Testing modern theories for correlated systems
Hao Tjeng
II. Physikalisches Institut
University of Cologne, Germany
[email protected]
• systems: LaTiO3, YTiO3, La1-xSrxTiO3+d, VO2, Ti2O3, V2O3, Ca2-xSrxRuO4
• theories: LDA, LDA+U, LDA+DMFT, LDA+CDMFT
• spectral weight transfer, metal-insulator transitions
• orbital occupations and spin-spin correlations
• dimers, H2-model
Motivation:
How does the spectral weight distribution change in
a Mott-Hubbard system as a function of U / W ??
non-correlated metal
which scenario ?
B-R
Hubbard
DMFT
Mott-insulator
from Ronald Hesper,
thesis Groningen 2001
Phys. Rev. B 54, 8446 (1996).
Bandwidth control: W vs. U
3d1 perovkites
Ca -- Sr:
• same valence
• no doping
 different bond-angles
 different band widths
Interesting proposition:
spectral weight transfer
near a Mott transition by
band width control
Remark:
both systems are on the
metal side of the MIT.
Bulk-sensitive PES
LDA+DMFT
There should be differences ! But too bad
that the differences are too small !
How about LaTiO3 versus YTiO3 ?!
• both are correlated systems
• La – Y : same valence
• different Ti-O-Ti bond-angles
• different band widths
consequences for spectral weight distributions ?!
Remark:
• both systems are on the insulating side of the MIT
• the band gaps are different but both are small
Phase diagram: YTiO3 - LaTiO3 with Ca, Sr, and O doping
Temp.
Mott-Insulator
antiferromag.
ferromag.
Radius R3+
x = 0.4
La
Metall
1-x Sr
x Ti
Isolator
O
3
3
x = 0.05
a T
x iO
1-x C
LaTiO3
TC
30 K
Y
145 K
N
RTiO3
YTiO3
x = 0.1
T
hole-doping : d1-x, d1-2d
Existing experimental data:
Fujimori et al., PRB 1992
LaTiO3
Morikawa et al., PRB 1996
hv = 48 eV
However:
• LaTiO3 and YTiO3 have very similar Ti-3d spectral weight !!
• O-2p spectrum does not agree with O-2p from LDA !!
• Something wrong with the data from literature ????
YTiO3
“
“
NOT TRUE!!
very different
lineshapes !!
OUR EXPERIMENTS: samples made by Holger Roth
Single Crystals LaTiO3 and YTiO3 : Ø 6 mm; 10-50 mm length
LaTiO3: TN = 148 K
YTiO3: TC = 29 K
OUR EXPERIMENTS: bulk-sensitive photoemission
[high photon energy; normal emission on cleaved single crystal surfaces]
Good agreement between “O-2p” band and LDA results !!
 Samples are of good quality; measurements are reliable !!
O-2p band not
so correlated
Close-up of the Ti-3d band region:
LaTiO3 and YTiO3 have different band widths indeed !!
The Ti-3d band region: comparison experiment with theories
GGA band structure calculations: much too narrow bands !!
The Ti-3d band region: comparison experiment with theories
GGA+U band structure calculations: even worse !!
The Ti-3d band region: comparison experiment with theories
LDA+DMFT seems to work well !! Surprising ?!?
The Ti-3d band region: comparison experiment with theories
Full GGA t2g band width works even better !! Very surprising ?!
How does the spectral weight distribution change in
a Mott-Hubbard system as a function of U / W ??
non-correlated metal
which scenario ?
B-R
Hubbard
DMFT
Mott-insulator
LaTiO3/YTiO3: t-J type of Mott-insulators ?!
• no dubbel occupation
• total effective band width given by total 1-electron band width
Phase diagram: YTiO3 - LaTiO3 with Ca, Sr, and O doping
Temp.
Mott-Insulator
antiferromag.
ferromag.
Radius R3+
x = 0.4
La
Metall
1-x Sr
x Ti
Isolator
O
3
3
x = 0.05
a T
x iO
1-x C
LaTiO3
TC
30 K
Y
145 K
N
RTiO3
YTiO3
x = 0.1
T
hole-doping : d1-x, d1-2d
Doping dependence:
excess oxygen: d1-2d
Sr doping: d1-x
Doping dependence:
excess oxygen: d1-2d
Sr doping: d1-x
• very rapid increase of ‘metallic’ peak with doping
• more rapid with Sr than with oxygen excess
Doping dependence:
excess oxygen: d1-2d
DMFT
Sr doping: d1-x
DMFT
• very rapid increase of ‘metallic’ peak with doping
• more rapid with Sr than with oxygen excess
Calculating electronic structure and spectral weights of correlated systems:
Dynamical Mean Field Theory:
See review: G. Kotliar and D. Vollhardt,
Physics Today, March 2004, page 53-59.
