photoelectron spectroscopyetsf.polytechnique.fr/system/files/users/francesco/merged2.pdf ·...

Photoelectron Spectroscopy

Claudia Rödl and Matteo Gatti

Laboratoire des Solides IrradiésEcole Polytechnique, Palaiseau, France

SOLEIL Theory Day, Gif-sur-Yvette

May 6th, 2014

Methods: Starting from First Principles

First-principles calculations

Density-functional

theory (DFT)

Many-body perturbation

theory (MBPT)

Ground state

structural properties

total energies

charge densities

One-particle excitations

band gaps

band structures

densities of states

Two-particle excitations:

optical absorption

excitonic states

loss spectra

Methods: Starting from First Principles

First-principles calculations

Density-functional

theory (DFT)

Many-body perturbation

theory (MBPT)

Ground state

structural properties

total energies

charge densities

One-particle excitations

band gaps

band structures

densities of states

DO

S /

inte

nsity

HSE06HSE06+G0W0

HSE06+GnW0

ExperimentGhijsen et al. (1988)

-20 -15 -10 -5 0 5 10 15energy [eV]

PBE+UPBE+U+G0W0

PBE+U+GnW0

ExperimentGhijsen et al. (1988)

Two-particle excitations:

optical absorption

excitonic states

loss spectra

Methods: Starting from First Principles

First-principles calculations

Density-functional

theory (DFT)

Many-body perturbation

theory (MBPT)

Ground state

structural properties

total energies

charge densities

One-particle excitations

band gaps

band structures

densities of states

DO

S /

inte

nsity

HSE06HSE06+G0W0

HSE06+GnW0

ExperimentGhijsen et al. (1988)

-20 -15 -10 -5 0 5 10 15energy [eV]

PBE+UPBE+U+G0W0

PBE+U+GnW0

ExperimentGhijsen et al. (1988)

Two-particle excitations:

optical absorption

excitonic states

loss spectra

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

20.1

0.3

0.5

0.7

0.9

energy [eV]

q

Cu 3p statesmain plasmon peak

plasmons

ETSF Photoemission Beamline

Systems:

insulators

semiconductors

metals

correlated oxides

Codes:

Dimensions:

bulk

surfaces

molecules

nanosystems

up to ∼ 100 atoms

ETSF Photoemission Beamline

Simulating photoemission

Spectral function Beyond spectral function

Quasiparticle (QP)

approximation:

QP band structures

QP band gaps

DOS

GW method

Beyond QPs:

lifetimes

satellites

(e.g. plasmon

satellites)

GW method

cumulant expansion

Towards reality:

matrix-element effects

(photoionization

cross-sections)

