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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
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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
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Angle-resolved photoelectron spectroscopy
non-interacting electrons
ARPES
band structure E(k)
interacting electrons
ARPES
spectral function
),(Im),( 1 kGkA
)(kE
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Photoemission: many-body effects
Ekin
h
electron-electron interaction
photoelectron: "loss" of kinetic energy due to excitation energy stored in the remaining interacting system !
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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
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Many-body theory of photoemission
Fermi´s Golden Rule for N-particle states:
)(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
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Many-body theory of photoemission
Fermi´s Golden Rule for N-particle states:
)(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
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Many-body theory of photoemission
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
?
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L. Åsbrink, Chem. Phys. Lett. 7, 549 (1970)
Many-body effects in photoemission
Example: Photoemission from the H2 molecule
Ekin
H2
E
g
u*
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L. Åsbrink, Chem. Phys. Lett. 7, 549 (1970)
Many-body effects in photoemission
Example: Photoemission from the H2 molecule
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*
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L. Åsbrink, Chem. Phys. Lett. 7, 549 (1970)
Many-body effects in photoemission
Example: Photoemission from the H2 molecule
Ekin
H2
E Franck-Condon principle
proton distance
ener
egy
v' = 0
v = 0
v = 1
v = 2
ħ0g
u*
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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 !
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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
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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
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Hubbard model
iii
jiji nnUcctH
,,
ˆt
U kinetic energy,itinerancy
local Coulomb energy,localization
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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
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Photoemission of a Mott insulator
TiOCl
O 2p / Cl 3p
Ti 3d1
U
d1 d0
LHBd1 d2
UHB
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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
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Photoemission of a correlated d1 metal
A. Fujimori et al., PRL 1992
LHBQP
O 2p V 3d1
incoherentweight coherent
excitations
LHB
QP
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Spectral evolution through the Mott transition
A. Fujimori et al., PRL 1992
DMFTphotoemission
QPLHB
QPLHB UHB
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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)
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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
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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)
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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
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Spectral function of a Fermi liquid
Fermi liquid
dressed quasiparticles
non-interacting electrons
bare particles
EF=0
k0 k
energy
k
kF
A(k,)
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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:
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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
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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 strong coupling U >> t
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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
J
strong coupling U >> t
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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
J
strong coupling U >> t
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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
J
strong coupling U >> t
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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
J
strong coupling U >> t
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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
J
strong coupling U >> t
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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
J
strong coupling U >> t
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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
J
strong coupling U >> t
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spinon holon
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
J
strong coupling U >> t
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Strongly coupled electrons: 1D Hubbard model
,ijji cctH t – hopping integral
iii nnU
U – local Coulomb energy
J t 2/U - magnetic exchange energy
J
J
J
strong coupling U >> t
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Strongly coupled electrons: 1D Hubbard model
,ijji cctH t – hopping integral
iii nnU
U – local Coulomb energy
J t 2/U - magnetic exchange energy
in D>1: heavy hole (quasiparticle)
strong coupling U >> t
QP
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Strongly coupled electrons: 1D Hubbard model
,ijji cctH t – hopping integral
iii nnU
U – local Coulomb energy
J t 2/U - magnetic exchange energy
spinon holon
in 1D: spin-charge separation
strong coupling U >> t
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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
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TTF-TCNQ: An organic 1D metal
strongly anisotropic conductivity b/a b/c ~1000
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-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 (?)
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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
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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:
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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)
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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
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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´
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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´
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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 ?
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TiOCl: Electronic origin of 1D physics
Seidel et al. (2003)Valenti et al. (2004)
band theory (LDA+U):
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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
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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)
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ARPES on Ti 3d band
e-
h
lightsource
analyzer
sample
e-
h
lightsource
analyzer
sample
PRB 72, 125127 (2005)
k
T = 370 K
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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
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TiOCl vs. TiOBr: effective dimensionality?
TiOCl:
Wb ~ 4 x Wa
Wa
Wb
TiOBr:
Wb ~ Wa
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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
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Doping a Mott insulator: TiOCl
Doping by intercalation
van der Waals-gapNa, K
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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]
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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
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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
?
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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)