Mott insulatorsMott insulators

• Mott-Hubbard type vs charge-transfer type • Cluster-model description • Chemical trend• Band theory• Self-energy correction• Electron-phonon interaction

Mott insulatorsMott insulators

• Mott-Hubbard type vs charge-transfer type

Correlated electron systems/Correlated electron systems/Complex materials Complex materials

La2-xSrxCuO4

Ga1-xMnxAs

Lattice models for transitionLattice models for transition--metal compounds metal compounds

p-d modelHubbard model

Transition metal ion (with d orbitals)

Non-metal anion (with p orbitals)

Lattice models for transitionLattice models for transition--metal compounds metal compounds

p-d model(degenerate) Hubbard model

t-J model

no double occopancy

Band gap excitations Band gap excitations -- relevant to relevant to charge transportcharge transport

Hubbard modelp-d model

U U

∆Charge transfer energy: ∆on-site Coulomb energy: U

Photoemission spectroscopyPhotoemission spectroscopy

MottMott--HubbardHubbard--type insulators type insulators vsvschargecharge--transfertransfer--type insulatorstype insulators

µ

Mott-Hubbard gap Charge-transfer gap~ U - W ~ ∆ - W

chemical potential Photoem

ission spectra

Inverse-photo-

emission

spectra

U < ∆ U > ∆

Charge transfer energy: ∆on-site Coulomb energy: UBand width: W

µchemical potentialW

W

UHB

LHB

Oxygen p band

Oxygen p band

UHB

LHB

UHB: upper Hubbard bandLHB: lower Hubbard band

Mott insulatorsMott insulators

• Cluster-model description

Cluster model for transitionCluster model for transition--metal oxidesmetal oxides

are treated as adjustable parameters

perovskite

AB2O4 spinel

L: ligand (p) hole

ABO3 perovskite

BO6 cluster model

ManyMany--electron energy levels electron energy levels vsvs singlesingle--particle energy level particle energy level

Photoemission

Inverse

photoemission

µ Photoem

ission spectra

Eg

Inverse-photoem

ission spectra

Ground state

EN

+1

EN

-1 µ : chemical potentialEg : band gap

MottMott--Hubbard type Hubbard type versusversus chargecharge--transfer type transfer type manymany--electron energy level scheme electron energy level scheme

Mott-Hubbard typeinsulator

NU < ∆

Charge-transfer typeinsulator

U > ∆

ConfigurationConfiguration--interaction clusterinteraction cluster--model model analysis of analysis of dd--electron photoemission satelliteselectron photoemission satellites

dn-1 final statednL final state

U - ∆

∆ - U

U > ∆

U < ∆

charge-transfer type

Mott-Hubbard type

ConfigurationConfiguration--interaction clusterinteraction cluster--model model analysis analysis vsvs LDA band theory for LDA band theory for NiONiO

G.A. Sawatzky and J.W. Allen, PRL ‘84A. Fujimori and F. Minami, PRB ‘83

T. Oguchi et al., PRB ‘83

satellite

LDA band calc.

O 2p

O 2peg↓

t2g↑

t2g↓

eg↑

Mott insulatorsMott insulators

• Chemical trend

Systematic variation of band gaps in Systematic variation of band gaps in transitiontransition--metal oxidesmetal oxides

Ueff, ∆eff

T. Arima et al., PRB ‘93

Systematic variation of band gaps in Systematic variation of band gaps in transitiontransition--metal oxidesmetal oxides

Ueff, ∆eff: Eestimated from ionic model

T. Arima et al., PRB ‘93

Systematic materials dependence of Systematic materials dependence of chargecharge--transfer energy transfer energy ∆∆

Z v

A.E. Bocquet et al., PRB ‘92

~ 23 eV, 22.5 eV for selenides, tellurides

Systematic materials dependence of Systematic materials dependence of onon--site Coulomb energy site Coulomb energy UU

Z v

A.E. Bocquet et al., PRB ‘92

Systematic materials dependence of Systematic materials dependence of pp--d d transfer integral transfer integral

Tpd ≡ √3(pdσ), 2(pdπ)

A.E. Bocquet et al., PRB ‘92

ZaanenZaanen--SawatzkySawatzky--Allen diagramAllen diagram

Mott-Hubbard regime

Mott-Hubbard regime

charge-transferregime

charge-transfer regime

nega

tive-

∆re

gim

e

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+

J. Zaanen, G.A. Sawatzky, J.W. Allen, PRL ‘85 A.E. Bocquet et al., PRB ‘96

MottMott--HubbardHubbard--type insulators type insulators vsvschargecharge--transfertransfer--type insulatorstype insulators

