Electronic structures and optical properties of rutile TiO 2 with different point defects from DFT + U calculations

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Physics Letters A

www.elsevier.com/locate/pla

Electronic structures and optical properties of rutile TiO ₂ with different point defects from DFT + ^U calculations

H.X. Zhu

^a^,^b

, P.X. Zhou

^a

, X. Li

^a

, J.-M. Liu

^a^,^∗

aLaboratoryofSolidStateMicrostructures,NanjingUniversity,Nanjing210093,China

bCollegeofPhysicalScienceandElectronicTechniques,YanchengNormalUniversity,Yancheng224002,China

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received7May2014

Receivedinrevisedform16July2014 Accepted17July2014

Availableonline22July2014 CommunicatedbyV.A.Markel

Keywords:

RutileTiO2 Electronicstructure First-principlescalculations Opticalproperties

The electronic states and formation energies of fourtypes oflattice point defects inrutile TiO2 are studied usingtheﬁrst-principles calculations.Theexistence ofoxygen vacancyleadsto adeepdonor defect levelinthe forbiddenband, whiletheTi interstitialformstwo localstates. It ispredicted that oxygen vacancy prefersto combine withTi-interstitial to formVO–Ti_i dimerby apartial 3d electron transfer from the Tii to its neighboring VO.The charge distributionbetween a Ti interstitialand its neighboringTiionspartiallyshieldstheCoulombinteractions.Lastly,opticalpropertiesofthesedefective latticesare discussed.

1. Introduction

Titaniumdioxide(TiO2) isoneoftheimportantindustrialma- terials, and it has been widely used for synthetic fibers, white pigments, plastics, etc. TiO2 also has high oxidizing power over similarmaterialsassemiconductorphotocatalyst.In requesttothe developmentofrenewableenergy,it hasbecomeoneofthemost promisingsemiconductingphotocatalysismaterials.Alltheseneeds drivecontinuousandsustainableinvestigationsonTiO₂ fromvari- ousphysicalandchemicalaspects.TiO2hasthreebasiccrystalline phases:brookite, anatase,andrutile,whiletheanataseandrutile phasesare mostconcerned [1–3]. For photocatalyst applications, however, the band gap of perfect TiO2 is too big (∼³.0 eV for rutilephase and∼³.2 eV foranatase phase), whilethepreferred bandgapshouldbe∼².0 eV.A largenumberofinvestigationson thisissue havebeen carriedout [4–8]. It iswell knownthat for photocatalyticapplications,thetechnique formassivesynthesisof TiO2 ineitherpowderorthinfilmformmustbesimpleandcost- competitive. It implies that the as-prepared materials have high densitystructuredefects,andinparticularthoseintrinsicpointde- fectsgeneratedduringtheconventionalsynthesisconditions.These pointdefects maybringintoadditional ingredientsinto theelec- tronicband structure, influencing substantially the transport and opticalbehaviors [9], some ofwhich maybe favoredin termsof practicalapplications.Thisisthemotivationforcomprehensivein- vestigationsonthedefectstatesinTiO₂.

*

Correspondingauthor.

Nevertheless,thereare debatesontheelectronicstructureand magnetic property induced by point defects. Recent work sug- gested that an oxygen vacancy (VO) can createa shallow defect state withintheenergygap, whichsuppressesthe localmagnetic state oftheneighboringTiion[10].Furthermore,it waseven re- vealed that an interstitial Ti as a defect is responsible for the anomaly in the infrared absorption data [11]. However, contro- versy issues on themicroscopic mechanism still exist.It was re- portedthatTiinterstitialsmaybelargelyresponsiblefortheband gapstates,as revealedby high-resolution scanningtunneling mi- croscopyandphotoelectronspectroscopymeasurements(PES)[12].

Alternatively, it was claimed that the band gap states are at- tributedtoOVsinthesamples,as revealedusingscanningtunnel- ing microscopy (STM)andultravioletphotoemission spectroscopy (UPS) [13].Recently, it wasreportedthat sputter/annealingcycles in ultrahigh vacuum (UHV) can lead to eﬃcient creation of VO andcondensationofTiinterstitials(Ti_i)neartheTiO2 surface[14].

