搜索
24620126PROGRESSINCHEMISTRYVol．24No．6Jun．20122012120123*No．2097300921173005＊＊Correspondingauthore-mailh．jiang@pku．edu．cn*＊＊100871。。“”“”。。GW。、。。Kohn-ShamGWO641.12A1005-281X201206-0910-18TheBandGapProblemtheStateoftheArtofFirst-PrinciplesElectronicBandStructureTheoryJiangHong＊＊BeijingNationalLaboratoryofMolecularSciencesStateKeyLaboratoryofRareEarthMaterialsChemistryandApplicationsInstituteofTheoreticalandComputationalChemistryCollegeofChemistryandMolecularEngineeringPekingUniversityBeijing100871ChinaAbstractElectronicbandstructureisoneofthemostfundamentalpropertiesofamaterialthatplaysacrucialroleinmanyimportantapplicationsanditsaccuratedescriptionhasbeenalong-standingchallengeforthefirst-principleselectronicstructuretheory．Kohn-Shamdensity-functionaltheoryKS-DFTwithinlocaldensityorgeneralized-gradientapproximationsLDA/GGAcurrentlythe“standardmodel”forfirst-principlescomputationalmaterialssciencesuffersfromthewell-knownbandgapproblem．AlotofeffortshavebeeninvestedtoimprovethedescriptionofbandgapswithintheframeworkofKohn-ShamDFToritsgeneralizedformalisms．Ontheotherhandmany-bodyperturbationtheoryMBPTbasedonGreen'sfunctionGGFprovidesadifferentandconceptuallymorerigorousframeworkforelectronicbandstructure．ThecentralingredientoftheGF-basedMBPTistheexchange-correlationself-energy#xcwhichcanbeformallyobtainedbysolvingasetofcomplicatedintegro-differentialequationsnamedHedin'sequations．TheGWapproximationinwhich#xcissimplyaproductofGandthescreenedCoulombinteractionWiscurrentlythemostaccuratefirst-principlesapproachtodescribe6·911·electronicbandpropertiesofextendedsystems．ComparedtoLDA/GGAthecomputationaleffortsrequiredforGWcalculationsaremuchheaviersothatitsapplicationshavebeenlimitedtorelativelysmallsystems．Inthisworkwereviewthebasicprincipleslatestdevelopmentsandremainingchallengesoffirst-principleselectronicbandstructuretheoryfrombothDFTandGF-basedMBPTperspectives．Itishopedthatnewideasonfurtherdevelopmentscanbeobtainedbysettinguptheconnectionbetweenthetwodifferenttheoreticalframeworks．Keywordselectronicbandstructurethebandgapproblemdensity-functionaltheorygeneralizedKohn-ShamGreen’sfunctionmany-bodyperturbationtheoryGWapproximationContents1IntroductionElectronicbandstructuresandthebandgapproblem1.1Experimentalmeasurementsofelectronicbandstructures1.2Theoreticaltreatmentsofelectronicbandstructuresandthebandgapproblem2Electronicbandstructuresfrommean-fieldapproaches2.1Hartreetheory2.2Hartree-Focktheory3Electronicbandstructuresfromdensityfunctionaltheory3.1DensityfunctionaltheoryandKohn-Shamequations3.2Thebandgapproblemanditsorigin3.3Optimizedeffectivepotentialandrelatedmethods3.4GeneralizedKohn-Shammethods4ElectronicbandstructuresfromGreen’sfunctionbasedmany-bodyperturbationtheory4.1Green’sfunction4.2Self-energyandquasi-particleequations4.3Hedin’sequationsandGWapproximation4.4G0W0approachandself-consistency5Concludingremarks1。