• Introduction • Cluster-model description • Chemical trend • Band description • Self-energy correction

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

2. • Introduction
Mott insulatorsMott insulators

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

4. Lattice models for transition-metal compoundsLattice 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 excitationBand gap excitation and localized excitation
Band gap excitation
energy: Eg = EN+1 + EN-1 - 2EN
EN-1 - EN EN+1 - EN
E*N - EN
Localized excitation
(d-d excitation, exciton, ...)
Relevant to charge transport
Photoemission Inverse-photoemission

6. Band gap excitations - relevant toBand gap excitations - relevant to
charge transportcharge transport
Excitation energy: Eg = EN+1 + EN-1 - 2EN
EN-1 - EN EN+1 - EN
U
∆
Charge transfer energy:
on-site Coulomb energy:
L: ligand (p) hole

7. Photoemission spectroscopyPhotoemission spectroscopy

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

9. Mott-Hubbard-type insulatorsMott-Hubbard-type insulators vsvs
charge-transfer-type insulatorscharge-transfer-type insulators
Charge-transfer energy:
On-site Coulomb energy:
Band width: W
µ
Mott-Hubbard gap Charge-transfer gap
~ U - W ~ ∆ - W
chemical potential
Photoemissionspectra
Inverse-
photo-
emission
spectra
L: ligand (p) hole
U < ∆ U > ∆
W
W

10. Resonant photoemissionResonant photoemission
discrete level
continuous level

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

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

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

14. • Cluster-model description
Mott insulatorsMott insulators

15. Cluster model for transition-metal oxidesCluster model for transition-metal oxides
treated as adjustable parameters
perovskite
AB2O4 spinel
BOBO66 cluster modelcluster model

16. atomicatomic dd andand pp orbitals, molecular orbitalsorbitals, molecular orbitals
on the clusteron the cluster
Atomic d orbitals
Crystal-field splitting

17. Molecular orbitals composed
of atomic p orbitals
Atomic d orbitals
atomicatomic dd andand pp orbitals, molecular orbitalsorbitals, molecular orbitals
on the clusteron the cluster

18. Many-electron energy level schemeMany-electron energy level scheme
for BOfor BO66 clustercluster
N
: Band gap
= EN+1 + EN-1 - 2EN
Ground state
Photoem
ission
Inverse
photoem
ission
Optical
absorptionMultiplet effects

19. Many-electron energy levelsMany-electron energy levels
vsvs single-particle energy levelsingle-particle energy level
Photoem
ission
Inverse
photoem
ission
µ
µ : chemical potential
Eg : band gap
Photoemission
spectra
Eg
Inverse-
photoemission
spectra
Ground state
EN+1EN-1

20. Mott-Hubbard-type insulatorsMott-Hubbard-type insulators vsvs
charge-transfer-type insulatorscharge-transfer-type insulators
Charge-transfer energy:
On-site Coulomb energy:
Band width: W
µ
Mott-Hubbard gap Charge-transfer gap
~ U - W ~ ∆ - W
chemical potential
Photoemissionspectra
Inverse-
photo-
emission
spectra
L: ligand (p) hole
U < ∆ U > ∆
W
W

21. Mott-Hubbard typeMott-Hubbard type versusversus charge-transfer typecharge-transfer type
many-electron energy level schememany-electron energy level scheme
Mott-Hubbard type
insulator
Charge-transfer type
insulator
N
U > ∆
U < ∆

22. Configuration-interaction cluster-modelConfiguration-interaction cluster-model
analysis ofanalysis of dd-electron photoemission spectra-electron photoemission spectra
Ground state
Photoem
ission
Ground state
Final states
Intensities
main
satellite

23. Configuration-interaction cluster-modelConfiguration-interaction cluster-model
analysis ofanalysis of dd-electron photoemission satellites-electron photoemission satellites
dn-1 final state
dnL final state
U - ∆
∆ - U
U > ∆
U < ∆
charge-transfer type
Mott-Hubbard type

24. G.A. Sawatzky and J.W. Allen, PRL ‘84
A. Fujimori and F. Minami, PRB ‘83
T. Oguchi et al., PRB ‘83
Configuration-interaction cluster-modelConfiguration-interaction cluster-model
analysisanalysis vsvs LDA band theory forLDA band theory for NiONiO
satellite
LDA band calc.
O 2p
O 2p
eg↓
t2g↑
t2g↓
eg↑

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

26. Configuration-interaction cluster-modelConfiguration-interaction cluster-model
analysis of core-level satelliteanalysis of core-level satellite
J. Park et al., PRB ‘88
G. van der Laan et al., PRB ‘86 A.E. Bocquet et al., PRB ‘92
Mn 2p3/2
∆ = 9 eV
2+
3+
4+
∆ = 4.5 eV
∆ = 3.2 eV
∆ = 6.5 eV
∆ = 4.5 eV
∆ = 2.0 eV
Mn 2p3/2Mn 2p1/2
satellite
satellite

