PhysRevB.95.245434

《PhysRevB.95.245434》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

PHYSICALREVIEWB95,245434(2017)Current-dependentpotentialfornonlocalabsorptioninquantumhydrodynamictheoryCristianCiracì*CenterforBiomolecularNanotechnologies(CBN),IstitutoItalianodiTecnologia(IIT),ViaBarsanti14,73010Arnesano(LE),Italy(Received18July2016;revisedmanuscriptreceived6June2017;published30June2017)Thequantumhydrodynamictheoryisapromisingmethodfordescribingmicroscopicdetailsofmacroscopicsystems.Thehydrodynamicequationcanbepartiallyobtainedfromasingle-particleKohn-Shamequationandimprovedbyaddingaviscoelastickinetic-exchange-correlationtensorterm,sothatbroadeningofcollectiveexcitationcanbetakenintoaccount,aswellasacorrectiontotheplasmondispersion.Theresultisanaccurateself-consistentandcomputationallyefficienthydrodynamicdescriptionofthefree-electrongas.Averyaccurateagreementwithfullquantumcalculationsisshown.DOI:10.1103/PhysRevB.95.245434I.INTRODUCTIONaccount.InconformitywiththerecentliteratureIwillrefertothismodeltoasthequantumhydrodynamictheory(QHT).PlasmonicnanosystemshaveenabledthepossibilitytoAself-consistentapproachbasedontheQHTcoupledtomacroscopicallyprobeeffectsthataregenerallyconfinedtotheMaxwellsequationswasrecentlyintroducedbyToscanomicroscopicrealm[18].Nonlocalelectronresponse[911]etal.,whoappliedittoshowsize-dependentblueshiftinsmallandquantumtunneling[1215]havebeenexperimentallynoblemetalnanowires[40].SystematiccomparisonsofQHTobservedinplasmonicsystemscharacterizedbysubnanometerwithDFTresults[41,42],however,pointedoutthatinorderdielectricgaps.Theadvancesoffabricationtechniquesallowtodescribewellbothnear-andfar-fieldfeaturesofplasmoniconetocontrolthefeaturesofsuchsystemsattheangstromsystems,onemightneedtosacrificetheself-consistencyofscale[1618].Inthiscontext,itbecomesveryimportanttothemethod[42].Aprecisepredictionofthefieldsnearthedevelopsimulationtechniquestotakeintoaccountquantumsurfaceisinfactextremelyimportantfortunnelingregimesandmicroscopicfeaturesatthescaleofbillionsofatoms.Densitynonlinearapplications.Moreover,initspresentformtheQHTfunctionaltheory(DFT)methodsaregenerallyunsuitable3doesnottakeintoaccountsize-dependentbroadeningofthebecausetheircomputationalcostgrowsasfastasO(Ne)suchplasmonicresonances,althougharecenteffortinthisdirectionthattheirreachislimitedtosystemswithfewthousands[43]suggestedtheuseofadensity-dependentdampingrate.Itofelectrons[6].Conversely,methodsbasedoneffectiveisworthnotingthatalthoughtheexacthydrodynamicformofdescriptionshavealsobeenproposed[19,20],althoughtheirthemany-bodytime-dependentSchrödingerequationhasbeenapplicabilitydependsonaprioricalculationsusuallyrelyingknown(inanonclosedform)foralongtime[44],itsactualonadifferentmethodanditislimitedtothelinearresponseimplementationhasbeenverydifficultduetothefactthattheregime.functionaldependenceofthestresstensoronthedensityandApromisingalternativeisgivenbyorbital-freetechniquesthecurrentsisstillunknown.Asforallorbital-freemethods,[2123]whereelectronenergyfunctionalsareexpressedinthechallengeistofindanappropriateapproximatefunctionaltermsoftheelectrondensity,n(r,t),ratherthanthesinglethatcorrectlyaccountsfortheelectrongastotalenergy.Inelectronicorbitals.Tothiscategorybelongsthehydrodynamicrecentyears,effortsinthisdirectionhaveincreased[4550].theory(HT),inwhichthequantumdynamicsissolvedviaInthispaper,IfirstshowthattheQHTequationinthemacroscopicobservablequantities,suchasnandthecurrentlimitofTF-vWapproximationcanbeformallyderivedfromdensityJ(r,t)orthevelocityv(r,t).TheHThasalonghistorythesingle-particleKohn-Sham(KS)equation.Thisderivation[2426]andithasbeenappliedtoavarietyofproblemsclearlyshowsthedegreeofapproximationsthataremade[2730]includingabsorptionofmetallicnanoparticles(NPs)whenusingtheQHT.Asthesecondstep,acorrectionofthe[31,32]andnonlinearoptics[3336].formofakineticviscoustensorisintroducedempiricallyinRecently,theHThasvigorouslyreemergedinthecontextofordertotakeintoaccountLandau-dampingeffects.Moreover,nanoplasmonics[3739],stronglyfueledbytheproliferationbecausetheQHTintrinsicallydescribesbothlongitudinalofself-assemblingcolloidalplasmonicstructures[9,14,16andtransversefields,itisthenpossibletoincludeinthe18].Becauseofthecomplexityofthesystemsinvolved,Hamiltonianacurrent-dependentexchange-correlation(XC)however,theHThasusuallybeenconsideredwithinthelimitvectorpotential.SuchpotentialhasbeendevelopedbyVignaleoftheThomas-Fermi(TF)approximationwiththeassumptionandKohn[51]inthecontextofthecurrentdensityfunctionalofaconstantgrounddensity,neglectingessentialeffectssuchtheory(CDFT).Thepeculiarityofthisfunctionalisthatitaselectronspill-outandquantumtunneling.TheTF-HT,hastheformofadivergenceofaviscoelasticstresstensor.however,canbegreatlyimprovedbyaddinga∇n-dependentTheresultisaself-consistenttheorythatcanbeappliedtocontributionthevonWeizsäcker(vW)correctiontothemuchlargerscaleproblemscomparedtoDFTtechniques,TFkineticenergyofthefree-electrongas.Inthisway,awithcomparableaccuracy.Thetheorycorrectlypredictsspace-dependentgrounddensitycanbeeasilytakenintosize-dependentplasmonenergiesandbroadening,aswellasnear-fieldproperties.ApplicationtoNPdimersalsoshowsgoodagreementwithDFTcalculationspreviouslypublished,*cristian.ciraci@iit.itdowntothetunnelingregime.2469-9950/2017/95(24)/245434(10)245434-1©2017AmericanPhysicalSociety

