高等固體物理中科大5關(guān)聯(lián)課件

第五章關(guān)聯(lián)5.1單電子近似的理論基礎(chǔ)5.2費米液體理論5.3強關(guān)聯(lián)體系多電子體系(AfterBorn-Oppenheimer絕熱近似):5.1單電子近似的理論基礎(chǔ)關(guān)聯(lián)：電子－電子相互作用弱：單電子近似，電子平均場1.Hartree方程(1928)連乘積形式：按變分原理，的選取E達到極小正交歸一條件單電子方程Hartree方程中的勢：第二項是全部電子在r處形成的勢，與相抵消第三項是須扣除的自作用，與j有關(guān)，但如取r為計算原點：所以對凝膠模型，Hartree方程：相互作用→沒有相互作用電子＋正電荷背景→自由電子氣3.Hartree-Fock方程(1930)Hartree方程不滿足Pauli不相容原理電子：費米子單電子波函數(shù)f：→N電子體系的總波函數(shù)：

不涉及自旋－軌道耦合時：N電子體系能量期待值：1.第二項j,j'可以相等，自相互作用2.自相互作用嚴格相消（通過第二，三項）3.第三項為交換項，同自旋電子通過變分:么正變換:單電子方程：與Hartree方程的差別：第三項對全體電子，第四項新增，交換作用項。求和只涉及與j態(tài)自旋平行的j’態(tài)，是電子服從Fermi統(tǒng)計的反映。4.Koopmann定理（1934)單電子軌道能量等于N電子體系從第j個軌道上取走一個電子并保持N－1個電子狀態(tài)不不變的總能變化值。定性討論：假設(shè)Fermihole:與某電子自旋相同的其余鄰近電子在圍繞該電子形成總量為1的密度虧欠域energyasafunctionoftheoneelectrondensity,nuclear-electronattraction,electron-electronrepulsionThomas-FermiapproximationforthekineticenergySlaterapproximationfortheexchangeenergy6.密度泛函理論(Densityfunctionaltheory)

(1)Thomas-Fermi-DiracModel(2)TheHohenberg-KohnTheorem

propertiesareuniquelydeterminedbytheground-stateelectron

In1964,HohenbergandKohnprovedthatmolecularenergy,wavefunction

andallothermolecularelectronic

probabilitydensity

namely,Phys.Rev.136,13864(1964)

.”Densityfunctionaltheory(DFT)attemptstoandotherground-statemolecularproperties

fromtheground-stateelectrondensity

“Formoleculeswitha

nondegenerate

groundstate,theground-state

calculate

Nowweneedtoprovethattheground-stateelectronprobabilitydensitythenumberofelectrons.

theexternalpotential(exceptforanarbitraryadditiveconstant)

a)Sincedeterminesthenumberofelectrons.b)Toseethatdeterminestheexternalpotential,wesupposethatthisisfalseandthattherearetwoexternalpotentialsand(differingbymorethanaconstant)thateachgiveriseto

thesameground-stateelectrondensity.determinestheexactground-statewavefunctionandenergyoftheexactground-statewavefunctionandenergyofLetSinceanddifferbymorethanaconstant,andmustbe

differentfunctions.Proof:Assumethusthuswhichcontradictsthegiveninformation.function,theexactground-statewavefunction

stateenergy

foragivenHamiltonianIfthegroundstateisnondegenerate,thenthereisonlyonenormalizedthatgivestheexactgroundLetbeafunctionofthespatialcoordinatesofelectroni,thenUsingtheaboveresult,wegetSimilarly,ifwegothroughthesamereasoning

withaandbinterchanged,wegetByhypothesis,thetwodifferentwavefunctionsgivethesameelectron.Puttingandaddingtheabovetwoinequalitiesdensity:

yieldpotentialscouldproducethesameground-stateelectrondensitymustbefalse.