It issuggestedthatthemid-gapstatesmayarisefromthecoexis- tenceofVOandTi_idefects.TheseresultsseemtoadvisethatTiO2 accommodates point defects more than OVs. On the other hand thesedifferenttypesofpointdefectsmayinteractwitheachother, addingcomplexitytotheelectronicstructure[15].

Therefore,a comprehensiveknowledgeontheelectronicstruc- tureof TiO2 withvarious types ofpoint defects seems appealed.

Although relevant theoretical investigations are available in literature [16–21], these works focus more on OVs than others. De- tailed theoreticalstudyoftheelectronic structuresofthevarious typesofpointdefectsinTiO2 isstilllacking.In particular,speciﬁc http://dx.doi.org/10.1016/j.physleta.2014.07.029

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It isknown that the standard DFT oftenunderestimate the band gap[24–26].In somecases,it isalsoinadequateforthestandard DFT methodusedin someelectronic characteristics ofTiO2,such asfailuretodescribethelocalizedstates,defectorimpuritystates inTiO2[27].WithrespecttothestandardDFTscheme,theDFT+Û schemeandHSEschememaybemoreaccurateinthetreatmentof theseproblems[28–30].UsuallytheHSEmethodisveryaccurate butalso requires a huge amount of computational resources be- yondthecapacityofourcomputers.In ourwork,usingtheDFT+Û approachisagoodchoicebecauseitalsocanobtainrelativelyac- curateresults[15,31].TheDFT+Ûâpproachîntroducesânôn-site correction in order to describe systems with localized d and f electrons, which can predict band gaps comparablewith experi- mentalresults.ThismethodhasbeensuccessfullyappliedtoTiO2 systemswhere thestandard DFT incorrectly predicts thedelocal- izedstates[32–35].

Here,we mainlyusetheDFT+Û^scheme^to^calculate^theêlec- tricstructuresand optical propertieson thedefective rutile TiO2 systems.While,forcalculationoftheformationenergywestilluse pureDFT methodto calculatebecauseDFT+Û^schemeôften^dis- tortenergyofsystem.

ThecalculationsareperformedwiththeViennaabinitiosimu- lationpackage(VASP)code[36,37],usingthegeneralizedgradient approximation (GGA)Perdew–Becke–Erzenhof (PBE)function [38]

and the projector augmented wave (PAW) pseudopotential [39]

withthecutoffenergyof450 eV.Theon-site effectiveU parame- ter[22]isappliedtotheTi-3dtomodifystronglycorrelatedinter- actions. Because Ti⁴⁺-3dorbital isalmost empty, usually enough bigparameter Ushould be chosen to pull the band gap, such as UTi = ^{9.3 eV in}^Ref.^[5]^and^UTi =^{9.8 eV in} ^Ref.^[40].^{We choose} U=⁹.3 eV fortheTi-3delectrons. We fullyrelax theunit cellto obtain the lattice constants: a=^b=⁴.6961 Å andc=².9736 Å, consistentwithearlierreports[41].Thesimulateddefectsareem- beddedintheoptimizedcelltobuilda2×²×2 supercell(shown inFig. 1).ThekpointgridsamplingusingtheMonkhorst–Pack[42]

meshof3×³×^{5 is}ûsed^for^geometryoptimization,andthemesh of 5×⁵×^{7 is} ûsed ^for êlectronic ^properties calculations. The atomic positions are relaxed until the Hellmann–Feynman force actingoneachatomisreduceddowntolessthan1.0 meV/Å.

3. Resultsanddiscussion 3.1. Locallatticedistortions

Weﬁrstaddressthelocallatticedistortionarisingfromthede- fect embedding, which is taken as the lattice structure basis for understanding the electronic structure. For the lattice with a VO defect,theoptimizedatomicpositionsonthe (−¹¹⁰) planeare shown inFig. 2(b) where the original oxygen ion position is la- beledwithasmalldashedcircle.Theperfect(−¹¹⁰)latticeplane ispresentedinFig. 2(a)asareference.It isseenthatuponthere- movalof an Oion, the locallattice is distorted,characterized by

Fig. 1.(Coloronline.)Alatticestructureschematicofa2×²×^{2 rutile}^TiO2supercell.

