George Sawatzky Departments of Physics and Chemistry UBC …scs.sa.infn.it/TCXIV/Download/Sawatzky/Lecture 1.pdf · 2009-10-15 · experiment Lecture 1,2 George Sawatzky Departments
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Electronicstructureofcorrelatedelectronsystems:theoryand
experimentLecture1,2
GeorgeSawatzkyDepartmentsofPhysicsandChemistryUBCVancouver
Roughcontentof5lectures• Lecture1and2:Electronicstructureofcorrelatedandnarrowbandsystems.
• Lecture2and3:ElectronicstructureandtheoryoftransiLonmetaloxides(orbital,charge,spinandlaNcedegreesoffreedom)
• ImportanceofnonuniformpolarizabiliLesSurfaces,interfacesofstronglycorrelatedOxides.
• BasicelectronicstructureandtheoreLcalmodelsofFePnicLdes
ContentLecture1,2
• Electronicstructureofcorrelatedelectronsystems– WhyareTMcompoundsandrareearthsspecial– Quasiatomicvsbandstructureapproaches– AbitaboutDFT,LDA+U,DMFT,ModelHexactdiagonalizaLon
– AbitaboutAugerandARPES– SpectralweighttransferarguablythemostdirectevidenceforstrongcorrelaLon
– Modelsandcuriosity’s
SomeOldHistoricalmilestones
• 1929‐1931BlochWilsontheoryofsolids• 1937DeBoerandVerwey(NiO‐CoObreakdownofbandtheory
• 1937Peierls3delectronsavoideachother(basicallytheHubbardmodel)
• 1950JonkervanZanten‐ZenerPervoskitesdoubleexchange
• 1959Andersonsuperexchange(U>>W)• 1964Hubbardmodel‐,HohenbergKohnDFT‐KohnSham,GoodenoughTransiLonmetalcompounds
Everyone claiming to work on real materials should be familiar with this
WidediversityofproperLes
• Metals:CrO2,Fe3O4T>120K• Insulators:Cr2O3,SrTiO3,CoO• Semiconductors:Cu2O• Semiconductor–metal:VO2,V2O3,Ti4O7• Superconductors:La(Sr)2CuO4,LiTiO4,YBCO• PiezoandFerroelectric:BaTiO3• Catalysts:Fe,Co,NiOxides• FerroandFerrimagnets:CrO2,gammaFe2O3• AnLferromagnets:alfaFe2O3,MnO,NiO‐‐‐• Ionicconductors(baheries)LixNi1‐xO• OxidefuelcellsuseManganitesandcobaltates
Proper&esdependindetailoncomposi&onandstructure
Phase Diagram of La1-xCaxMnO3
Uehara, Kim and Cheong
R: Rombohedral
O: Orthorhombic (Jahn-Teller distorted)
O*: Orthorhombic (Octahedron rotated)
MizokawaetalPRB63,0244032001
Mn4+,d3,S=3/2,Noquadrupole;Mn3+,S=2,orbitaldegeneracy
Ordering in strongly correlated systems Stripes in Nd-LSCO
DQC ~ 1 e DQO ~ 0
DQ < 0.5 e
Charge inhomogeneity in Bi2212
Pan, Nature, 413, 282 (2001); Hoffman, Science, 295, 466 (2002)
DQ ~ 0.1 e
Quadrupole moment ordering
rivers of Charge— Antiferro/ Antiphase
It’stheoutermostvalenceelectronstatesthatdeterminetheproperLes;Boththe
occupiedandunoccupiedones
Coexistance‐‐‐‐‐Hybridiza&onKondo,Mixedvalent,Valencefluctua&on,localmoments,Semicond.‐metaltransi&ons,HeavyFermions,HighTc’s,Colossalmagnetoresistance,Spintronics,orbitronics
Twoextremesforatomicvalencestatesinsolids
Whereistheinteres&ngphysics?
