mott insulators

• 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