Transcript Document 7337866

Dynamical Mean Field Theory
(DMFT) Approach to Correlated
Materials
G. Kotliar
Physics Department and Center for
Materials Theory
Rutgers
Outline
• Introduction to the Dynamical Mean Field
ideas and techniques.
• Learning about materials with DMFT: (or
Mott physics is everywhere ).
• Kappa organics <sp>
• The Mott transition across the actinide
series , Pu- Am <5f>
• Ti2O3 -----LixCoO3----Fe-Ni <3d>
• Ce < 4f>
Schematic DMFT phase diagram and DOS of a
partially frustrated integer filled Hubbard
model and pressure driven Mott transition.
Physics Today Vol 57, 53
(2004)
Outline, Collaborators, References
• Introduction to extensions of DMFT for
applications to electronic structure. [ S.
Savrasov and Phys. Rev. B 69, 245101
(2004) ]
• C-DMFTstudy of the Mott transition in kappa
organics. [O. Parcollet G. Biroli and GK PRL,
92, 226402. (2004) ]
• The Mott transition in Actinides Pu [Xi Dai S.
Savrasov GK A Migliori H. Ledbetter E.
Abrahams Science 300, 953 (2003)] and Am[J.
C Griveaux J. Rebizant G. Lander and GK
][Sahana Murthy Ph. D].
•
MIT in Ti2O3[S. Poteryaev S.
Lichtenstein and GK cond-mat 0311319 ]
• Alpha Gamma transition in Cerium.
K. Haule S. Savrasov V. Udovenko and GK
cond-matt 2004.
Weakly correlated electrons. FLT and DFT, and what goes wrong in
correlated materials.
• Fermi Liquid . . Correspondence between a
system of non interacting particles and the full
Hamiltonian.
• A band structure is generated (Kohn Sham
system).and in many systems this is a good
starting point for perturbative computations of the
spectra (GW).
 [  (r) ,  (r) ]
A different paradigm: the area of
influence of a quantum critical point
Energy Landscape of a Correlated
Material and a top to bottom
approach to correlated materials.
Energy
T
Configurational Coordinate in the space of Hamiltonians
DMFT Cavity Construction: A. Georges and G. Kotliar
PRB 45, 6479 (1992). Figure from : G. Kotliar and D.
Vollhardt Physics Today 57,(2004)
http://www.physics.rutgers.edu/~kotliar/RI_gen.html
The self consistent impurity model is a new
reference system, to describe strongly
correlated materials.
H  H cluster  H cluster exterior  H exterior
H
H cluster  H cluster exterior  H exterior
Simpler "medium" Hamiltonian
Dynamical Mean Field
Theory (DMFT) Cavity
Construction: A. Georges
and G. Kotliar PRB 45, 6479
(1992).
EDMFT [H. Kajueter Rutgers Ph.D Thesis 1995 Si and Smith
PRL77, 3391(1996) R. Chitra and G. Kotliar PRL84,3678 (2000)]


i , j  ,
b
D0-
1
i
  Vij ni n j
i , j 
b
òò
0
(tij   ij )(ci† c j  c †j ci )  U  ni  ni 
co†s ( t )Go( t , t ')cos ( t ') + no­ no¯U d( t , t ')
+ Do(t , t ')n0n0
0
é
(iwn ) = ê
ê
ê
ë
- 1
å
k
ù
1
ú
Vk - P (iwn ) ú
ú
û
+ P (iwn )
S (iwn )[G0 ] = G0- 1 (iwn ) + [áca† (iwn )cb (iwn )ñS (G0 ) ]- 1
P (iwn )[G0 ] = D0- 1 (iwn ) + [án0 (iwn )n0(iwn )ñS () ]é
- 1
G0 (iwn ) = ê
ê
ê
ë
1
- 1
å
k
ù
1
ú
iwn - tk + m- S (iwn ) ú
ú
û
+ S (iwn )
Site Cell. Cellular DMFT. C-DMFT.
