Photoelectron Spectroscopyetsf.polytechnique.fr/system/files/users/francesco/merged2.pdf ·...
Transcript of Photoelectron Spectroscopyetsf.polytechnique.fr/system/files/users/francesco/merged2.pdf ·...
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Photoelectron Spectroscopy
Claudia Rödl and Matteo Gatti
Laboratoire des Solides IrradiésEcole Polytechnique, Palaiseau, France
SOLEIL Theory Day, Gif-sur-Yvette
May 6th, 2014
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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
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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
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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
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ETSF Photoemission Beamline
Systems:
insulators
semiconductors
metals
correlated oxides
Codes:
Dimensions:
bulk
surfaces
molecules
nanosystems
up to ∼ 100 atoms
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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
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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
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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
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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
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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
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Key Quantity: Green’s Function
Green’s function: G(r1t1, r2t2) =1i~
〈N| T ψ(r1t1)ψ+(r2t2) |N〉
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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
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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
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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)
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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)
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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)
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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
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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
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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, ...)
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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
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adapted from M. van Schilfgaarde et al., PRL 96 (2006)
Why more than DFT?
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LDA = “band theory” ?
Fujimori et al, PRL 69 (1992)
Why more than DFT?
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Quasiparticle Excitations
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Quasiparticle Excitations
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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
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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)
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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)
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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)
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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)
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adapted from M. van Schilfgaarde et al., PRL 96 (2006)
Why more than DFT?
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Fujimori et al, PRL 69 (1992)
LDA = “band theory” ?
Also missing physics!
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Mott insulators?
Fujimori et al, PRL 69 (1992)
Exchange + Correlation
LDA = “band theory” ?
Also missing physics!
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The role of nonlocal exchange
F. Iori, M. Gatti and A. Rubio, PRB 85 (2012)Exp. from H. Roth PhD thesis – Köln 2008
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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)
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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
![Page 36: Photoelectron Spectroscopyetsf.polytechnique.fr/system/files/users/francesco/merged2.pdf · Photoelectron Spectroscopy Claudia Rödl and Matteo Gatti Laboratoire des Solides Irradiés](https://reader036.fdocuments.in/reader036/viewer/2022071014/5fcc984d246afe23e648b1f0/html5/thumbnails/36.jpg)
Electron Correlation ... 3-Particle Problem
Romaniello et al., PRB 85, 155131 (2012)
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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)
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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)
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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)
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Fujimori et al, PRL 69 (1992)
Correlated metals:Satellites?
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Adding/removing electrons: reaction of all the others
Correlation = coupling of excitations
Quasiparticle Plasmons(waves of charge)
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Metal-Insulator transition in VO2
Dynamical correlation: VO2
M. Gatti, G. Panaccione, and L. Reining (submitted)Spectral function
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-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
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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
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Ralf Hambach, PhD thesis, Ecole Polytechnique (2010)
Dynamical correlation: VO2
See Francesco's talk
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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
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hν
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hν
Matrixelements
Background
Interference &Extrinsic effects
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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)
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Matrix elements: V2O
3
E. Papalazarou, et al., PRB 80 (2009)
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Getting closer to what is really measured
Spectral function
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Getting closer to what is really measured
+ Background
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Getting closer to what is really measured
+ Matrix elements
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Getting closer to what is really measured
+ Interference/extrinsic effects
Valencebands
Satellites
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M. Guzzo, et al, PRL 107 (2011)
Valencebands
Satellites
Spectral function Photocurrent
Getting closer to what is really measured
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Adding/removing electrons: reaction of all the others
Correlation = coupling of excitations
Quasiparticle Plasmons(waves of charge)
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Adding/removing electrons: reaction of all the others
Correlation = coupling of excitations
Quasiparticle Plasmons(waves of charge)
ExcitonsOrbitonsMagnonsPhonons
…
More couplings...
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Springer, Aryasetiawan, Karlsson, PRL 80, 2389 (1998)
6 eV satellite in Ni: 2-hole bound state
Outlook: coupling with other excitations
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Back up slides
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Interacting particles
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Interacting particles
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Interacting particles
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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〉
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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)
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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)