Fig. 2.(Coloronline.)Optimized(−¹¹⁰)planeatomicstructureswithnopoint defect (a),one VO defect (b),oneVTi defect (c),and oneTii defect (d).Thered (smallsolid),andblue(biggrey)ballsaretheOandTiions,respectively.

the stretched distance of the Ti–Ti pair rightabove the VO from 0.307 nmto0.308 nm,andthestretcheddistanceoftheTi–Tipairs aside the VO from 0.365 nm to 0.364 nm.Those surrounding Ti ions becomecloserto theneighboringoxygen ions.Theseresults are consistent with earlier calculations [10], conﬁrming that the present calculationsmake sense. For V_Ti, one sees from Fig. 2(c) thelocallatticedistortionupontheremovaloftheTiion.Thesix oxygen ionssurrounding theV_Ti shiftoutwardsfromthe original positions. The two apical oxygen ions are displaced for 0.18 nm away each other outward, while the four equatorial oxygen ions are displacedby0.21–0.23 nmawayfromtheVTi.Forthecaseof embeddingaTi_i,we insertaTiatomintothecenterofaTioctahe- dral.ThelatticeconﬁgurationisshowninFig. 2(d)withthegreen ballrepresentingtheinterstitialTi.TheTiionssurroundingtheTi_i shift outward,andthe fourequatorialoxygen ions arepulled in- wardtotheTiiwhilethetwoapicaloxygenionsrelaxoutward.

At last, we consider the lattice distortion associated with the VO–Ti_idimer,whereVOiscreated bytakingawayoneoxygenion

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Fig. 3.(Coloronline.)(a)SchematicofthelatticestructurewithVOdefectdenotedbyanopendashedcircle,andthesixpossiblepositionsforaTiinterstitial.(b) Thetotal energyofthelatticewiththeTiinterstitiallocatedrespectivelyatoneofthesixpositions,andtheblueandredlinesrepresenttheenergycalculatedbythemethodofDFT andDFT+U,respectively.

withinthe(−¹¹⁰)plane(theposition 3ofFig. 3(a)byadashed circle).The Ti_i isinserted one byone into positionsfrom 1 to 6, as markedinFig. 3(a),i.e.thecentersofsixneighboringTioctahe- dra.The totalenergyvaluesforthesixdifferentcasesareplotted inFig. 3(b),indicatingthattheposition 1isthemoststableloca- tionfortheTiiintheVO–Tiidimersincetheenergyatthiscaseis thelowest,andobviouslythemostunfavored siteisaroundposi- tion 3.

3.2.Defectformationenergy

Furthermore,inordertoexplorethethermodynamicsrelatedto theformationofthesepointdefectsunderdifferentgrowthcondi- tions,we calculatetheformationenergyofthesedefects.ForTiO2, theformationenthalpycanbewrittenas:

H

(

TiO2

) =

μ

_Ti

₋ μ

_Ti

_[

Bulk

] +

²

μ

O

− μ

O

[

^O2

/

2

]

μ

O[^O2/2]^is^the^chem- icalpotentialofOelementwhichequalstohalfofthetotalenergy ofanoxygenmolecule.

O[^O2/2]^.^The^calculated^formation^en- thalpyforrutileTiO2is−¹⁰.18 eV undertheTirichcondition,and

−¹⁰.19 eV undertheOrichcase,whichisclosetomeasuredvalue (−⁹.6±⁰.8 eV)reportedinRef.[43].

Consequently,the formation energy for a point defect can be written as:

E

=

^E

[

^defect

] −

^E

[

^perfect

] +

ⁿO

μ

O

+

ⁿTi

μ

_Ti

,

(2) whereE[defect] is the total energy of a supercell containing the defects, and E[perfect] is the total energy of the corresponding perfectsupercell.Here,n_Oandn_Ti denotethe numbersofO and Tiatomstobe takenfromorinsertedinto thesupercellinorder totakeaccount ofpointdefectgeneration,whereni (i=^{O or}^Ti) is negativeifa corresponding atom isinserted into thesupercell and positive if such an atom is taken away from the supercell.

Table 1

CalculatedformationenergiesforthefourtypesofpointdefectsinrutileTiO2lat- tice.

