高等固体物理中科大5关联课件

第五章关联5.1单电子近似的理论基础5.2费米液体理论5.3强关联体系多电子体系(AfterBorn-Oppenheimer绝热近似):5.1单电子近似的理论基础关联：电子－电子相互作用弱：单电子近似，电子平均场1.Hartree方程(1928)连乘积形式：按变分原理，的选取E达到极小正交归一条件单电子方程Hartree方程中的势：第二项是全部电子在r处形成的势，与相抵消第三项是须扣除的自作用，与j有关，但如取r为计算原点：所以对凝胶模型，Hartree方程：相互作用→没有相互作用电子＋正电荷背景→自由电子气3.Hartree-Fock方程(1930)Hartree方程不满足Pauli不相容原理电子：费米子单电子波函数f：→N电子体系的总波函数：

不涉及自旋－轨道耦合时：N电子体系能量期待值：1.第二项j,j'可以相等，自相互作用2.自相互作用严格相消（通过第二，三项）3.第三项为交换项，同自旋电子通过变分:么正变换:单电子方程：与Hartree方程的差别：第三项对全体电子，第四项新增，交换作用项。求和只涉及与j态自旋平行的j’态，是电子服从Fermi统计的反映。4.Koopmann定理（1934)单电子轨道能量等于N电子体系从第j个轨道上取走一个电子并保持N－1个电子状态不不变的总能变化值。定性讨论：假设Fermihole:与某电子自旋相同的其余邻近电子在围绕该电子形成总量为1的密度亏欠域energyasafunctionoftheoneelectrondensity,nuclear-electronattraction,electron-electronrepulsionThomas-FermiapproximationforthekineticenergySlaterapproximationfortheexchangeenergy6.密度泛函理论(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.沈吕九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费米液体理论费米体系费米温度：均匀的无相互作用的三维系统，费米温度：费米简并系统：费米子系统的温度通常运运低于费米温度

室温下金属中的传导电子费米温度给出了系统中元激发存在与否的标度在费米温度以下，系统的性质由数目有限的低激发态决定。有相互作用和无相互作用的简并费米子系统中，低激发态的性质具有较强的对应性。2.费米液体金属中电子通常是可迁移的，称为电子气，电子动能：电子势能：在高密度下，电子动能为主，自由电子气模型是较好的近似。在低密度下，电子之间的势能或关联变得越来越重要，电子可能由于这种关联作用进入液相甚至晶相。较强关联下，电子系统被称为电子液体或费米液体或Luttinger液体(1D)相互作用：(1)单电子能级分布变化(势的变化);(2)电子散射导致某一态上有限寿命(驰豫时间)3.朗道费米液体理论单电子图象不是一个正确的出发点，但只要把电子改成准粒子或准电子，就能描述费米液体。准粒子遵从费米统计，准粒子数守恒，因而费米面包含的体积不发生变化。假设激发态用动量表示朗道费米液体理论的适用条件：(1).必须有可明确定义的费米面存在(2).准粒子有足够长的寿命FermiLiquidTheorySimplePictureforFermiLiquid朗道费米液体理论是处理相互作用费米子体系的唯象理论。在相互作用不是很强时，理论对三维液体正确。二维情况下，多大程度上成立不知道。一维情况下，不成立。luttinger液体一维：低能激发为自旋为1/2的电中性自旋子和无自旋荷电为的波色子的激发。非费米液体行为：与费米液体理论预言相偏离的性质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强关联体系窄能带现象金属与绝缘体之分：(1)能带框架下的区分：导带导带价带价带(2)无序引起的Anderson转变：局域态扩展态局域态局域态局域态扩展态EFEF(3)电子间关联导致的Mott金属－绝缘体转变(a).MnO:5个3d未满3d带；O2-2p是满带不与3d能带重叠能带论MnO的3d带将具有金属导电性实际上，MnO是绝缘体！(b).ReO3:能带论绝缘体。实际上是金属。(c).一些过渡金属氧化物当温度升高时会从绝缘体金属f电子或d电子波函数的分布范围是否和近邻产生重叠，是电子离域还是局域化的基本判据l壳层体积与Winger-Seitz元胞体积的比值：4f最小，5f次之，3d,4d,5d…多电子态的局域化强度的顺序：4f>5f>3d>4d>5d______________能带宽度上升另外，从左往右穿过周期表，部分填充壳层的半径逐步降低，关联重要性增加。4f，5f元素和3d,4d,5d元素的壳层体积与Winger-Seitz元胞体积的比值YScSmith和Kmetko准周期表窄带区域重费米子强铁磁性超导体离域性局域性另一类窄带现象：来自能带中的近自由电子与溶在晶格中具有3d,5f或4f壳层电子的溶质原子相互作用

Friedel与Anderson稀土元素或过渡金属化合物中的能隙不可能仅用“电荷转移能”、“杂化能隙”、“有效库仑相关能”三者之一来描述，而应该说三者同时发挥作用。稀土化合物部分存在混价“mixedvalence”。混价的作用导致在Fermi面附近存在非常窄的能带(部分填充f能带或f能级），电子可以在4f能级和离域化能带之间转移，对固体基态性质产生显著影响。2.窄能带现象的理论模型选择经验参数的模型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:Schrödingerequation

Expectationvalues

Orthonormalityandclosurerelation

Thebra-ketnotationBasistransformation,mathematicallyAbasistransformationcanbesimplyperformed:Anequationistransformedthesameway:SingleparticleequationsParticleinapotential:

Periodicpotentials:

Solutionforweakcouplingtopotential:

BlochwaveSingleparticleequationsDispersionrelationforfreeelectrons(dashedline):DispersionrelationforBlochelectrons(quasi-free)(solidline):Theenergiesat arenolongerdegenerated.Twoeigenenergiesatthosepoints.GraphfromGerdCzycholl,„TheoretischeFestkörperphysik“,Vieweg-VerlagSingleparticleequationsWannierstatesproduceanorthonormalbaseoflocalizedstates;atomicwavefunctionswouldalsobelocalized,buttheyarenotorthonormal.Strongerlatticepotential:couplingtolatticepointsoccurs;amodifiedBlochwaveisused,e.g.WannierstatesresultingfromtheTight-Binding-Model:ComparisonbetweenthetwonewwavefunctionsBlochwavefunctionWannierwavefunction(w-part)GraphfromGerdCzycholl,„TheoretischeFestkörperphysik“,Vieweg-VerlagGraphfromGerdCzycholl,„TheoretischeFestkörperphysik“,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