Oxygen vacancies on TiO2 (110) from first principles calculations

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 ARTICLES YOU MAY BE INTERESTED IN Adso p ion, di usion, and dissocia ion o molecula oxygen a de ec ed A densi y unc ional heo y s udy The Jou nal o Chemical Physics 120, 988 (2004); h ps://doi.o g/10.1063/1.1631922 Fi s -p inciples calcula ions o de ec s and impu i ies: Applica ions o III-ni ides Jou nal o Applied Physics 95, 3851 (2004); h ps://doi.o g/10.1063/1.1682673 A DFT s udy o wa e adso p ion on u ile TiO2 (110) su ace: The e ec s o su ace s eps The Jou nal o Chemical Physics 145, 044702 (2016); h ps://doi.o g/10.1063/1.4958969 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 No. MAT2002-576. 1U. Diebold, Su . Sci. Rep. 48,53共2003兲. 2V. E. Hen ich and P. A. Cox, The Su ace Science o Me al Oxides 共Cam- b idge Uni e si y P ess, Camb idge, 1996兲. 3R. Schaub, E. Wahls o ¨m, A. Rønnau, E. Laegsgaa d, I. S ensgaa d, and F. Besenbache , Science 299, 377 共2003兲. 4E. Wahls o ¨m, E. K. Ves e gaa d, R. Schaub, A. Rønnau, M. Ves e gaa d, E. Laegsgaa d, I. S ensgaa d, and F. Besenbache , Science 303,511 共2004兲. 5S. P. Ba es, G. K esse, and M. J. Gillan, Su . Sci. 385, 386 共1997兲. 6A. Vijay, G. Mills, and H. Me iu, J. Chem. Phys. 118,6536共2003兲. 7J. A. Rod ı ´guez, J. H bek, Z. Chang, J. D o ak, T. Ji sak, and A. Mai i, Phys. Re . B 65, 235414 共2002兲. 8J. A. Rod ı ´guez, G. Liu, T. Ji sak, J. H bek, Z. Chang, J. D o ak, and A. Mai i, J. Am. Chem. Soc. 124, 5242 共2002兲. 9P. J. D. Lindan, N. M. Ha ison, M. J. Gillan, and J. A. Whi e, Phys. Re . B55, 15919 共1997兲. 10J. Lecon e, A. Ma ko i s, M. K. Skalli, C. Mino , and A. Belmajdoub, Su . Sci. 497, 194 共2002兲. 11 T. B edow and G. Pacchioni, Chem. Phys. Le . 355,417共2002兲. 12J. A. Rod ı ´guez, T. Ji sak, G. Liu, J. H bek, J. D o ak, and A. Mai i, J. Am. Chem. Soc. 123, 9597 共2001兲. 13M. D. Rasmussen, L. M. Molina, and B. Hamme , J. Chem. Phys. 120, 988 共2004兲. 14D. C. So escu and J. T. Ya es, J ., J. Phys. Chem. B 102, 4556 共1998兲. 15M. Ramamoo hy, D. Vande bil , and R. D. King-Smi h, Phys. Re . B 49, 16721 共1994兲. 16R. Schaub, P. Thos up, N. Lo ´pez, E. Laegsgaa d, I. S ensgaa d, J. K. Nø sko , and F. Besenbache , Phys. Re . Le . 87, 266104 共2001兲. 17S. P. Ba es, M. J. Gillan, and G. K esse, J. Phys. Chem. B 102, 2017 共1998兲. 18S. P. Ba es, G. K esse, and M. J. Gillan, Su . Sci. 409, 336 共1998兲. 19W. C. Mack od , E. A. Simson, and N. M. Ha ison, Su . Sci. 384,192 共1997兲. 20U. Diebold, J. Lehman, T. Mahmoud, M. Kuhn, G. Leona delli, W. He- bens ei , M. Schmid, and P. Va ga, Su . Sci. 411, 137 共1998兲. 21E. L. D. Hebens ei , W. Hebes ei , and U. Diebold, Su . Sci. 461,87 共2000兲. 22E. L. D. Hebens ei , W. Hebes ei , and U. Diebold, Su . Sci. 470, 347 共2001兲. 23J. O iedo and M. J. Gillan, Su . Sci. 467,35共2000兲. 24M. C. Payne, M. P. Te e , D. C. Allan, T. A. A ias, and J. D. Joannopoulos, Re . Mod. Phys. 64, 1045 共1992兲. FIG. 7. Side iews o elaxed sub-b idging oxygen de ec i e su aces using 共a兲共1⫻2兲and 共b兲共4⫻1兲models. 7432 J. Chem. Phys., Vol. 121, No. 15, 15 Oc obe 2004 O iedo, San Miguel, and Sanz 25P. Hu, D. A. Kings, S. C ampin, M. H. Lee, and M. C. Payne, Su . Sci. 230,501共1994兲. 26J. P. Pe dew, Elec onic S uc u e in Solids ’91, edi ed by P. Ziesche and H. Esch ig 共Akademie, Be lin, 1991兲. 27J. P. Pe dew, J. A. Che a y, S. H. Vosko, K. A. Jackson, M. R. Pede son, D. J. Singh, and C. Fiolhais, Phys. Re . B 46, 6671 共1992兲. 28G. K esse and J. Ha ne , Phys. Re . B 47, 558 共1993兲. 29G. K esse, Ph.D. hesis, Technische Uni e si a Wien, 1993. 30G. K esse and J. Fu hmu ¨lle , Compu . Ma e . Sci. 6,15共1996兲. 31G. K esse and J. Fu hmu ¨lle , Phys. Re . B 54, 11169 共1996兲. 32D. Vande bil , Phys. Re . B 41, 7892 共1990兲. 33H. Monkho s and J. Pack, Phys. Re . B 13, 5188 共1976兲. 34P. Vine , J. Fe an e, J. R. Smi h, and J. H. Rose, J. Phys. C 19, L467 共1986兲. 35D. F. Cox, T. B. F ybe ge , and S. Semancik, Phys. Re . B 38,2072 共1988兲. 36J. O iedo, M. A. San Miguel, and J. F. Sanz 共unpublished兲. 7433J. Chem. Phys., Vol. 121, No. 15, 15 Oc obe 2004 Oxygen acancies on TiO2(100) om i s p inciples calcula ions