G. Kotliar,S.. Savrasov,
G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)
tˆ(K) hopping expressed in the superlattice notations.
•Other cluster extensions (DCA Jarrell Krishnamurthy,
Katsnelson and Lichtenstein periodized scheme, Nested
Cluster Schemes Schiller Ingersent ), causality issues, O.
Parcollet, G. Biroli and GK cond-matt 0307587 (2003)
Two paths for ab-initio
calculation of electronic
structure of strongly
correlated materials
Crystal structure +Atomic
positions
Model Hamiltonian
Correlation Functions Total
Energies etc.
DMFT ideas can be used in both cases.
LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin
and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). A Lichtenstein and M.
Katsnelson PRB 57, 6884 (1988).
• The light, SP (or SPD) electrons are extended,
well described by LDA .The heavy, D (or F)
electrons are localized treat by DMFT.
• LDA Kohn Sham Hamiltonian already contains an
average interaction of the heavy electrons, subtract
this out by shifting the heavy level (double counting
term)
Kinetic energy is provided by the Kohn Sham
Hamiltonian (sometimes after downfolding ). The U
matrix can be estimated from first principles of
viewed as parameters. Solve resulting model using
DMFT.
Functional formulation. Chitra and Kotliar (2001), Savrasov and Kotliarcondmatt0308053 (2003).
+
Ir>=|R, >
G = - < y ( Rr ') y † ( Rr ') >
1
f ( x)VC - 1 ( x, x ')f ( x ') +
ò
ò
2
ò if ( x)y
†
( x )y ( x )
< f ( Rr ')f ( Rr ) > - < f ( Rr ') > < f ( Rr ) > = W
1
1
[G,W ]  TrLnG  Tr[G01  G 1 ]G  TrLnW  Tr[VC1  W 1 ]W  Ehartree  [G,W ]
2
2
[G,W ]  EDMFT [Gloc ,Wloc , Gnonloc  0,Wnonloc  0]
Double loop in Gloc and Wloc
Next Step: GW+EDMFT S. Savrasov and GK.(2001). in New
Theoretical Approaches to Strongly Correlated Systems, A.M. Tsvelik
Ed., Kluwer Academic Publishers 259-301, (2001))
W
W
.P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002)
Phys. Rev. Lett. 91, 037209 (2003)
Biermann et.al. PRL 90,086402 (2003)
Impurity model representability of
spectral density functional.
LDA+DMFT Self-Consistency loop. S. Savrasov and G.
Kotliar (2001) and cond-matt 0308053
c ka | ­ Ñ 2 + Vxc (r ) | c ka = H LMTO ( k )
Impurity
Solver
G
0
E
G
U

S.C.C.
DMFT
r (r) = T
å
iw
G( r, r, iw)e
iw0+
nHH = T
å
iw
+
GHH ( r , r , iw)eiw 0
Impurity Solvers.
• Hubbard I.
• Fye Hirsch Quantum Montecarlo.
• Interpolative schemes for the self energy.
H. Kajueter and G. Kotliar PRL (1996).
cond-mat/0401539 V. Oudovenko, K.
Haule, S. Savrasov D. Villani and G. Kotliar.
• Extensions of NCA. Th. Pruschke and N.
Grewe, Z. Phys. B: Condens. Matter 74,
439, 1989. SUNCA K. Haule, S. Kirchner, J.
Kroha, and P. W¨olfle, Phys. Rev. B 64,
155111, (2001). K. Haule et. al. (2004)
How good is the local approximation ?
• It becomes exact as the coordination
number increases or in the limit of infinite
dimensions introduced by Metzner and
Vollhardt. PRL 62,34, (1989).
• How good is it in low dimensions ?
Promising recent developments from
theory and experiments.
One dimensional Hubbard model .
Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats,
[V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][ [M.
CaponeM.Civelli V Kancharla C.Castellani and GK Phys. Rev. B 69, 195105
U/t=4.
(2004) ]
Applications of DMFT to materials :
Goals of the research
• Computations develop a first principles method ,
based on DMFT, capable of predicting physical
properties of correlated materials.