1CRISTIANCIRACÌPHYSICALREVIEWB95,245434(2017)II.QUANTUMHYDRODYNAMICEQUATIONsothatwehaveφj=φ+φj.Analogously,forthevelocitiesTheHTisformallyexactsolelyforasingle-ortwo-particlewehavevj=v+v˜j,wherev=J/(−en)−eA/m(notethatbyconstructionφ2v˜=0).Afterthenewdefinitions,systeminwhichtheparticleslayinoneidenticalstate,j∈occjjEq.(3)canbewrittenasandinthiscaseitsequationscanbeeasilyderivedfromSchrödingersequation[24].Suchaprocedurecannotbe∂Jne2eappliedformany-particlesystems,andinthiscasetheHT=E−J×(B+∇×AXC)∂tmmequationsareusuallyapproximatedeitherbyderivingthefirstne∂AXCneδTWNmomentsofthecollisionlessBoltzmannequations[5254]+∇vXC−e+∇m∂tmδnorassumingacertainexpressionforthetotalenergyofthesysteminavariationalformulation[29,40,55].Here,wewould1JJe+∇·J+J·∇+∇·,(5)liketoderivetheQHTequationsfromthesingle-particleKSennmequation.Inordertodoso,letusconsiderasystemofNeδTh¯2∇n·∇n∇2nwhereW=(−2)isthevWkineticpotential,noninteractingparticlesinthepresenceofanelectromagneticδn8mn2nfieldgeneratedbythescalarandvectorpotentialsve(r,t)andtheremainingpartofthemomentumfluxtensorisandAm(r,t),andinthepresenceoftheXCpotentialenergyh¯2∂φ∂φ∂φ∂φ∂φ∂φjjjjvXC(r,t)andvectorpotentialAXC(r,t).Thesystemisdescribedμν=++m∂rμ∂rν∂rμ∂rν∂rμ∂rνbyasetoftime-dependentKSequationsforthesingleorbitalsj∈occϕj(r,t),j=1...Ne:2+mφjvμ,jvν,j.(6)∂ϕ(ih¯∇−eA)2j∈occjih¯=−eve+vXCϕj,(1)∂t2mItisworthnotingthatnoapproximationshavebeenmadeuptothispoint.Inparticular,thevWpotentialhasbeenexactlywhere¯histhereducedPlanckconstant,mandearethederivedfromEq.(1).Thisallowsonetoexactlydefinetheelectronmassandcharge(inabsolutevalue),respectively,andprefactorλthatusuallyprecedesthevWterm.ThisnumberA=Am+AXC.Theelectromagneticpotentialsarerelatedto∂Amvariesintheliteraturefrom1/9to1.ItisclearfromthetheusualelectricandmagneticfieldsasE=−−∇veand∂tpresentedderivationthattherightchoiceshouldbeλ=1asB=∇×Am,respectively.alreadysuggestedinpreviouspublications[41,42]bydirectWithoutlossofgeneralitywecanwritethecomplexiχcomparisonswithDFTresults.eigenfunctionsasϕ=φejwithφ(r,t)andχ(r,t)purelyjjjjMoreover,forthesimplecaseofN=2(singleorbital)erealfunctionsofspaceandtime[24].Ourgoalistoexpressiteasytoshowthatμν=0.InfactbecausethereisoneEq.(1)asafunctionoftheglobalmacroscopicvariablesn(r,t)√occupiedorbitalφ1=n/2,theelectronsmoveinphase,andJ(r,t),whicharedefinedashencev1=0.Equation(5)isthenanexacthydrodynamicne2descriptionofthetwo-electronsystem.Solvingthisequationn=φ2,J=−eφ2v−A,(2)jjjmwouldgivetheexactsameresultasEq.(1).Ingeneral,j∈occj∈occhowever,=0andEq.(5)cannotbesolvedwithoutμνwherethesumisperformedoveralloccupiedstatesandknowingthesingleorbitalsφj.v=h¯∇χ.MultiplyingEq.(1)byϕ∗,summingovertheOurnextstepisthenfindinganapproximationforjmjjμνoccupiedstatesandseparatingtherealpartoftheresultingthatwillallowitsevaluationwithouthavingtosolvefortheequationgives,afterusingthedefinitionofEq.(2),theorbitalsφj.Letmeanticipatethatsuchapproximationisinfollowingequation:facttheTFcontributiontothekineticenergy.Inparticular,Iwillshowthat∇·n∇δTTF,whereT=cn5/3,withδnTFTF∂Jne2e2c=h¯3(3π)2/3.=E−J×(B+∇×AXC)TFm10∂tmmSinceweareinterestedindescribingstructuresthatarene∂AXCeconstitutedbyalargenumberofelectrons,infirstapproxima-+∇vXC−e+∇·,(3)tionwecanconsidertheelectronicsystemasahomogeneousm∂tmeik·relectrongaswhoseorbitalsareϕk=V1/2,withVtheoccupiedwherethemomentumfluxtensorisgivenbyvolumeinrealspace.Itiseasytoidentify¯hk=mvj(without⎛⎞2thenetcontributioninducedbytheexternalfields),sothattheh¯⎝−δμν2∂φj∂φj⎠sumoverjbecomesasumofk:μν=∇n+22m2∂rμ∂rνj∈occ∂∂2¯h2∂μν2eemφjvμ,jvν,jkμkν,(7)+mφ2v+Av+A,(4)∂rν∂rνmV∂rνjμ,jμν,jνj∈occk∈occmmj∈occwherethefirstsuminEq.(6)iszerobecausetheamplitudeswithsubscriptsμandνspanningtheCartesiandirectionsandassociatedtotheorbitalsϕkareconstant.Note,however,thatδμνbeingtheKroneckerdelta.evenifeveryterminEq.(7)seemstobeconstant,takingtheItisusefultoextractfromthesumsinEq.(4)theknowndivergenceofthesumwillnotgiveequallyzero,because,asquantities.Todoso,letuswritethesingleorbitalsasthe√willbeclearlater,thenumberofoccupiedstateswilldependdifference,φ,withrespecttoanaverageorbitalφ=n,onthelocaldensityn(r).Sincetherearemanyoccupiedstates,jNe245434-2

2CURRENT-DEPENDENTPOTENTIALFORNONLOCAL...PHYSICALREVIEWB95,245434(2017)wecanreplaceasusualthesumbyanintegral:tensor[53]andsinceitdependsonlyontheinducedvelocityvitrepresentsadynamicalcorrectiontothekineticenergy2¯h2∂kμkνfunctional.InTokatlystheorythecorrectioninEq.(12)wasmV∂rνderivedforahomogeneouselectrongasanditislimitedtok∈occ2kFπ2πlowfrequencies;hereitisassumedthatsuchexpressionholdsh¯∂k4dkkksinθdθdφ,(8)alsoathighfrequenciestoaccountfortheextracollisionsdue3μνm4π∂rν000tothepresenceofsurfacestates.ThisassumptionissupportedV21/3bydirectcomparisontoTD-DFTresultsinSec.V.whereweuseddn=8π3dkandkF(r)=(3πn).EvaluatingItcanbeusefultoregroupthetotalforceexertedonthetheintegralsgivesvolumeelementduetothekineticenergyash¯21∂10∇·δk5=cn2/3∇n.(9)δTTF[n]δTW[n,∇n]2μνFTFF=−n∇++∇·σ(k).(13)mπ15∂rν9δnδnItiseasynowtoshowthatthelastterminthepreviousequationδTTFItisinterestingtonoticethatinthelimitofauniformelectronisexactlyequalton∇,aswasanticipated.