Photoelectron Spectroscopyetsf.polytechnique.fr/system/files/users/francesco/merged2.pdf · Photoelectron Spectroscopy Claudia Rödl and Matteo Gatti Laboratoire des Solides Irradiés

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



Density-functional

theory (DFT)


theory (MBPT)

Ground state


total energies

charge densities


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



optical absorption

excitonic states

loss spectra



Density-functional

theory (DFT)


theory (MBPT)

Ground state


total energies

charge densities


band gaps

band structures

densities of states

DO

S /

inte

nsity

HSE06HSE06+G0W0

HSE06+GnW0


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

PBE+UPBE+U+G0W0

PBE+U+GnW0



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


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



E(N)0 + Eph = E



(N)0

︸︷︷︸

EB

+Ekin + Φ

EVBM ECBM

DO

S

EFEvac

EB φ

Eph

Ekin



E(N)0 + Eph = E



(N)0

︸︷︷︸

EB

+Ekin + Φ

EVBM ECBM

DO

S

EFEvac

EB φ

Eph

Ekin

Ekin

inte

nsity DOS

DOS weighted bycross sections

background



E(N)0 + Eph = 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〉



〈N| T ψ(r1t1)ψ+(r2t2) |N〉

Electron propagator:

〈N|ψ(r1t1)ψ+(r2t2) |N〉

electron addition

inverse photoemission



〈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)





ectr

al fu

nctio

n-20 -15 -10 -5 0

energy [eV]

inte

nsity

weak correlations


strong correlations

weak satellite


QP peak

CuO photoemission






ectr

al fu

nctio

n-20 -15 -10 -5 0

energy [eV]

inte

nsity

weak correlations


strong correlations

weak satellite


QP peak

CuO photoemission





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:


(photoionization

cross-sections)

extrinsic and


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



−~

2

2m0

∆+ Vext(r) + VH(r, [n]) + VXC(r, [n])

]


density: n(r) =

occ∑

nk

|ϕnk(r)|2


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, ...)



−~

2

2m0

∆+ Vext(r) + VH(r, [n]) + VXC(r, [n])

]


density: n(r) =

occ∑

nk

|ϕnk(r)|2


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 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


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)




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

Hole propag.: 〈N|ψ+(r2t2)ψ(r1t1) |N〉

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









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




W (r1r2, ω) =∫

dr3 ε−1(r1r3, ω) v(r3 − r2)






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




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?



Also missing physics!

Mott insulators?


Exchange + Correlation


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)




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:


(photoionization

cross-sections)

extrinsic and


background

Electron Correlation ... 3-Particle Problem

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


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~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)


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)


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


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'|)


Ralf Hambach, PhD thesis, Ecole Polytechnique (2010)


See Francesco's talk




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:


(photoionization

cross-sections)

extrinsic and


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


+ Background


+ Matrix elements


+ Interference/extrinsic effects

Valencebands

Satellites

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

Valencebands

Satellites

Spectral function Photocurrent








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



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〉



Electrons


scopy (EELS)

low q

X-rays

non-resonant


large q




loss function − Im ε−1(q, ω)

ε−1(q, ω) = 1 + v(q)χ(qq, ω)



Electrons


scopy (EELS)

low q

X-rays

non-resonant


large q




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)

Documents

Photoelectron Spectroscopyetsf.polytechnique.fr/system/files/users/francesco/merged2.pdf · Photoelectron Spectroscopy Claudia Rödl and Matteo Gatti Laboratoire des Solides Irradiés