CharacterisLcsofsolidswith2extremevalenceorbitals
R>>D
• electronsloseatomicidenLty
• Formbroadbands• Smallelectronelectron
interacLons
• Lowenergyscale–chargefluctuaLons
• NonorweaklymagneLc• ExamplesAl,Mg,Zn,Si
R<<D
• ValenceElectronsremainatomic
• Narrowbands• Largeelectronelectron
interacLons(onsite)
• Lowenergyscale‐spinfluctuaLons
• MagneLc(Hunds’rule)• Gd,CuO,SmCo3
ManysolidshavecoexisLngR>>DandR<<Dvalenceorbitalsi.e.rareearth4fand5d,CuOCu3dandO2p,HeavyFermions,Kondo,HighTc,s,met‐insul.transiLons
Whyarethevalence3dand4forbitalsintransi&onmetalandrareearthcompoundsspecial
• Lowestprincipleq.n.forthatlvalue• Largecentrifugalbarrierl=2,3• Smallradialextent,noradialnodes,orthogonaltoallothercoreorbital’sviaangularnodes(snuggleuptothenucleus)
• HighkineLcenergy(angularnodes)compensatesforthestrongpotenLalenergy
• RelaLvisLceffects• Looklikecoreorb.Buthavehighenergyandformopenshellslikevalenceorb.
SpecialplacefortransiLonmetalandrareearths
Notethatthe4fstatesareNotfulloremptyfortherareEarthsandyettheyarewellInsideotheroccupiedorbitals!!
Hubbard For 4f U as large as 12eV
BandStructureapproachvsatomic
Bandstructure• DelocalizedBlochstates• Fillupstateswithelectrons
starLngfromthelowestenergy
• NocorrelaLoninthewavefuncLondescribingthesystemofmanyelectrons
• Atomicphysicsisthereonlyonameanfieldlikelevel
• SingleSlaterdeterminantstates
Atomic• Localatomiccoulomband
exchangeintegralsarecentral
• HundsrulesfortheGroundstate‐Maximizetotalspin‐Maximizetotalangularmomentum‐totalangularmomentumJ=L‐S<1/2filledshell,J=L+Sfor>1/2filledshell
• MostlymagneLcgroundstates
RecallthatthegroundStatehasfewproperLesItistheexcitedstatesthatDeterminetheresponsetoExternalperturbaLonssuchasfields.
SingleSlaterdet.OfOneelectronBlochStates.NocorrelaLonInthewavefuncLon
Interaction between two Bloch wave electrons = U/N ~0 So is correlation negligible?
SurelyalaNceofHatomsseparatedbysay1cmwouldnotbehavelikeametal
Whathaveweforgohen?Theelectronelectronrepulsive
interacLon
ExperimentalevidenceforatomiclikebehaviourintransiLonmetal
compoundsandrareearthsPhotoemission/inverse
photoemissionandAugerspectrscopy
Highresolu&onangularresolvedphotoelectronspectroscopy
Exampleofasimplemetalinoneelectrontheory
Exampleofametalinwhichelectronsaredressed
ARPESCu
3dbands
4s,4p,band
Cuisd10soonedholeHasnootherdholestoCorrelatewithso1part.TheoryworksiftheonlyImportantinteracLonisThed‐dinteracLon.GreatagreementwithDFT
Points–exp.Lines‐DFT
Angularresolvedphotoelectronspectroscopy(ARPES)ofCumetalThiryetal1979
We note that for Cu metal with a full 3d band in the ground state one particle theory works well to describe the one electron removal spectrum as in photoelectron spectroscopy this is because a single d hole has no other d holes to correlated with. So even if the on site d-d coulomb repulsion is very large there is no phase space for correlation.
The strength of the d-d coulomb interaction is evident if we look at the Auger spectrum which probes the states of the system if two electrons are removed from the same atom
If the d band had not been full as in Ni metal we would have noticed the effect of d-d coulomb interaction already in the photoemission spectrum as we will see.