extrinsic and

interference effects

background

Photoemission and the Many-Body Problem

probe the electronic structure

E(N)0 + Eph = E

(N−1)n + Ekin + Φ

Eph = E(N−1)n − E

(N)0

︸ ︷︷ ︸

EB

+Ekin + Φ

Photoemission and the Many-Body Problem

probe the electronic structure

E(N)0 + Eph = E

(N−1)n + Ekin + Φ

Eph = E(N−1)n − E

(N)0

︸ ︷︷ ︸

EB

+Ekin + Φ

EVBM ECBM

DO

S

EFEvac

EB φ

Eph

Ekin

Photoemission and the Many-Body Problem

probe the electronic structure

E(N)0 + Eph = E

(N−1)n + Ekin + Φ

Eph = E(N−1)n − E

(N)0

︸ ︷︷ ︸

EB

+Ekin + Φ

EVBM ECBM

DO

S

EFEvac

EB φ

Eph

Ekin

Ekin

inte

nsity DOS

DOS weighted bycross sections

background

Photoemission and the Many-Body Problem

probe the electronic structure

E(N)0 + Eph = E

(N−1)n + Ekin + Φ

Eph = E(N−1)n − E

(N)0

︸ ︷︷ ︸

EB

+Ekin + Φ

EVBM ECBM

DO

S

EFEvac

EB φ

Eph

Ekin

Ekin

inte

nsity DOS

DOS weighted bycross sections

background

requires full solution of the interacting many-body problem: E(N)0 , E

(N−1)n

Key Quantity: Green’s Function

Green’s function: G(r1t1, r2t2) =1i~

〈N| T ψ(r1t1)ψ+(r2t2) |N〉

Key Quantity: Green’s Function

Green’s function: G(r1t1, r2t2) =1i~

〈N| T ψ(r1t1)ψ+(r2t2) |N〉

Electron propagator:

〈N|ψ(r1t1)ψ+(r2t2) |N〉

electron addition

inverse photoemission

Key Quantity: Green’s Function

Green’s function: G(r1t1, r2t2) =1i~

〈N| T ψ(r1t1)ψ+(r2t2) |N〉

Electron propagator:

〈N|ψ(r1t1)ψ+(r2t2) |N〉

electron addition

inverse photoemission

Hole propagator:

〈N|ψ+(r2t2)ψ(r1t1) |N〉

electron removal

photoemission

Spectral Function and Quasiparticles

real space reciprocal space

Spectral function: A(ω) = Tr |Im G(ω)|

Density of states: DOS(ω) ∼ A(ω)sp

ectr

al fu

nctio

n-20 -15 -10 -5 0

energy [eV]

inte

nsity

weak correlations

non-interacting particles

strong correlations

weak satellite

dominating satellite

QP peak

CuO photoemission

Experimental data:Ghijsen et al., PRB 38, 11322 (1988)

Spectral Function and Quasiparticles

real space reciprocal space

Spectral function: A(ω) = Tr |Im G(ω)|

Density of states: DOS(ω) ∼ A(ω)sp

ectr

al fu

nctio

n-20 -15 -10 -5 0

energy [eV]

inte

nsity

weak correlations

non-interacting particles

strong correlations

weak satellite

dominating satellite

QP peak

CuO photoemission

Experimental data:Ghijsen et al., PRB 38, 11322 (1988)

Spectral Function and Quasiparticles

real space reciprocal space

Spectral function: A(ω) = Tr |Im G(ω)|

Density of states: DOS(ω) ∼ A(ω)sp

ectr

al fu

nctio

n-20 -15 -10 -5 0

energy [eV]

inte

nsity

weak correlations

non-interacting particles

strong correlations

weak satellite

dominating satellite

QP peak

CuO photoemission

Experimental data:Ghijsen et al., PRB 38, 11322 (1988)

ETSF Photoemission Beamline

Simulating photoemission

Spectral function Beyond spectral function

Quasiparticle (QP)

approximation:

QP band structures

QP band gaps

DOS

GW method

Beyond QPs:

lifetimes

satellites

(e.g. plasmon

satellites)

GW method

cumulant expansion

Towards reality:

matrix-element effects

(photoionization

cross-sections)

extrinsic and

interference effects

background

Density-Functional Theory (DFT)

Kohn-Sham equations: Kohn, Sham, Phys. Rev. 140, A1133 (1965)[

−~

2

2m0

∆+ Vext(r) + VH(r, [n]) + VXC(r, [n])

]

ϕnk(r) = εnk ϕnk(r)

density: n(r) =

occ∑

nk

|ϕnk(r)|2

self-consistent solution

Density-Functional Theory (DFT)

Kohn-Sham equations: Kohn, Sham, Phys. Rev. 140, A1133 (1965)[

−~

2

2m0

∆+ Vext(r) + VH(r, [n]) + VXC(r, [n])

]

ϕnk(r) = εnk ϕnk(r)

density: n(r) =

occ∑

nk

|ϕnk(r)|2

self-consistent solution

Exchange-correlation potential VXC?

local potential VXC(r, [n]): LDA, PBE, ...

non-local potential VXC(rr′, [n]):

hybrid functionals (PBE0, HSE06, ...)

on-site term U for localized orbitals (LDA+U, ...)

