封东来4-Lecture2 Simple models

1、Physics of strongly correlated systemsLecture 2Simple models Donglai Feng “Physics of Strongly Correlated Systems”Model HamiltoniansModel HamiltoniansGoal Set up a Hamiltonian with a minimal set of parameters and terms but which will describe the qualitative physics.Exchange2 electrons in 1 atom (He

2、)Direct exchangeHelium atom in large U limit Donglai Feng “Physics of Strongly Correlated Systems”Many body effect of atoms and exchange interactions Donglai Feng “Physics of Strongly Correlated Systems”Construct the basis Donglai Feng “Physics of Strongly Correlated Systems”Diagonalize the Hamilton

3、ian21122121|2|2baibaabbaibaabHJHC Donglai Feng “Physics of Strongly Correlated Systems”Eigen wavefunctions Donglai Feng “Physics of Strongly Correlated Systems”Effective Hamiltonian Donglai Feng “Physics of Strongly Correlated Systems”Summary for direct exchangep Exchange interaction is derived from

4、 Coulomb interactions. and quantum mechanical indistinguishablility.p Eigenstates are linear combination of many determinantsp for more than 3 electrons, its impossible to construct a totally antisymmetric state. i.e. cannot be simple spin and orbital dependencep For orthogonal states, the lowest en

5、ergy states is a triplet.pOn-site coulomb U enforces orbital orthogonality favor triplet or Ferro-spin arrangement (Hunds rule#1) Donglai Feng “Physics of Strongly Correlated Systems”Orthogonal orbitals favor FM2 electrons, 2 atoms (H2)Kinetic exchange for non-orthogonal orbitalsp Wigners theorem: w

6、hen electron occupies non-orthogonal orbitals, such as H2 molecule, and He atom (identical orbital or overleaping orbital) ground state is singlet. Donglai Feng “Physics of Strongly Correlated Systems”SetupIn nature. many AF materials rise from Non-ortho orbitals, as they are from different atoms. c

7、onsider H2 molecule, 2 orbitals Donglai Feng “Physics of Strongly Correlated Systems”A) Heiter-London SchemeTwo atoms are far apart. only consider neutral configuration. Donglai Feng “Physics of Strongly Correlated Systems”they are eigenstates, because they have different symmetry properties. Dongla

8、i Feng “Physics of Strongly Correlated Systems”B) kinetic exchange : Including Ionized configurations Donglai Feng “Physics of Strongly Correlated Systems”for sure its Singlet ground state Donglai Feng “Physics of Strongly Correlated Systems”Energy diagram Donglai Feng “Physics of Strongly Correlate

9、d Systems”Molecule orbital, vs. Heitler-LondonNote: construction of Hamiltonian is different from construct of the wavefunctions, both routes are taken by theorists. They are related of course.pHeitler-London = MO + remove the high polarity termspHeitler-London model + configuration interaction MO D

10、onglai Feng “Physics of Strongly Correlated Systems”Molecular orbitalOne electron molecular orbital theory Band theory Donglai Feng “Physics of Strongly Correlated Systems”Molecular orbitalTherefore, its an good and easy-to-use approximation when U is small. Donglai Feng “Physics of Strongly Correla

11、ted Systems”MO vs. H-L Donglai Feng “Physics of Strongly Correlated Systems”Practical approachFor low energy scale, only spin degree of freedom dominates.Neglect this due to energy degeneracy, for convenience Donglai Feng “Physics of Strongly Correlated Systems”Practical approachNew Model Hamiltonia

12、n with just one parameter : Donglai Feng “Physics of Strongly Correlated Systems”Principle for ground state spin configurationGround state:Lower the kinetic energy, while keeping the potential energy of the intermediate state as low as possible Donglai Feng “Physics of Strongly Correlated Systems”Su

13、per-exchangeSuper-exchange Donglai Feng “Physics of Strongly Correlated Systems”Super-exchange Donglai Feng “Physics of Strongly Correlated Systems”Contd Donglai Feng “Physics of Strongly Correlated Systems”Corner type super-exchangeOxygenTM1TM2Involve two p orbitals Donglai Feng “Physics of Strongl

14、y Correlated Systems”Double exchange, d4+d3 Donglai Feng “Physics of Strongly Correlated Systems”d4 case Donglai Feng “Physics of Strongly Correlated Systems”Goodenough-Kanamori RuleGoodenough-Kanamori 1st rule: TM d orbital # 5 AFMGoodenough-Kanamori 2nd rule: TM d orbital # 5 FM Donglai Feng “Phys

