高等固體物理中科大5關(guān)聯(lián)

高等固體物理中科大5關(guān)聯(lián)第1頁/共155頁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第2頁/共155頁動能原子核對電子形成的勢能其余N-1個電子對j電子的庫侖作用能自洽求解，H2,He計算與實驗相符。26個電子的Fe原子，運算要涉及1076個數(shù)，對稱簡化1053個整個太陽系沒有足夠物質(zhì)打印這個數(shù)據(jù)表！2.凝膠模型(jelliummodel)為突出探討相互作用電子系統(tǒng)的哪些特征是區(qū)別于不計其相互作用者，可人為地簡化假定電子是沉浸在空間密度持恒的正電荷背景之中(不考慮離子的周期性)。正電荷的作用體現(xiàn)于在相互作用電子體系的Hamiltonian中出現(xiàn)一個維持系統(tǒng)聚集的附加項金屬體系，設(shè)電子波函數(shù)：第3頁/共155頁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→自由電子氣第4頁/共155頁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，同自旋電子第5頁/共155頁通過變分: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。第6頁/共155頁推廣：系統(tǒng)中一個電子由狀態(tài)j轉(zhuǎn)移到態(tài)i而引起系統(tǒng)能量的變化5.交換空穴(Fermihole)

將H-F方程改寫為：其中：第7頁/共155頁定性討論：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的密度虧欠域第8頁/共155頁energyasafunctionoftheoneelectrondensity,nuclear-electronattraction,electron-electronrepulsionThomas-FermiapproximationforthekineticenergySlaterapproximationfortheexchangeenergy6.密度泛函理論(Densityfunctionaltheory)

(1)Thomas-Fermi-DiracModel第9頁/共155頁(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

第10頁/共155頁Proof:TheelectronicHamiltonianisitisproduced

bychargesexternaltothesystemofelectrons.InDFT,

iscalledtheexternalpotentialactingonelectroni,sinceOncetheexternalpotential

theelectronicwavefunctionsandallowedenergiesofthemoleculeare

andthenumberofelectronsnarespecified,

determinedasthesolutionsoftheelectronicSchr?dingerequation.

第11頁/共155頁Nowweneedtoprovethattheground-stateelectronprobabilitydensitythenumberofelectrons.

theexternalpotential(exceptforanarbitraryadditiveconstant)

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

thesameground-stateelectrondensity.determines第12頁/共155頁theexactground-statewavefunctionandenergyoftheexactground-statewavefunctionandenergyofLetSinceanddifferbymorethanaconstant,andmustbe

differentfunctions.第13頁/共155頁Proof:Assumethusthuswhichcontradictsthegiveninformation.function,theexactground-statewavefunction

stateenergy

foragivenHamiltonianIfthegroundstateisnondegenerate,thenthereisonlyonenormalizedthatgivestheexactground第14頁/共155頁Accordingtothevariationtheorem,supposethatIfthenisanynormalizedwell-behavedtrialvariationfunction.

NowuseasatrialfunctionwiththeHamiltonianthenSubstitutinggives第15頁/共155頁Letbeafunctionofthespatialcoordinatesofelectroni,thenUsingtheaboveresult,wegetSimilarly,ifwegothroughthesamereasoning

withaandbinterchanged,weget第16頁/共155頁Byhypothesis,thetwodifferentwavefunctionsgivethesameelectron.Puttingandaddingtheabovetwoinequalitiesdensity:

yieldpotentialscouldproducethesameground-stateelectrondensitymustbefalse.

energy)

andalsodeterminesthenumberofelectrons.Thisresultisfalse,soourinitialassumptionthattwodifferentexternalpotential(towithinanadditiveconstantthat

simplyaffectsthezerolevel

ofHence,the

ground-stateelectronprobabilitydensity

determinestheexternal第17頁/共155頁probabilitydensityandotherproperties”emphasizesthedependenceoftheexternalpotential

differs

fordifferentmolecules.“Forsystemswithanondegenerategroundstate,theground-stateelectrondeterminestheground-statewavefunctionandenergy,,whichHowever,thefunctionalsareunknown.isalsowrittenasThefunctionalindependentoftheexternalonispotential.第18頁/共155頁(3)TheHohenberg-kohnvariationaltheorem“Foreverytrialdensityfunctionthatsatisfiesandforall,thefollowinginequalityholds:,isthetrueground–stateenergy.”Proof:LetsatisfythatandHohenberg-Kohntheorem,determinestheexternalpotential

andthisinturndeterminesthewavefunctiondensity

.Bythe,thatcorrespondstothe

.where第19頁/共155頁withHamiltonian.AccordingtothevariationtheoremLetususethewavefunctionasatrialvariationfunctionforthe

moleculeSincethelefthandsideofthisinequalitycanberewrittenasOnegetsstates.Subsequently,Levyprovedthetheoremsfordegenerategroundstates.