27. • Chemical trend
Mott insulatorsMott insulators

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

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

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

31. Systematic materials dependence ofSystematic materials dependence of
p-dp-d transfer integraltransfer integral
A.E. Bocquet et al., PRB ‘92
Tpd ≡ √3(pdσ),
2(pdπ)

32. Zaanen-Sawatzky-Allen diagramZaanen-Sawatzky-Allen diagram
A.E. Bocquet et al., PRB ‘96J. Zaanen, G.A. Sawatzky, J.W. Allen, PRL ‘85
Mott-Hubbard
regime
Mott-Hubbard
regime
charge-transfer
regime
charge-transfer
regime
negative-∆
regime
Eg ~ ∆ − W
Eg ~ U - W
p-metal
d-metal
U = W
∆ = W
4+
3+
3+
2+
3+
3+
3+ 3+
3+
3+3+
3+
2+
2+
2+
2+
4+
4+
4+
5+

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

34. Multiplet corrections for Mott-Hubbard gapMultiplet corrections for Mott-Hubbard gap
and charge-transfer gapand charge-transfer gap
T. Saitoh et al., PRB ‘95
Correction for
charge-transfer energy: ∆ → ∆eff
Correction for
on-site Coulomb energy: U → Ueff
Multiplet corrections for ∆ and U
d5
d4
M-H and CT gap
is enhanced
CT gap is
reduced

35. Multiplet corrections for Mott-Hubbard gapMultiplet corrections for Mott-Hubbard gap
and charge-transfer gapand charge-transfer gap
T. Saitoh et al., PRB ‘95T. Arima et al., PRB ‘93
Calculated band gapsOptical gaps
d3
d3

36. Zaanen-Sawatzky-Allen diagramZaanen-Sawatzky-Allen diagram
A.E. Bocquet et al., PRB ‘96J. Zaanen, G.A. Sawatzky, J.W. Allen, PRL ‘85
Mott-Hubbard
regime
Mott-Hubbard
regime
charge-transfer
regime
charge-transfer
regime
negative-∆
regime
Eg ~ ∆ − W
Eg ~ U - W
p-metal
d-metal
U = W
∆ = W
4+
3+
3+
2+
3+
3+
3+ 3+
3+
3+3+
3+
2+
2+
2+
2+
4+
4+
4+
5+

37. Negative-Negative-∆∆ (covalent) insulator(covalent) insulator
T. Mizokawa et al., PRL ‘94
Ex.) NaCu3+(d8)O2
ground state:
cf) Covalent insulator: S. Nimkar et al., PRB ‘93
p-p gap
determined by
p-d hybridization strength
Modified Zaanen-Sawatzky-Allen diagram

38. • Band description
Mott insulatorsMott insulators

39. Hartree-Fock and LDA+Hartree-Fock and LDA+UU band calculationsband calculations
- failure of LDA in- failure of LDA in NiONiO
Local-density-approximation (LDA)
band calc.
O 2p
O 2p
eg↓
t2g↑
t2g↓
eg↑
eg↓
eg↓O 2p
O 2p
t2g↑
t2g↓
eg↑
t2g↑ t2g↓
eg↑
LDA+U band calc.
Hartree-Fock band calc.
T. Oguchi et al., PRB ‘83V.I. Anisimov et al., PRB ‘91
T. Mizokawa and A.F., PRB ‘96
Eg ~ 4 eV
Eg ~ 4 eV
Eg ~ 0.2 eV
CoO, FeO: metallic !

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

41. Orbital magnetic moments in FeOrbital magnetic moments in Fe33OO44
T. Koide et al., PRB ‘91
Fe 3p MCD
Fe 2p MCD
D.J. Huang et al., unpublished
Fe3+ (d5 : t2g↑
3 eg↑
2 ) <LZ> = 0
Fe2+ (d6 : t2g↑
3 eg↑
2 t2g↓ ) <LZ> ~ -1

42. Magnetic circularMagnetic circular dichroismdichroism (MCD) in(MCD) in
core-level absorptioncore-level absorption

43. Orbital ordering inOrbital ordering in
perovskite-type ABOperovskite-type ABO33 compoundscompounds
orbital 1
orbital 2
ex) LaMn3+O3
d4: t2g↑
3 eg↑
Jahn-Teller
distortion

44. Charge and orbital ordering in RCharge and orbital ordering in R0.50.5AA0.50.5MnOMnO33
Jahn-Teller
distortion
Breathing-type
distortion
T．Mizokawa and A.F., PRB ‘97
3+ 2+
Mn3.5+ (d3.5 : t2g↑
3 eg↑
0.5 )

45. • Self-energy correction
Mott insulatorsMott insulators

46. Hartree-Fock band calculation +Hartree-Fock band calculation +
self-energy correctionself-energy correction Σ(ω)Σ(ω)
T. Mizokawa and A. Fujimori, PRB ‘96
calculated with 2nd order perturbation
Hartree-Fock eigenvalue
exptexpt
Spectral function:Green’s function:

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