energy)

andalsodeterminesthenumberofelectrons.Thisresultisfalse,soourinitialassumptionthattwodifferentexternalpotential(towithinanadditiveconstantthat

simplyaffectsthezerolevel

ofHence,the

ground-stateelectronprobabilitydensity

determinestheexternalprobabilitydensityandotherproperties”emphasizesthedependenceoftheexternalpotential

differs

fordifferentmolecules.“Forsystemswithanondegenerategroundstate,theground-stateelectrondeterminestheground-statewavefunctionandenergy,,whichHowever,thefunctionalsareunknown.isalsowrittenasThefunctionalindependentoftheexternalonispotential.withHamiltonian.AccordingtothevariationtheoremLetususethewavefunctionasatrialvariationfunctionforthe

moleculeSincethelefthandsideofthisinequalitycanberewrittenasOnegetsstates.Subsequently,Levyprovedthetheoremsfordegenerategroundstates.

HohenbergandKohnprovedtheirtheoremsonlyfornondegenerateground(4)TheKohn-Shammethod

Ifweknowtheground-stateelectrondensity

molecularpropertiesfromfunction.,theHohenberg-Kohntheoremtellsusthatitispossibleinprincipletocalculatealltheground-state,withouthavingtofindthemolecularwave

1965,KohnandShamdevisedapracticalmethodforfinding

andforfinding

from.[Phys.Rev.,140,A1133(1965)].Theirmethod

iscapable,inprinciple,ofyieldingexactresults,butbecausetheequationsof

theKohn-Sham(KS)methodcontainanunknownfunctionalthatmustbeapproximated,theKSformationofDFTyield

approximateresults.沈呂九electronsthateachexperiencethesameexternalpotential

theground-stateelectronprobabilitydensity

equaltotheexactofthemoleculeweareinterestedin:.KohnandShamconsideredafictitiousreferencesystemsofnnoninteractingthatmakesofthereferencesystemSincetheelectronsdonot

interactwithoneanotherinthereferencesystem,theHamiltonianofthereferencesystemiswhereistheone-electronKohn-ShamHamiltonian.

RememberthatWiththeabovedefinitions,

canbewrittenasDefinetheexchange-correlationenergyfunctionalbyNowwehaveside

are

easytoevaluatefromgetagoodapproximationto

totheground-stateenergy.

Thefourthquantity

accurately.

ThekeytoaccurateKSDFT

calculationofmolecular

propertiesisto

Thefirstthreetermsontherightisarelativelysmallterm,butisnoteasytoevaluate

andtheymakethe

maincontributionsThusbecomes.Nowweneedexplicitequationstofindtheground-stateelectrondensity.sameelectrondensityasthatinthegroundstateofthemolecule:isreadilyprovedthatSincethefictitioussystemofnoninteractingelectronsisdefinedtohavethe,it(6)Variousapproximatefunctionals

DFcalculations.Thefunctionalandacorrelation-energyfunctionalAmongvariousCommonlyusedandPW91(PerdewandWang’s1991functional)Lee-Yang-Parr(LYP)functionalareusedinmolecularapproximations,gradient-corrected

exchangeandcorrelationenergyfunctionalsarethemostaccurate.PW86(PerdewandWang’s1986functional)B88(Becke’s1988functional)P86(the

Perdew1986correlationfunctional)

(7)NowadaysKSDFTmethodsaregenerallybelievedtobebetterthantheHFmethod,andinmostcasestheyareevenbetterthanMP2

iswrittenasthesumofanexchange-energyfunctional

XLocalexchangeApproximatedensityfunctionaltheoriesforexchangeandcorrelationX:

LocalexchangefunctionalofthehomogeneouselectrongasLDALocalexchange+localcorrelationGGALocalexchange+localcorrelation+gradientcorrections3rdGenerationoffunctionalsLDA:Localexchangefunctional+localcorrelationfunctionalofthehomogeneouselectrongasGGA:SameasLDA+“non-local”gradientcorrectionstoexchangeandcorrelation3rdGenerationoffunctionals:SameasGGA+instilationof“exact-exchange”and+2ndderivativesofthedensitycorrectionsTermsinDensityFunctionalsr Localdensityrs Seitzradius=(3/4pr)1/3kF Fermiwavenumber=(3p2r)1/3t Densitygradient=|gradr|/2fksrz Spinpolarization=(rup-rdown)/rf Spinscalingfactor=[(1+z)2/3+(1-z)2/3]/2ks

