Mott insulators. Introduction Cluster-model description Chemical trend Band description Self-energy correction

Transcription

1 Mott insulators Introduction Cluster-model description Chemical trend Band description Self-energy correction

2 Introduction Mott insulators

3 Lattice models for transition-metal compounds Hubbard model Anderson-lattice model or p-d model Transition metal ion (with d orbitals) Non-metal anion (with p orbitals)

4 Lattice models for transition-metal compounds (degenerate) Hubbard model Anderson-lattice or p-d model t-j model no double occopancy

5 Band gap excitation and localized excitation Photoemission Inverse-photoemission E N-1 - E N E N+1 - E N E* N - E N Band gap excitation energy: E g = E N+1 + E N-1-2E N Localized excitation (d-d excitation, exciton,...) Relevant to charge transport

6 Band gap excitations - relevant to charge transport Excitation energy: E g = E N+1 + E N-1-2E N U E N-1 - E N E N+1 - E N Charge transfer energy: on-site Coulomb energy: L: ligand (p) hole

7 Photoemission spectroscopy

8 Lattice models for transition-metal compounds Hubbard model Anderson-lattice model or p-d model Transition metal ion (with d orbitals) Non-metal anion (with p orbitals)

9 Mott-Hubbard-type insulators vs charge-transfer-type insulators chemical potential µ W Inversephotoemission spectra W U < U > Photoemission spectra Mott-Hubbard gap Charge-transfer gap ~ U - W ~ - W Charge-transfer energy: On-site Coulomb energy: Band width: W L: ligand (p) hole

10 Resonant photoemission discrete level continuous level

11 Resonant photoemission discrete level Fano line shape continuous level q = [g.st.-discr.]/[discr.-contin.] Effectively enhances the 3d photoionization cross-section

12 Photoemission spectra of NiO XPS spectrum main peaks LDA band calc. satellite Ligand-field theory T. Oguchi et al., PRB 83

13 Resonant photoemission spectra of NiO satellite main peaks Ni 3p core abs. S.-J. Oh et al., PRB 82

14 Mott insulators Cluster-model description

15 Cluster model for transition-metal oxides BO 6 cluster model AB 2 O 4 spinel perovskite treated as adjustable parameters

16 atomic d and p orbitals, molecular orbitals on the cluster Atomic d orbitals Crystal-field splitting

17 atomic d and p orbitals, molecular orbitals on the cluster Molecular orbitals composed of atomic p orbitals Atomic d orbitals

18 Many-electron energy level scheme for BO 6 cluster N Inverse photoemission Optical absorption Ground state Photoemission : Band gap = E N+1 + E N-1-2E N Multiplet effects

19 Many-electron energy levels vs single-particle energy level Photoemission Inverse photoemission µ µ : chemical potential E g : band gap Inversephotoemission spectra Photoemission E spectra N+1 E N-1 E g Ground state

20 Mott-Hubbard-type insulators vs charge-transfer-type insulators chemical potential µ W Inversephotoemission spectra W U < U > Photoemission spectra Mott-Hubbard gap Charge-transfer gap ~ U - W ~ - W Charge-transfer energy: On-site Coulomb energy: Band width: W L: ligand (p) hole

21 Mott-Hubbard type versus charge-transfer type many-electron energy level scheme Charge-transfer type insulator U > Mott-Hubbard type insulator U < N

22 Configuration-interaction cluster-model analysis of d-electron photoemission spectra Ground state Final states Photoemission satellite Ground state main Intensities

23 Configuration-interaction cluster-model analysis of d-electron photoemission satellites d n-1 final state U - d n L final state charge-transfer type U > - U Mott-Hubbard type U <

24 Configuration-interaction cluster-model analysis vs LDA band theory for NiO LDA band calc. t 2g satellite O 2p e g t 2g e g O 2p T. Oguchi et al., PRB 83 G.A. Sawatzky and J.W. Allen, PRL 84 A. Fujimori and F. Minami, PRB 83

25 Configuration-interaction cluster-model analysis of core-level satellite satellite Ground state with core hole main Final states e photoemission hν without core hole Intensities ground state I.H. Inoue et al., PRB 92 G. van der Laan et al., PRB 81

26 Configuration-interaction cluster-model analysis of core-level satellite Mn 2p 3/2 2+ Mn 2p 1/2 Mn 2p 3/2 satellite = 9 ev satellite = 6.5 ev = 4.5 ev 3+ = 4.5 ev = 3.2 ev 4+ = 2.0 ev J. Park et al., PRB 88 G. van der Laan et al., PRB 86 A.E. Bocquet et al., PRB 92