δndensity,Eq.(13)reducestothegeneralizednonlocaltermpresentedinRef.[59],withanextracontributionproportionalIII.BEYONDTHETHOMAS-FERMIKINETICENERGYto∇2J,inadditiontotheterm∇∇·J.Themaindifference,Thederivationperformedintheprevioussectiondemon-however,isthattheauthorsofRef.[59]attributesuchstratesthattheerrorcommittedusingtheQHTissolelygivencorrectiontoasomewhatarguablediffusionmechanism,ratherbytheapproximation(9).Wecanwritethenthanamuchmoreintuitiveviscouslikebehavioroftheelectrongas.δTTF∇·=n∇+∇·,(10)δnIV.CURRENT-DEPENDENTXCPOTENTIALwithbeingatensorthataccountsforacorrectiontotheItremainsnowtogiveanexplicitexpressionfortheXCTFpotential.Thistensorcanbefurtherdecomposedinapotentials.ForthescalarpotentialvXCtheusuallocaldensitycontributionPthatdependsontheinstantaneousdensity,asapproximationisassumed[60].Thatis,weapproximatethewellasadynamicalterm:XCscalarpotentialvXCasafunctionoftheinstantaneouslocal=P(n)+(n,ω).(11)density:PisimportantinthestaticcasesinceitguaranteestheδvXC[n(r,t)]=[nεXC(n)],(14)calculationoftheexactground-statedensity,whileensuresaδnproperfrequency-dependentresponse.UnfortunatelyitisstillwhereεXC=εx+εcistheXCenergyperparticleofaamajorissuebeingabletoderivegoodexpressionsforPandhomogeneouselectrongaswithdensitynandtheexchangewithoutknowingtheexactorbitals[5658].InthispaperIandcorrelationenergiesaredefined,respectively,as[38]willfocusonthedynamicalcomponentandconsiderP=0forsimplicity.1/333Animportanteffectduetoisthebroadeningoftheεx(n)=−(Eha0)n,4πplasmonicresonanceinboundedsystems,suchasmetallicnanoparticles.Inparticular,thebroadeningistheresultofacln(rs)+bc+ccrsln(rs)+dcrs,rs<1excitingsingleparticlesintoelectron-holepairs.Infact,theεc(n)=Eh√α,r1,1+β1rs+β2rssmyriadofstatesexistingatthesurfaceofsmallparticlesprovidesabroadrangeofpossibletransitionsthattranslate(15)intoabroadeningofthecollectiveplasmonresonance.Inthe2whereE=h¯istheHartreeenergy,aistheBohrradius,HTelectronsareassumedtolayinidenticalstatesandsingle-hma200particleexcitationscannotbeexactlytakenintoaccount.31/3a0rs=(4πn)istheWigner-Seitzradius,andthecoefficientsItispossibletoassume,however,thatthebroadeningofareac=0.0311,bc=−0.048,cc=0.002,dc=−0.0116,thecollectiveresonanceisduetotheincreaseofeffectiveα=−0.1423,β1=1.0529,andβ2=0.3334.Equation(15),collisionsintheelectrongas.Thesenewcollisionsmaybeaswellasotherformulasinthispaper,areinS.I.units;thoughtintuitivelyastheresultofthedifferentphasevelocityexpressionsinatomicunits(a.u.)canalsobeeasilyobtainedbetweenelectronsindifferentstates.AnexpressionforbyconsideringthatEh=a0=m=h¯=1.withintheHThasbeenproposedbyTokatly,whoderivedTheXCpotential,however,isanintrinsicallynonlocalageneralizedHTbyexpandingthekineticequationforthefunctionalofthedensity,namely,itdoesnotadmitagradientdistributionfunctiontohigh-ordermoments[53].Tokatlyexpansioninnwithoutsacrificingsomebasicsymmetriesobtainedanexpressionoftheform[51,61].Fortunately,VignaleandKohnhaveshowninthe(k)∂vμ∂vν2contextofCDFTthatalocalgradientexpansionisstillpossibleμν=−σμν=−ηk+−δμν∇·v,(12)intermsofJ,andhaveprovidedanexplicitapproximated∂rν∂rμ3expressionfortheXCvectorpotentialAXC[51].ItwaslaterwhereηkisaphenomenologicalparameterthatisingeneralashownthattheXCvectorpotentialcanbearrangedintoamorefunctionofthedensityn.σ(k)hastheformofaviscousstressintuitiveform[61],sothatitisexpressedasthedivergenceofμν245434-3

3CRISTIANCIRACÌPHYSICALREVIEWB95,245434(2017)theviscoelasticstresstensor,namely,contributingtothelifetimeofcollectiveexcitations[6769],althoughitisusuallysmallcomparedtothecontributionof∂AXC1(xc)thekineticcomponent.Moreimportantly,Eq.(20)addsan=∇·σ,(16)elasticlikebehaviortotheelectrongas(notpresentinthe∂tenviscouslikekineticterm)thatpartiallyaffectsthepositionofthecollectiveresonance.whereσ(xc)istheclassicalviscoelasticstresstensor:V.RESULTS(xc)∂vμ∂vν2σμν=ηxc+−δμν∇·v+ζxcδμν∇·v,Beforeshowingsomeapplicationsofthemodelobtained,∂rν∂rμ3letussummarizealltheelementsintoasingleequation.By(17)combiningEq.(5)withEqs.(11)and(12)andEq.(20)weobtainwithηxcandζxcgeneralizedcomplexviscositiesthatdepend2∂JneneδGeonthedensitynandthefrequencyω,andcanberelatedto=E+∇+∇·σ(kxc)thek→0limitoftheXClongitudinalandtransversekernel∂tmmδnmfunctionsfXC,L(T)(ω,k)[61].Thecaveatisthatthekernel1JJe+∇·J+J·∇−J×(B+∇×AXC),functionsarenotknownexactly,althoughseveralinterpolationennmformulashavebeendeveloped[6265].(21)InthisworktheinterpolationproposedbyContiandVignale(CV)isused[66].LetuswritethecomplexviscositieswhereδG=δTTF+δTW+v,andσ(kxc)=σ(k)+σ(xc)isδnδnδnXCastheviscoelastickinetic-XCtensor.Equation(21)coupledtoMaxwellsequationsdescribesself-consistentlythelinearandμCV(r)nonlinearelectromagneticresponseofafree-electrongas.Aηxc(ω,r)=ηCV(r)−,iωsimilarequationcanalsobeobtainedfromEulersequationKCV(r)[33,35]wherethetotalinternalenergytermislimitedtoaζxc(ω,r)=ζCV(r)−,(18)scalarpressure(i.e.,TFapproximation)andtheviscoelasticiωtensorisneglected.Wecanidentify,infact,ontheright-handsideofEq.(21),theCoulombforceterm(∝nE),theconvectivewhereallthecoefficientsarerealquantitiesindependentterms(∝J∇·JandJ·∇J),andtheLorentzforcetermofthefrequency,whoseinterpolationformulasintheCV(∝E×B).Thesetermsareimportantinthestudyoftheapproximationarenonlinearopticalresponseofmetallicsystems[33,35,36].Inadditiontopreviouslypublishedresults,Eq.