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arXiv:cond-mat/0201483v1[cond-mat.mes-hall]25Jan2002Time-dependentcurrentdensityfunctionaltheoryforthelinearresponseofweaklydisorderedsystemsC.A.UllrichDepartmentofPhysics,UniversityofMissouri-Rolla,Rolla,Missouri65409G.VignaleDepartmentofPhysicsandAstronomy,UniversityofMissouri-Columbia,Columbia,Missouri65211(Dated:February1,2008)Time-dependentdensityfunctionaltheory(TDFT)providesawayofcalculating,inprincipleexactly,thelinearresponseofinteractingmany-electronsystems,andthusallowsonetoobtaintheirexcitationenergies.Forextendedsystems,thereexistexcitationsofacollectivenature,suchasbulk-andsurfaceplasmonsinmetalsorintersubbandplasmonsindopedsemiconductorquantumwells.Thispaperdevelopsaquantitativelyaccuratefirst-principlesdescriptionforthefrequencyandthelinewidthofsuchexcitationsininhomogeneousweaklydisorderedsystems.Afinitelinewidthingeneralhasintrinsicandextrinsicsources.Atlowtemperaturesandoutsidetheregionwhereelectron-phononinteractionoccurs,theonlyintrinsicdampingmechanismisprovidedbyelectron-electroninteraction.ThiskindofintrinsicdampingcanbedescribedwithinTDFT,butoneneedstogobeyondtheadiabaticapproximationandincluderetardationeffects.Ithasbeenshown[G.Vignale,C.A.Ullrich,andS.Conti,Phys.Rev.Lett.79,4878(1997)]thatadensity-functionalresponsetheorythatislocalinspacebutnonlocalintimehastobeconstructedintermsofthecurrents,ratherthanthedensity.Thistheorywillbereviewedinthefirstpartofthispaper.Forquantitativelyaccuratelinewidths,extrinsicdissipationmechanisms,suchasimpuritiesordisorder,havetobeincludedintheresponsetheory.Inthesecondpartofthispaper,wediscusshowextrinsicdissipationcanbedescribedwithintheso-calledmemoryfunctionformalism.Thisformalismswillfirstbeintroducedandreviewedforhomogeneoussystems.WewillthenpresentasynthesisofTDFTwiththememoryfunctionformalismforinhomogeneoussystems,whichallowsonetoaccountsimultaneouslyforintrinsicandextrinsicdampingofcollectiveexcitations.Asanexamplewherebothsourcesofdissipationareimportantandwherehigh-qualityexperimentaldataisavailableforcomparison,wediscussintersubbandplasmonsina40nmwideGaAs/Al0.3Ga0.7Asquantumwell.PACSnumbers:71.15.Mb;71.45.Gm;73.21.Fg;78.67.DeI.INTRODUCTIONThecalculationofexcitationenergiesandlinewidthsofcollectiveexcitationsinextendedelectronicsys-temsisoneoftheoutstandingproblemsinmany-bodytheory.Time-dependentdensityfunctionaltheory(TDFT)1,2,3,4,5,6offersapowerfulandelegantapproachtothisdifficultproblem.Tosetthestageforthedevelop-mentsthataretofollow,weshallbeginthispaperwithasummaryofthekeyelementsofTDFT.LetˆH0=Xip2i2m+v0(ri)+12Xi6=jU(|ri−rj|)(1)betheHamiltonianofamany-electronsystem,whereriandpiarethecanonicalcoordinatesandmomentaoftheithelectron,misitsmass,v0(r)isastaticexternalpotential,whichincludescontributionsfromrandomlydistributedimpuritiesandothersourcesofdisorder,andU(|ri−rj|)istheCoulombinteractionpotential.Tocal-culatetheexcitationenergies,2,5oneaddstoˆH0asmalltime-dependentperturbationoftheformˆH1(t)=Zd3rv1(r,t)ˆn(r),(2)wherev1(r,t)=v1(r,ω)e−iωt+c.c.isaperiodicpoten-tial(v1≪v0)thatcoupleslinearlytothedensityoper-atorˆn(r)=Piδ(r−ri).Onethencomputesthetime-dependentdensityofthesystem,which,inthelinearapproximation,willbegivenbyn(r,t)=n0(r)+n1(r,ω)e−iωt+c.c.,(3)wheren0(r)istheground-statedensity,andn1(r,ω)islinearlyrelatedtov1(r,ω)vian1(r,ω)=Zd3rχ(r,r′,ω)v1(r′,ω).(4)Thedensity-densityresponsefunctionχ(r,r′,ω)containstheessentialinformationaboutthoseexcitedstatesofthesystemthatarecoupledtothegroundstatebytheperturbationˆH1(t).Morespecifically,inafinitesystem(atomormolecule)thisresponsefunctionhasadiscretesetofpolesontherealfrequencyaxis,correspondingtothediscreteexcitationenergiesofthesystem.Inanex-tendedsystem,thepolesmergeintoacontinuousbranchcutalongtherealaxis.However,isolatedpolescanariseinthelowerhalfofthecomplexfrequencyplane:theycorrespondtocollectiveexcitationsofthesystem,wheretheimaginarypartofthefrequencydefinesthecharac-teristiclifetimeoftheexcitation.2ThebasicTDFTstrategyforcalculatingχ(r,r′,ω)istoconstructanoninteractingsystemthathasthesameground-statedensityn0(r),andyieldsthesamedensityresponsen1(r,ω)astheinteractingsystemunderstudy.Thedynamicsofthisnoninteractingsystemiscontrolledbyaneffectivesingle-particlepotentialwhichiswrittenasthesumofthetotalexternalpotentialv0(r)+v1(r,t)plustheHartreepotentialvH(r,t)=e2Zd3r′n(r′,t)|r−r′|(5)plusaremainder,whichisknownasthe“exchange-correlation”(xc)potentialvxc(r,t).Itisnotatallobvi-ousthatsuchapotentialvxccanbeconstructed,but,ifitcan,thenRungeandGross1showedthatitisauniquefunctionalofthetime-dependentdensityuptowithinanadditivefunctionoftime.Theformofthexcpotentialdepends,ingeneral,ontheinitialstateofthesystem,butthisdependencedisappearsifoneassumes,aswedohere,thatthesystemisinitiallyinitsgroundstate.7TheeffectivenoninteractingHamiltonian(alsoknownastheKohn-ShamHamiltonian)thatyieldstheexactdensityisthengivenbyˆHKS(t)=Xip2i2m+v0(ri)+vH,0(ri)+vxc,0(ri)+Zd3r[v1(r,t)+vH,1(r,t)+vxc,1(r,t)]ˆn(r),(6)whereboththeHartreeandthexcpotentialhavebeenwrittenasthesumofstaticpartsvH,0,vxc,0associ-atedwiththegr