• Develop a physical picture of the f and spd
electrons in Ce and Pu.
• Test the theory against experiments.
• Bring theory to the point that it plays an equal
role in the field of correlated electron materials.
Combining theory and experiment.
Experimental verifications
• Finding the QP , the Hubbard band and
the transfer of spectral weight between
them in optics and photoemission in
different materials.
• Exploring the various regimes of the phase
diagram, including the Mott endpoint using
transport probes.
Recent Experiments support qualitative
single site DMFT predictions
Mo et al., Phys. Rev.Lett.
90, 186403 (2003).
Limelette et. al.(2003)
Ito et. al. (1995)
Outline
• Introduction to the Dynamical Mean Field
ideas and techniques.
• Learning about materials with DMFT: (or
Mott physics is everywhere ).
• Kappa organics <sp>
• The Mott transition across the actinide
series , Pu- Am <5f>
• Ti2O3 -----LixCoO3----Fe-Ni <3d>
• Ce < 4f>
ET =
k-(ET)2X are across Mott transition
Insulating
anion layer
X-
Ground
State
U/t
t’/t
Cu2(CN)3
Mott
insulator
8.2
1.06
Cu[N(CN)2]Cl
Mott
insulator
7.5
0.75
Cu[N(CN)2]Br
SC
7.2
0.68
Cu(NCS)2
SC
6.8
0.84
Cu(CN)[N(CN)2 SC
]
Ag(CN)2 H2O
SC
6.8
0.68
6.6
0.60
I3
6.5
0.58
X-1
conducting
ET layer
[(ET)2]+1
modeled to triangular lattice
t’
t
SC
Prof. Kanoda U. Tokyo
Mott transition in layered organic conductors
cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)
S Lefebvre et al.
• Theoretical issue: is there a Mott transition
in the integer filled Hubbard model, and is it
well described by the single site DMFT ?
Double Occupancy vs U
• CDMFT Parcollet,
Biroli GK PRL (2004)
Study frustrated t
t’ model
t’/t=.9
Evolution of the spectral
function at low frequency.
A(  0, k )vs k
Ek=t(k)+Re( k ,   0)  
 k = Im( k ,   0)
A( k ,   0) 
k
 k 2  Ek 2
If the k dependence of the self energy is
weak, we expect to see contour
lines corresponding to Ek = const
and a height increasing as we
approach the Fermi surface.
Evolution of the k resolved
Spectral Function at zero A(  0, k )vs k
frequency. (Parcollet Biroli and GK)
U/D=2
Uc=2.35+-.05, Tc/D=1/44
U/D=2.25
Near the transition k
dependence is strong.
• Qualitative effect, formation of hot regions!
• D wave gapping of the single particle
spectra as the Mott transition is
approached. New paradigm for thinking
about the approach to the Mott insulator.
• Square symmetry is restored as we
approched the insulator.
• Experimental predictions! Photoemission
?
Lattice and cluster self energies
Mechanism for hot spot formation: nn
self energy ! General phenomena.
Conclusion.
• Mott transition survives in the cluster
setting. Role of magnetic frustration.
• Surprising result: formation of hot and cold
regions as a result of an approach to the
Mott transition. General result ?
• Unexpected role of the next nearest
neighbor self energy.
• CDMFT a new window to extend DMFT to
lower temperatures.
Mott transition in the actinide series
(Smith-Kmetko phase diagram)
Total Energy as a function of volume for Pu
(Savrasov, Kotliar, Abrahams, 2001,410,793, 2001)
DMFT Phonons in fcc -Pu
C11 (GPa)
C44 (GPa)
C12 (GPa)
C'(GPa)
Theory
34.56
33.03
26.81
3.88
Experiment
36.28
33.59
26.73
4.78
( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)
(experiments from Wong et.al, Science, 22 August 2003)
Mott transition in the actinide series
(Smith-Kmetko phase diagram)
Am
At room pressure a localised 5f6 system;j=5/2. S = -L = 3: J = 0
J. Smith & R. Haire, Science (1978)
J. Smith, J. Phys. (1979)
Mott transition into an open (right)
and closed (left) shell systems.