(21)introduces−3/2−1−2/3−1/3−1severalnew(nonlinear)termsthatarehiddeninthetotalηCV(r)=h¯60rs+80rs−40rs+62rsn,internalenergyexpression,whichisingeneralanonlineard2K(r)=(E)n2[nε(n)],functionoftheelectrondensityandintheviscoelasticterm,CVh2XCdnasidefromtheLorentz-liketermassociatedtotheXCvector.ζCV(r)=0,Thesetermsareexpectedtobeimportantinthenonlinearresponseofnanogapplasmonicsystems[16,70,71].c−bμ(r)=Ear−2+br−1+n,(19)Moreimportantly,theviscoelastictermcontainsaneffec-CVhssrs+20tivek-dependentmechanismfordissipatingenergy.Thisisassociatedtoanincreaseofelectron-electroncollisionsnearwitha=1(9π)2/3,b=1(3)2/3,andc=0.12.theparticlesurfacewhere,infact,thegradientofthevelocity54102πExpression(17)withthecomplexcoefficients(18)isonlyislarger.Thistermisnotonlyexpectedtostronglyimpactvalidifatime-harmonicdependenceisconsidered.Ingeneral,thespectralwidthoflinearresponseresonancesbutalsothehowever,thetensor(17)canbedecomposedintoaviscousandopticalnonlinearsignalsgeneratedatthemetalsurface.anelastictermsothatwehaveItisworthnotingthatwhiletheXCcomponentoftheviscoelastictensorwasexplicitlygiveninEqs.(19)and(20),thecoefficientηkassociatedwiththekineticviscoustensoris(xc)∂vμ∂vν2σμν=ηCV+−δμν∇·vyettobespecified.Findingarigorousexpressionforηkasa∂rν∂rμ3functionofthedensitynisaverychallengingproblemanda∂uμ∂uν2morerigorousstudyisremittedtofutureworks.Forsimplicity,+μCV∂r+∂r−3δμν∇·u+KCVδμν∇·u,inthispaper,letusassumethatηk∝ηCV.Itisfoundthatνμtheparticularchoiceofηk=14ηCVgivesagreatdegreeof(20)predictabilityintermsofnear-andfar-fieldproperties,asisshownbelow.withubeingthedisplacementvectorsuchthatu˙=v.AsafirstapplicationofEq.(21),letusconsiderthelinearEquation(20)isverysimilartoEq.(12)anddoesnotopticalresponseofsinglejelliumnanospheres.Bylinearizingaddanydegreeofcomplexitytotheproblem.TheXCEq.(21),couplingittoMaxwellsequations,andrememberingvectorpotentialprovidesafurtherdissipationmechanismthat∂P/∂t=Jandv=J/(−en),weobtaininthefrequency245434-4

4CURRENT-DEPENDENTPOTENTIALFORNONLOCAL...PHYSICALREVIEWB95,245434(2017)13.4affectedbythefinitenessofthesimulationdomainsize.With(a)(b)QHT(c)120.5TD-DFTQHTnind3.3theintroductionoftheviscosity,thesestatesarenolonger0508TD-DFT)-0.5supportedaswouldbeexpectedforjelliumspheres[39].100204060803.249121dInFig.1(b)theQHT-inducedchargedensitiescorrespond-10743.10.58nindingtoplasmonenergyarecomparedtothetime-dependent36440-0.53.0(TD)DFTresults.Althoughoscillationsappearinginthe020406080E(eV)v1.5=+F6288612.90RTD-DFTcaseinthebulkregionarenotreproduced,themain20480.52048nind1inducedpeakisverywelldescribedbytheQHT.Notethatthis02.84(eV)1502-0.5isnotnecessarilythecasefortheapproachusedinRef.[43].0204060800.52.710741−5/626384570Becauseγ∝ndivergesneartheparticlesurface,the0.52.600438ind02468186n0Diameter(nm)induceddensityresultsprematurelydampedatthesurface.040-0.52.523402040608002468InFig.1(c)theplasmonresonancesobtainedwiththeE(eV)r(bohr)Spherediameter(nm)presentQHTmodelarecomparedagainstTD-DFTresultsFIG.1.Propertiesofjelliumspheres(rs=4a.u.)ofdifferent[42],forNPdiametersDrangingfrom∼0.85to∼7.25nmsizes:(a)absorptioncross-section;(b)imaginarypartofthenormal-(Ne=8toNe=5032).ForD>3nm(Ne>398)QHTizedinducedchargedensity;(c)plasmonresonanceasafunctionofreproducesDFTplasmonenergieswithgreataccuracy,withthespherediameter;intheinset,thebroadeningoftheresonanceQHTresonancesmarkingalmostexactlythemeantrajectoryforQHT.PeakpositionsandwidthswerecalculatedbyfittingtheofDFTdata.AlsostrikingisthecomparisonofthebroadeningspectrawithaLorentzian-shapedfunction;TD-DFTdataaretakenoftheresonanceshownintheinset.ThereferencecurveinthisfromRef.[42].caseisgivenbytheknownformula[74]γ=γ0+vF/RwherevFistheFermivelocityforthehomogeneouselectrongasanddomainthefollowingsystemofequations:R=D/2.Theagreementisverygoodforallthediametersexceptthesmallestonesforwhichtheanalyticalformulaisω2∇×∇×E−E=ω2μP,notexpectedtohold.20cneδGene2−0∇+∇·σ(kxc)−(ω2+iωγ)P=0E,A.Nanoparticledimermδn1mmAnotherimportantsystemtoconsideristheNPdimer.As(22)thedistancebetweentwocloselyspacedNPsreduces,fourdifferenteffectscomesimultaneouslyintoplay[1,6,9,12]:wheren0(r)istheequilibriumdensity,μ0andcarethe(i)hybridizationoftheplasmonicmodes;(ii)nonlocalitymagneticpermeabilityandthespeedoflightinvacuum,δGcharacteroftheopticalresponse;(iii)broadeningoftherespectively,()1isthefirst-ordertermforthepotentialδnresonance,whichisintrinsicallyduetononlocalabsorption(whoseexplicitexpressionscanbefoundinRef.[42]).In(sincethesizeofthespheresremainsunchanged);and(iv)writingthesecondEq.(22)thephenomenologicaldampingtunnelingeffectsduetothebondingoftheelectrondensityrateγhasbeenintroducedinordertotakeintoaccountlossestails.TheNPdimerrepresentsthenanimportanttestoftheoccurringinthebulkregions.QHTpresentedhere.Thegroundstaten0canbecalculatedself-consistently(seeLetusconsideradimerofNaspheresofD3nm(Ne=AppendixA1fordetails)bysolvingthefollowingdifferential398)andseparatedbyadistancegthatgoesfrom2to0nm.equation[40]:ThedimerisexcitedbyaplanewavepropagatingorthogonallyδG[n]e2tothedimeraxiswhoseelectricfieldispolarizedalongz,as∇2+(n−n+)=0,(23)δn0depictedinFig.2(a).Theground-statechargedensityhasn=n00beenself-consistentlycalculatedusingEq.