HohenbergandKohnprovedtheirtheoremsonlyfornondegenerateground第20頁/共155頁(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.沈呂九第21頁/共155頁electronsthateachexperiencethesameexternalpotential

theground-stateelectronprobabilitydensity

equaltotheexactofthemoleculeweareinterestedin:.KohnandShamconsideredafictitiousreferencesystemsofnnoninteractingthatmakesofthereferencesystemSincetheelectronsdonot

interactwithoneanotherinthereferencesystem,theHamiltonianofthereferencesystemiswhereistheone-electronKohn-ShamHamiltonian.

第22頁/共155頁Thus,theground-statewavefunctionofthereferencesystemis:isaspinfunctionorbitalenergies.areKohn-ShamForconvenience,thezerosubscriptonisomittedhereafter.Defineasfollows:ground-state

electronickineticenergysystemofnoninteractingelectrons.(either)isthedifferenceintheaveragebetweenthemoleculeand

thereference

Thequantityrepulsionenergy.units)

for

theelectrostaticinterelectronicistheclassicalexpression(inatomic第23頁/共155頁RememberthatWiththeabovedefinitions,

canbewrittenasDefinetheexchange-correlationenergyfunctionalbyNowwehaveside

are

easytoevaluatefromgetagoodapproximationto

totheground-stateenergy.

Thefourthquantity

accurately.

ThekeytoaccurateKSDFT

calculationofmolecular

propertiesisto

Thefirstthreetermsontherightisarelativelysmallterm,butisnoteasytoevaluate

andtheymakethe

maincontributions第24頁/共155頁Thusbecomes.Nowweneedexplicitequationstofindtheground-stateelectrondensity.sameelectrondensityasthatinthegroundstateofthemolecule:isreadilyprovedthatSincethefictitioussystemofnoninteractingelectronsisdefinedtohavethe,it第25頁/共155頁ground-stateenergybyvaryingtominimizethefunctional

canvarytheKSorbitals

minimizetheaboveenergyexpressionsubjecttotheorthonormalityconstraint:TheHohenberg-Kohnvariationaltheoremtellusthatwecanfindthe

soas.Equivalently,insteadofvaryingweThus,theKohn-Shamorbitalsarethosethatwiththeexchange-correlationpotential

definedby(Ifisknown,itsfunctionalderivative

isalsoknown.)第26頁/共155頁CommentsontheDFTmethods:(1)TheKSequationsaresolvedinaself-consistentfashion,liketheHFequations.(2)ThecomputationtimerequiredforaDFTcalculationformallyscalesthe

third

power

ofthenumberofbasisfunctions.(3)ThereisnoDFmolecularwavefunction.(4)TheKSorbitalscanbeusedinqualitativeMOdiscussions,liketheHF

orbitals.(5)Koopmans’theorem

doesn’t

holdhere,exceptTheKSoperatorexchangeoperatorsintheHFoperatorarereplacedbytheeffectsofbothexchangeandelectroncorrelation.isthesameastheHFoperator

exceptthatthe,whichhandles第27頁/共155頁(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

第28頁/共155頁XLocalexchangeApproximatedensityfunctionaltheoriesforexchangeandcorrelationX:

LocalexchangefunctionalofthehomogeneouselectrongasLDALocalexchange+localcorrelationGGALocalexchange+localcorrelation+gradientcorrections3rdGenerationoffunctionalsLDA:Localexchangefunctional+localcorrelationfunctionalofthehomogeneouselectrongasGGA:SameasLDA+“non-local”gradientcorrectionstoexchangeandcorrelation3rdGenerationoffunctionals:SameasGGA+instilationof“exact-exchange”and+2ndderivativesofthedensitycorrections第29頁/共155頁TermsinDensityFunctionalsr 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).第30頁/共155頁LocalDensityApproximation第31頁/共155頁LocalSpinDensityApproximation第32頁/共155頁LocalSpinDensityCorrelationFunctionalNotforthefaintofheart:第33頁/共155頁GeneralizedGradientApproximationFunctionals第34頁/共155頁第35頁/共155頁第36頁/共155頁第37頁/共155頁TheNobelPrizeinChemistry1998“forhisdevelopmentofthedensity-functionaltheory"WalterKohn(1923-)第38頁/共155頁第39頁/共155頁第40頁/共155頁第41頁/共155頁第42頁/共155頁第43頁/共155頁第44頁/共155頁第45頁/共155頁第46頁/共155頁第47頁/共155頁第48頁/共155頁第49頁/共155頁第50頁/共155頁第51頁/共155頁第52頁/共155頁第53頁/共155頁第54頁/共155頁第55頁/共155頁第56頁/共155頁第57頁/共155頁第58頁/共155頁第59頁/共155頁第60頁/共155頁第61頁/共155頁第62頁/共155頁第63頁/共155頁第64頁/共155頁第65頁/共155頁第66頁/共155頁第67頁/共155頁第68頁/共155頁第69頁/共155頁第70頁/共155頁第71頁/共155頁第72頁/共155頁第73頁/共155頁第74頁/共155頁第75頁/共155頁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(dǎo)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。第76頁/共155頁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)第77頁/共155頁相互作用：(1)單電子能級分布變化(勢的變化);(2)電子散射導(dǎo)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)用動量表示第78頁/共155頁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第79頁/共155頁FermiLiquidTheory第80頁/共155頁第81頁/共155頁第82頁/共155頁第83頁/共155頁第84頁/共155頁第85頁/共155頁第86頁/共155頁SimplePictureforFermiLiquid第87頁/共155頁第88頁/共155頁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(yù)言相偏離的性質(zhì)第89頁/共155頁THEPHYSICS

OFLUTTINGERLIQUIDSFERMISURFACEHASONLYTWOPOINTSfailureofLandau′sFermiliquidpictureELECTRONSFORMAHARMONICCHAINATLOWENERGIES

Coulomb+PauliinteractionTHELUTTINGERLIQUID:INTERACTINGSYSTEMOF1DELECTRONSATLOWENERGIEScollectiveexcitationsarevibrationalmodes第90頁/共155頁REMARKABLEPROPERTIESAbsenceofelectron-likequasi-particles(onlycollectivebosonicexcitations)Spin-chargeseparation(spinandchargearedecoupledandpropagatewithdifferentvelocities)AbsenceofjumpdiscontinuityinthemomentumdistributionatPower-lawbehaviorofvariouscorrelationfunctionsandtransportquantities.Theexponentdependsontheelectron-electroninteraction第91頁/共155頁OUTLINEWhatisaFermiliquid,andwhytheFermiliquidconceptbreaksin1DTheTomonaga-LuttingermodelTheTL-HamiltoniananditsbosonizationDiagonalizationBosonicfieldsandelectronoperatorsLocaldensityofstatesTunnelingintoaLuttingerliquidLuttingerliquidwithasingleimpurityPhysicalrealizationsofLuttingerliquids第92頁/共155頁LITERATURE

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/9807366第93頁/共155頁SHORTLYABOUTFERMILIQUIDSLandau1957-1959Alsocollectiveexcitationsoccur(e.g.zerosound)atfiniteenergiesLowenergyexcitationsofasystemofinteractingparticlesdescribedintermsof``quasi-particles``(single-particleexcitations)Keypoint:quasi-particleshavesamequantumnumbersasthecorrespondingnon-interactingsystem(adiabaticcontinuity)StartfromappropriatenoninteractingsystemRenormalizationofasetofparameters(e.g.effectivemass)第94頁/共155頁FERMILIQUIDSIIPauliexclusionprinciple

onlystateswithinkTaroundFermisphereavailablequasiparticlestatesnearFermispherescatteronlyweaklyQUASI-PARTICLEPICTUREISAPPLICABLEIN3DEffectofCoulombinteractionistoinduceafinitelife-timet3D第95頁/共155頁FERMILIQUIDSIIIcollectiveexcitations(plasmons)single-particleexcitations12340132DISPERSIONOFEXCITATIONSIN3D0nointeractingT=0FinitejumpinmomentumdistributionZZquasi-particleweight第96頁/共155頁LIFETIMEOF``QUASI-PARTICLES′′scatteringoutofstatekscatteringintostatekspinscreenedCoulombinteractionenergyconservationIn3Danintegrationoverangulardependencetakescareofd-functionFermi′sgoldenruleyieldsforthelifetimetT=0第97頁/共155頁LIFETIMEOF``QUASI-PARTICLES′′IIIn1Dk,k′arescalars.Integrationoverk′yieldsWhataboutthelifetimetin1D?formally,itdivergesatsmallqbutwecaninsertasmallcut-offAtsmallTi.e.,thisratiocannotbemadearbitrarilysmallasin3D第98頁/共155頁BREAKDOWNOFLANDAUTHEORYIN1D12340132DISPERSIONOFEXCITATIONSIN1D

