LDA+U第一性原理方法

IntroductiontoLDA+Umethodandapplicationstotransition-metaloxides:

Importanceofon-siteCoulombinteractionU鄭弘泰國家理論中心(清華大學)28Aug,2003,東華大學OutlineDFT,LDA(LDA,LSDA,GGA)InsufficienciesofLDAHowtoimproveLDASelf-interactioncorrection(SIC)LDA+UmethodApplicationsofLDA+Uonvarioustransition-metaloxidesConclusionsDensityFunctionalTheory(DFT)

Hohenberg-KohnTheorem,PR136(1964)B864Theground-stateenergyofasystemofidenticalspinlessfermionsisauniquefunctionaloftheparticledensity.Thisfunctionalattainsitsminimumvaluewithrespecttovariationoftheparticledensitysubjecttothenormalizationconditionwhenthedensityhasitscorrectvalues.Localdensityapproximation(LDA)

Kohn-ShamschemePR140(1965)A1133InsufficienciesofLDAPooreigenvalues,PRB23,5048(1981)LackofderivativediscontinuityatintegerN,PRL49,1691(1982)Gapstoosmallornogap,PRB44,943(1991)Spinandorbitalmomenttoosmall,PRB44,943(1991)EspeciallyfortransitionmetaloxidesPRB23(1981)5048PRL49(1982)1691PRB44(1991)943AttemptsonimprovingLDASelf-interactioncorrection(SIC)PRB23(1981)5048,PRL65(1990)1148Hartree-Fock(HF)method,PRB48(1993)5058GWapproximation(GWA),PRB46(1992)13051,PRL74(1995)3221LDA+HubbardU(LDA+U)method,PRB44(1991)943,PRB48(1993)16929Localdensityapproximation(LDA)

Kohn-ShamschemePR140(1965)A1133Self-interactioncorrection(SIC)

PerdewandZunger,PRB23(1981)5048BasicideaofLDA+UPRB44(1991)943,PRB48(1993)169Delocalizedsandpelectrons:LDALocalizeddorfelectrons:+U usingon-sited-dCoulombinteraction(Hubbard-liketerm) UΣi≠jninj insteadofaveragedCoulombenergy UN(N－1)/2HubbardUforlocalizeddorbital:nnn+1n-1UeLDA+Uenergyfunctional:LDA+Upotential:LDA+Ueigenvalue:Foroccupiedstateni=1,Forunccupiedstateni=0,peEexact(H)=－1.0RyELDA(H)=－0.957Ry～Eexact(H)eLDA(H)=－0.538Ry<<Eexact(H)TakeHydrogenforexample:pe2Eexact(H)=－2.0RypEexact(H+)=0.0RyepeEexp(H-)=－1.0552RyeESIC(H-)=－1.0515RyELDA(H-):noboundstateU=E(H+)+E(H-)－2E(H)=0.9448RypeeLDA(H)=－0.538RyU=0.9448RyOccupied(H)state:eLDA+U(H)=eLDA(H)－U/2=－1.0104Ry～－1RyUnoccupied(H+)state:eLDA+U(H+)=eLDA(H)＋U/2=－0.0656Ry～0RyWhentouseLDA+USystemsthatLDAgivesbadresultsNarrowbandmaterials:U≧WTransition-metaloxidesLocalizedelectronsystemsStronglycorrelatedmaterialsInsulators…..HowtocalculateUandJPRB39(1989)9028Constraineddensityfunctionaltheory+Supper-cellcalculationCalculatetheenergysurfaceasafunctionoflocalchargefluctuationsMappedontoaself-consistentmean-fieldsolutionoftheHubbardmodelExtracttheCoulombinteractionparameterUfrombandstructureresultsWheretofindUandJPRB44(1991)943:3datomsPRB50(1994)16861:3d,4d,5datomsPRB58(1998)1201:3datomsPRB44(1991)13319:Fe(3d)PRB54(1996)4387:Fe(3d)PRL80(1998)4305:Cr(3d)PRB58(1998)9752:Yb(4f)NotesonusingLDA+UThemagnitudeofUisdifficulttocalculateaccurately,thedeviationcouldbeaslargeas~2eVForthesameelement,Udependsalsoontheionicityindifferentcompounds:thehigherionicity,thelargerUOnethusvariesUinthereasonablerangetoobtainbetterresultsOnemightvariesUinamuchlargerrangetoseetheeffectofU(qualitatively)→Self-consistentLDA+U(muchmoredifficult)VariousLDA+UmethodsHubbardmodelinmeanfieldapprox.(HMF)LDA+U:PRB44(1991)943(WIEN2K,LMTO)Approximateself-interactioncorrection(SIC)LDA+U:PRB48(1993)16929(WIEN2K)Aroundthemeanfield(AMF)LDA+U:PRB49(1994)14211(WIEN2K)RotationallyinvariantLDA+U:PRB52(1995)R5468(VASP4.6,LMTO)SimplifiedrotationallyinvariantLDA+U:PRB57(1998)1505(VASP4.6,LMTO)OriginalLDA+U(HMF):PRB44(1991)943LDA+U(SIC):PRB48(1993)16929LDA+U(AMF):PRB49(1994)14211RotationallyinvariantLDA+U:

PRB52(1995)R5468

SlaterintegralSimplifiedrotationallyinvariantLDA+U:PRB57(1998)1505ApplicationsofLDA+Uontransition-metaloxidesPyrochlore:Cd2Re2O7(VASP)Rutile:CrO2(FP-LMTO)Doubleperovskite:Sr2FeMoO6,Sr2FeReO6,Sr2CrWO6(FP-LMTO)Cubicinversespinel:high-temperaturemagnetite(Fe3O4,CoFe2O4,NiFe2O4)(FP-LMTO,VASP,WIEN2K)Low-temperaturecharge-orderingFe3O4(VASP,LMTO)PyrochloreCd2Re2O7Latticetype:fcc88atomsincubicunitcellSpacegroup:Fd3ma=10.219A,x=0.316Ionicmodel:Cd+2(4d10),Re+5(5d2),O-2(2p6)

U(Cd)=5.5eV,U(Re)=3.0eVJ=0eV(nospinmoment)PyrochlorestructurePRB66(2002)12516O-2pCd-4dRe-5d-t2gPRB66(2002)12516PyrochloreCd2Re2O7LDAunoccupiedDOSagreewellwithXASdatafromK.D.Tsuei,SRRCCd-4dbandfromLDAis~3eVhigherthanphotoemissionspectrum(PRB66(2002)1251)Cd-4dbandfromLDA+Uagreewellwithphotoemissionspectrum(PRB66(2002)1251)Cd-4dorbitalisclosetolocalizedelectronpicture,whereastheotherorbitalsaremoreorlessitinerantRutileCrO2Half-metal,moment=2μBLatticetype:bct6atomsinbctunitcellSpacegroup:P42/mnma=4.419A,c=2.912A,u=0.303Ionicmodel:Cr+4(3d2),O-2(2p6)

U=3.0eV,J=0.87eVRutilestructurePRB56(1997)15509SpinandorbitalmagneticmomentsofCrO2(uB)SpinmomentOrbitalmomentCrOTotalCrOLDA1.89-0.0422.00-0.037-0.0011LDA+U1.99-0.0792.00-0.051-0.0025Exp.2-0.05*-0.003**D.J.Huangetal,SRRCRutileCrO2LDA+UenhancesthegapandtheexchangesplittingattheFermilevelLDA+UalsogiveslargerspinandorbitalmagneticmomentU≦W,orbitalmomentquenched,→strongerhybridization,strongercrystalfield,→closetoitinerantpictureDoubleperovskites:Sr2FeMoO6,Sr2FeReO6,Sr2CrWO6Half-metal,moment=4,3,2μBLatticetype:tet,fcc,fcc40atomsintet,fcc,fccunitcellSpacegroup:I4/mmm,Fm3m,Fm3ma=7.89,7.832,7.878A,c/a=1.001,1,1Ionicmodel:Fe+3(3d5),Cr+3(3d3),Mo+5(4d1),Re+5(5d2),W+5(5d1)U(Fe,Cr)=4,3eV,J(Fe,Cr)=0.89,0.87eVDoubleperovskitestructureLDALDA+USFMOSFROSCWOSFMOSFROSCWOspinmomentorbitalmomentSr2FeMoO6

GGAGGA+UFeMototal3.80-0.334.003.96-0.434.00FeMo0.0430.0320.0470.045Sr2FeReO6GGAGGA+UFeRetotal3.81-0.853.003.98-0.963.00FeRe0.0700.230.0660.27Sr2CrWO6GGAGGA+UCrWtotal2.30-0.332.002.46-0.452.00CrW-0.0070.10-0.0070.15GGAm=-2-10+1+2Fe3d↑Fe3d↓Re5d↑

Re5d↓0.950.920.980.920.950.170.200.170.200.200.270.210.280.200.22

0.350.420.260.580.43GGA+Um=-2-10+1+2Fe3d↑Fe3d↓Re5d↑

Re5d↓0.960.930.980.940.960.150.170.160.160.180.260.200.270.190.21

0.360.430.260.610.45OccupationnumberofdorbitalinSr2FeReO6GGAm=-2-10+1+2Cr3d↑Cr3d↓W5d↑W5d↓0.550.860.200.870.520.140.130.160.140.150.180.160.180.150.15

0.200.220.180.300.24GGA+Um=-2-10+1+2Cr3d↑Cr3d↓W5d↑W5d↓0.550.890.180.900.520.120.090.160.100.140.170.140.180.130.14

0.210.230.180.340.25OccupationnumberofdorbitalinSr2CrWO6Doubleperovskites:Sr2FeMoO6,Sr2FeReO6,Sr2CrWO6LDA+UhassignificanteffectsonDOS,butexperimentaldataisnotavailableOrbitalmomentof3dand4delementsareallquenchedbecauseofstrongcrystalfield5delementsexhibitlargeunquenchedorbitalmomentbecauseofstrongspin-orbitinteractionin5dorbitalsMagnetite(hightemperature):Fe3O4,CoFe2O4,NiFe2O4Half-metal,insulator,moment=4,3,2μBLatticetype:fccSpacegroup:Fd3m56atomsinfccunitcella=8.394,8.383,8.351AIonicmodel:Fe+3(3d5),Fe+2(3d6),Co+2(3d7),Ni+2(3d8)U(Fe+3,Fe+2,Co+2,Ni+2)=4.5,4.0,7.8,8.0eVJ(Fe,Co,Ni)=0.89,0.92,0.95eVSpinelstructureLDA+U,U(Fe)=4.5eV,U(Co)=7.8eVLDACoFe2O4LDA+U,U(Fe)=4.5eV,U(Ni)=8eVLDANiFe2O4Magnetite(hightemperature):Fe3O4,CoFe2O4,NiFe2O4LDA+UgivesbetterDOSforFe3O4LDA+UgivesinsulatinggroundstatesforCoFe2O4andNiFe2O4SpinmomentofFe3O4fromLDA+UagreesbetterwithexperimentalvalueOn-sit