physics of correlated electron materials: experiments with photoelectron spectroscopy ralph claessen...

Physics of correlated electron materials:Experiments with photoelectron spectroscopy

Ralph Claessen

U Würzburg, Germany

e-h

Summer School on Ab-initio Many-Body Theory, San Sebastian, 25-07-2007

Outline:

• Photoemission of interacting electron systems

• Mott-Hubbard physics in transition metal oxides

• Correlation effects in 1D

• TiOCl: Challenges for ab initio many-body theory

e-h

Angle-resolved photoelectron spectroscopy

non-interacting electrons

ARPES

band structure E(k)

interacting electrons

ARPES

spectral function

),(Im),( 1 kGkA

)(kE

Photoemission: many-body effects

Ekin

h

electron-electron interaction

photoelectron: "loss" of kinetic energy due to excitation energy stored in the remaining interacting system !

Many-body theory of photoemission

Fermi´s Golden Rule for N-particle states:

with

N-electron ground state of energy EN, 0

N-electron excited state of energy EN, s,

consisting of N-1 electrons in the solid and one free photoelectron of momentum and energy

in second quantization

)(ˆ),( 0,,2

0,, hEEkI NsNs

isf

0,0, Ni

sNksf ,1,,

k

if kkif

N

iii ccMprA

1

)(ˆ

one-particle matrix element )( fi kk



)(ˆ),( 0,,2

0,, hEEkI NsNs

isf

sNksf ,1,,

sNck ,1

SUDDEN APPROXIMATION:

Factorization !

photoelectron sth eigenstate of remaining N-1 electron system

Physical meaning:photoelectron decouples from remaining system immediately after photoexcitation, before relaxation sets in



)(0,,1),( 0,,1

22 hEENcsNMkI NsN

skif

sNksf ,1,,

sNck ,1

SUDDEN APPROXIMATION:

Factorization !

photoelectron sth eigenstate of remaining N-1 electron system

Physical meaning:photoelectron decouples from remaining system immediately after photoexcitation, before relaxation sets in


If additionally Mif ~ const in energy and k-range of interest:

)(),(

)(0,,1),( 0,,12

hfhkA

hEENcsNkI NsNs

k

The ARPES signal is directly proportional to the

single-particle spectral function ),(Im1

),(

kGkA

single-particle Green´s function

),( kI

otherelectrons

phonons

spin excitations

?

L. Åsbrink, Chem. Phys. Lett. 7, 549 (1970)

Many-body effects in photoemission

Example: Photoemission from the H2 molecule

Ekin

H2

E

g

u*




Ekin

H2

Eelectrons couple to proton dynamics !

photoemission intensity:

electronic-vibrational eigenstates of H2+:

2,

1,

0,,

1

1

12

v

v

vsH

)(0,ˆ,)( 0,,

2

22 22HsH

s

EEHcsHI g

u*




Ekin

H2

E Franck-Condon principle

proton distance

ener

egy

v' = 0

v = 0

v = 1

v = 2

ħ0g

u*

Caveat: Effect of photoelectron lifetime

ARPES intensity actually convolution of photohole and photoelectron spectral function

),,(),,(),(

kkAhkkAdkkI eh llllll

h

h

e

ener

gy

k

slope

k

v hh

ev

ee

hhtot v

v

tot

assuming Lorentzian lineshapes the total width is given by

~ meV~ eV

spectrum dominated by photo-electron linewidth unless

1

e

h

v

v low-dim systems !

Outline:





e-h

Transition metal oxides

oxides of the 3d transition metals: M = Ti, V, … ,Ni, Cu

basic building blocks: MO6 octahedra

electronic configuration: O 2s2p6 = [Ne]

M 3dn

cubic perovskites perovskite-like anatas rutile spinel

O2-

quasi-atomic,strongly localized

Hubbard model

iii

jiji nnUcctH

,,

ˆt

U kinetic energy,itinerancy

local Coulomb energy,localization

k-integrated spectral function for limiting cases (non-interacting bandwidth W t ):

U/W << 1

U/W >> 1

Hubbard model with half-filled band (n=1)

iii

jiji nnUcctH

,,

ˆ

d1 configuration (Ti3+, V4+)

A()

one-electron conduction band: metal

U

atomic limit: Mott insulator

W

Photoemission of a Mott insulator

TiOCl

O 2p / Cl 3p

Ti 3d1

U

d1 d0

LHBd1 d2

UHB

Bandwidth-controlled Mott transition

dynamical mean-field theory

band metal

insulator

evolution of quasiparticle peak for local self-energy ()

correlated metal

dynamical mean-field theory of the Hubbard model

Photoemission of a correlated d1 metal

A. Fujimori et al., PRL 1992

LHBQP

O 2p V 3d1

incoherentweight coherent

excitations

LHB

QP

Spectral evolution through the Mott transition

A. Fujimori et al., PRL 1992

DMFTphotoemission

QPLHB

QPLHB UHB

Surface effects in photoemission

photoelectron mean free path (Ekin)

