Physics of correlated electron materials: Experiments with photoelectron spectroscopy Ralph Claessen...
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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
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
Angle-resolved photoelectron spectroscopy
non-interacting electrons
ARPES
band structure E(k)
interacting electrons
ARPES
spectral function
),(Im),( 1 kGkA
)(kE
Photoemission: many-body effects
Ekin
h
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:
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
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
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
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
?
L. Åsbrink, Chem. Phys. Lett. 7, 549 (1970)
Many-body effects in photoemission
Example: Photoemission from the H2 molecule
Ekin
H2
E
g
u*
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*
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*
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 !
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
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
Hubbard model
iii
jiji nnUcctH
,,
ˆt
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)
iii
jiji nnUcctH
,,
ˆ
d1 configuration (Ti3+, V4+)
A()
one-electron conduction band: metal
U
atomic limit: Mott insulator
W
Photoemission of a Mott insulator
TiOCl
O 2p / Cl 3p
Ti 3d1
U
d1 d0
LHBd1 d2
UHB
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
LHBQP
O 2p V 3d1
incoherentweight coherent
excitations
LHB
QP
Spectral evolution through the Mott transition
A. Fujimori et al., PRL 1992
DMFTphotoemission
QPLHB
QPLHB UHB
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)
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
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:
• 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
Spectral function of a Fermi liquid
Fermi liquid
dressed quasiparticles
non-interacting electrons
bare particles
EF=0
k0 k
energy
k
kF
A(k,)
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:
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
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
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
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
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
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
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
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
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
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
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
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
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
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
TTF-TCNQ: An organic 1D metal
strongly anisotropic conductivity b/a b/c ~1000
-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 (?)
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
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:
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:
• 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
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´
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´
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
Ti 3d
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
e-
h
lightsource
analyzer
sample
e-
h
lightsource
analyzer
sample
PRB 72, 125127 (2005)
k
T = 370 K
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
TiOCl vs. TiOBr: effective dimensionality?
TiOCl:
Wb ~ 4 x Wa
Wa
Wb
TiOBr:
Wb ~ Wa
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
Doping a Mott insulator: TiOCl
Doping by intercalation
van der Waals-gapNa, K
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]
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
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)