Thomas-Fermiscreeningwavenumber =(4kF/pa0)1/2s Anotherdensitygradient=|gradr|/2kFrJ.Chem.Phys.,100,1290(1994);PRL77,3865(1996).LocalDensityApproximationLocalSpinDensityApproximationLocalSpinDensityCorrelationFunctionalNotforthefaintofheart:GeneralizedGradientApproximationFunctionalsTheNobelPrizeinChemistry1998“forhisdevelopmentofthedensity-functionaltheory"WalterKohn(1923-)5.2費米液體理論費米體系費米溫度：均勻的無相互作用的三維系統(tǒng)，費米溫度：費米簡并系統(tǒng)：費米子系統(tǒng)的溫度通常運運低于費米溫度

室溫下金屬中的傳導電子費米溫度給出了系統(tǒng)中元激發(fā)存在與否的標度在費米溫度以下，系統(tǒng)的性質(zhì)由數(shù)目有限的低激發(fā)態(tài)決定。有相互作用和無相互作用的簡并費米子系統(tǒng)中，低激發(fā)態(tài)的性質(zhì)具有較強的對應(yīng)性。2.費米液體金屬中電子通常是可遷移的，稱為電子氣，電子動能：電子勢能：在高密度下，電子動能為主，自由電子氣模型是較好的近似。在低密度下，電子之間的勢能或關(guān)聯(lián)變得越來越重要，電子可能由于這種關(guān)聯(lián)作用進入液相甚至晶相。較強關(guān)聯(lián)下，電子系統(tǒng)被稱為電子液體或費米液體或Luttinger液體(1D)相互作用：(1)單電子能級分布變化(勢的變化);(2)電子散射導致某一態(tài)上有限壽命(馳豫時間)3.朗道費米液體理論單電子圖象不是一個正確的出發(fā)點，但只要把電子改成準粒子或準電子，就能描述費米液體。準粒子遵從費米統(tǒng)計，準粒子數(shù)守恒，因而費米面包含的體積不發(fā)生變化。假設(shè)激發(fā)態(tài)用動量表示朗道費米液體理論的適用條件：(1).必須有可明確定義的費米面存在(2).準粒子有足夠長的壽命FermiLiquidTheorySimplePictureforFermiLiquid朗道費米液體理論是處理相互作用費米子體系的唯象理論。在相互作用不是很強時，理論對三維液體正確。二維情況下，多大程度上成立不知道。一維情況下，不成立。luttinger液體一維：低能激發(fā)為自旋為1/2的電中性自旋子和無自旋荷電為的波色子的激發(fā)。非費米液體行為：與費米液體理論預言相偏離的性質(zhì)THEPHYSICS

OFLUTTINGERLIQUIDSFERMISURFACEHASONLYTWOPOINTSfailureofLandau′sFermiliquidpictureELECTRONSFORMAHARMONICCHAINATLOWENERGIES

Coulomb+PauliinteractionTHELUTTINGERLIQUID:INTERACTINGSYSTEMOF1DELECTRONSATLOWENERGIEScollectiveexcitationsarevibrationalmodesREMARKABLEPROPERTIESAbsenceofelectron-likequasi-particles(onlycollectivebosonicexcitations)Spin-chargeseparation(spinandchargearedecoupledandpropagatewithdifferentvelocities)AbsenceofjumpdiscontinuityinthemomentumdistributionatPower-lawbehaviorofvariouscorrelationfunctionsandtransportquantities.Theexponentdependsontheelectron-electroninteractionOUTLINEWhatisaFermiliquid,andwhytheFermiliquidconceptbreaksin1DTheTomonaga-LuttingermodelTheTL-HamiltoniananditsbosonizationDiagonalizationBosonicfieldsandelectronoperatorsLocaldensityofstatesTunnelingintoaLuttingerliquidLuttingerliquidwithasingleimpurityPhysicalrealizationsofLuttingerliquidsLITERATURE

K.FlensbergLecturenotesontheone-dimensionalelectrongasandthetheoryofLuttingerliquids