Density-Functional Theory (DFT)

Kohn-Sham equations: Kohn, Sham, Phys. Rev. 140, A1133 (1965)[

−~

2

2m0

∆+ Vext(r) + VH(r, [n]) + VXC(r, [n])

]

ϕnk(r) = εnk ϕnk(r)

density: n(r) =

occ∑

nk

|ϕnk(r)|2

self-consistent solution

Problems:

Kohn-Sham band structure has no physical meaning (strictly speaking)

band-gap problem: strong underestimation of band gaps

electron addition and removal properties wrong

adapted from M. van Schilfgaarde et al., PRL 96 (2006)

Why more than DFT?

LDA = “band theory” ?

Fujimori et al, PRL 69 (1992)

Why more than DFT?

Quasiparticle Excitations

Quasiparticle Excitations

Quasiparticle Excitations

Quasiparticle equation:[

−~

2

2m0

∆+ Vext(r) + VH(r)

]

ϕnk(r) +

∫

dr′ Σ(r r

′, ω = εnk/~)ϕnk(r′) = εnk ϕnk(r)

Problems:

exchange and correlation: non-local and frequency-dependent self-energy Σ

self-consistent solution

no explicit expression for the self-energy

Hedin’s GW Approximation

Self-energy: Σ ≈ i~G W

Green’s function G(r1t1, r2t2)

Hedin, Phys. Rev. 139, A796 (1965)

Hedin, Lundqvist, Solid State Phys. 23, 1 (1969)

Dynamically screened

Coulomb interaction W (r1r2, ω)

Hedin’s GW Approximation

Self-energy: Σ ≈ i~G W

Green’s function G(r1t1, r2t2)

Electron propag.: 〈N|ψ(r1t1)ψ+(r2t2) |N〉

Hole propag.: 〈N|ψ+(r2t2)ψ(r1t1) |N〉

Hedin, Phys. Rev. 139, A796 (1965)

Hedin, Lundqvist, Solid State Phys. 23, 1 (1969)

Dynamically screened

Coulomb interaction W (r1r2, ω)

Hedin’s GW Approximation

Self-energy: Σ ≈ i~G W

Green’s function G(r1t1, r2t2)

Electron propag.: 〈N|ψ(r1t1)ψ+(r2t2) |N〉

Hole propag.: 〈N|ψ+(r2t2)ψ(r1t1) |N〉

Hedin, Phys. Rev. 139, A796 (1965)

Hedin, Lundqvist, Solid State Phys. 23, 1 (1969)

Dynamically screened

Coulomb interaction W (r1r2, ω)

W (r1r2, ω) =∫

dr3 ε−1(r1r3, ω) v(r3 − r2)

Hedin’s GW Approximation

Self-energy: Σ ≈ i~G W

Green’s function G(r1t1, r2t2)

Electron propag.: 〈N|ψ(r1t1)ψ+(r2t2) |N〉

Hole propag.: 〈N|ψ+(r2t2)ψ(r1t1) |N〉

Hedin, Phys. Rev. 139, A796 (1965)

Hedin, Lundqvist, Solid State Phys. 23, 1 (1969)

Dynamically screened

Coulomb interaction W (r1r2, ω)

W (r1r2, ω) =∫

dr3 ε−1(r1r3, ω) v(r3 − r2)

main contribution:

dynamical screened exchange(Hartree-Fock: ε−1(r1r3, ω) ≡ 1)

adapted from M. van Schilfgaarde et al., PRL 96 (2006)

Why more than DFT?

Fujimori et al, PRL 69 (1992)

LDA = “band theory” ?

Also missing physics!

Mott insulators?

Fujimori et al, PRL 69 (1992)

Exchange + Correlation

LDA = “band theory” ?

Also missing physics!