Realistic LDA+DMFT calculations:
good results for:
• a-g transition in Cerium
• d-phase Plutonium
Summary: LaTiO3, YTiO3, La1-xSrxTiO3+d - perovskite d1 systems:
• LDA, LDA+U : fail completely
• LDA+DMFT : good results for photoemission !
•
: inverse photoemission: untested !!
Metal insulator transitions
3d1 system: VO2
Metal insulator transition in VO2 at 340 K.
T > 340K: metal, rutile
M. Marezio et. al.,
Phys. Rev. B 5, 2541 (1972)
log-scale
linear scale
P. B. Allen et. al.,
Phys. Rev. B 48,
4359 (1993)
T < 340K: insulator, monoclinic,
dimerized zig-zag chain
p. 3389
p. 3042
rutile – metallic phase
monoclinic – insulating phase
Band theory allways produces a metal
No agreement with UPS spectrum of Goering et al., Phys. Rev. B 55, 4225 (1997)
cond-mat/0310216v1 9 Oct 2003
UPS:VO2 thin film
LDA+DMFT
No agreement with UPS of
LDA+DMFT produces a metal for both
rutile and monoclinic structure, using
"realistic" values for U (= 4 eV).
K. Okazaki et al.,
Phys. Rev. B 69, 165104 (2004)
Unfortunately: also no agreement between experiments
Sawatzky and Post, PRB 20, 1546 (1979)
hn = 1486 eV
Shin et al., PRB 41, 4993 (1990)
hn = 21.2 eV
Okazaki et al., PRB 69, 165104 (2004)
Our experiment: bulk sensitive photoemission on VO2 single crystals
monoclinic, insulating
rutile, metallic
V-3d
"prominent"
quasi-particle
peak
O-2p
hn = 700 eV [ESRF-ID08, DE=0.15 eV]
cleaved single crystal, flat surface,
normal emission max. probing depth
"incoherent"
peak
Comparison I:
Band theory
Experiment
monoclinic – insulating phase
bulk sensitive photoemission
monoclinic, insulating
rutile, metallic
V-3d
O-2p
rutile – metallic phase
hn = 700 eV [ESRF-ID08, DE=0.15 eV]
cleaved single crystal, normal emission
No gap between 1.0 and 2.0 eV region
Eyert, Ann. Phys. 11, 650 (2002)
Comparison II:
Experiment
bulk sensitive photoemission
LDA+DMFT
Liebsch et al., PRB 71, 085109
monoclinic, insulating
rutile, metallic
V-3d
O-2p
hn = 700 eV [ESRF-ID08, DE=0.15 eV]
cleaved single crystal, normal emission
• LDA+DMFT is too metallic
• U = 4 eV is too small ?!
• position of insulating peak is okay
Comparison III:
bulk sensitive photoemission
monoclinic, insulating
rutile, metallic
LDA + cluster DMFT

Experiment
1,8
1,6
rutile=metallic
U = 4 eV !!
monoclinic=
insulating !!!
1,4
1,2
V-3d
O-2p
1,0
0,8
0,6
0,4
hn = 700 eV [ESRF-ID08, DE=0.15 eV]
cleaved single crystal, normal emission
0,2
0,0
-4
-2
0
2
4
6
(eV)
LDA + cluster DMFT:
S. Biermann, A. Poteryaev, A.
Lichtenstein, A. Georges,
Phys. Rev. Lett. 94, 026404 (2005)
Valence Band
What is the underlying physics??  Orbitals in VO2: 3d1 - (t2g)1
Soft-X-Ray Absorption Spectroscopy:
powerful in combination with theory
EFermi
V 3d
hn 510 eV
O 2p
hn 530 eV
V 2p3/2
2p1/2
O 1s
Spectrum (hn=Sfie.rf² d(hn - Ef + Ei)
i = initial state, f = final state
e.r = dipole transition
• use of core levels  local transitions 
element and site specific
• involves most relevant orbitals:
2p-3d (TM), 3d-4f (RE), 1s-2p (O,N,C)
• dipole allowed  very strong intensities
• dipole selection rules + multiplet structure
give extreme sensitivity to symmetry of
initial state: charge, spin and orbital
theory:
TM 2p-3d: Cluster calculations with
full atomic multiplet theory
O 1s-2p : LDA+U calculations
Technique developed in late 1980‘s:
• Fink, Sawatzky, Fuggle
• Thole, van der Laan
• Chen, Sette
Exp
Theory
All multiplet structures can be reproduced !!
