A new approach to the modelling of local defects in crystals

This article is concerned with the derivation and the mathematical study of a new mean-field model for the description of interacting electrons in crystals with local defects. We work with a reduced Hartree-Fock model, obtained from the usual Hartree-Fock

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002 rpA 71 3v170207/0hp-htam:viXraANEWAPPROACHTOTHEMODELINGOFLOCALDEFECTS

INCRYSTALS:

THEREDUCEDHARTREE-FOCKCASE

ERIC´CANCES,`AMELIE´DELEURENCE,ANDMATHIEULEWIN

Abstract.Thisarticleisconcernedwiththederivationandthemathematicalstudyofanewmean- eldmodelforthedescriptionofinteractingelectronsincrystalswithlocaldefects.WeworkwithareducedHartree-Fockmodel,obtainedfromtheusualHartree-Fockmodelbyneglectingtheexchangeterm.First,werecallthede nitionoftheself-consistentFermiseaoftheperfectcrystal,whichisobtainedasaminimizerofsomeperiodicproblem,aswasshownbyCatto,LeBrisandLions.Wealsoprovesomeofitspropertieswhichwerenotmentionedbefore.

Then,wede neandstudyindetailanonlinearmodelfortheelectronsofthecrystalinthepresenceofadefect.WeuseformalanalogiesbetweentheFermiseaofaperturbedcrystalandtheDiracseainQuantumElectrodynam-icsinthepresenceofanexternalelectrostatic eld.ThelatterwasrecentlystudiedbyHainzl,Lewin,S´er´eandSolovej,basedonideasfromChaixandIracane.Thisenablesustode nethegroundstateoftheself-consistentFermiseainthepresenceofadefect.

Weendthepaperbyprovingthatourmodelisinfactthethermodynamiclimitoftheso-calledsupercellmodel,widelyusedinnumericalsimulations.

Describingtheelectronicstateofcrystalswithlocaldefectsisamajorissueinsolid-statephysics,materialsscienceandnano-electronics[25,17,33].

Inthisarticle,wedevelopatheorybasedonformalanalogiesbetweentheFermiseaofaperturbedcrystalandthepolarizedDiracseainQuantumElectrodynam-icsinthepresenceofanexternalelectrostatic eld.Recently,thelattermodelwasextensivelystudiedbyHainzl,Lewin,S´er´eandSolovejintheHartree-Fockapprox-imation[10,11,13,12],basedonideasfromChaixandIracane[6](seealso[7,1]).Thiswassummarizedinthereview[14].Usingandadaptingthesemethods,weareabletoproposeanewmathematicalapproachfortheself-consistentdescriptionofacrystalinthepresenceoflocaldefects.

WefocusinthisarticleonthereducedHartree-Fock(rHF)modelinwhichtheso-calledexchangetermisneglected.Tofurthersimplifythemathematicalformulas,wedonotexplicitlytakethespinvariableintoaccountandweassumethatthehostcrystaliscubicwithasingleatomofchargeZperunitcell.Theargumentsbelowcanbeeasilyextendedtothegeneralcase.

Inthewholepaper,themainobjectofinterestwillbetheso-calleddensitymatrixoftheelectrons.Thisisaself-adjointoperator0≤γ≤1actingontheone-bodyspaceL2(R3).Whenγhasa niterank,itmodelsa nitenumberof

electrons.Intheperiodiccase,thegroundstatedensitymatrixγ0

describesin nitelymanyelectrons)andcommuteswithperhasanin niterank(itthetranslationsofthelattice.Wewillseeinthesequelthatthegroundstatedensitymatrixofa

crystalwithalocaldefectcanbewrittenasγ=γ0

per+Q,whereQisacompact

perturbationoftheperiodicdensitymatrixγ0

perofthereferenceperfectcrystal.

This article is concerned with the derivation and the mathematical study of a new mean-field model for the description of interacting electrons in crystals with local defects. We work with a reduced Hartree-Fock model, obtained from the usual Hartree-Fock

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`A.DELEURENCE,ANDM.LEWINE.CANCES,

Ineachoftheabovethreecases( nitenumberofelectrons,perfectcrystal,

defectivecrystal),thegroundstatedensitymatrixcanbeobtainedbyminimizingsomenonlinearenergyfunctionaldependingonasetofadmissibledensitymatrices.Inthecaseofacrystalwithalocaldefect,theperturbationQisaminimizerofsomenonlinearminimizationproblemsetinthewholespaceR3,withapossiblelackofcompactnessatin nity.ThemainunusualfeaturecomparedtostandardvariationalproblemsisthatQisaself-adjointoperatorofin niterank.Thiswasalreadythecasein[10,11,12,13].

Thepaperisorganizedasfollows.InSection1,werecall

thede nitionofthereducedHartree-Fockmodelfora nitenumberofelectrons,whichservesasabasisforthetheoriesofin nitelymanyelectronsina(possiblyperturbed)periodicnucleardistribution.Section2isdevotedtothede nitionofthemodelforthein niteperiodiccrystal,followingmainly[4,5](butweprovidesomeadditionalmaterialcomparedtowhatwasdonein[4,5]).InSection3,wede neamodelforthecrystalwithlocaldefectswhichtakestheperfectcrystalasreference.InSection4,weprovethatthismodelisthethermodynamiclimitofthesupercellmodel.

Fortheconvenienceofthereader,wehavegatheredalltheproofsinSection5.Often,theproofsfollowthesamelinesasthosein[10,11,12,13]andweshallnotdetailidenticalarguments.Buttherearemanydi cultiesassociatedwiththeparticularmodelunderstudywhichdonotappearinpreviousworksandwhichareaddressedindetailhere.

1.ThereducedHartree-FockmodelforNelectrons

Westartbyrecallingthede nitionofthereducedHartree-Fockmodel[31]fora nitenumberofelectrons.NotethatthereducedHartree-FockmodelshouldnotbeconfusedwiththerestrictedHartree-Fockmodelcommonlyusedinnumericalsimulations(seee.g.[8]).WeconsiderasystemcontainingNnonrelativisticquan-tumelectronsandasetofnucleihavingadensityofchargeρnuc.IfforinstancethereareKnucleiofchargesz1,...,zK∈N\{0}locatedatR1,...,RK∈R3,then

ρnuc(x):=

k=1K

zkmk(x Rk),

Wehavechosenasystemofunitssuchthat =m=e=

wherem1,...,mKarepositivemeasuresonR3oftotalmassone.Point-likenuclei

wouldcorrespondtomk=δ(theDiracmeasure)butforconvenienceweshalldealwithsmearednucleiinthesequel,i.e.we assumethatforallk=1...K,mkisasmoothnonnegativefunctionsuchthatR3mk=1.Thetechnicaldi cultiesarisingwithpoint-likenucleiwillbedealtwithelsewhere.

TheenergyofthewholesysteminthereducedHartree-Fockmodelreads[31,5]

1rHF

D(ργ ρnuc,ργ ρnuc).(γ)=Tr (1.1)Eρnuc

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1

denotestheFouriertransformoff.Inthismean- eldmodel,thestateofwheref

theNelectronsisdescribedbytheone-bodydensitymatrixγ,whichisanelement

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