J.vonDelftandH.SchoellerBosonizationforbeginnersrefermionizationforexperts,cond-mat/9805275J.VoitOne-dimensionalFermiliquids,Rep.Prog.Phys.58,977(1995)H.J.Schulz,G.CunibertiandP.PieriFermiliquidsandLuttingerliquids,cond-mat/9807366SHORTLYABOUTFERMILIQUIDSLandau1957-1959Alsocollectiveexcitationsoccur(e.g.zerosound)atfiniteenergiesLowenergyexcitationsofasystemofinteractingparticlesdescribedintermsof``quasi-particles``(single-particleexcitations)Keypoint:quasi-particleshavesamequantumnumbersasthecorrespondingnon-interactingsystem(adiabaticcontinuity)StartfromappropriatenoninteractingsystemRenormalizationofasetofparameters(e.g.effectivemass)FERMILIQUIDSIIPauliexclusionprinciple

onlystateswithinkTaroundFermisphereavailablequasiparticlestatesnearFermispherescatteronlyweaklyQUASI-PARTICLEPICTUREISAPPLICABLEIN3DEffectofCoulombinteractionistoinduceafinitelife-timet3DFERMILIQUIDSIIIcollectiveexcitations(plasmons)single-particleexcitations12340132DISPERSIONOFEXCITATIONSIN3D0nointeractingT=0FinitejumpinmomentumdistributionZZquasi-particleweightLIFETIMEOF``QUASI-PARTICLES′′scatteringoutofstatekscatteringintostatekspinscreenedCoulombinteractionenergyconservationIn3Danintegrationoverangulardependencetakescareofd-functionFermi′sgoldenruleyieldsforthelifetimetT=0LIFETIMEOF``QUASI-PARTICLES′′IIIn1Dk,k′arescalars.Integrationoverk′yieldsWhataboutthelifetimetin1D?formally,itdivergesatsmallqbutwecaninsertasmallcut-offAtsmallTi.e.,thisratiocannotbemadearbitrarilysmallasin3DBREAKDOWNOFLANDAUTHEORYIN1D12340132DISPERSIONOFEXCITATIONSIN1D

collectiveexcitationsareplasmonswith(RPA)singleparticlegaplessplasmon

COLLECTIVEAND

SINGLE-PARTICLEEXCITATIONNONDISTINCT

nolongerdivergesat(noangularintegrationoverdirectionofasin3D)THETOMONAGA-LUTTINGERMODELEXACTLYSOLVABLEMODELFORINTERACTING1DELECTRONSATLOWENERGIESDispersionrelationislinearizednear(bothcollectiveandsingle-particleexcitationshavelineardispersion)ModelbecomesexactwhenlinearizedbranchesextendfromAssumptions:OnlysmallmomentaexchangesareincludedTOMONAGA-LUTTINGERHAMILTONIANFreepart

freepartinteraction

fermionicannihilation/creationoperatorsIntroducerightmoving

k>0,andleftmovingk<0electronsTLHAMILTONIANIIInteractions

freepartinteractionbackscatteringforwardumklappforwardBOSONIZATIONBOSONIZATION:EXPRESSFERMIONICHAMILTONIANINTERMSOFBOSONICOPERATORSconstructbosonicHamiltonianwiththesamespectrun(a)(b)(c)(d)(a)and(b)havesamespectrumbutdifferentgroundstateEXCITEDSTATECANBEWRITTENINTERMSOFCHARGEEXCITATIONS,ORBOSONICELECTRON-HOLEEXCITATIONSSTEP1WHICHOPERATORSDOTHEJOB?Introducethedensityoperators(createexcitationofmomentumq)andconsidertheircommutationrelations

nearlybosonic

commutationrelationsSTEP1:PROOFConsidere.g.algebraoffermionicoperatorsoccupationoperatorSTEP2ExaminenowBOSONIZEDHAMILTONIANSTATESCREATEDBYAREEIGENSTATESOFWITHENERGY

andinteractionsSTEP2:PROOFExample:STEP3IntroducethebosonicoperatorsyieldingDIAGONALIZATIONSPIN-CHARGESEPARATIONandinteraction(satisfyingSU2symmetry)Ifweincludespin,itgetsslightlymorecomplicated...andinterestingIntroducethespinandchargedensitiesHamiltoniandecoupleintwoindependentspinandchargeparts,withexcitationspropagatingwithvelocitiesSPACEREPRESENTATIONLongwavelengthlimit(interactions)AppropriatelinearcombinationsP,qofthefieldr(x)canbedefined.ThenonefindswhereLuttingerparameterg<1repulsiveinteractionBOSONICREPRESENTATIONOFYFermionicoperatorWheree.g.Expressyintheformofabosonicdisplacementoperator

