e. Ene gies om di ec calcula ions a e shown wi h ci cles
and he a e age ene gy be ween slabs wi h odd and e en numbe o laye s
a e shown wi h iangles.
7428 J. Chem. Phys., Vol. 121, No. 15, 15 Oc obe 2004 O iedo, San Miguel, and Sanz
in he (11
¯
0) di ec ion. The la ges uni cell used in his s udy
is a 共4⫻1兲model ha co esponds o
␪
⫽0.25 and con ains
acancies sepa a ed by ⬃6 Å. The ene gy om his calcula-
ion cons i u es ou bes app oxima ion o he o ma ion en-
e gy o an isola ed acancy.
I has been es ablished expe imen ally1,20–22 ha b idg-
ing acancies 共i.e., emo ing b idging oxygen a oms兲a e he
mos s able. Howe e , compa isons among ea lie heo e ical
epo s, e en using he same pe iodic DFT schemes, a e di -
icul because he e a e a conside able numbe o echnical
pa ame e s o se up such as k-poin sampling, ene gy cu o ,
acuum wid h, slab hickness, cell shape, acancy concen a-
ion, acancy a angemen , spin pola iza ion, e c. We can
easily cope wi h he i s h ee, howe e , he in luence o he
es o he pa ame e s is mo e sub le and has been analyzed
in de ail in his wo k.
An addi ional choice o se up when c ea ing acancies
is o emo e he a om om only one o he slab sides o om
bo h o hem. Al hough bo h al e na i e ways should con-
e ge o he same esul a an in ini e slab hickness, due o
he limi ed slab size o ou models some disc epancies may
occu . We ha e chosen he i s op ion because he c ea ion
o a couple o acancies may in oduce some di icul ies in
he elec onic con e gence o iple s a es. We ha e made
some es s and checked ha his pa icula choice leads o
small di e ences in acancy o ma ion ene gies ha become
smalle han 0.1 eV o slabs hicke han ou laye s.
The acancy o ma ion ene gy is de ined as he ene gy
equi ed o emo e an oxygen a om om he su ace and o
c ea e molecula oxygen in he gas phase,
E ac⫽ETinO2n⫺1⫹1
2EO2⫺ETinO2n,
whe e ETinO2n⫺1and ETinO2na e he o al ene gies o he de-
ec i e and s oichiome ic slabs, espec i ely, and EO2is he
o al ene gy o an isola ed O2molecule in i s g ound iple
s a e. We should emphasize ha in o de o ge meaning ul
esul s we ha e made calcula ions o ETinO2n⫺1and ETinO2nin
exac ly he same echnical condi ions in o de o ge an e -
ec i e cancella ion o e o s.
In Fig. 4 we ha e plo ed he acancy o ma ion ene gy
o b idging acancies in hei spin pola ized single and ip-
le s a es o
␪
⫽0.50. F om he igu e we can d aw some
conclusions. Fi s o all, he ene gy shows an oscilla ing be-
ha io wi h he numbe o laye s as we ha e no iced ea lie .
This e ec could o igina e an impo an disag eemen , e en
highe han 1 eV, be ween ene gies aken om calcula ions
made wi h hin slabs 共i.e., h ee o less laye s兲and hose wi h
slabs ha a e hick enough. The acancy o ma ion ene gy
inc eases 共dec eases兲 o slabs wi h an odd 共e en兲numbe o
FIG. 3. Schema ic cha ha shows he models we use in he p esen wo k,
wi h he nomencla u e used in he ex . The uni cell, ha is made o wo o
mo e p imi i e su ace uni cells, is ema ked in bold lines. P imi i e su ace
uni cells ha con ain de ec s a e ep esen ed by c osses and s oichiome ic
ones by ci cles.
FIG. 4. Vacancy o ma ion ene gy o b idging acancies e sus he numbe
o laye s. Resul s o he 共1⫻2兲and 共2⫻1兲models in hei single and iple
s a es a e included.
7429J. Chem. Phys., Vol. 121, No. 15, 15 Oc obe 2004 Oxygen acancies on TiO2(100) om i s p inciples calcula ions
laye s and a ull con e gence can only be eached o slabs
wi h a conside able numbe o laye s. I is wo h no icing, as
we men ioned abo e, ha he a e age alues con e ge ap-
idly, and only he ene gy o 3.5 laye s is sligh ly lowe han
he con e ged numbe . F om hese esul s we sugges he use
o a e aged alues highe han 4.5 laye s o ob ain ep esen-
a i e ene gy alues.
