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• Introduction• Cluster-model description• Chemical trend• Band description• Self-energy correction
Mott insulatorsMott insulators
• Introduction
Mott insulatorsMott insulators
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
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
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
Band gap excitations - relevant toBand gap excitations - relevant tocharge 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
Photoemission spectroscopyPhotoemission spectroscopy
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
Mott-Hubbard-type insulators Mott-Hubbard-type insulators vsvscharge-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 Photoemission spectra
Inverse-photo-
emission
spectra
L: ligand (p) hole
U < ∆ U > ∆
W
W
Resonant photoemissionResonant photoemission
discrete level
continuous level
Resonant photoemissionResonant photoemission
Fano line shape
Effectively enhances the 3d photoionization cross-section
discrete level
continuous level
q = [g.st.-discr.]/[discr.-contin.]
Photoemission spectra of Photoemission spectra of NiONiO
satellite
Ligand-field theory
T. Oguchi et al., PRB ‘83
LDA band calc.XPS spectrum
main peaks
S.-J. Oh et al., PRB ‘82
Resonant photoemission spectra of Resonant photoemission spectra of NiONiO
satellite
Ni 3p core abs.
main peaks
• Cluster-model description
Mott insulatorsMott insulators
Cluster model for transition-metal oxidesCluster model for transition-metal oxides
treated as adjustable parameters
perovskiteAB2O4 spinel
BOBO66 cluster model cluster model
atomic atomic dd and and pp orbitals, molecular orbitals orbitals, molecular orbitalson the clusteron the cluster
Atomic d orbitals
Crystal-field splitting
Molecular orbitals composed of atomic p orbitals
Atomic d orbitals
atomic atomic dd and and pp orbitals, molecular orbitals orbitals, molecular orbitalson the clusteron the cluster
Many-electron energy level schemeMany-electron energy level schemefor BOfor BO66 cluster cluster
N
: Band gap= EN+1 + EN-1 - 2EN
Ground state
Photoemission
Inverse
photoemission
Opt
ical
ab
sorp
tion
Multiplet effects
Many-electron energy levelsMany-electron energy levelsvsvs single-particle energy level single-particle energy level
Photoemission
Inverse
photoemission
µ
µ : chemical potentialEg : band gap
Photoemission
spectra
Eg
Inverse-photoem
ission spectra
Ground state
EN
+1
EN
-1
Mott-Hubbard-type insulators Mott-Hubbard-type insulators vsvscharge-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 Photoemission spectra
Inverse-photo-
emission
spectra
L: ligand (p) hole
U < ∆ U > ∆
W
W
Mott-Hubbard type Mott-Hubbard type versusversus charge-transfer type charge-transfer typemany-electron energy level schememany-electron energy level scheme
Mott-Hubbard typeinsulator
Charge-transfer typeinsulator
N
U > ∆
U < ∆
Configuration-interaction cluster-modelConfiguration-interaction cluster-modelanalysis of analysis of dd-electron photoemission spectra-electron photoemission spectra
Ground state
Photoemission
Ground state
Final states
Intensities
main
satellite
Configuration-interaction cluster-modelConfiguration-interaction cluster-modelanalysis of analysis of dd-electron photoemission satellites-electron photoemission satellites
dn-1 final statednL final state
U - ∆
∆ - U
U > ∆
U < ∆
charge-transfer type
Mott-Hubbard type
G.A. Sawatzky and J.W. Allen, PRL ‘84A. Fujimori and F. Minami, PRB ‘83
T. Oguchi et al., PRB ‘83
Configuration-interaction cluster-modelConfiguration-interaction cluster-modelanalysis analysis vsvs LDA band theory for LDA band theory for NiONiO
satellite
LDA band calc.
O 2p
O 2peg↓
t2g↑
t2g↓
eg↑
I.H. Inoue et al., PRB ‘92G. van der Laan et al., PRB ‘81
Configuration-interaction cluster-modelConfiguration-interaction cluster-modelanalysis 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
Configuration-interaction cluster-modelConfiguration-interaction cluster-modelanalysis of core-level satelliteanalysis of core-level satellite
J. Park et al., PRB ‘88G. 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
• Chemical trend
Mott insulatorsMott insulators
Systematic variation of band gaps inSystematic variation of band gaps intransition-metal oxidestransition-metal oxides
T. Arima et al., PRB ‘93
Ueff, ∆eff: Eestimated from ionic model
Ueff, ∆eff
Systematic materials dependence ofSystematic materials dependence ofcharge-transfer energy charge-transfer energy ∆∆
A.E. Bocquet et al., PRB ‘92
Z v
~ 23 eV, 22.5 eV for selenides, tellurides
Systematic materials dependence ofSystematic materials dependence ofon-site Coulomb energy on-site Coulomb energy UU
A.E. Bocquet et al., PRB ‘92
Z v
Systematic materials dependence ofSystematic materials dependence of p-d p-d transfer integraltransfer integral
A.E. Bocquet et al., PRB ‘92
Tpd ≡ √3(pdσ), 2(pdπ)
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 regimecharge-transfer
regime
nega
tive-∆
regi
me
Eg ~ ∆ − W
Eg ~ U - Wp-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+
Systematic variation of band gaps inSystematic variation of band gaps intransition-metal oxidestransition-metal oxides
T. Arima et al., PRB ‘93
Ueff, ∆eff: Eestimated from ionic model
Ueff, ∆eff
Multiplet corrections for Mott-Hubbard gapMultiplet corrections for Mott-Hubbard gapand 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
Multiplet corrections for Mott-Hubbard gapMultiplet corrections for Mott-Hubbard gapand charge-transfer gapand charge-transfer gap
T. Saitoh et al., PRB ‘95T. Arima et al., PRB ‘93
Calculated band gapsOptical gaps
d3
d3
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 regimecharge-transfer
regime
nega
tive-∆
regi
me
Eg ~ ∆ − W
Eg ~ U - Wp-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+
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
• Band description
Mott insulatorsMott insulators
Hartree-Fock and LDA+Hartree-Fock and LDA+UU band calculations band calculations - failure of LDA in - failure of LDA in NiONiO
Local-density-approximation (LDA) band calc.
O 2p
O 2peg↓
t2g↑
t2g↓eg↑
eg↓
eg↓O 2p
O 2pt2g↑
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 !
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
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
Magnetic circular Magnetic circular dichroism dichroism (MCD) in(MCD) incore-level absorptioncore-level absorption
Orbital ordering inOrbital ordering inperovskite-type ABOperovskite-type ABO3 3 compoundscompounds
orbital 1
orbital 2
ex) LaMn3+O3
d4: t2g↑3 eg↑
Jahn-Teller distortion
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 )
• Self-energy correction
Mott insulatorsMott insulators
Hartree-Fock band calculation +Hartree-Fock band calculation +self-energy correction self-energy correction Σ(ω)Σ(ω)
T. Mizokawa and A. Fujimori, PRB ‘96calculated with 2nd order perturbation
Hartree-Fock eigenvalue
exptexpt
Spectral function: Green’s function:
CI cluster model, Hartree-Fock band theoryCI cluster model, Hartree-Fock band theoryand photoemission spectraand photoemission spectra
Experimental input
band gapsmagnetic momenthybridization strength