collectiveexcitationsareplasmonswith(RPA)singleparticlegaplessplasmon

COLLECTIVEAND

SINGLE-PARTICLEEXCITATIONNONDISTINCT

nolongerdivergesat(noangularintegrationoverdirectionofasin3D)第99頁/共155頁THETOMONAGA-LUTTINGERMODELEXACTLYSOLVABLEMODELFORINTERACTING1DELECTRONSATLOWENERGIESDispersionrelationislinearizednear(bothcollectiveandsingle-particleexcitationshavelineardispersion)ModelbecomesexactwhenlinearizedbranchesextendfromAssumptions:Onlysmallmomentaexchangesareincluded第100頁/共155頁TOMONAGA-LUTTINGERHAMILTONIANFreepart

freepartinteraction

fermionicannihilation/creationoperatorsIntroducerightmoving

k>0,andleftmovingk<0electrons第101頁/共155頁TLHAMILTONIANIIInteractions

freepartinteractionbackscatteringforwardumklappforward第102頁/共155頁BOSONIZATIONBOSONIZATION:EXPRESSFERMIONICHAMILTONIANINTERMSOFBOSONICOPERATORSconstructbosonicHamiltonianwiththesamespectrun(a)(b)(c)(d)(a)and(b)havesamespectrumbutdifferentgroundstateEXCITEDSTATECANBEWRITTENINTERMSOFCHARGEEXCITATIONS,ORBOSONICELECTRON-HOLEEXCITATIONS第103頁/共155頁STEP1WHICHOPERATORSDOTHEJOB?Introducethedensityoperators(createexcitationofmomentumq)andconsidertheircommutationrelations

nearlybosonic

commutationrelations第104頁/共155頁STEP1:PROOFConsidere.g.algebraoffermionicoperatorsoccupationoperator第105頁/共155頁STEP2ExaminenowBOSONIZEDHAMILTONIANSTATESCREATEDBYAREEIGENSTATESOFWITHENERGY

andinteractions第106頁/共155頁STEP2:PROOFExample:第107頁/共155頁STEP3IntroducethebosonicoperatorsyieldingDIAGONALIZATION第108頁/共155頁SPIN-CHARGESEPARATIONandinteraction(satisfyingSU2symmetry)Ifweincludespin,itgetsslightlymorecomplicated...andinterestingIntroducethespinandchargedensitiesHamiltoniandecoupleintwoindependentspinandchargeparts,withexcitationspropagatingwithvelocities第109頁/共155頁SPACEREPRESENTATIONLongwavelengthlimit(interactions)AppropriatelinearcombinationsP,qofthefieldr(x)canbedefined.ThenonefindswhereLuttingerparameterg<1repulsiveinteraction第110頁/共155頁BOSONICREPRESENTATIONOFYFermionicoperatorWheree.g.Expressyintheformofabosonicdisplacementoperator

B

from

decreasesthenumberofelectronsbyonedisplacesthebosonconfigurationforthatstateBOSONIZATIONIDENTITYifac-numberUladderoperator,qbosonic第111頁/共155頁LOCALDENSITYOFSTATESi)Localdensityofstatesatx=0ndensityofstatesofnon-interactingsystematT=0ii)LocaldensityofstatesattheendofaLuttingerliquidatT=0cut-offenergyG

gammafunction第112頁/共155頁MEASURINGTHELDOS

Measurementofthelocaldensityofstatessystem1system2couplingIVbytunnelingSeee.g.carbonnanotubeexperimentbyBockrathetal.Nature,397,598(1999)第113頁/共155頁MEASURINGTHELDOSIItunnelingrateitojTunnelingcurrentcanbeevaluatedbyuseofFermi′sgoldenruleconstant

LLtoLLLLtometal第114頁/共155頁SINGLEIMPURITYAgaintunnelingcurrentcanbeevaluatedbyuseofFermi′sgoldenrule

endtoendWeaklinkx=0However,nowistunnelingfromtheendofaLLChargedensitywaveispinnedattheimpurity第115頁/共155頁PHYSICALREALIZATIONS

SemiconductingquantumwiresEdgestatesinfractionalquantumHalleffectSingle-walledmetalliccarbonnanotubesEFEnergymetallic1Dconductorwith

