Physics of correlated electron materials: Experiments with photoelectron spectroscopy

transcript

Physics of correlated electron materials:Experiments with photoelectron spectroscopy

Ralph Claessen

U Würzburg, Germany

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

Angle-resolved photoelectron spectroscopy

non-interacting electrons

band structure E(k)

interacting electrons

spectral function

),(Im),( 1 kGkA

Photoemission: many-body effects

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:

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

0,0, Ni

sNksf ,1,,

if kkif

iii ccMprA

one-particle matrix element )( fi kk

)(ˆ),( 0,,2

0,, hEEkI NsNs

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

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

hEENcsNkI NsNs

The ARPES signal is directly proportional to the

single-particle spectral function ),(Im1

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

Eelectrons couple to proton dynamics !

photoemission intensity:

electronic-vibrational eigenstates of H2+:

)(0,ˆ,)( 0,,

22 22HsH

EEHcsHI g

E Franck-Condon principle

proton distance

v' = 0

Caveat: Effect of photoelectron lifetime

ARPES intensity actually convolution of photohole and photoelectron spectral function

),,(),,(),(

kkAhkkAdkkI eh llllll

hhtot v

assuming Lorentzian lineshapes the total width is given by

~ meV~ eV

spectrum dominated by photo-electron linewidth unless

v low-dim systems !

Outline:

Transition metal oxides

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

basic building blocks: MO6 octahedra

electronic configuration: O 2s2p6 = [Ne]

cubic perovskites perovskite-like anatas rutile spinel

quasi-atomic,strongly localized

Hubbard model

jiji nnUcctH

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)

jiji nnUcctH

d1 configuration (Ti3+, V4+)

one-electron conduction band: metal

atomic limit: Mott insulator

Photoemission of a Mott insulator

O 2p / Cl 3p

Ti 3d1

LHBd1 d2

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

O 2p V 3d1

incoherentweight coherent

excitations

Spectral evolution through the Mott transition

A. Fujimori et al., PRL 1992

DMFTphotoemission

QPLHB UHB

Surface effects in photoemission

photoelectron mean free path (Ekin)

Ekin ~ h

A. Sekiyama et al., PRL 2004

40 eV 275 eV

900 eV

surface

(Ekin)

Surface effects in photoemission

A. Sekiyama et al., PRL 2004

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:

Spectral function of a Fermi liquid

Fermi liquid

dressed quasiparticles

non-interacting electrons

bare particles

energy

charge

Electron-electron interaction in 1D metals

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

ii nnU

U – local Coulomb energy

J t 2/U - magnetic exchange energy

ii nnU

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

ii nnU

strong coupling U >> t

ii nnU

spinon holon

ii nnU

iii nnU

in D>1: heavy hole (quasiparticle)

iii nnU

spinon holon

in 1D: spin-charge separation

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

ii nnU

,ijji cctH

spinon holon

charge

momentum

-kF kF 3kF

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

k|| (Å-1)

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

T = 61 K

ARPES @ kF

Binding energy (eV)

• 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 ?

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:

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:

TiOCl: A low-dimensional Mott insulator

configuration: Ti 3d1

1e-/atom: Mott insulator

local spin s=1/2

(a) (b)

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 ?

(a) (b)

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

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

lightsource

analyzer

sample

lightsource

analyzer

sample

PRB 72, 125127 (2005)

T = 370 K

ARPES on Ti 3d band

PRB 72, 125127 (2005)

lightsource

analyzer

sample

lightsource

analyzer

sample

1D Hubbard model

DDMRGH. Benthien, E. Jeckelmann

TiOCl vs. TiOBr: effective dimensionality?

TiOCl:

Wb ~ 4 x Wa

TiOBr:

Wb ~ Wa

Doping a Mott insulator

Oxide-based electronics

SrTiO3

LaTiO3

High-Tc superconductors

field effect transistor (FET)

doping x

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

new states in the Mott gap

-10 -8 -6 -4 -2 0

minutesNa exposure

energy relative to µexp

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

k|| k||

pristine TiOCl Na-doped

multiorbital and/or lattice (polaronic) effects ?

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

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