(23)foreachvaluewhereistheelectricpermittivityandn+isthehomogeneousofthedistanceg.ThemapofFig.2(b)showstheabsorption0iondensity.Notethatsinceσ(kxc)affectsonlythedynamicalspectrumofthedimerasafunctionofthegapsize.Astheresponse,Eq.(23)andthepropertiesofitssolutionn0remaingapshrinkstheplasmonresonanceundergoesaredshiftuptounchangedwithrespecttoRef.[42].thepoint(g0.4nm)wheretunnelingeffectskickinandThesystemofEqs.(22)andEq.(23)arenumericallytheresonancebroadensandtheshiftpushesbacktohighersolvedwithacommercialsoftwarebasedonthefinite-elementenergies.Notethatwithoutthekinetic-XCviscositytheQHTmethod,COMSOLMULTIPHYSICS[72].Inparticular,the2.5Dwouldhavepredictedanunnoticeablebroadening.technique[73]hasbeenused,whichallowsonetoefficientlyInFig.2(c)arereportedtheequilibriumchargedensityn0,computeabsorptionspectraforaxissymmetricstructures(seetheinduceddensityn1,andtheelectricfieldnormdistribution,AppendixA2).respectively,forthreecriticalsituationslabeledinthemap[inAbsorptionspectrafordifferentjelliumNa(rs=4a.u.)Fig.2(b)].TheseresultscanbedirectlycomparedtoresultsofnanospheresareshowninFig.1(a).ThefirstthingtonoticeisRef.[75]inwhichTD-DFTcalculationsforthesamejelliumthatastheparticlesizeshrinksthebroaderaretheirplasmonicNadimerarereported.Itcanbeseenthatallquantitiesareresonances.OneimportantdifferencewithpreviousQHTwellreproduced.ItisworthnotingthatinDFTthereisnoresults[42]istheabsenceofhigherenergyresonances.TheseintrinsicbroadeningmechanismforeachspectrallineandaresonancesaretheanalogofRydbergstatesforatoms.Theyphenomenologicalvalueofγ(usuallymuchlargerthantheareassociatedwithverydelocalizedstatesandarenumericallybulkvalue)hastobetakenintoaccountinordertoproducea245434-5

5CRISTIANCIRACÌPHYSICALREVIEWB95,245434(2017)4(a)(a)z(b)0.6NaIII91.2nm3I(N=10)7EekII0.42nmE(eV)20.21000.51.01.52.0g(nm)(c)IIIIIInnn000QHT-VK527(b)@1.825eV0n1n1n1454HT-TFz-direction@1.890eV0ELocal456@1.825eVEEEx-direction(d)-3@1.860eV1004(c)2g=1(a.u.)0n0-100-50050100-3104g=0.4(a.u.)20n0Im(n1)-100-50050100-3104g=0.01(a.u.)20n0-100-50050100FIG.3.PropertiesofadimerofNaspheresconstitutedbyNe=zdirection(bohr)107electronseach,placedatadistanceof2nm.In(a),extinctionspectraobtainedusingdifferentmethods.Themapsin(b)depicttheFIG.2.DimerofNaspheresconstitutedbyNe=398electronsfieldenhancementaroundthestructure,withamagnificationoftheeach.(a)Schemeofthesystem;(b)absorptionefficiencyspectraasgapregionshownforeachmethod.(c)Electricfieldnormalongafunctionoftheinterparticledistanceg;(c)near-fieldpropertiesthegapforthevariousmodels.Theimaginarypartoftheinducedcorrespondingtothepointsdepictedin(b)(g=1.0,0.4,0nm).ThechargedensityfortheQHTisalsoplotted.densitiesn0andn1areinatomicunits,while|E|isnormalizedtotheincidentfieldamplitude.(d)Equilibriumdensitytakenalongthedimeraxisatdistancesg=1.0,0.4,0.01nm.B.A“macroscopic”systemInordertoshowthefullpotentialoftheQHTmethodletusconsiderastructurewhosesizemakesitunapproachablebyDFTtechniques.LetusconsideradimerofNaspherescon-continuousspectrum.InFig.2(d)theequilibriumdensitynstitutedbyN=107electronseach(diameterof∼91.2nm)0etakenalongthedimeraxisforgapsapproachingandenteringplacedatadistanceof2nmasdepictedintheinsetofFig.3(a).thetunnelingregimeisreported.ItcanbeclearlyseenfortheTheQHTcalculationiscomparedtolocalresponseandHT-TFsmallestgapthattheresultingdensityisnotsimplyasumofcalculations.Aswouldbeexpected,theenergyofthemainthesingle-particledensitiesaswouldbeexpected.plasmonicpeakisslightlyredshiftedwithrespecttofully245434-6

6CURRENT-DEPENDENTPOTENTIALFORNONLOCAL...PHYSICALREVIEWB95,245434(2017)classicalcalculations.Onthecontrary,theHT-TFpredictsACKNOWLEDGMENTaslightblueshift.TheauthorthanksDr.FabioDellaSalaforfruitfuldiscus-Figure3(b)showsthedifferentnear-fieldfeaturesinsions.theregionnearbythegapandtherelativemaximumfieldenhancement.ItisinterestingtonoticethatwhilethelocalandHT-TFcalculationspredictthesameamountofmaximumAPPENDIX:NUMERICALIMPLEMENTATIONfieldenhancementwithrespecttotheincidentamplitude1.Ground-statedensity|E|/E0455,theQHTpredictsa∼15%strongerfieldinsideThestaticquantumhydrodynamicequationforthecalcu-thegapregion,|E|/E0=527asishighlightedinFig.3(c).lationofthegrounddensitycanbeobtainedfromEq.(21):Thisresultmightbesurprising,sincethepresenceofnonlocalorquantumeffectsisusuallyassociatedtoadetrimentaleffectδG[n]onthefieldenhancement[6,9,15].Theresultis,however,in∇−eE0=0,(A1)agreementwithpreviouslypublishedsystematicstudiesonδnn=n0closelyspacednanowires[8].Teperiketal.haveshown,inwherethesubscript0indicatesthezerofrequencydepen-fact,thatbychangingtheeffectivegapsizeitispossibletodence.Inordertobesolved,thisequationmustbecoupledtoretrievethespectralfeaturesofvariousmethods.Inparticular,Gaussslaw:forlocalorTF-FTmodelstoreproduceTD-DFTresultsfore∇·E=−(n−n+),(A2)Nacouplednanowires,itisnecessarytoreducetheeffective00ε0gapsize.Thisisinaccordwithanincreaseofthemaximumwheren+isthepositivebackgroundcharge,whichisassumedfieldssinceasmallergapwouldcorrespondtoahigherlocalfieldenhancement.Teperikandco-workersfoundthatfortobeconstantinsidethejelliumspherewhileitabruptlydropssmallwiresthevariationofthegapisoftheorderof2δ,tozerooutsidethejelliumedge.Bytakingthedivergenceofwhereδ0.9A.Althoughthiswouldseemasmallcorrection,˚Eq.