B

from

decreasesthenumberofelectronsbyonedisplacesthebosonconfigurationforthatstateBOSONIZATIONIDENTITYifac-numberUladderoperator,qbosonicLOCALDENSITYOFSTATESi)Localdensityofstatesatx=0ndensityofstatesofnon-interactingsystematT=0ii)LocaldensityofstatesattheendofaLuttingerliquidatT=0cut-offenergyG

gammafunctionMEASURINGTHELDOS

Measurementofthelocaldensityofstatessystem1system2couplingIVbytunnelingSeee.g.carbonnanotubeexperimentbyBockrathetal.Nature,397,598(1999)MEASURINGTHELDOSIItunnelingrateitojTunnelingcurrentcanbeevaluatedbyuseofFermi′sgoldenruleconstant

LLtoLLLLtometalSINGLEIMPURITYAgaintunnelingcurrentcanbeevaluatedbyuseofFermi′sgoldenrule

endtoendWeaklinkx=0However,nowistunnelingfromtheendofaLLChargedensitywaveispinnedattheimpurityPHYSICALREALIZATIONS

SemiconductingquantumwiresEdgestatesinfractionalquantumHalleffectSingle-walledmetalliccarbonnanotubesEFEnergymetallic1Dconductorwith

2linearbandsk5.3強關(guān)聯(lián)體系窄能帶現(xiàn)象金屬與絕緣體之分：(1)能帶框架下的區(qū)分：導帶導帶價帶價帶(2)無序引起的Anderson轉(zhuǎn)變：局域態(tài)擴展態(tài)局域態(tài)局域態(tài)局域態(tài)擴展態(tài)EFEF(3)電子間關(guān)聯(lián)導致的Mott金屬－絕緣體轉(zhuǎn)變(a).MnO:5個3d未滿3d帶；O2-2p是滿帶不與3d能帶重疊能帶論MnO的3d帶將具有金屬導電性實際上，MnO是絕緣體！(b).ReO3:能帶論絕緣體。實際上是金屬。(c).一些過渡金屬氧化物當溫度升高時會從絕緣體金屬f電子或d電子波函數(shù)的分布范圍是否和近鄰產(chǎn)生重疊，是電子離域還是局域化的基本判據(jù)l殼層體積與Winger-Seitz元胞體積的比值：4f最小，5f次之，3d,4d,5d…多電子態(tài)的局域化強度的順序：4f>5f>3d>4d>5d______________能帶寬度上升另外，從左往右穿過周期表，部分填充殼層的半徑逐步降低，關(guān)聯(lián)重要性增加。4f，5f元素和3d,4d,5d元素的殼層體積與Winger-Seitz元胞體積的比值YScSmith和Kmetko準周期表窄帶區(qū)域重費米子強鐵磁性超導體離域性局域性另一類窄帶現(xiàn)象：來自能帶中的近自由電子與溶在晶格中具有3d,5f或4f殼層電子的溶質(zhì)原子相互作用