Ti rich (eV) O rich (eV)

VO 0.60 5.68

VTi 14.51 4.28

Tii −1.29 8.90

VO+^Tii −⁰.76 14.55

The as-calculated formation energy values for the four types of point defectsarelisted inTable 1.The formationenergyofV_O is 0.6 eV,whichisveryclosetothevalue0.8 eVcalculatedby Mor- ganet al.[44].Quiteobviously,theformationofVO,Tii,andVO–Tii dimer can be much easier in the Ti-rich circumstance than that intheO-richambient,whileV_Ti issurely hardtoforminthe Ti- richcircumstance.Andalso,we notethatVO–Tiidimerisfavorable thantheisolatedpointdefectsby0.07 eVduetotheirinteraction.

Forasingle VO–Tiidimer,0.07 eVmaybe relativelysmall, butfor theentirecrystal,theenergyofthesystemmayreducegreatly.

Insequenceofthethermodynamicsforthesedefectsformation, in the Ti-rich circumstance, STi has the lowest formation energy, followedbytheV_O–Ti_idimerandV_O.In fact,theTi_i givesrise to somechargecompensationaroundVO,allowingapartialshielding ofthe Coulombinteraction betweenthetwo neighboringTi ions.

It implies that an oxygen vacancy prefers to combine with a Ti interstitial,formingaVO–Ti_idimer.

3.3. Electronicstructures

The bandstructuresofthe rutileTiO₂ withdefectsare shown inFig. 4(a)–(d).ThereddashedlinesrepresenttheFermilevelEF. We studythebandstructureofthepurerutileTiO2 andtheband gap is ∼².86 eV. This value is similar to the measured value of

∼³.0 eV[45].Fig. 4(a) istheband diagram ofthe supercellwith oneV_O.Thebandgapis∼².77 eV whichisabout0.09 eVsmaller than the pure case, andthe red linebelow theconduction band minimum (CBM) about 1.2 eV represents the defect state. It is noted that this defect level is fully occupied and the local state consistofTi-3d.It hasbeenrevealedthatthereductionofTi⁴⁺ to Ti³⁺occursinTiO2 withoxygenvacancyduetothechargeimbal- ance[46].Moreover,thepositionoftheE_FrevealthatV_Oresultin ann-typesemiconductingbehaviorsinTiO2.Whenthesupercellis accommodated withaVTi,theFermilevelshiftsdown totheva- lencebandmaximum(VBM),as showninFig. 2(b).Theband gap become∼².72 eV.A Tiinterstitial(Tii)createsadeepdonorlevel and a shallowdonor level at ∼¹.25 eV and 0.28 eV respectively belowtheCBMin theforbiddenband, as shownin Fig. 4(c).The

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Fig. 4.(Coloronline.)CalculatedbandstructurealongthehighsymmetrylinesoftheBrillouinzoneforasupercellwithoneVOdefect (a),oneVTi defect (b),oneTii defect (c),andoneVO+Tiidimer (d).

impuritystatesuppressespartiallytheeffectivebandgapandthus maychangetheinfraredabsorption[10].InFig. 4(d)isshownthe band structureofthesupercellwitha V_O–Ti_i dimer.Three defect energylevelsinthebandgapareobserved.Thereasonliesinthe fact that the presence ofoxygen vacancy breaks the crystalﬁeld ofoxygenoctahedra,loweringthedegeneration.Thisdegeneration loweringallowsadditionalenergylevelsplitting.TheTiinterstitial state intheenergygapsplits furtherbesides theoxygen vacancy induceddefectlevel.

To understand the main characteristics, we look into the total andpartial densityofstates.For allthese lattices, it isfound that the conductionbandsare mainly of theTi-3d character and the valence bands consist of the O-2p states mostly. We care- fullycomparedthetotaldensityofstates(TDOS)ofaperfectTiO2 lattice andthose with the fourtypes of point defects (shown in Fig. 5).ThecalculatedTDOSneartheFermilevelforthesupercells withdefectVO orVO–Ti_i dimerare similar inshape. Thedefects VO,Tii,andVO–Tii dimerallbehaveasthen-typedonor,allowing theFermilevelshiftingtotheCBM.However, defectVTiactsasa p-typedonor,andthustheFermilevelshiftsintothevalenceband.