(A1)andusingEq.(A2)itispossibletoobtainonesingleIhaveverifiedthattheincreaseinthefieldenhancementnonlineardifferentialequation:correspondingtothelocalmodelinwhichthegapisreducedδG[n]e2by2δgivessimilarresultsastheQHTmodelforthesystem∇2+(n−n+)=0.(A3)0considered.δnn=n0ε0ItisworthnotingthattheincreaseofthelocalfieldIsolvedthisequationusingacommerciallyavailablesoft-enhancementobservedinNajelliumspheredimersmightnotwarebasedonthefinite-elementmethod(FEM),COMSOLbegenerallyvalid,inparticularfornoblemetalsystems.ForMULTIPHYSICS[72].Itwasfoundthatconvergenceismoresilverandgold,infact,theeffectivescreeningchargeisinsideeasilyachievedbysolvingforthetransformedvariableξ=thejelliumedgeandtheeffectivegapsizemightresultaslarger√n0,andsinceFEMtechniquesusuallyrequireonetowritethantheactualseparationbetweenthejelliumedges[8].thedifferentialproblemintoitsweakform,bymultiplyingEq.(A3)bythetestfunctionξandintegratingovertheVI.CONCLUSIONsimulationdomainweobtainthefollowingweakform:IhavepresentedaQHTmodelthatisabletoaccuratelyandδG[ξ2]e2−∇·∇ξ+(ξ2−n+)ξdV=0,self-consistentlydescribefar-fieldandnear-fieldpropertiesofδnξ=√nε00plasmonicsystemsinmostextremeconditions.Thismodelrepresentsageneraltheorythatisalsovalidinthenonlinear(A4)regime[61]andcouldbeusedfortheinvestigationofopticalwhereitisassumedzerocontributionfromtheintegralovernonlinearsurfaceeffects.theboundariesofthedomain.IsolveEq.(A4)iterativelyAlthoughtheapproximationintroducedforthekineticusingCOMSOLsbuilt-innonlinearsolverbasedonNewtonsstresstensorlacksananalyticalmicroscopicderivation,themethod.Theinitialguessvalueforn0,inthecaseofasinglemodeloffersagreatdegreeofpredictability,eveninsituationssphere,wastakenoftheformn(r)=nbwithnthe01+exp(κr−R)bwherequantumtunnelingcannotbeneglected,anditmightbulkdensityandRtheradiusofthesphere.Forthedimercaseplayanimportantroleinthedevelopmentofthefield.IusedthesolutionforthesinglesphereandslowlyreducedFinally,Ibelievethisworkoffersavalidandefficientthegapfromg=2to0.01nm.solutionforstudyingindetailelectrondynamicsofmesoscopicstructures.ThecomputationalscalinginfactdoesnotdependonthenumberofelectronsNeinthesystem(asinDFT)but2.ThelinearresponseratheronthecomplexityoftheequilibriumdensityfunctionForthelinearresponseIsolvedthesystemofEqs.(22)inwhichisbeingdiscretized.TheQHTself-consistentdensitythemaintext.Inparticular,Iusedforthepolarizationequationresultsareconstantinthebulkregionandonlyfast-varyingatthefollowingweakexpression:themetalboundaries.Thecomputationalcostwouldthenscaleinprincipleasafunctionoftheboundaryarea,ratherthantheeδGiω∂Pμ(n0∇·P˜+∇n0·P˜)+σμνvolume,providinganextremelyefficientcomputationaltool.Vmδn1mμ,ν∂rνThisgivesaccesstoanunparalleledregimeoflight-matterne2interactions,whichinturnmightleadtonovelandunexploited2+(−ω+iωγ)P−E·P˜dV=0,(A5)effects.m245434-7

7CRISTIANCIRACÌPHYSICALREVIEWB95,245434(2017)wherePisthetestfunctionandtheviscoelasticstresstensorσasafunctionoftheequilibriumdensityn0andthepolarizationvectorPis∂Pμ∂Pν2∂Pl∂Plσμν=η+−δμν+ζδμν.(A6)∂rνn0∂rμn03∂rln0∂rln0NotethatbecausewedistributedthederivativestothetestfunctionsP˜wedonotneedtoevaluatethegradientofthefunctional(δG),northedivergenceoftheviscoelasticstresstensorσ.Explicitexpressionsfor(δG)canbefoundinRef.[42].δn1δn1Inordertotakeadvantagefromthesymmetryofthegeometry,Ihaveimplementedourequationsassuminganazimuthaldependenceoftheforme−imφwithm∈Z.Thatis,foravectorfieldV,wehaveV(ρ,φ,z)=V(m)(ρ,z)e−imφ.Maxwellsm∈Zequationandthepolarizationequationarewrittenassumingthefollowingdefinitions:1∂im∂V(m)∇·V(m)≡+V(m)−V(m)+z,ρφρ∂ρρ∂z∂V(m)∂V(m)∂V(m)V(m)∂V(m)(m)φm(m)ρzφφm(m)∇×V≡ρ−−iVz+φ−+z++iVρ,∂zρ∂z∂ρρ∂ρρ⎛∂V(m)∂V(m)(m)⎞ρφ∂Vz⎜∂ρ∂ρ∂ρ⎟(m)⎜V(m)V(m)V(m)V(m)(m)⎟∇V≡⎜−imρ−φ−imφ+ρ−imVz⎟.(A7)⎝ρρρρρ⎠∂V(m)∂V(m)(m)ρφ∂Vz∂z∂z∂zThetestfunctionsareassumedtohaveadependenceoftheformeimφsothatthederivativewithresecttoφgivesafactor+im.Itispossiblethentoreducetheinitiallythree-dimensionalprobleminto2mmax+1two-dimensionalproblems.Foreachmthesystemofequationstosolvereads2π(∇×E(m))·(∇×E˜(m))−k2E(m)+μω2P(m)·E˜(m)ρdρdz=0,00(m)eδGiω−2π(n∇·P˜(m)+∇n·P˜(m))+σ[n,P(m)]·∇P(m)000mδn1m−(ω2−iγω)P(m)+εω2E(m)+E(m)·P˜(m)ρdρdz=0,(A8)0pincForthecaseofanincidentplanewavepropagatingalongthezaxis,onehastosolvetheproblemjustform=±1.Moreoverbytakingintoaccountfieldparities,thesolutionform=1canberelatedtothesolutionform=−1,sothatasingletwo-dimensionalcalculationbecomesnecessary[4,55].Forthedimerstructure,ontheotherhand,theincidentwavepropagatesperpendicularlytothezaxis,sothattheproblemhastobesolvedform=0...mmax.Because,however,thesystemisdeeplysubwavelengthonlyfewterms(m2)giveanon-negligiblecontribution.Notethatsincetheexpressionoftheenergyfunctionalcontainssecond-orderderivatives,workingvariablesassociatedtoextraequationsmustbeintroducedsothatthesystemonlycontainsfirst-orderderivatives[42].[1]S.Raza,G.Toscano,A.P.Jauho,M.Wubs,andN.A.Mortensen,[6]J.Zuloaga,E.Prodan,andP.Nordlander,QuantumdescriptionUnusualresonancesinnanoplasmonicstructuresduetononlocaloftheplasmonresonancesofananoparticledimer,NanoLett.response,Phys.Rev.B84,121412(2011).9,887(2009).