Ekin ~ h

A. Sekiyama et al., PRL 2004

CaVO3

40 eV 275 eV

900 eV

LHB

QP

surface

bulk

h

(Ekin)

Surface effects in photoemission

A. Sekiyama et al., PRL 2004

CaVO3

LHB

QP

at surface reduced atomic coordination

effective bandwidth smaller:Wsurf < Wbulk

surface stronger correlated:U / Wsurf >U / Wbulk

Surface versus bulk: V2O3

S.K. Mo et al., PRL 90, 186403 (2003)

unit celld ~ 8 Å

(40 eV) ~ 5 Å

surface

(800 eV) ~ 15 Å

(6 keV) ~ 50 Å

G. Panaccione et al., PRL 97, 116401 (2006)

soft x-ray PES (h ~ several 100 eV)

hard x-ray PES (h ~ several keV)

Outline:





e-h

Spectral function of a Fermi liquid

Fermi liquid

dressed quasiparticles

non-interacting electrons

bare particles

EF=0

k0 k

energy

k

kF

A(k,)

E0(k)

k

EF

kF

charge

spin

Electron-electron interaction in 1D metals

EF

de

nsity

of

sta

tes

0.125

21.5

1

0.5

= ~

chargespin

Voit (1995)Schönhammer and Meden (1995)

Tomonaga-Luttinger model:

Strongly coupled electrons: 1D Hubbard model

,ijji cctH t – hopping integral

1D atomic (or molecular) chain

i

ii nnU

U – local Coulomb energy

J t 2/U - magnetic exchange energy

t t

U-J




i

ii nnU


J t 2/U - magnetic exchange energy strong coupling U >> t




i

ii nnU



t

J

strong coupling U >> t

spinon holon




i

ii nnU



J




iii nnU



J

J

J




iii nnU



in D>1: heavy hole (quasiparticle)


QP



iii nnU



spinon holon

in 1D: spin-charge separation


1D Hubbard-Model: spectral function A(k,)en

ergy

rel

ativ

e to

EF

i

ii nnU

,ijji cctH

spinon holon

charge

~O(t)

~O(J)

spin

momentum

-/2

-kF kF 3kF

/20

0

K. Penc et al. (1996): tJ-modelJ.M.P. Carmelo et al. (2002 / 2003): Bethe ansatzE. Jeckelmann et al. (2003): dynamical DMRG

TTF-TCNQ: An organic 1D metal

strongly anisotropic conductivity b/a b/c ~1000

-0.2 0.0 0.2 0.4

-0.8

-0.6

-0.4

-0.2

0.0

E-E

F (

eV)

k|| (Å-1)

a

d

b

c

TCNQ-band: ARPES versus 1D Hubbard model

band theory

photoemission model

dynamical DMRG E. Jeckelmann et al., PRL 92, 256401 (2004)

model parameters forTCNQ band:

n = 0.59 (<1)

U/t = 4.9

t 2tLDA (?)

TTF-TCNQ: low energy behavior ?

0.4 0.3 0.2 0.1 0.0 -0.1

h = 25 eVE = 60 meV = ±1°k = k

F

T = 61 K

Inte

nsity

(a.u

.)

ARPES @ kF

Binding energy (eV)

~E1/8

• Tomonaga-Luttinger model:

• power law exponent for 1D Hubbard model: α 1/8 (~0.04)

• experiment: α ~ 1

electron-phonon interaction ?

long-range Coulomb interaction ?

)(A

TCNQ-band: non-local interaction L. Cano-Cortés et al.,Eur. Phys. J. B 56, 173 (2007)

on-site Coulomb energy U (screened): 1.7 eV

Hubbard model fit of PES data: 1.9 eV

BUT: nearest neighbor interaction V: 0.9 eV

extended Hubbard model:

i ij

jiiiij

ji nnVnnUcctH

,

V induces larger "band width",i.e. mimicks larger t !

also: Maekawa et al, PRB (2006)

local spectral function:

Spin-charge separation in 1D Mott insulators

B.J. Kim et al., Nature Physics 2, 397 (2006)

ARPES on SrCuO2 1D Hubbard model (n=1)

H. Benthien and E. Jeckelmann, in Phys. Rev. B 72, 125127 (2005)

Outline:





e-h

TiOCl: A low-dimensional Mott insulator

configuration: Ti 3d1

1e-/atom: Mott insulator

local spin s=1/2

TiOCl

ab

c

b

a

(a) (b)

t

t´

TiOCl: A low-dimensional Mott insulator

?

configuration: Ti 3d1

1e-/atom: Mott insulator

local spin s=1/2

frustrated magnetism, resonating valence bond (RVB) physics ?