Exp
Theory
holes are
in-plane
Photon energy (eV)
polarization dependence in VO2 : experiment and best fits
Orbital occupation in VO2: insulating and metallic phase
insulating phase
XAS
metallic phase
XAS
MIT in VO2
• orbital occupation: from isotropic (metal) to s-polarized (insulator)
• electronic structure: from 3-dimensional to effectively 1-dimensional
•
 more susceptible to Peierls transition: dimerization
• dramatic switching only possible if close to Mott limit.
• “collaborative” Mott-Peierls transitions
Calculating electronic structure and spectral weights of correlated systems:
LDA + cluster DMFT:
S. Biermann, A. Poteryaev, A. Lichtenstein, A. Georges,
Phys. Rev. Lett. 94, 026404 (2005)
Switching of orbital occupation – XAS:
Haverkort et al., Phys. Rev. Lett. 95, 196404 (2005)
Spectral weight transfer – Photoemission:
Koethe et al., Phys. Rev. Lett. 97, 116402 (2006)
Summary: Metal-insulator transition in d1 system: VO2
• LDA
• LDA+U
• LDA+DMFT
• LDA+CDMFT
•
: fail completely
: ???
: not good enough !
: good results, also for photoemission !
: inverse photoemission: untested !!
Dimers have great impact in VO2
• from XAS
• from LDA+CDMFT
- but not so from PES
How about Ti2O3 and V2O3 ?
• dimers important ?
• can we observe them with PES ?
Role of dimers in V2O3 and Ti2O3?
• Corundum structure
• MIT in V2O3 and Ti2O3
• V-V and Ti-Ti pairs in c-direction
V 2 O3
Wei Bao et al., Phys. Rev. Lett. 78, 507 (1997)
“Classic” Ansatz for V2O3
• V3+
: 3d2, S=1
• V3+-V3+ pairs: a1g molecular singlet formation  effectively S=1/2
• low T AF: 1.2µB/V [R. M. Moon, PRL 25, 527 (1970)]  taken as evidence for S=1/2!
same Ansatz for Ti2O3
• Ti3+
: 3d1, S=1/2
• Ti3+-Ti3+ pairs: a1g molecular singlet formation  effectively S=0
Comparison:
Experiment
vs.XAS
Theory
Orbital occupation
in Ti2O3 from
EC
E II C
diff.
Experiment
T= 300 K
Theory
a1ga1g
EC
E II C
diff.
Intensity (arb. units)
Intensity (arb. units)
455
460
465
Energy (eV)
470
455
460
465
470
Energy (eV)
At insulating state: dimers are formed in Ti2O3!
Insulating state: Ti3+-Ti3+ c-axis dimers are electronically formed
orbital occupation= a1ga1g !
Comparison:
Experiment
vs.XAS
Theory
Orbital occupation
in Ti2O3 from
EC
E II C
diff.
Experiment
T= 300 K
Theory
a1ga1g
EC
E II C
diff.
Intensity (arb. units)
Intensity (arb. units)
455
460
465
Energy (eV)
470
455
460
465
470
Energy (eV)
At insulating state: dimers are formed in Ti2O3!
Insulating state: Ti3+-Ti3+ c-axis dimers are electronically formed
orbital occupation= a1ga1g !