、。。。、、、。。。1.11。photoemissionspectroscopyPES2。。。PESinversephotoemissionspectroscopyIPS。PES/IPS。PESIPSNN－1N+1。Landauquasi-particleQP3PESIPS。。·912·24“”Landau。PESIPSfundamentalbandgap。1abPESIPS。1abcFig．1Schematicillustrationsofphotoemissionspectroscopyainversephotoemissionspectroscopybandopticalabsorptionspectroscopyc11c。opticalbandgap。PESIPS。exciton-PESIPS。SiGaAs-。。。。。。1。。1.2。IA4Eg=I－A1I≡EN－1－ENA≡EN－EN+12ENN-。。。“”local-densityapproximationLDAgeneralized-gradientapproximationGGAKohn-ShamKSdensity-functionaltheoryDFT5—7。Kohn-ShamDFT-Exc8。LDA/GGAKohn-ShamDFT9。Kohn-ShamDFT。Kohn-Sham。Kohn-Sham。SiGaAsLDA/GGAKohn-ShamDFT10。GeInNLDA/GGA。LDA/GGA。6·913·DFTKohn-ShamgeneralizedKohn-ShamGKS111213Hartree-FockLDA/GGAKohn-Sham14。、DFT。DFTone-bodyGreen’sfunction31015—17。Kohn-ShamDFT--15。-Hedin1718。Hedin。GW17GW。PESIPS。GWHamiltonianH0G0W019。LDAGGAG0W01980101620。LDA/GGADFTGW。GW1821。GW1822。、GW。DFT2324。。。2NBorn-OppenheimerHamiltonian6H^=Σi－12$2i+Vextri+Σi＜j1ri－rj≡T^+V^ext+V^ee3Vextr。mePlanckh。H^-e-eHamiltonianHamiltonian25。2.1HartreeHartree26Hartree-Fock。HartreeN-N。Ψx1x2…xN=∏iψixi4Pauli。x≡rσ。－12$2+Vextx+ViHx[]ψix=εiψixViHx≡ΣNj≠i∫vxx'ψjx'2dx'5Hamiltoniani。vxx'≡·914·241r－r'。HartreeHamiltonian。Hartree－12$2+Vextx+VHx[]ψix=εiψixVHx≡ΣNj∫vxx'ψjx'2dx'≡∫vxx'ρx'dx'6VHxρxHartree。Hartree。Hartree。———26。Hartree。26。。。2.2Hartree-FockHartreeHartree。N-NSlaterΦx1x2…xN=1N槡detψ1ψ2…ψN7Hartree-FockHF－12$2+Vextx+VHx[]ψix+∫VHFxxx'ψix'dx'=εiψix8HFVHFxxx'≡－ΣNjvxx'ψjxψ*jx'9HFEHF0=Σi∈occεi－12Σij∈occ〈ijij〉－〈ijji〉10〈ijkl〉≡∫∫ψ*ixψ*jx'ψkxψlx'r－r'dxdx'11HF。Koopmansεi=EN－E'N－1ii∈occupiedεa=E'N+1a－ENa∈unoccupied12E'N－1iiN－1HFE'N+1aaN+1HF。HF。、HF。HF。HFHF。HF。HF。SlaterconfigurationinteractionMoller-Plessetcoupled-clusterpost-HF27。“”。。33.1Kohn-ShamThomas、FermiDirac6·915·5。HFSlater20SlaterX%28HF。Hohenberg-Kohn29。KohnShamKS8。Etotρr=Σifi〈ψi|－12$2|ψi〉+∫ρxVextxdx+EHρ+Excρ13KS-HartreeEHρ≡12∫∫ρxρx'r－r'dxdx'14。KSKS－12$2+Vsx[]ψix=εiψix15Vsx≡Vextx+VHx+Vxcx16Kohn-ShamVxcx≡δExcρ/δρx17。1KSKS2KSHFKS-DFTHF3KSSlaterKSΦΨKS。Excρr≡〈Ψ|T^|Ψ〉－〈Φ|T^|Φ〉+〈Ψ|V^ee|Ψ〉－EHρr18adiabaticconnection1230。&Kohn-ShamV^λee=λV^eeVλxρr。&=1Vλ=1x=Vextx&=0Kohn-ShamVλ=0x=Vsx。Hellmann-Feynman-Excρr=∫10Uxcλρ