Second, ene gies a e ema kably independen om he
spin s a e. In an ea lie s udy Lindan e al. ound a signi ican
di e ence in ene gy acancy o ma ion be ween single and
iple s a es.9Howe e , he same au ho has ecen ly
claimed ha hey used a nonadequa e pseudopo en ial which
led o un ealis ic esul s.13
Thi d, he o ma ion ene gy is no sensi i e o he wo
di e en a angemen s conside ed. Ou esul s show a small
ene gy di e ence 共⬍0.2 eV兲be ween he 共2⫻1兲and 共1⫻2兲
econs uc ions ha ag ees well wi h p e ious heo e ical
wo ks.7,9 This migh be an indica ion o a weak in e ac ion
among acancies.
One aspec ha has usually been neglec ed in p e ious
calcula ions is a sys ema ic s udy o he o ma ion o di e -
en kind o acancies compa ed o he b idging ones. We
ha e ex ended ou models o explo e his possibili y and con-
side ed he c ea ion o acancies wi hin he ou e mos laye s
共see Fig. 1 o acancy iden i ica ion兲. In Fig. 5 we show
acancy o ma ion ene gies e sus numbe o laye s. We
ha e conside ed an 共1⫻2兲a angemen o wo di e en spin
s a es. F om he igu e, we i s no ice he same oscilla ing
beha io o models wi h odd/e en numbe o laye s and a
slow con e gence, in pa icula , o acancies in he ou e -
mos laye s 共i.e., b idging, in-plane and sub-b idging acan-
cies兲. This ac is a clea wa ning ha one should be cau ious
when de i ing conclusions om single model calcula ions
共i.e., in-plane acancies a e almos as s able as b idging a-
cancies o h ee laye slabs兲. In any case, i is clea om he
con e ged ene gy alues ha i is easie o emo e oxygen
a oms om he i s laye and, among he a oms in he same
laye , he less demanding p ocess is o c ea e b idging and
sub-b idging acancies. This is in disag eemen wi h esul s
om7,12 ha concluded ha an in-plane acancy in he sec-
ond laye could be app oxima ely as s able as a b idging one.
Howe e , his conclusion was indi ec ly d awn om calcu-
la ions wi h ou -laye slabs, pa ially elaxed, and/o in he
p esence o adso bed molecules. I may be possible ha he
FIG. 5. Vacancy o ma ion ene gy o se e al kind o acancies s he
numbe o laye s. Resul s o he 共1⫻2兲model in i s single and iple s a es
a e shown.
FIG. 6. Vacancy o ma ion ene gy o se e al kind o acancies s he
numbe o laye s. Resul s o he 共2⫻1兲model in i s single s a e a e shown.
TABLE I. Vacancy o ma ion ene gy o
␪
⫽0.25 using a 共4⫻1兲model.
Calcula ions we e pe o med o ou - and i e-laye slabs in he single
s a e o se e al kind o acancies: b idging, sub-b idging, in plane wi hin
he i s laye , and b idging and in plane in he second laye . 共see Fig. 1 o
nomencla u e兲.
No. laye s B idging In-plane Sub-b idging B idg2 In2
4 3.77 5.14 4.16 5.32 4.93
5 2.36 4.69 2.94 5.40 4.23
4.5 3.07 4.92 3.55 5.36 4.58
7430 J. Chem. Phys., Vol. 121, No. 15, 15 Oc obe 2004 O iedo, San Miguel, and Sanz

p esence o adso ba es could al e he ela i e s abili y o
acancies. I can also be seen ha plo s look e y simila o
bo h spin s a es as no ed be o e.
In Fig. 6, we ha e included g aphs o a 共2⫻1兲model in
he single s a e 共 esul s o iple s a es a e simila and a e
no shown兲. The numbe s and beha io a e compa able o he
共1⫻2兲model al hough he ene gy luc ua ions o in-plane
oxygen a oms a e smalle .
In addi ion, we ha e pe o med calcula ions a a lowe
acancy densi y 共i.e.,
␪
⫽0.25兲using a 共4⫻1兲model. Table I
epo s he ene gy alues o his model using ou - and i e-
laye slabs in he single s a e. We ha e also included a e age
alues ha we hink a e mo e ep esen a i e o compa i-
sons. We did addi ional calcula ions in some selec ed cases
and checked ha esul s o iple s a es a e again qui e simi-
la o hose p esen ed he e. We ecall ha hese alues ep-
esen ou bes e o o ep oduce expe imen al indings. The
acancy o ma ion ene gy alues 共⬇3eV兲a e highe han
hose ound o he 共110兲su ace in SnO2(⬇2.3eV).23 This
is expec ed since i has been de e mined expe imen ally ha
he oxygen acancy densi y is conside ably highe in SnO2
han in TiO2共Re . 35兲wha implies ha he Ti-O bonds a e
s onge han he Sn-O bonds.