[2]A.I.Fernández-Domínguez,P.Zhang,Y.Luo,S.A.Maier,[7]W.Zhu,R.Esteban,A.G.Borisov,J.J.Baumberg,P.Nord-F.J.García-Vidal,andJ.B.Pendry,Transformation-opticslander,H.J.Lezec,J.Aizpurua,andK.B.Crozier,Quantuminsightintononlocaleffectsinseparatednanowires,Phys.Rev.mechanicaleffectsinplasmonicstructureswithsubnanometreB86,241110(R)(2012).gaps,Nat.Commun.7,11495(2016).[3]G.Toscano,S.Raza,A.-P.Jauho,N.A.Mortensen,andM.[8]T.V.Teperik,P.Nordlander,J.Aizpurua,andA.G.Borisov,Wubs,ModifiedfieldenhancementandextinctionbyplasmonicRobustSubnanometricPlasmonRulerbyRescalingofthenanowiredimersduetononlocalresponse,Opt.Express20,NonlocalOpticalResponse,Phys.Rev.Lett.110,2639014176(2012).(2013).[4]C.Ciracì,Y.A.Urzhumov,andD.R.Smith,Effectsofclas-[9]C.Ciracì,R.T.Hill,J.J.Mock,Y.A.Urzhumov,A.I.Fernandez-sicalnonlocalityontheopticalresponseofthree-dimensionalDominguez,S.A.Maier,J.B.Pendry,A.Chilkoti,andplasmonicnanodimers,J.Opt.Soc.Am.B30,2731(2013).D.R.Smith,Probingtheultimatelimitsofplasmonicenhance-[5]W.YanandN.A.Mortensen,Nonclassicaleffectsinplasmonics:ment,Science337,1072(2012).Anenergyperspectivetoquantifynonclassicaleffects,Phys.[10]C.Ciracì,X.Chen,J.J.Mock,F.McGuire,X.Liu,S.-H.Oh,Rev.B93,115439(2016).andD.R.Smith,Film-couplednanoparticlesbyatomiclayer245434-8

8CURRENT-DEPENDENTPOTENTIALFORNONLOCAL...PHYSICALREVIEWB95,245434(2017)deposition:Comparisonwithorganicspacinglayers,Appl.Phys.[29]E.ZarembaandH.C.Tso,ThomasFermiDiracvonLett.104,023109(2014).Weizsäckerhydrodynamicsinparabolicwells,Phys.Rev.B[11]S.Raza,S.Kadkhodazadeh,T.Christensen,M.DiVece,M.49,8147(1994).Wubs,N.A.Mortensen,andN.Stenger,Multipoleplasmons[30]G.Manfredi,Howtomodelquantumplasmas,arXiv:quant-andtheirdisappearanceinfew-nanometresilvernanoparticles,ph/0505004.Nat.Commun.6,8788(2015).[31]R.Ruppin,Plasmonfrequenciesofsmallmetalspheres,J.Phys.[12]K.J.Savage,M.M.Hawkeye,R.Esteban,andA.G.Borisov,Chem.Solids39,233(1978).Revealingthequantumregimeintunnelingplasmonics,Nature[32]A.BanerjeeandM.K.Harbola,Hydrodynamicalapproachto(London)491,574(2012).collectiveoscillationsinmetalclusters,Phys.Lett.A372,2881[13]J.A.Scholl,A.García-Etxarri,A.L.Koh,andJ.A.Dionne,(2008).Observationofquantumtunnelingbetweentwoplasmonic[33]J.E.Sipe,V.C.Y.So,M.Fukui,andG.I.Stegeman,Analysisnanoparticles,NanoLett.13,564(2013).ofsecond-harmonicgenerationatmetalsurfaces,Phys.Rev.B[14]L.Lin,M.Zapata,M.Xiong,Z.Liu,S.Wang,H.Xu,21,4389(1980).A.G.Borisov,H.Gu,P.Nordlander,J.Aizpurua,andJ.Ye,[34]A.ChizmeshyaandE.Zaremba,Second-harmonicgenerationNanoopticsofplasmonicnanomatryoshkas:ShrinkingthesizeatmetalsurfacesusinganextendedThomasFermivonWeiz-ofacoreshelljunctiontosubnanometer,NanoLett.15,6419sackertheory,Phys.Rev.B37,2805(1988).(2015).[35]M.Scalora,M.A.Vincenti,D.deCeglia,V.Roppo,M.Centini,[15]G.Hajisalem,M.S.Nezami,andR.Gordon,ProbingtheN.Akozbek,andM.J.Bloemer,Second-andthird-harmonicquantumtunnelinglimitofplasmonicenhancementbythirdgenerationinmetal-basedstructures,Phys.Rev.A82,043828harmonicgeneration,NanoLett.14,6651(2014).(2010).[16]J.B.Lassiter,X.Chen,X.Liu,C.Ciracì,T.B.Hoang,S.[36]C.Ciracì,E.Poutrina,M.Scalora,andD.R.Smith,Second-Larouche,S.-H.Oh,M.H.Mikkelsen,andD.R.Smith,Third-harmonicgenerationinmetallicnanoparticles:Clarificationharmonicgenerationenhancementbyfilm-coupledplasmonicoftheroleofthesurface,Phys.Rev.B86,115451striperesonators,ACSPhoton.1,1212(2014).(2012).[17]S.Kheifets,A.Simha,K.Melin,T.Li,andM.G.Raizen,[37]C.Ciracì,J.B.Pendry,andD.R.Smith,HydrodynamicmodelObservationofBrownianmotioninliquidsatshorttimes:forplasmonics:AmacroscopicapproachtoamicroscopicInstantaneousvelocityandmemoryloss,Science343,1493problem,ChemPhysChem14,1109(2013).(2014).[38]S.Raza,S.I.Bozhevolnyi,M.Wubs,andN.A.Mortensen,[18]R.Chikkaraddy,B.deNijs,F.Benz,S.J.Barrow,O.A.Nonlocalopticalresponseinmetallicnanostructures,J.Phys.:Scherman,E.Rosta,A.Demetriadou,P.Fox,O.Hess,andCondens.Matter27,183204(2015).J.J.Baumberg,Single-moleculestrongcouplingatroom[39]L.Stella,P.Zhang,F.J.García-Vidal,A.Rubio,andP.temperatureinplasmonicnanocavities,Nature(London)535,Garcia-Gonzalez,Performanceofnonlocalopticswhenapplied127(2016).toplasmonicnanostructures,J.Phys.Chem.C117,8941[19]W.Yan,M.Wubs,andN.A.Mortensen,ProjectedDipole(2013).ModelforQuantumPlasmonics,Phys.Rev.Lett.115,137403[40]G.Toscano,J.Straubel,A.Kwiatkowski,C.Rockstuhl,F.(2015).Evers,H.Xu,N.A.Mortensen,andM.Wubs,Resonance[20]R.Esteban,A.G.Borisov,P.Nordlander,andJ.Aizpurua,shiftsandspill-outeffectsinself-consistenthydrodynamicBridgingquantumandclassicalplasmonicswithaquantum-nanoplasmonics,Nat.Commun.6,7132(2015).correctedmodel,Nat.Commun.3,825(2012).[41]W.Yan,Hydrodynamictheoryforquantumplasmonics:Linear-[21]A.Domps,P.G.Reinhard,andE.Suraud,Time-Dependentresponsedynamicsoftheinhomogeneouselectrongas,Phys.Thomas-FermiApproachforElectronDynamicsinMetalRev.B91,115416(2015).Clusters,Phys.Rev.Lett.80,5520(1998).[42]C.CiracìandF.DellaSala,Quantumhydrodynamictheoryfor[22]H.Xiang,X.Zhang,D.Neuhauser,andG.Lu,Size-dependentplasmonics:Impactoftheelectrondensitytail,Phys.Rev.B93,plasmonicresonancesfromlarge-scalequantumsimulations,205405(2016).J.Phys.Chem.Lett.5,1163(2014).