Temperature
Orbital occupation
in Ti2O3: dependence
temperature-dependence
Experiment
EC
E II C
Theory
EC
E II C
575 K
71.5%
a1ga1g
500 K
77.8%
a1ga1g
458 K
a1ga1g
300 K
455
460
465
Energy (eV)
470
455
460
465
Energy (eV)
470
Intensity (arb. units)
Intensity (arb. units)
48.7%
a1ga1g
Orbital occupation in Ti2O3: temperature-dependence
“dimer”
MIT in Ti2O3:
π)= 0.96 : 0.04
1g
gradual
n(a
):n(egtransition
 ~101 change in ρ
“isotropic”
Ti2O3
LDA:
in metallic phase
L. F. Mattheiss, 1996
~101 phase
LDA+DMFT: n(egπ)=0.09 insulating phase
0.15 metallic
n(a1g) : n(egp)
A. I. Poteryaev et al., 2004
LDA (Mattheis 1996)
(M) 0.96 : 0.04
Three-band
model:
DMFT
(Poteryaev 2005) Hubbard
(I)
0.90 : 0.10
π)= 0.90 : 0.1 at 10K
(M)1g):n(eg
0.85 : 0.15
n(a
Cluster (Tanaka 2004)
(I)
0.90 : 0.10
π)= 0.61 : 0.39 at 800K
n(a
):n(eg
A. Tanaka,2004
500 K
(M) 1g 0.61 : 0.39
300 K
Bulk sensitive photoemission on Ti2O3 single crystals
Bulk sensitive photoemission on Ti2O3 single crystals
U/t = 0
U/t = 1
antibonding
U/t = 5
bonding
U/t = 10
U/t = 100
Two-peak structure like in a H2 molecule model 
(relative weights according to quantum mechanical interference effect)
H2 molecule model
S
anti-bonding
M
M
bonding
bonding
EF
S
anti-bonding
Comparison experiment vs. theory
Chang, Koethe et al. (Cologne)
Poteryaev, Lichtenstein, Kotliar.,
Phys. Rev. Lett. 93, 086401 (2004).
2t = 1.7 eV
antibonding
bonding
too low intensity of anti-bonding peak ?!
“Classic” Ansatz for V2O3
• V3+
: 3d2, S=1
• V3+-V3+ pairs: a1g molecular singlet formation  effectively S=1/2
• low T AF: 1.2µB/V [R. M. Moon, PRL 25, 527 (1970)]  taken as evidence for S=1/2!
p. 11506
J.-H. Park, L.H. Tjeng, A. Tanaka, J.W. Allen, C.T. Chen, P. Metcalf, J.M. Honig, F.M.F. de Groot, G.A. Sawatzky
Experiment:
• S = 1!
• rejects the existence of the
claimed molecular orbital
singlet formation (= the dimer)
I. S. Elfimov, T. Saha-Dasgupta, and M. A. Korotin
t1 = -0.25eV
t2 = t3 = t4 = 0
t3 = -0.15eV
t2 = t4 = 0
no electronic sign
for a dimer
t2
t4
t3
t4 = -0.06eV t2 = 0
t2 = -0.03ev
Where to find and not to find the dimers?
V2O3
Cr2O3
c-axis pair bond length
below TMIT above TMIT
Ti2O3
V2O3
Ti2O3
Cr2O3
2.579 Å
(300K)
2.761 Å
(15K)
2.725 Å
(780K)
2.709 Å
(300K)
2.650 Å
(300K)
c-axis dimers are present structurally in corundum structures
but exist electronically only in Ti2O3 and Cr2O3.
Metal insulator transitions
3d2 system: V2O3
“k-dependence of the self-energy”
metal-insulator transitions (MIT) in V2O3
Phys. Rev. B 22, 2626 (1980)
Phys. Rev. B 7, 1920 (1973)
classical example for a Mott-transition, i.e. beyond band structure effects
(note: big resistivity jumps by themself do not make the MIT special)
p. 105
TMIT 140 K
metallic
V2O3
Egap = 0.66 eV
metal-insulator transitions (MIT) in V2O3:
• enormous transfer of spectral weight
• kBTMIT << Egap
insulating
V2O3
extreme case = best test case
for new theories
Enormous transfer of spectral weight across MIT in V2O3
J.-H. Park, thesis, Univ. of Michigan, 1994
AFI
AFI
photoemission
PI
PM
Photoemission: AFI very different from PM, but AFI very similar to PI
(neutrons: AFI very different from PM and PI, but PM similar to PI)
Issues to be addressed :
metal-insulator transitions (MIT) in V2O3:
• enormous transfer of spectral weight
• Egap / kBTMIT > 10 - 40
In contrast to weak coupling, e.g. BCS:
• 2D / kBTc 3.5
• Why ??
• Which entropy drives the transition ??