We mus no ice ha he ene gy o c ea e a sub-b idging
acancy is su p isingly simila o he ene gy o a b idging
acancy, and he s abili y is e en e e sed o a h ee laye s
model. We es ima e he ene gy di e ence be ween he wo
kinds o acancies o be abou 0.1–0.2 eV o
␪
⫽0.5 and he
gap inc eases o abou 0.4–0.5 eV o
␪
⫽0.25. In expe imen-
al condi ions, acancy o ma ion is s imula ed by annealing
a empe a u es highe han 1000 K, ollowed by cooling o
oom empe a u e o measu emen s. P elimina y calcula-
ions36 show ha ansi ions ba ie s a e low enough o allow
acancy ans o ma ions be ween b idging and sub-b idging
acancies a ypical annealing empe a u es. We specula e
ha i a he modynamic equilib ium is eached a such em-
pe a u es, he dis ibu ion o acancies would ollow a Bol -
zmann dis ibu ion, and would be main ained a oom em-
pe a u e, whe e ansi ions a e no u he possible. In ha
case, we es ima e he p opo ion o sub-b idging acancies o
be abou 1%. To he bes o ou knowledge he e is no posi-
i e expe imen al iden i ica ion o subb idging acancies.
Howe e , his ac migh be ela ed o he low acancy den-
si y and/o a simila i y wi h he p ope ies o he b idging
acancies.
C. De ec i e su ace elaxa ion
We ha e also analyzed he ionic displacemen s in he
de ec i e su aces o he elaxed s uc u es. Tables II and III
epo he displacemen s compu ed as a di e ence in zcoo -
dina e espec o he equi alen ions in he elaxed s oichio-
me ic su ace, and includes alues aken o se e al models.
Posi i e alues indica e ha a oms mo e ou wa ds and nega-
i e alues co espond o inwa d elaxa ion. We ha e only
conside ed he b idging and he sub-b idging acancies
since, as we discussed abo e, a e he mos s able and he
only ones ha migh be expe imen ally obse ed.
A. B idging acancy
When a b idging oxygen a om is emo ed, he Ti a om
which was p e iously bonded o i mo es inwa ds in o de o
inc ease i s coo dina ion numbe and he oxygen a oms in
plane elax ou wa ds. We ind he same gene al end using
any model, bu di e en displacemen s o he sub-b idging
oxygen a oms, which mo e downwa ds in 共1⫻2兲and up-
wa ds in 共2⫻1兲and 共4⫻1兲. This disc epancy is no su p ising
because as can be seen om Fig. 3 he acancy a angemen
in he 共1⫻2兲model, is d as ically dis inc including al e na -
ing whole ows o acancies. This a angemen ep esen s
TABLE II. Ionic displacemen s in de ec i e su aces wi h b idging acancies. Calcula ions we e pe o med o
ou - and i e-laye slabs in he single s a e. The displacemen s a e compu ed as a di e ence in zcoo dina e
espec o he equi alen ions in he elaxed s oichiome ic su ace. Only displacemen o some selec ed ions
a e included 共see ex 兲.
A om
共1⫻2兲共2⫻1兲共4⫻1兲
Fou laye s Fi e laye s Fou laye s Fi e laye s Fou laye s Fi e laye s
Ti ⫺0.52 ⫺0.47 ⫺0.28 ⫺0.20 ⫺0.37 ⫺0.31
O in plane ⫹0.24 ⫹0.26 ⫹0.11 ⫹0.16 ⫹0.04 ⫹0.08
O sub-b idging ⫺0.16 ⫺0.09 ⫹0.11 ⫹0.32 ⫹0.02 ⫹0.07
TABLE III. Ionic displacemen s in de ec i e su aces wi h sub-b idging acancies. Calcula ions we e pe -
o med o ou - and i e-laye slabs in he single s a e. The displacemen s a e compu ed as a di e ence in z
coo dina e espec o he equi alen ions in he elaxed s oichiome ic su ace. Only displacemen o some
selec ed ions a e included 共see ex 兲.
