Implementation of the Projector Augmented Wave LDA

arXiv:cond-mat/0003182v1[cond-mat.str-el]10Mar2000ImplementationoftheProjectorAugmentedWaveLDA+UMethod:ApplicationtotheElectronicStructureofNiOO.Bengone1,2,M.Alouani1,P.Bl¨ochl3,andJ.Hugel21InstitutdePhysiqueetChimiedesMat´eriauxdeStrasbourg(IPCMS),23rueduLoess,F-67037StrasbourgCedex,France2Univesit´edeMetz,IPC-LPLIGroupedeStructureElectroniquedesMilieuxdenses,1bdArago,F-57078MetzCedex3,France3IBMResearch,ZurichResearchLaboratory,CH-8803R¨uschlikon,Switzerland(February1,2008)AbstractTheso-calledlocaldensityapproximationplusthemulti-orbitalmean-fieldHubbardmodel(LDA+U)hasbeenimplementedwithintheall-electronpro-jectoraugmented-wavemethod(PAW),andthenusedtocomputethein-sulatingantiferromagneticgroundstateofNiOanditsopticalproperties.TheelectronicandopticalpropertieshavebeeninvestigatedasafunctionoftheCoulombrepulsionparameterU.WefindthatthevalueobtainedfromconstrainedLDA(U=8eV)isnotthebestpossiblechoice,whereasaninter-mediatevalue(U=5eV)reproducestheexperimentalmagneticmomentandopticalpropertiessatisfactorily.AtintermediateU,thenatureofthebandgapisamixtureofchargetransferandMott-Hubbardtype,andbecomesalmostpurelyofthecharge-transfertypeathighervaluesofU.Thisisduetotheenhancementoftheoxygen2pstatesnearthetopofthevalencestateswithincreasingUvalue.71.15.Ap,71.20.-b,71.20.Be,78.20.CiTypesetusingREVTEX1I.INTRODUCTIONFormanymaterials,thedensityfunctionaltheory(DFT)1inthelocal-spin-densityap-proximation(LSDA)2providesagooddescriptionoftheirground-stateproperties.However,problemsarisewhentheDFT-LSDAapproachisappliedtomaterialswithionsthatcontainincompleted-orf-shells,suchastransition-metaloxidesorheavyfermionsystems.Forex-ample,mosttransition-metaloxidesarewide-gapantiferromagneticinsulators,3–9andtheDFT-LSDApredictsthemtobeeithermetals(FeOandCoO)orsmall-gapsemiconductors(MnOandNiO).10ThefailureoftheDFT-LSDAcanbetracedtothemean-fieldcharacteroftheKohn-Shamequationsaswellastothepoordescriptionofstrongcorrelationeffectswithinthehomogeneouselectrongas.ThestrongcorrelationeffectsareresponsibleforthebreakdownoftheDFT-LSDAdescriptionoftheelectronicstructureofthesecompounds.Inordertoprovideabetterdescriptionoftheseeffects,theMott-Hubbardpicturehasbeenintroduced.11,12IntheMott-HubbardpictureofNiO,thed-dCoulombinteractionsplitstheNidsub-bandsintotheso-calledlowerandupperHubbardbands.TheupperHubbardbandhasmostlyNi3d9character,whilethetopofthevalencebandisof3d8character,leadingtoaMott-Hubbardgapofd-dtype.However,O1sx-rayabsorption13aswellasx-rayphotoemissionandbremsstrahlungisochromatspectroscopies9onLixNi1−xOhaveshownthattheadditionalholehasmainlyoxygencharacter.IncontrasttotheMott-Hubbardmodel,theenergybandgapcausedbytheNi3dcorrelationsisthereforeofthecharge-transfertypebetweentheoccupiedoxygen2pandtheNi3demptystates.Ontheotherhand,localizedapproaches,14–20suchasthelocalclusterschemebasedontheconfigurationinteractionmethodortheAndersonimpuritymodel,inwhichtransition-metalionsaretreatedlikeanimpurityinanoxygen2phost,predictawell-definedbandgapof5.0eV.However,theoxygen2pbanddispersionobservedinangle-resolvedphotoe-missionspectroscopy21cannotbedescribedbythesemethodsbecausethelatticeeffectsareneglected.SeveralattemptshavebeenmadetoincludethemissingcorrelationeffectsinDFT-LSDA.Thegeneralizedgradientapproximation(GGA),22whichtakesintoaccounttheradialandangulargradientcorrections,canonlyopenasmallbandgap.23Theself-interactioncorrection(SIC)24–26eliminatesthespuriousinteractionofanelectronwithitselffromtheconventionalDFT-LSDAmethod.ComparedtoLSDA,thebandgapandthemagneticmomentsaresignificantlyincreased.However,thebandgapstillistoosmall,andtheSIC-LSDAmethodpredictsalargerenergybandgapforNiOthanforFeOandCoO,incontradictiontoexperiment.3Thecrystal-fieldorbitalpolarizationintroducedbyNorman27todeterminethemagnetismandinsulatingbandgapoftransition-metaloxidesispromisingbutunderestimatesboththespinmagneticmomentandthebandgap.Anotherpromisingapproachforcorrelatedmaterialsistheso-calledlocaldensityap-proximation(LDA)plusthemulti-orbitalmean-fieldHubbardmodel(LDA+U)28–34whichincludestheon-siteCoulombinteractionintheLSDAHamiltonian.Afteraddingtheon-siteCoulombinteractiontotheLSDAHamiltonian,thepotentialbecomesspin-andorbital-dependent.Becausealargerenergycostisassociatedwithfluctuationsofthed-occupancy,theorbital-dependentpotentialreducesthefluctuationsofthed-occupancy,resultinginabetterjustificationofamean-fieldapproach.LDA+U,althoughitisamean-fieldapproach,2hastheadvantageofdescribingboththechemicalbondingandtheelectron-electroninter-action.ThequestionregardingthebestvaluefortheCoulombrepulsionparameterUis,however,stillunderdebate.TheUparameterforNiOobtainedfromaconstrainedLDAcalculationisabout7to8eV,andthisisthevaluegenerallyusedinLDA+Ucalculations.AsimilarvalueofUhasbeenobtainedfromaconstrainedLDAcalculationforbulkFe,29eventhoughamuchsmallervaluehadbeenexpectedbecauseofthemetallicscreeninginFe.Theauthorsargued29thatthehighervaluecouldbeanartifactduetothepoorscreeningwithintheatomicsphereapproximation(ASA),andthatwithinafull-potentialcalculationamuchsmallerv