It isnotedthattheVO,Ti_i andVO–Ti_idimerhavetheisolatedde- fectstatebelowtheCBM,whichreduceseffectivelytheenergyfor electrontransitionfromtheVBMtothe CBM.Thistransitionwas observedintheinfraredabsorption[13,47–50].

Inordertofurtherunderstandthedefectstateinthebandgap, we take the supercell with a Tii defect as an example and the partialdensity ofstates(PDOS) ofthisTi interstitial ispresented inFig. 6. It isseen that the isolated defect state mainly exhibits the Ti_i 3d character which further splits into the eg-levels and t2g-levels dueto the crystalﬁeld effect ofthe oxygen octahedra.

It is clearthatthe orbitalcharacter oftheisolated defectstateis mostly attributedtothe d_x2−^y² orbitofthee_g electronof theTi interstitial,as showninFig. 6(b).

Finally,we lookatthechargedensitydistributionforthedefect statesinthesupercells withtheVO,theTiiandVO–Tiidimer, respectively.TheevaluateddataareplottedinFig. 7.Thedataforthe VO intheenergy range4.6–5.1 eV,which coversdefect-state en-

2(

ω

) in lattice with theTi_i hasonemainpeak at0.3 eV.ForthecasewiththeVO–Ti_i dimer,it has twomain peaks,one at0.63 eV,whichcloseto the valueobservedbythenormalizedIRspectruminRef.[7],andthe other at1.53 eV.Theresultsshow thattheinfraredabsorption in thelatticewiththeVO–Tiidimerisconsistentwiththeexperimen- talIRabsorptionpeaks(0.75 eVand1.18 eV)[11].

3.5. Discussion

Tothisstage,it isgenerallyseenthattheelectronicstructureof rutileTiO2 canbemodulatedremarkablybythedefects,evensin- glepointdefect.Onceadimer-likedefectsuchastheVO–Ti_idimer considered here,is introducedintothe lattice,thecharge density distributioncanberemarkablyextended,allowingremarkableim-

Fig. 7.(Color online.) Calculated partial charge density distributions around the VOdefect (a), the Tiidefect (b) and the VO+^Tiidimer (c).

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Fig. 8.(Coloronline.)ImaginarypartsofthedielectricfunctionofrutileTiO2with defects.

pactonthetransportbehaviorsandoptical response.Thepresent workstarts fromthreetypesofsinglepoint defectsandone kind of dimer defects. Thesedefects induced ﬂuctuations inthe elec- tronicstatesandbandstructuresareevident.

Nevertheless,it shouldbenotedthataconsiderationofisolated point defects may be over-simpliﬁed. The interactions between thesedefectsandstructuralclusteringoftheminpracticalmateri- alsinservicemayalsoplaynon-negligiblerolesinmodulatingthe electronicstructure,whicharenotdealtwithinthiswork.In fact, relatedinvestigationsontheinﬂuenceofV_OandTi_ihavebeenre- ported in literature [14,53]. It is expected that such interactions and coupling between particulars centered atthese defects may excite much more features in the transport andexcitation spectroscopy.

4. Conclusion

Insummary, we havecarefullyinvestigated theformation energies of the four types of point defects in rutile TiO₂ and the electronic structures associated with the four types of point de- fectsusingtheﬁrst-principlesdensityfunctionalcalculations. Our results show that both the oxygen vacancy (VO) andTi interstitial(Ti_i)cancreatedonordefectlevelsintheelectronicstructure.

In particular, the VO–Ti_i dimer is preferred when an oxygen vacancy hasaTi interstitial neighbor inthelattice. The chargedis- tributionbetweentwoneighboringTiionsisobservedtopartially shieldtheCoulombinteractions. Ourresultsshowthat theV_O–Ti_i dimerdefectismorefavoredthermodynamicallythanisolatedVO inrutileTiO2,especiallyintheTi-richcase.Therefore,theeffectof VO–Tiidimerontheelectronicstructureandconsequenttransport andcatalyticbehaviorsshouldbetakenintoaccount.

Acknowledgements

ThisworkwassupportedbytheNational973ProjectsofChina (Grant No. 2011CB922101),the NationalNatural Science Founda- tion ofChina (Grants No. 11234005 and No. 51332006), andthe PriorityAcademic ProgramDevelopmentofJiangsuHigherEduca- tionInstitutions,China.

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