Whatifweremove2‐delectrons?TwoholestatewithAugerspectroscopy
3d
2p 932eV
PhotonPhotoelectronAugerelectron
E(photon)‐E(photoelectr)=E(2p),E(2‐dholes)=E(2p)‐E(3d)‐E(Auger)
U=E(2‐dholes)‐2xE(1‐dhole)
ExampleisforCuwithAfullyoccupied3dband
AugerspectroscopyofCumetalAtomicmulLpletsLookslikegasphaseU>W
Hund’sruleTripletFisLowest
Antonidesetal1977 Sawatzkytheory1977
TheL3M45M45AugerspectrumofCumetali.efinalstatehas2‐3dholesontheAtomthatstartedwitha2phole.Solidlineistheexperiment.DashedlineisoneElectronDFTtheory,verLcalbarsandlablesarethefreeatommulLpletsfor8‐3delectronsonaCuatom.EfdesignatestheposLonoftheFermilevelintheDFT.
Twoholeboundstates
Ladder approximation Is exact for only two particles
DshellsarecomplicatedbymulLpletstructure
• Atomicphysics–dorbitalis5folddegeneratenotincludingthespinandneglecLngthespinorbitcoupling.
• Twodelectronsorholeswithorbitalangularmomentum=2andspinof½cancoupleintototalangularmomentumstatesLwithtotalspin1or0asfollows;singletS,singletG,singletDandtripletPandtripletF
• TheenergyseparaLonsintheCuAugerspectrumarefromatomiccoulombintegralswithtripletFasthelowestenergystatefor8delectronsasgivenbyHunds’rule
ForU>>WandinthepresenceofunfilledbandstheoneparLcleremovalspectrumwillbeverydifferent
fromthatofafilledband
ComparethePESofCumetalwithafulldbandtothatofNiwithontheaverage0.6holesinthe3dband
The relative weights Of the d9 and d8 (satellite) configurations depend on Initial d occupation and i.e. mixing i.e. band width Both in the initial and final states
For a LDA/DMFT try see Lichtenstein et al PRL 067205 (2001)
J.Ghijsenetal
Phys.Rev.B.42,(1990)2268.PhotoemissionspectrumofCuOCuind9S=1/2state
EnergybelowEfineV
Note the atomic like multiplet structure as for the rare earths
WewillcomebacktothePhotoemissionandZhangRice
singletslater.
TransiLonmetald‐dinteracLons
• ThesatelliteposiLongivesanesLmateoftheF0SlaterintegralorU.Moredetailsbelow
• ImportantisthatthemulLpletspreadisgivenbyonlyslightlydecreasedgasphaseatomicvaluesi.e.F2andF4SlaterintegralsorJhund=(1/14)(F2+F4)isreducedbyatmost20%fromtheatomicvalues.
• F0ontheotherhandisreducedfromtheatomic>20eVtoabout7eV!!!!!
LangBaerandCoxJPhysF11,121(1981)
• Photoemissionandinversephotoemissionofalltherareearthmetals
• DemonstratestheatomicmulLpletsofthe4felectronremovalandaddiLonstates
• IntensiLesgivenbyatomiccoefficientsoffracLonalparentagestarLngfromtheHunds’rulegroundstate
U = 12 eV
MOREONRAREEARTHS
• TheHubbardU;asclearlydemonstrated,itsdefiniLondependsonwhichmulLpletsyoutakeanddependsstronglyontheelement.ConvenLonistoeithertakethemulLpletaverageortheSlaterF0integral.