Our hypothesis: at the MIT, not only gap closes but also spin
and orbital structures change with consequences for the band
width.
Are there new theoretical developments to address these issues?
Maybe ! But must be beyond single-site approaches ?!
How good is single-site DMFT with spectral weights across MIT ??
J.-H. Park, thesis
Univ. of Michigan, 1994
Phys. Rev. Lett. 86, 5345 (2001)
AFI
PM
photoemission
AFI
PI
Single-site DMFT: fast transfer of spectral weight, but not enough!
• AFI to PM : one-electron band width changes with ~ 10%
• needed
: larger change in effective band width ~ 30% or more
p. 11506
J.-H. Park, L.H. Tjeng, A. Tanaka, J.W. Allen, C.T. Chen, P. Metcalf, J.M. Honig, F.M.F. de Groot, G.A. Sawatzky
Experiment:
• orbital occupation changes in
going from AFI to PM to PI
Experimental observations:
V2 O3 :
orbital occupation of the V 3d2 ions significantly changes
across the AFI-PM, AFI-PI, and PM-PI transitions
J.-H Park et al., Phys. Rev. B 61, 11506 (2000)
V2O3:
dramatic switching of magnetic short-range exhange interactions
across the AFI-PM and AFI-PI transitions
W. Bao et al., Phys. Rev. Lett. 78, 507 (1997)

k-dependence of the self-energy ?!!
…… inter-site spin and/or orbital correlations
• changes in orbital occupation  changes in S(,k)
• changes in orbital occupation  changes in exchange interactions:
- short range, nearest neighbor
- Goodenough-Kanamori-Anderson rules
• changes in exchange interactions  changes in S(,k)
Optical transitions : excitonic
ferromagnetic-cluster
antiferromagnetic-cluster
hn
hn
hn = U0 – 2 JH
hn = U0
Note: JH 0.7 eV hardly screened from atomic values, Antonides et al., PRB 15, 1669 (1977)
• changes in orbital occupation  changes in S(,k)
• changes in orbital occupation  changes in exchange interactions:
- short range, nearest neighbor
- Goodenough-Kanamori-Anderson rules
• changes in exchange interactions  changes in S(,k)
Optical transitions : excitonic
ferromagnetic-cluster
antiferromagnetic-cluster
hn
hn
hn = U0 – 2 JH
hn = U0
Note: JH 0.7 eV hardly screened from atomic values, Antonides et al., PRB 15, 1669 (1977)
Photoemission, Inverse Photoemission, Conductivity gap
hn
ferromagnetic-cluster
far left
PES
ferromagnetic-cluster
far right
IPES
t
WN-1= t
t
Egap= U0-2JH-2t
hn
antiferromagnetic-cluster
far left
PES
antiferromagnetic-cluster
far right
IPES
t
WN-1= t2/JH
WN+1= t
t
Egap U0-2JH-0.9t
WN+1= t2/JH
Photoemission, Inverse Photoemission, Conductivity gap
hn
ferromagnetic-cluster
far left
PES
ferromagnetic-cluster
far right
IPES
t
WN-1= t
t
Egap= U0-2JH-2t
hn
antiferromagnetic-cluster
far left
PES
antiferromagnetic-cluster
far right
IPES
t
WN-1= t2/JH
WN+1= t
t
Egap U0-2JH-0.9t
WN+1= t2/JH
Influence of intersite spin correlations on electronic structure:
• J.-H-Park, L.H. Tjeng et al., Phys. Rev. B 61, 11506 (2000)
Spin and orbital occupation and phase transitions in V2O3
[with example of ferro/antiferro-cluster]
• A. Tanaka, J. Phys. Soc. Jpn. 73, 152 (2004)
On the metal-insulator transition in VO2 and Ti2O3 from a unified viewpoint
• L.N. Bulaevvskii and D.I. Khomskii, Sov. Phys.- Solid State 9, 2422 (1968)
Insulator-metal transitions in antiferromagnets
For intersite spin correlations to have strong impact on band width,
it is required that the correlations on a short range scale are changed
--- Orbital occupation changes will trigger this in a natural manner
k-dependence of the self-energy:
• crucial part of MIT in dn systems (degeneracy  JH)
• important for inverse photoemission on d1 systems (degeneracy  JH)
Photoemission / inverse photoemission: the technique for measuring
short-range exchange correlations