µ

Mott-Hubbard gap Charge-transfer gap~ U - W ~ ∆ - W

chemical potential Photoem

ission spectra

Inverse-photo-

emission

spectra

U < ∆ U > ∆

Charge transfer energy: ∆on-site Coulomb energy: UBand width: W

µchemical potentialW

W

UHB

LHB

Oxygen p band

Oxygen p band

UHB

LHB

UHB: upper Hubbard bandLHB: lower Hubbard band

Systematic variation of band gaps in Systematic variation of band gaps in transitiontransition--metal oxidesmetal oxides

Ueff, ∆eff

Ueff, ∆eff: Eestimated from ionic model

T. Arima et al., PRB ‘93

Multiplet corrections for MottMultiplet corrections for Mott--Hubbard gap Hubbard gap and chargeand charge--transfer gaptransfer gap

Correction for charge-transfer energy: ∆ → ∆eff

Correction for on-site Coulomb energy: U → Ueff

d5

d4

M-H and CT gap is enhanced

CT gap is reduced

Multiplet corrections for ∆ and U

T. Saitoh et al., PRB ‘95

Multiplet corrections for MottMultiplet corrections for Mott--Hubbard gap Hubbard gap and chargeand charge--transfer gaptransfer gap

Optical gaps

d3

d3

Calculated band gaps

T. Arima et al., PRB ‘93 T. Saitoh et al., PRB ‘95

Mott insulatorsMott insulators

• Band theory

HartreeHartree--Fock and LDA+Fock and LDA+UU band calculationsband calculations-- failure of LDA in failure of LDA in NiONiO --

Hartree-Fock band calc.

O 2p(6)

O 2p(6)eg↓(2)t2g↑(3)

t2g↓(3)

eg↑(2)

eg↓

eg↓O 2p

O 2pt2g↑

t2g↓

eg↑

t2g↑ t2g↓

eg↑LDA+U band calc.

V.I. Anisimov et al., PRB ‘91

T. Mizokawa and A.F., PRB ‘96

Eg ~ 4 eV

Eg ~ 4 eV

Eg ~ 0.2 eV

eg(4)

t2g(6)

6 8 2

6 8 2

Local-density-approximation (LDA) band calc.- NiO (14 valence electrons)

CoO, FeO: metallic !

T. Oguchi et al., PRB ‘83

Failure of LDA in Mott insulatorsFailure 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

Orbital ordering in Orbital ordering in perovskiteperovskite--type ABOtype ABO3 3 compoundscompounds

Jahn-Teller distortion

orbital 1

orbital 2

ex) LaMn3+O3

d4: t2g↑3 eg↑

Mott insulatorsMott insulators

• Self-energy correction

HartreeHartree--Fock band calculation + Fock band calculation + selfself--energy correction energy correction Σ(ω)Σ(ω)

T. Mizokawa and A. Fujimori, PRB ‘96calculated with 2nd order perturbation

Hartree-Fock eigenvalue

expt

Spectral function: Green’s function:

V 3dO 2p

expt

Ni 3dO 2p

Mott-Hubbard type Charge-transfer type

O 2p band V 3d band

CI cluster model, HartreeCI cluster model, Hartree--Fock band theory Fock band theory and photoemission spectraand photoemission spectra

Experimental input

band gapsmagnetic momenthybridization strength

Mott insulatorsMott insulators

• Electron-phonon interaction

Discrepancy of spectral line shapes between Discrepancy of spectral line shapes between band theory and photoemission spectraband theory and photoemission spectra

Mott-Hubbard type insulator VO25

4

3

2

1

0

DO

S (s

tate

s/eV

f.u.

)

2.5 2.0 1.5 1.0 0.5 0

Binding Energy (eV)

band calc. spectrum (280K)

VO2/TiO2(001)

O 2p band V 3d band

10-5

10-4

10-3

10-2

10-1

100

101

Res

istiv

ity (

ohm

cm

)

400350300250200

Temperature (K)

Ts = 553 K

Ts = 583 K

Ts = 643 K

TiO2(001)

c

PO2 = 1 Pa TMI bulk

metallic phase

insulating phase

LDA+U band-structure calculation, X. Hunag, et al., cond-mat/98

Y. Muraoka et al.

20 15 10 5 0Energy /ω0

zero-phonon line

multiple-phonon lines

ElectronElectron--phonon interaction phonon interaction in the insulating phase of VOin the insulating phase of VO22

Simulations using independent-boson model

K. Okazaki et al. PRB ‘04

20 15 10 5 0Energy /ω0

zero-phonon line

multiple-phonon lines

Local co-ordinateEle

ctro

n +

latti

ce e

nerg

y

Zero-phonon line

N-electron

N-1-electron Frank-Condontransition

ElectronElectron--phonon interaction phonon interaction in the insulating phase of VOin the insulating phase of VO22

Simulations using independent-boson model