The role of nonlocal exchange

F. Iori, M. Gatti and A. Rubio, PRB 85 (2012)Exp. from H. Roth PhD thesis – Köln 2008

Bin

din

g e

ner

gy

ARPES vs. band structure: ZnO

Exp from M. Kobayashi, et al. Proceedings of the 29th International Conference on the Physics of Semiconductors (2008)André Schleife PhD thesis, University of Jena (2010)

ETSF Photoemission Beamline

Simulating photoemission

Spectral function Beyond spectral function

Quasiparticle (QP)

approximation:

QP band structures

QP band gaps

DOS

GW method

Beyond QPs:

lifetimes

satellites

(e.g. plasmon

satellites)

GW method

cumulant expansion

Towards reality:

matrix-element effects

(photoionization

cross-sections)

extrinsic and

interference effects

background

Electron Correlation ... 3-Particle Problem

Romaniello et al., PRB 85, 155131 (2012)

Electron Correlation ... 3-Particle Problem

Σ = i~GW

plasmon satellite series

Langreth, PRB 1, 471 (1970)

Aryasetiawan et al.,

PRL 77, 2268 (1996)

Guzzo et al.,

PRL 107, 166401 (2011)

Romaniello et al., PRB 85, 155131 (2012)

Electron Correlation ... 3-Particle Problem

Σ = i~GW

plasmon satellite series

Langreth, PRB 1, 471 (1970)

Aryasetiawan et al.,

PRL 77, 2268 (1996)

Guzzo et al.,

PRL 107, 166401 (2011)

Σ = i~GT

hole-hole or electron-hole

T matrix

6 eV satellite in NiSpringer et al., PRL 80, 2389 (1998)

Romaniello et al., PRB 85, 155131 (2012)

Spectral Function ... Plasmon Satellites

CuO

-20 -15 -10 -5 0 5energy [eV]

00.10.20.30.40.50.60.70.80.9

11.11.21.31.41.51.61.7

- Im

ε-1

(ω)

/ XP

S [a

rb. u

.]

Cu 2p1/2

Cu 2p3/2

valence

valence

EELS

satellite QP

XPS

satellite: response of the material

to the photoemission hole

remaining in the system

plasmon excitations

prominent loss peaks shouldappear in satellite structure

Experimental data:Siemons, PhD thesis (2008)Ghijsen et al., PRB 38, 11322 (1988)Shen et al., PRB 42, 8081 (1990)

Fujimori et al, PRL 69 (1992)

Correlated metals:Satellites?

Adding/removing electrons: reaction of all the others

Correlation = coupling of excitations

Quasiparticle Plasmons(waves of charge)

Metal-Insulator transition in VO2

Dynamical correlation: VO2

M. Gatti, G. Panaccione, and L. Reining (submitted)Spectral function

-Im ε-1(q,ω)

W(r,r',ω) = ε-1(r,r',ω)v(|r-r'|)

Metal-Insulator transition in VO2IXS/EELS

Spectral function

Dynamical correlation: VO2

Metal-Insulator transition in VO2

Spectral function

IXS/EELS-Im ε-1(q,ω)

W(r,r',ω) = ε-1(r,r',ω)v(|r-r'|)

Dynamical correlation: VO2

Ralf Hambach, PhD thesis, Ecole Polytechnique (2010)

Dynamical correlation: VO2

See Francesco's talk

ETSF Photoemission Beamline

Simulating photoemission

Spectral function Beyond spectral function

Quasiparticle (QP)

approximation:

QP band structures

QP band gaps

DOS

GW method

Beyond QPs:

lifetimes

satellites

(e.g. plasmon

satellites)

GW method

cumulant expansion

Towards reality:

matrix-element effects

(photoionization

cross-sections)

extrinsic and

interference effects

background

hν

hν

Matrixelements

Background

Interference &Extrinsic effects

TCOs: Including Photoionization Cross-Sections

ZnO

-25 -20 -15 -10 -5 0 5 10 15 20

DO

S (a

rb. u

.)