2linearbandsk第116頁/共155頁5.3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(dǎo)帶導(dǎo)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第117頁/共155頁(3)電子間關(guān)聯(lián)導(dǎo)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(dǎo)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。第118頁/共155頁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YSc第119頁/共155頁Smith和Kmetko準周期表窄帶區(qū)域重費米子強鐵磁性超導(dǎo)體離域性局域性第120頁/共155頁另一類窄帶現(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(dǎo)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。第121頁/共155頁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模型第122頁/共155頁TheHubbardModelFromsimplequantummechanicstomany-particleinteractioninsolids-ashortintroduction第123頁/共155頁HistoricalfactsHubbardModelwasfirstintroducedbyJohnHubbardin1963.WhowasHubbard?Hewasbornin1931anddied1980.Theoreticianinsolidstatephysics,fieldofwork:Electroncorrelationinelectrongasandsmallbandsystems.HeworkedattheA.E.R.E.,Harwell,U.K.,andattheIBMResearchLabs,SanJosé,USA.Picturetakenfrom:PhysicsToday,Vol.34,No4,1981第124頁/共155頁What,ingeneral,istheHM?

Hubbardmodelisaquantumtheoreticalmodelformany-particleinteractioninandwithaperiodiclatticeItisbasedonaninteractionHamitonian,sometransformationsandassumptionstobeabletotreatcertainproblems(e.g.magneticbehaviourandphasetransitions)withsolidstatetheory第125頁/共155頁QuantummechanicsBasics:Schr?dingerequation

Expectationvalues

Orthonormalityandclosurerelation

Thebra-ketnotation第126頁/共155頁Basistransformation,mathematicallyAbasistransformationcanbesimplyperformed:Anequationistransformedthesameway:第127頁/共155頁SingleparticleequationsParticleinapotential:

Periodicpotentials:

Solutionforweakcouplingtopotential:

Blochwave第128頁/共155頁SingleparticleequationsDispersionrelationforfreeelectrons(dashedline):DispersionrelationforBlochelectrons(quasi-free)(solidline):Theenergiesat arenolongerdegenerated.Twoeigenenergiesatthosepoints.GraphfromGerdCzycholl,?TheoretischeFestk?rperphysik“,Vieweg-Verlag第129頁/共155頁SingleparticleequationsWannierstatesproduceanorthonormalbaseoflocalizedstates;atomicwavefunctionswouldalsobelocalized,buttheyarenotorthonormal.Strongerlatticepotential:couplingtolatticepointsoccurs;amodifiedBlochwaveisused,e.g.WannierstatesresultingfromtheTight-Binding-Model:第130頁/共155頁ComparisonbetweenthetwonewwavefunctionsBlochwavefunctionWannierwavefunction(w-part)GraphfromGerdCzycholl,?TheoretischeFestk?rperphysik“,Vieweg-VerlagGraphfromGerdCzycholl,?TheoretischeFestk?rperphysik“,Vieweg-Verlag第131頁/共155頁WavefunctionformanyparticlesWavefunctionisnotsimplytheproductofallsingleparticlewavefunctions;ParticlescannotbedifferedFermionsmustobeyPauliprincipleAnsatz:Slaterdeterminante第132頁/共155頁SecondQuantizationforFermionsCreationanddistructionoperatorscreateordestroystates:第133頁/共155頁SecondQuantizationTheoperatorsfulfillthecommutatorrelation:Thisisamust,otherwiseonewoulddisturbclosurerelationandorthonormalityofwavefunctionsdescribedbysecondquantization第134頁/共155頁HamiltonianformanyparticlesSummationoverallsingleparticlesHamiltonians+interactionHamiltonian:interactionpotentialuistherepulsiveCoulombinteraction第135頁/共155頁Operatorsinsecondquantization第136頁/共155頁Operatorsinsecondquantization第137頁/共155頁HamiltonianinsecondquantizationIstransformedliketheone-particleoperatorA(1)andthetwo-particleoperatorA(2)第138頁/共155頁HamiltonianinsecondquantizationNow:Matrixelement mustbedetermined.Herefore,awavefunctionhastobechosen.Example:Bloch-wave第139頁/共155頁ComingclosertoHubbard...EvaluationofmatrixelementswithWannierwavefunctions:第140頁/共155頁FinalAssumptionsNow:onlydirectneighborinteractions,restrictiontooneband.第141頁/共155頁Meaningofmatrixelementst:singleparticlehoppingU:Hubbard-U,describesonsite-CoulombinteractionV:Nearest-neighbor(density)interactionX:conditionalhoppinginteraction第142頁/共155頁TheHubbardModels