• ThemulLpletspliNngisveryclosetotheatomicvaluelihleSCREENINGOFTHEHUNDSRULESINTERACTIONSI.E.SLATERF2,F4,F6INTERACTIONS
Notetheatomicphysicsneededtodescribetherareearth4felectronremoval
andaddiLonspectrum
Forthe3dtransiLonmetalcompoundsthingsarealotmoresubtle.InsomecasesweneedtheatomicapproachesandinothersoneparLcletheoryseemstowork
verywell
TheholecanfreelyPropagateleadingtoAwidth
TheelectroncanfreelyPropagateleadingtoawidth
LargestcoulombInteracLonisonsiteU
SimplestmodelsinglebandHubbardRowofHatoms1sorbitalsonly
Egap=12.9eV‐W
TheactualmoLonoftheParLcleswillturnouttobemorecomplicated
ForlargeU>>Wand1electronpersite
• ‐‐‐‐Insulator• LowenergyscalephysicscontainsnochargefluctuaLons
• SpinfluctuaLonsdeterminethelowenergyscaleproperLes
• Canweprojectoutthehighenergyscale?
HeisenbergSpinHamiltonian
We should be a bit careful about decoupling spin and charge degrees of freedom even in this case
ThechargedistribuLonfortheanLparrallelandparallelnnspinstatesaredifferent!Forthesinglet
thereisadmixtureofdoublyoccupiedsites.Fortripletsthereis
not!Hasstrongconsequences!
TemperaturedependentOpLcalconducLvity
Tsvetkov et al PRB 69, 075110 (2004)
Spin order dependent Optical transitions
BeforewegoonletslookataspecificpropertyoftheHubbardmodelwhichismeasureablefora
“doped”MHsystem
SeamusDavistheSTSasymmetryCTShenXrayabsorpLonindoped
Cuprates
Spectralweighttransfer
TherealsignatureofstrongcorrelaLoneffects
H.EskesetalPRL67,1035(1991)Meindersetal,PRB48,3916(1993)
N N
EFPES PES
U
EF
N‐1 N‐12
EF
N+1N‐1
2
DopingaMoh–Hubbardsystem
(1‐x)/2x
H.EskesetalPRL67,1035(1991)Meindersetal,PRB48,3916(1993
x=0.0
x=0.1
x=0.2
x=0.3
x=0.4
x=0.5
x=0.6
x=0.7
x=0.8
x=0.9
Meindersetal,PRB48,3916(1993)
ThesestateswouldbevisibleinatwoparLcleaddiLonspectralfuncLon
10 site Hubbard 1 D periodic U-10,t=1
TheseparLclesblock2ormorestates
Bosons–block0statesFermions–block1state
These–block2statesonThelowenergyscale
Note the even larger slop for finite hopping integrals Dynamic spectral weight transfer
Phillip Phillips uses this to Define “Mottness” Stanescu ,phillips PRB 69 245104 (2004)
Eskes et al PRL 67, (1991) 1035 Meinders et al PRB 48, (1993) 3916
Elfimovunpublished Whatwouldameanfieldtheorygiveyou?
Note that there is no spectral weight transfer and a gap closing with doping From half filled . Both opposite to the real situation
WecomebacktospectralweighttransferlaterforthetransiLon
metalcompounds
Hubbardmodelisnotexactlysolvableexceptin1DbuteventhenthespectralfuncLonsare
difficulttoextract
Lieb and Wu PRL 20, 1445, (1968)
AbitmoreaboutsimplemodelsandsomepeculiarproperLesin1and2dimensionsofthesimple
models
Don’t know of a rigorous Proof of Hubb---t,J (U>>w)
SpinchargeseparaLonin1D
Antiphase Domain wall
Now the charge is free to move
Magnonsandspinonsin1D
Magnon S=1
Two spinons
Spinons propagate via J Si+S-
1+1
InelasLcNeutronscahering
Similar is some sense to the 1D case it is proposed that one has 2D rivers of charge separating anti-phase domain walls. Charges can now fluctuate from left to right without costing J
Anisimov, Zaanen ,Andersen, Kivelson,Emery-----
Pleaseciteas:
GeorgeSawatzky:LecturedeliveredattheXIVTrainingCourseinthePhysics ofStronglyCorrelatedSystems,VietrisulMare(Salerno) Italy,October5–16,2009.
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