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

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.