15、ics of Strongly Correlated Systems”Orbital ordering and exchangeLow energy intermediate state principle Donglai Feng “Physics of Strongly Correlated Systems”Orbital vs SpinIn plane: Orbital AF Spin FZ direction: Orbital F Spin AFOrbital AFCrystal fieldSolid (Ligand field theory, or molecular orbital

16、 theory) Donglai Feng “Physics of Strongly Correlated Systems” Donglai Feng “Physics of Strongly Correlated Systems”Ions in SolidsFor free ions:1. Filled electronic shells are not magnetic (the spins are paired; ms = 1/2) 2. Only unfilled shells may possess a magnetic moment 3. The magnetic moment i

17、s given by m = gBJ, where J represents the total angular momentum. For a given configuration the values of g and J in the ground state are given by Hunds rules When the ion is embedded in a solid,1. the crystal field interaction is important, and the third point is modified 2. Orbital angular moment

18、um for 3d ions is quenched. The spin only moment is m gBS, with g = 2. 3. Magnetocrystalline anisotropy appears, making certain crystallographic axes easy directions of magnetization. Donglai Feng “Physics of Strongly Correlated Systems”The HamiltonianHoHspin-orbitHcrystal-fieldHzeeman3d15 eV10100 m

19、eV100 meV0.1 * BTmeV4f1060 eV100500 meV30 meVso Hspin-orbit must be considered before Hcrystal-field for 4f ions, and the converse for 3d ions. Hence J is a good quantum number for 4f ions, but S is a good quantum number for 3d ions. The 4f electrons are generally localized, and 3d electrons are loc

20、alized in oxides and other ionic compounds. Donglai Feng “Physics of Strongly Correlated Systems” Donglai Feng “Physics of Strongly Correlated Systems”Note that the z-component of angular momentum; lz is zero for these wavefunctions. Hence the orbital angular momentum is quenched. Donglai Feng “Phys

21、ics of Strongly Correlated Systems”Orbitals in the crystal field Donglai Feng “Physics of Strongly Correlated Systems”Crystal field splitting Donglai Feng “Physics of Strongly Correlated Systems”px,py Donglai Feng “Physics of Strongly Correlated Systems”One-electron energy diagramsfield-free octahed

22、ral tetragonal trigonal monoclinic ion Oh D4h C3v C2 Donglai Feng “Physics of Strongly Correlated Systems”Multi-electron statesIn insulators, the electrons in an unfilled shell interact strongly with each other giving rise to a series of sharp energy levels which are determined by the action of the

23、crystal field on the orbital terms of the free atom. The spacing of theses levels may be determined by spectroscopy, and the crystal-field determined.the effect of a cubic crystal field on the Hunds rule ground state term. Since a half-filled shell has spherical symmetry, the cases dn and d5+n are e

24、quivalent. Also, since a hole is the absence of an electron, the cases dn and d10-n are related.d5 octahedral or tetrahedral Donglai Feng “Physics of Strongly Correlated Systems”Orgel Diagramsd3 , d8 octahedral (d2 , d7 tetrahedral) d2 , d7 octahedral (d3 , d8 tetrahedral)d4, d9 octahedral(d1, d6 te

25、trahedral)d1, d6 octahedral(d4, d9 tetrahedral) Donglai Feng “Physics of Strongly Correlated Systems”High-spin and low-spin statesAn ion is in a high-spin state or a low spin state, depending on whether the Coulomb interaction U leading to Hunds first rule (maximize S) is greater or less than the cr

26、ystal-field splitting D D. Donglai Feng “Physics of Strongly Correlated Systems”More examples Donglai Feng “Electronic Structure of Strongly Correlated Systems”“Correlated” band Insulator: LaCoO3Y. Tokura et. al. Phys. Rev. B 58 (1998) R1699.Crossover to magnetic insulator LS to IS and eventually HS

27、 Donglai Feng “Physics of Strongly Correlated Systems”Spin states and Cation radii in oxidesThe radius of the O2- anion is 140 pm* low spin values are in parentheses.High spin usually occupies more orbitals, U wins over D D Donglai Feng “Physics of Strongly Correlated Systems”Tanabe-Sugano diagramTh

28、ese show the splitting of the ground state and higher terms by the crystal field. The high-spin low spin crossover is seen. Diagrams shown are for d-ions octahedral environments.Jahn-Teller distortion Donglai Feng “Physics of Strongly Correlated Systems”Jahn-Teller distortion, competition between fr