Friedel與Anderson稀土元素或過渡金屬化合物中的能隙不可能僅用“電荷轉(zhuǎn)移能”、“雜化能隙”、“有效庫侖相關(guān)能”三者之一來描述，而應(yīng)該說三者同時發(fā)揮作用。稀土化合物部分存在混價“mixedvalence”?；靸r的作用導致在Fermi面附近存在非常窄的能帶(部分填充f能帶或f能級），電子可以在4f能級和離域化能帶之間轉(zhuǎn)移，對固體基態(tài)性質(zhì)產(chǎn)生顯著影響。2.窄能帶現(xiàn)象的理論模型選擇經(jīng)驗參數(shù)的模型Hamilton量方法Hubbard模型和Anderson模型TheHubbardModelFromsimplequantummechanicstomany-particleinteractioninsolids-ashortintroductionHistoricalfactsHubbardModelwasfirstintroducedbyJohnHubbardin1963.WhowasHubbard?Hewasbornin1931anddied1980.Theoreticianinsolidstatephysics,fieldofwork:Electroncorrelationinelectrongasandsmallbandsystems.HeworkedattheA.E.R.E.,Harwell,U.K.,andattheIBMResearchLabs,SanJosé,USA.Picturetakenfrom:PhysicsToday,Vol.34,No4,1981What,ingeneral,istheHM?

Hubbardmodelisaquantumtheoreticalmodelformany-particleinteractioninandwithaperiodiclatticeItisbasedonaninteractionHamitonian,sometransformationsandassumptionstobeabletotreatcertainproblems(e.g.magneticbehaviourandphasetransitions)withsolidstatetheoryQuantummechanicsBasics:Schr?dingerequation

Expectationvalues

Orthonormalityandclosurerelation

Thebra-ketnotationBasistransformation,mathematicallyAbasistransformationcanbesimplyperformed:Anequationistransformedthesameway:SingleparticleequationsParticleinapotential:

Periodicpotentials:

Solutionforweakcouplingtopotential:

BlochwaveSingleparticleequationsDispersionrelationforfreeelectrons(dashedline):DispersionrelationforBlochelectrons(quasi-free)(solidline):Theenergiesat arenolongerdegenerated.Twoeigenenergiesatthosepoints.GraphfromGerdCzycholl,?TheoretischeFestk?rperphysik“,Vieweg-VerlagSingleparticleequationsWannierstatesproduceanorthonormalbaseoflocalizedstates;atomicwavefunctionswouldalsobelocalized,buttheyarenotorthonormal.Strongerlatticepotential:couplingtolatticepointsoccurs;amodifiedBlochwaveisused,e.g.WannierstatesresultingfromtheTight-Binding-Model:ComparisonbetweenthetwonewwavefunctionsBlochwavefunctionWannierwavefunction(w-part)GraphfromGerdCzycholl,?TheoretischeFestk?rperphysik“,Vieweg-VerlagGraphfromGerdCzycholl,?TheoretischeFestk?rperphysik“,Vieweg-VerlagWavefunctionformanyparticlesWavefunctionisnotsimplytheproductofallsingleparticlewavefunctions;ParticlescannotbedifferedFermionsmustobeyPauliprincipleAnsatz:SlaterdeterminanteSecondQuantizationforFermionsCreationanddistructionoperatorscreateordestroystates:SecondQuantizationTheoperatorsfulfillthecommutatorrelation:Thisisamust,otherwiseonewoulddisturbclosurerelationandorthonormalityofwavefunctionsdescribedbysecondquantizationHamiltonianformanyparticlesSummationoverallsingleparticlesHamiltonians+interactionHamiltonian:interactionpotentialuistherepulsiveCoulombinteractionOperatorsinsecondquantizationOperatorsinsecondquantizationHamiltonianinsecondquantizationIstransformedliketheone-particleoperatorA(1)andthetwo-particleoperatorA(2)HamiltonianinsecondquantizationNow:Matrixelement mustbedetermined.Herefore,awavefunctionhastobechosen.Example:Bloch-waveComingclosertoHubbard...EvaluationofmatrixelementswithWannierwavefunctions:FinalAssumptionsNow:onlydirectneighborinteractions,restrictiontooneband.Meaningofmatrixelementst:singleparticlehoppingU:Hubbard-U,describesonsite-CoulombinteractionV:Nearest-neighbor(density)interactionX:conditionalhoppinginteractionTheHubbardModelssimpleHubbardmodelextendedHubbardmodelandanycombinationofmatrixelements...Mott-Hubbardtransition,insulating(Mott)phaseCase1:Strongcoupling,U/t>>1:Mottinsulating

stateforahalf-filledsystem.Thedensityofstates(availablestatesforaddingorremovingparticle)consitsof