-10 -8 -6 -4 -2 0 2

TheoryExp.

-30 -25 -20 -15 -10 -5 0Energy (eV)

-6 -4 -2 0 2

Phot

oem

issi

on in

tens

ity (

arb.

u.)

Zn 4sO 2s Zn 4pO 2ptotal

Zn 3d

UPS (21.2 eV)

XPS (1486.6 eV)

(a) ZnO

SnO2

-30 -25 -20 -15 -10 -5 0 5 10 15 20

DO

S (a

rb. u

.)

-10 -8 -6 -4 -2 0 2

TheoryExp.

-28 -24 -20 -16 -12 -8 -4 0Energy (eV)

-12 -10 -8 -6 -4 -2 0 2Ph

otoe

mis

sion

inte

nsity

(ar

b. u

.)

Sn 5sO 2s

Sn 5pO 2p

total

Sn 4d

UPS (40 eV)

XPS (1486.6 eV)

(c) SnO2

Rödl et al., Phys. Stat. Solidi A 211, 74 (2014)

Matrix elements: V2O

3

E. Papalazarou, et al., PRB 80 (2009)

Getting closer to what is really measured

Spectral function

Getting closer to what is really measured

+ Background

Getting closer to what is really measured

+ Matrix elements

Getting closer to what is really measured

+ Interference/extrinsic effects

Valencebands

Satellites

M. Guzzo, et al, PRL 107 (2011)

Valencebands

Satellites

Spectral function Photocurrent

Getting closer to what is really measured

Adding/removing electrons: reaction of all the others

Correlation = coupling of excitations

Quasiparticle Plasmons(waves of charge)

Adding/removing electrons: reaction of all the others

Correlation = coupling of excitations

Quasiparticle Plasmons(waves of charge)

ExcitonsOrbitonsMagnonsPhonons

…

More couplings...

Springer, Aryasetiawan, Karlsson, PRL 80, 2389 (1998)

6 eV satellite in Ni: 2-hole bound state

Outlook: coupling with other excitations

Back up slides

Interacting particles

Interacting particles

Interacting particles

Probing Neutral Electronic Excitations

Inelastic scattering of

Electrons

electron-energyloss spectro-

scopy (EELS)

low q

X-rays

non-resonant

inelastic x-rayscattering (IXS)

large q

charge density fluctuations (interband transitions, excitons, plasmons)

charge-density response function χ(r1r2, t1 − t2)

χ(r1r2, t1 − t2) =1i~〈N| T ∆n(r1, t1)∆n(r2, t2) |N〉

Probing Neutral Electronic Excitations

Inelastic scattering of

Electrons

electron-energyloss spectro-

scopy (EELS)

low q

X-rays

non-resonant

inelastic x-rayscattering (IXS)

large q

charge density fluctuations (interband transitions, excitons, plasmons)

charge-density response function χ(r1r2, t1 − t2)

χ(r1r2, t1 − t2) =1i~〈N| T ∆n(r1, t1)∆n(r2, t2) |N〉

loss function − Im ε−1(q, ω)

ε−1(q, ω) = 1 + v(q)χ(qq, ω)

Probing Neutral Electronic Excitations

Inelastic scattering of

Electrons

electron-energyloss spectro-

scopy (EELS)

low q

X-rays

non-resonant

inelastic x-rayscattering (IXS)

large q

charge density fluctuations (interband transitions, excitons, plasmons)

charge-density response function χ(r1r2, t1 − t2)

χ(r1r2, t1 − t2) =1i~〈N| T ∆n(r1, t1)∆n(r2, t2) |N〉

loss function − Im ε−1(q, ω)

ε−1(q, ω) = 1 + v(q)χ(qq, ω)

dynamic structure factor S(q, ω)

S(q, ω) ∼ χ(qq, ω)