29、eedoms Donglai Feng “Physics of Strongly Correlated Systems”Jahn-Teller distortionA system with a single electron (or hole) in a degenerate level will tend to distort spontaneously. The effect is particularly strong for d4 and d9 ions in octahedral symmetry (Mn3+, Cu2+) which can lower their energy

30、by distorting the crystal environment. This is the Jahn-Teller effect. If the local strain is e e, the energy change DE -Ae DE -Ae +Be+Be2 2, where the first term is the crystal-field stabilization energy D Dcfse and the second term is the increased elastic energy.The J-T distortion may be static or

31、 dynamic. Donglai Feng “Physics of Strongly Correlated Systems”Cu2+ in cupratesMott insulator Donglai Feng “Electronic Structure of Strongly Correlated Systems”56Mott insulator vs. Band insulatora) Band insulator; Si, Ge, E-p-pp pkBand gap (EG) Gcryst2EVGconduction bandvalence bandnumber of electron

32、s: Nenumber of atoms: NNe=2Nb) Mott insulator; Transition metal oxides(La2CuO4, LaTiO3, )Ne=NEG0Uexcited stateground stateatomGECoulomb interaction between electrons (U)new concept !BandMottkconductionvalencekupper Hubbardlower HubbardE Ee eFE Ee eFBand insulator vs. Mott insulatorelectronholeV(r)ex

33、citon pictureV(r) Donglai Feng “Electronic Structure of Strongly Correlated Systems”Mott Hubbard Insulator and Charge Transfer Insulator:Zaanen-Sawatzky-AllenMott-HubbardCharge TransferLHBUHBLHBUHBCharge tranfer energyCharge tranfer energyzoology of correlated insulator in transition metal oxides Do

34、nglai Feng “Electronic Structure of Strongly Correlated Systems”Mott vs CT: systematics on periodic tableUdd increase D Dpd decrese from Ti to Cu Donglai Feng “Electronic Structure of Strongly Correlated Systems” Donglai Feng “Electronic Structure of Strongly Correlated Systems”TMOTransition MetalOx

35、ygensRare earth or Alkaline earth ions3d: can find Mott Ins.3d: can find Mott Ins.4d, 5d: hard to find Mott Ins.4d, 5d: hard to find Mott Ins.ABO3A2BO4 (layered) Donglai Feng “Electronic Structure of Strongly Correlated Systems”Guide Map of 113+214 perovskitesABO3A2BO4Pr-EuMetalMetalMetalMetalMetalB

36、and InsBand InsBand InsMott Insulatorp-d hybridizationLa(RE) more ins. than Sr less distorted2D more insulating Donglai Feng “Electronic Structure of Strongly Correlated Systems”H lattice (Hubbard model) Donglai Feng “Physics of Strongly Correlated Systems” Donglai Feng “Physics of Strongly Correlat

37、ed Systems” Donglai Feng “Physics of Strongly Correlated Systems” Donglai Feng “Physics of Strongly Correlated Systems” Donglai Feng “Physics of Strongly Correlated Systems” Donglai Feng “Physics of Strongly Correlated Systems” Donglai Feng “Physics of Strongly Correlated Systems” Donglai Feng “Phys

38、ics of Strongly Correlated Systems” Donglai Feng “Physics of Strongly Correlated Systems”（S=1/2）（+e） Donglai Feng “Electronic Structure of Strongly Correlated Systems”74One-dimensionalSpin-charge separationTwo-dimensional Donglai Feng “Electronic Structure of Strongly Correlated Systems”75Experiment

39、 on Sr2CuO3 chain systemC. Kim, preprint Donglai Feng “Physics of Strongly Correlated Systems”（S=1/2）（S=1/2） Donglai Feng “Physics of Strongly Correlated Systems” Donglai Feng “Electronic Structure of Strongly Correlated Systems”79Neutron scattering of CuGeO3 Donglai Feng “Electronic Structure of St

40、rongly Correlated Systems”80Spin-charge separation by optical excitationhn n(+)(+)(- -)(- -)kskhspinonholonkskdspinondoublonelectroncharge degreespin degreeholon (- -e) + doublon (+e)spinon (S=1/2) + spinon (S=- -1/2)（+e）（-e）Absorption Profile (Q | Chain)M. Z. Hasan et al.: PRL 88, 177403 (2002)(c.f. : H. Suga : 27pXF-6)Sr2CuO3 Donglai Feng “Electronic Structure of Strongly Correlated Systems”83RIXS in insulating cuprates:1-D Sr2CuO32-D Ca2CuO2Cl21-D2-D Donglai Feng “Physics of Strongly Correlated Systems” Donglai Feng “Phy