TiOCl

ab

c

b

a

(a) (b)

t

t´

Magnetic susceptibility: 1D physics

High T Bonner-Fisher behavior

characteristic for 1D AF spin ½ chains

Low T spin gap

formation of spin singlets due to a spin-Peierls transition ?

TiOCl: Electronic origin of 1D physics

Seidel et al. (2003)Valenti et al. (2004)

band theory (LDA+U):

Valence band: Photoemission vs. theory

PRB 72, 125127 (2005)with T. Saha-Dasgupta, R. Valenti et al.

O 2p / Cl 3p

Ti 3d

T = 370 K

Ti 3d PDOS: photoemission vs. theory

PRB 72, 125127 (2005)

cluster = Ti dimer

T. Saha-Dasgupta, R. Valenti, A. Lichtenstein et al., submitted

T = 370 K

(QMC, T=1400K)

ARPES on Ti 3d band

e-

h

lightsource

analyzer

sample

e-

h

lightsource

analyzer

sample

PRB 72, 125127 (2005)

k

T = 370 K

ARPES on Ti 3d band

PRB 72, 125127 (2005)

e-

h

lightsource

analyzer

sample

e-

h

lightsource

analyzer

sample

1D Hubbard model

DDMRGH. Benthien, E. Jeckelmann

TiOCl vs. TiOBr: effective dimensionality?

TiOCl:

Wb ~ 4 x Wa

Wa

Wb

TiOBr:

Wb ~ Wa

Doping a Mott insulator

Oxide-based electronics

2DEG

SrTiO3

LaTiO3

High-Tc superconductors

field effect transistor (FET)

doping x

tem

pera

ture

metal

insulator

e.g., La2-xSrxCuO4

Doping a Mott insulator: TiOCl

Doping by intercalation

van der Waals-gapNa, K

doped Hubbard model

In situ doping of TiOCl with Na

U

LHB

LHB

UHB

UHB

QP

U

LHB

LHB

UHB

UHB

QP

new states in the Mott gap

-10 -8 -6 -4 -2 0

minutesNa exposure

inte

nsi

ty (

arb

. un

its)

energy relative to µexp

(eV)

5

60

50

10

15

20

25

55

40

30

0

Na exposure[min]

In situ doping of TiOCl with Na

• new states in theMott gap

• but not metallic (?)

3.0 3.0

2.5 2.5

2.0 2.0

1.5 1.5

1.0 1.0

0.5 0.5

0.0 0.0

-0.5 -0.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

-0.5

ener

gy r

elat

ive

to µ

chem

(eV

)

XXXX

k|| k||

en

erg

y r

ela

tiv

e t

o c

he

m.

po

ten

tia

l (e

V)

pristine TiOCl Na-doped

ARPES

multiorbital and/or lattice (polaronic) effects ?

t2g

U

cf

U + cf - JH

Summary

Photoemission of interacting electron systems

- (AR)PES probes single-particle excitation spectrum -Im G(k,) (generalized Franck-Condon effect)

- required: Sudden Approximation, low dimensionality, constant matrix elements

- pitfalls: surface effects, charging

Transition metal oxides:

- Hubbard model good starting point

Correlation effects in 1D:

- spin-charge separation on high energy scale

Additional challenges for real materials:

- orbital degrees of freedom

- electron/spin-lattice coupling

- magnetic frustration

- doping of Mott insulators ( oxide-based electronics, FET,…)

otherelectrons

phonons

spin excitations

?

Reading

Photoemission of interacting electron systems: Theory

• L. Hedin and S. LundqvistEffects of electron-electron and electron-phonon interactions on the one-elecron states of solidsVol. 23 of Solid State PhysicsAcademic Press (1970)

• C.-O. Almbladh and L. HedinBeyond the one-electron model / Many-body effects in atoms, molecules and solidsin Vol. 1 of Handbook on Synchrotron RadiationNorth-Holland (1983)

Photoemission of interacting electrons systens: Examples

• S. Hüfner (ed.)Very High Resolution Photoelectron SpectroscopySpringer (2007)

physics of correlated electron materials: experiments with photoelectron spectroscopy ralph claessen...

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