27 Chemical trend Mott insulators

28 Systematic variation of band gaps in transition-metal oxides U eff, eff U eff, eff : Eestimated from ionic model T. Arima et al., PRB 93

29 Systematic materials dependence of charge-transfer energy Z v ~ 23 ev, 22.5 ev for selenides, tellurides A.E. Bocquet et al., PRB 92

30 Systematic materials dependence of on-site Coulomb energy U Z v A.E. Bocquet et al., PRB 92

31 Systematic materials dependence of p-d transfer integral T pd 3(pdσ), 2(pdπ) A.E. Bocquet et al., PRB 92

32 Zaanen-Sawatzky-Allen diagram = W E g ~ W charge-transfer regime 3+ charge-transfer regime p-metal d-metal Mott-Hubbard regime E g ~ U - W U = W 4+ negative- regime Mott-Hubbard regime J. Zaanen, G.A. Sawatzky, J.W. Allen, PRL 85 A.E. Bocquet et al., PRB 96

33 Systematic variation of band gaps in transition-metal oxides U eff, eff U eff, eff : Eestimated from ionic model T. Arima et al., PRB 93

34 Multiplet corrections for Mott-Hubbard gap and charge-transfer gap Correction for charge-transfer energy: eff Multiplet corrections for and U Correction for on-site Coulomb energy: U U eff CT gap is reduced d 4 d 5 M-H and CT gap is enhanced T. Saitoh et al., PRB 95

35 Multiplet corrections for Mott-Hubbard gap and charge-transfer gap Optical gaps Calculated band gaps d 3 d 3 T. Arima et al., PRB 93 T. Saitoh et al., PRB 95

36 Zaanen-Sawatzky-Allen diagram = W E g ~ W charge-transfer regime 3+ charge-transfer regime p-metal d-metal Mott-Hubbard regime E g ~ U - W U = W 4+ negative- regime Mott-Hubbard regime J. Zaanen, G.A. Sawatzky, J.W. Allen, PRL 85 A.E. Bocquet et al., PRB 96

37 Negative- (covalent) insulator p-p gap determined by p-d hybridization strength Modified Zaanen-Sawatzky-Allen diagram Ex.) NaCu 3+ (d 8 )O 2 ground state: T. Mizokawa et al., PRL 94 cf) Covalent insulator: S. Nimkar et al., PRB 93

38 Band description Mott insulators

39 Hartree-Fock and LDA+U band calculations - failure of LDA in NiO Hartree-Fock band calc. t 2g e e g g t 2g O 2p E g ~ 4 ev Local-density-approximation (LDA) band calc. O 2p t 2g t 2g e g e g E g ~ 0.2 ev T. Mizokawa and A.F., PRB 96 O 2p e g LDA+U band calc. t 2g t 2g O 2p E g ~ 4 ev e g CoO, FeO: metallic! V.I. Anisimov et al., PRB 91 T. Oguchi et al., PRB 83

40 Failure of LDA in Mott insulators Hartree-Fock potential energy (also for LDA+U) : occupation number of orbital i orbital-dependent self-consistent potential positive feedback toward orbital polarization Local-density approximation (LDA) potential energy : total occupation number (local density) spherically averaged potential, unphysical self-interaction orbital polarization suppressed

41 Orbital magnetic moments in Fe 3 O 4 Fe 3+ (d 5 : t 2g 3 e g 2 ) <L Z > = 0 Fe 2+ (d 6 : t 2g 3 e g 2 t 2g ) <L Z > ~ -1 Fe 3p MCD Fe 2p MCD D.J. Huang et al., unpublished T. Koide et al., PRB 91

42 Magnetic circular dichroism (MCD) in core-level absorption

43 Orbital ordering in perovskite-type ABO 3 compounds orbital 1 orbital 2 ex) LaMn 3+ O 3 Jahn-Teller distortion d 4 : t 2g 3 e g

44 Charge and orbital ordering in R A 0.5 MnO MnO 3 Mn 3.5+ (d 3.5 : t 2g 3 e g 0.5 ) Jahn-Teller distortion Breathing-type distortion T Mizokawa and A.F., PRB 97

45 Mott insulators Self-energy correction

46 Hartree-Fock band calculation + self-energy correction Σ(ω) expt expt Green s function: Spectral function: calculated with 2nd order perturbation Hartree-Fock eigenvalue T. Mizokawa and A. Fujimori, PRB 96

47 CI cluster model, Hartree-Fock band theory and photoemission spectra Experimental input band gaps magnetic moment hybridization strength