[43]X.Li,H.Fang,X.Weng,L.Zhang,X.Dou,A.Yang,[23]H.Xiang,M.Zhang,X.Zhang,andG.Lu,UnderstandingandX.Yuan,Electronicspill-outinducedspectralbroadeningquantumplasmonicsfromtime-dependentorbital-freedensityinquantumhydrodynamicnanoplasmonics,Opt.Express23,functionaltheory,J.Phys.Chem.C120,14330(2016).29738(2015).[24]E.Madelung,Quantentheorieinhydrodynamischerform,[44]I.V.Tokatly,Time-dependentcurrentdensityfunctionaltheoryZ.Phys.40,322(1927).viatime-dependentdeformationfunctionaltheory:Acon-[25]F.Bloch,Bremsvermögenvonatomenmitmehrerenelektronen,strainedsearchformulationinthetimedomain,Phys.Chem.Z.Phys.81,363(1933).Chem.Phys.11,4621(2009).[26]D.BohmandJ.P.Vigier,Modelofthecausalinterpretationof[45]I.V.Tokatly,Quantummany-bodydynamicsinaLagrangianquantumtheoryintermsofafluidwithirregularfluctuations,frame:I.Equationsofmotionandconservationlaws,Phys.Rev.Phys.Rev.96,208(1954).B71,165104(2005).[27]A.Eguiluz,S.Ying,andJ.Quinn,Influenceoftheelectron[46]I.V.Tokatly,Quantummany-bodydynamicsinaLagrangiandensityprofileonsurfaceplasmonsinahydrodynamicmodel,frame:II.Geometricformulationoftime-dependentdensityPhys.Rev.B11,2118(1975).functionaltheory,Phys.Rev.B71,165105(2005).[28]C.SchwartzandW.L.Schaich,Hydrodynamicmodelsof[47]I.V.Tokatly,Time-dependentdeformationfunctionaltheory,surfaceplasmons,Phys.Rev.B26,7008(1982).Phys.Rev.B75,125105(2007).245434-9

9CRISTIANCIRACÌPHYSICALREVIEWB95,245434(2017)[48]J.Tao,X.Gao,G.Vignale,andI.V.Tokatly,LinearContinuum[61]G.Vignale,C.A.Ullrich,andS.Conti,Time-DependentMechanicsforQuantumMany-BodySystems,Phys.Rev.Lett.DensityFunctionalTheoryBeyondtheAdiabaticLocalDensity103,086401(2009).Approximation,Phys.Rev.Lett.79,4878(1997).[49]X.Gao,J.Tao,G.Vignale,andI.V.Tokatly,Continuum[62]E.K.U.GrossandW.Kohn,LocalDensity-FunctionalTheorymechanicsforquantummany-bodysystems:LinearresponseofFrequency-DependentLinearResponse,Phys.Rev.Lett.55,regime,Phys.Rev.B81,195106(2010).2850(1985).[50]T.Gould,G.Jansen,I.V.Tokatly,andJ.F.Dobson,Quantum[63]N.IwamotoandE.K.U.Gross,Correlationeffectsonthethird-continuummechanicsmadesimple,J.Chem.Phys.136,204115frequency-momentsumruleofelectronliquids,Phys.Rev.B(2012).35,3003(1987).[51]G.VignaleandW.Kohn,Current-DependentExchange-[64]R.Nifosi,S.Conti,andM.P.Tosi,Dynamicexchange-CorrelationPotentialforDynamicalLinearResponseTheory,correlationpotentialsfortheelectrongasindimensionalityD=3Phys.Rev.Lett.77,2037(1996).andD=2,Phys.Rev.B58,12758(1998).[52]J.C.Nickel,J.V.Parker,andR.W.Gould,Resonance[65]Z.QianandG.Vignale,Dynamicalexchange-correlationpo-OscillationsinaHotNonuniformPlasmaColumn,Phys.Rev.tentialsforanelectronliquid,Phys.Rev.B65,235121(2002).Lett.11,183(1963).[66]S.ContiandG.Vignale,Elasticityofanelectronliquid,Phys.[53]I.TokatlyandO.Pankratov,HydrodynamictheoryofanelectronRev.B60,7966(1999).gas,Phys.Rev.B60,15550(1999).[67]P.Hessler,J.Park,andK.Burke,SeveralTheoremsinTime-[54]I.V.TokatlyandO.Pankratov,HydrodynamicsbeyondlocalDependentDensityFunctionalTheory,Phys.Rev.Lett.82,378equilibrium:Applicationtoelectrongas,Phys.Rev.B62,2759(1999).(2000).[68]R.DAgostaandG.Vignale,RelaxationinTime-Dependent[55]A.EguiluzandJ.Quinn,HydrodynamicmodelforsurfaceCurrent-Density-FunctionalTheory,Phys.Rev.Lett.96,016405plasmonsinmetalsanddegeneratesemiconductors,Phys.Rev.(2006).B14,1347(1976).[69]C.A.UllrichandG.Vignale,Collectiveintersubbandtransitions[56]D.Neuhauser,S.Pistinner,A.Coomar,X.Zhang,andG.inquantumwells:Acomparativedensity-functionalstudy,Phys.Lu,Dynamickineticenergypotentialfororbital-freedensityRev.B58,15756(1998).functionaltheory,J.Chem.Phys.134,144101(2011).[70]D.Yoo,N.-C.Nguyen,L.MartínMoreno,D.A.Mohr,[57]S.Laricchia,E.Fabiano,L.A.Constantin,andF.DellaSala,S.Carretero-Palacios,J.Shaver,J.Peraire,T.W.Ebbesen,Generalizedgradientapproximationsofthenoninteractingki-andS.-H.Oh,High-throughputfabricationofresonantmeta-neticenergyfromthesemiclassicalatomtheory:Rationalizationmaterialswithultrasmallcoaxialaperturesviaatomiclayeroftheaccuracyofthefrozendensityembeddingtheoryforlithography,NanoLett.16,2040(2016).nonbondedinteractions,J.Chem.TheoryComput.7,2439[71]X.Chen,N.C.Lindquist,D.J.Klemme,P.Nagpal,D.J.Norris,(2011).andS.-H.Oh,Split-wedgeantennaswithsub-5nmgapsfor[58]S.Laricchia,L.A.Constantin,E.Fabiano,andF.DellaplasmonicNanofocusing,NanoLett.16,7849(2016).Sala,Laplacian-levelkineticenergyapproximationsbasedon[72]COMSOLMULTIPHYSICS,http://www.comsol.com.thefourth-ordergradientexpansion:Globalassessmentand[73]C.Ciracì,Y.A.Urzhumov,andD.R.Smith,Far-fieldanalysisapplicationtothesubsystemformulationofdensityfunctionalofaxiallysymmetricthree-dimensionaldirectionalcloaks,Opt.theory,J.Chem.TheoryComput.10,164(2014).Express21,9397(2013).[59]N.A.Mortensen,S.Raza,M.Wubs,T.Søndergaard,and[74]U.KreibigandM.Vollmer,OpticalPropertiesofMetalClusters,S.I.Bozhevolnyi,Ageneralizednon-localopticalresponseSpringerSeriesinMaterialsScienceVol.25(SpringerSciencetheoryforplasmonicnanostructures,Nat.Commun.5,3809&BusinessMedia,Berlin/Heidelberg,2013).(2014).[75]M.Barbry,P.Koval,F.Marchesin,R.Esteban,A.G.Borisov,J.[60]J.P.PerdewandA.Zunger,Self-interactioncorrectiontoAizpurua,andD.Sánchez-Portal,Atomisticnear-fieldnanoplas-density-functionalapproximationsformany-electronsystems,monics:Reachingatomic-scaleresolutioninnanooptics,NanoPhys.Rev.B23,5048(1981).Lett.15,3410(2015).245434-10