A om
共1⫻2兲共2⫻1兲共4⫻1兲
Fou laye s Fi e laye s Fou laye s Fi e laye s Fou laye s Fi e laye s
Ti ⫹0.75 ⫹0.97 ⫹0.18 ⫹0.23 ⫹0.09 ⫹0.10
Ti second laye ⫺0.52 ⫺0.43 ⫺0.50 ⫺0.38 ⫺0.40 ⫺0.43
O b idging ⫹0.68 ⫹0.91 ⫺0.08 ⫺0.08 ⫺0.31 ⫺0.36
O in plane ⫹0.03 ⫹0.12 ⫹0.02 ⫹0.06 0.00 ⫹0.01
7431J. Chem. Phys., Vol. 121, No. 15, 15 Oc obe 2004 Oxygen acancies on TiO2(100) om i s p inciples calcula ions
qui e an un ealis ic model o eal su aces since he low den-
si y o acancies on hem makes a he imp obable such a
acancy alignmen .
Absolu e numbe s a e simila o he 共2⫻1兲and 共4⫻1兲
models. Howe e , a
␪
⫽0.25, he Ti a oms mo e inwa ds
mo e signi ican ly han a
␪
⫽0.5, while he in-plane and sub-
b idging O a oms mo e ou mo e app eciably a
␪
⫽0.5. In
gene al, a a lowe acancy concen a ion we ha e go a less
s ained su ace and he elaxa ion is mainly es ic ed o he
acancy su oundings.
B. Sub-b idging acancy
The emo al o an oxygen a om om he su ace can
modi y mo e o less s ongly he posi ions o he neighbo ing
a oms depending on he model used. This is pa icula ly seen
when c ea ing a sub-b idging acancy. In Table III we show
he ionic displacemen s o se e al models. Fo 共1⫻2兲model
he ionic displacemen s a e signi ican ly di e en om hose
o he o he models. Figu e 7 shows a iew o he elaxed
su ace using 共1⫻2兲and 共4⫻1兲models. In he 共1⫻2兲model
bo h he Ti a om ow and he b idging oxygen ow elax
mo ing ou abo e he missing sub-b idging O a oms. How-
e e , in 共4⫻1兲model he su ace is s abilized when b idging
O a oms elax inwa ds. Thus, he Ti a om abo e he acancy
is seen o elax ou wa ds by 0.97 Å in 共1⫻2兲bu only 0.23 Å
in 共2⫻1兲and 0.10 Å in 共4⫻1兲. Fu he mo e, he O b idging
bonded o ha Ti mo es ou by 0.91 Å, while, i mo es
inwa ds o he o he models 关0.08 Å in 共2⫻1兲and 0.36 Å in
共4⫻1兲兴.
These esul s show ha one has o be cau ious when
choosing limi ed models in o de o mimic de ec i e su -
aces, no only because we canno compu a ionally a o d a
low acancy densi y bu also because esul s could be
s ongly dependen on he pa icula a angemen o acan-
cies.
IV. CONCLUSIONS
In he p esen wo k we ha e ca ied ou a sys ema ic
s udy o oxygen acancy o ma ion on he TiO2共110兲su -
ace by means o plane-wa e pseudopo en ial densi y unc-
ional heo y calcula ions. We ha e ocused in compa ing he
o ma ion ene gy o se e al kind o acancies in he ou e -
mos laye s o he su ace and analyzing he in luence ha
di e en echnical pa ame e s and model size could make on
he esul s.
We ha e no iced ha ene gies ha a e based on slab
calcula ions show an oscilla ing beha io wi h he numbe o
laye s, he e o e i is ecommended o use a e age alues
be ween slabs wi h odd and e en numbe o laye s. In pa -
icula , ene gies a e almos con e ged when using a e age
alues highe han 4.5. Ene gies om calcula ions wi h
h ee-laye slabs could con ain e o s as high as 1 eV.
We ha e pe o med calcula ions wi h se e al a ange-
men s o acancies. We ha e no ound signi ican di e -
ences be ween single and iple s a es. Fu he mo e, he o -
ma ion ene gies o 共1⫻2兲and 共2⫻1兲models a e simila ,
which could imply a weak in e ac ion among acancies.
Among he se e al acancies we ha e es ed, we ind
ha he mos s able is he b idging acancy ha ag ees wi h
expe imen al da a. Howe e , he o ma ion o subb idging
acancies migh be possible a mode a e empe a u es.
Rega ding a omic displacemen s caused by su ace e-
laxa ions, we ind ha one has o be specially cau ious when
de i ing conclusions. Fo example, a 共1⫻2兲model ep o-
duces well he o ma ion ene gy o a b idging acancy bu
could lead o qui e an un ealis ic s uc u e o he educed
su ace. Since su ace s ess is a long ange e ec , in o de o
ep oduce ealis ic displacemen s a ound a acancy, he use
o e y la ge uni cells migh be necessa y.
ACKNOWLEDGMENT
This wo k was unded by he Spanish DGESIC, P ojec
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