S
.5  T2
Log[2J+1]
S
???
Uc
U
S=0
 ~1/(Uc-U)
U
• Approach the Mott transition, if the
localized configuration has an OPEN shell
the mass increases as the transition is
approached.
Consistent theory, entropy increases
monotonically as U  Uc .
• Approach the Mott transition, if the
localized configuration has a CLOSED
shell. We have an apparent paradox. To
approach the Mott transitions the bands
have to narrow, but the insulator has not
entropy.. SOLUTION: superconductivity
intervenes.
Mott transition in systems
with close shell.
• Resolution: as the Mott transition is
approached from the metallic side,
eventually superconductivity intervenes to
for a continuous transition to the localized
side.
• DMFT study of a 2 band model for
Buckminster fullerines Capone et. al.
Science 2002.
• Mechanism is relevant to Americium.
Am under pressure. Lindbaum
et.al. PRB 63,2141010(2001)
ITU [J.C. Griveaux J. Rebizant G.
Lander]
Overview of rho (p, T)
of Am
• Note strongly
increasing
resistivity as f(p)
at all T. Shows
that more
electrons are
entering the
conduction band
• Superconducting
at all pressure
• IVariation of rho
vs. T for
increasing p.
DMFT study in the fcc structure. S.
Murthy and G. Kotliar
fcc
LDA+DMFT spectra. Notice the
rapid occupation of the f7/2 band.
One electron spectra. Experiments (Negele) and LDA+DFT
theory (S. Murthy and GK )
Conclusion Am
• Crude LDA+DMFT calculations describe the crude
energetics of the material, eq. volume, even p vs V .
• Superconductivity near the Mott transition.
Tc increases first and the decreases as we approach the
Mott boundary.
Dramatic effect in the f bulk module.
 What is going on at the Am I- Am II boundary ??? Subtle
effect (bulk moduli do not change much ), but crucial
modifications at low energy.
 Mott transition of the f7/2 band ? Quantum critical point
?:
Conclusion: Cerium
• Qualitatively good agreement with existing
experiment.
• Some quantitative disagreement, see
however .
• Experiments should study the
temperature dependence of the optics.
• Optics + Theory can provide a simple
resolution of the Mott vs K-Collapse
conundrum.
Conclusion
• DMFT mapping onto “self consistent
impurity models” offer a new “reference
frame”, to think about correlated materials
and compute their physical properties.
• Treats atomic excitations and band-like
quasiparticle on the same footing. Can
treat electrons near a localization
delocalization boundary.
Conclusions
• Essential for many materials. New
physics. Case studies.
• Kappa organics , hot –cold regions.
• Parcollet Biroli and Kotliar PRL 2004.
• Mott transition across the actinide series.
Pu and Am.
• Interplay with dimerization, and Coulomb
interactions. The two impurity model and
Ti2O3.
Conclusions.
• Alpha Gamma Cerium. Optics and Kondo scales. The
mechanism revealed.
• LixCoO3. Mott transition in your cell phone.[ C.
Marianetti, G. Kotliar and G. Ceder Nature materials in
press ]
• Itinerant ferromagnets at high and low temperatures, Ni
and Fe crossover from atomic to band physics.
• Doping driven Mott transition in three dimensional
materials: LaSrTiO3.
• Other work, other materials, ENS, Stuttgart Augsburg,
Ekaterinburg…………….
Ti2O3 : Coulomb or Pauling
C.E.Rice et all, Acta Cryst B33, 1342 (1977)

LTS 250 K, HTS 750 K.
Ti2O3.
• Isostructural to V2-xCrxO3. Al lot of the
qualitative physics of the high temperature part
of the phase diagram of V2O3 can be
understood within single site DMFT. Is this true
in Ti2O3?
• Band Structure Calculations good metal. L.F.
Mattheiss, J. Phys.: Condens. Matter 8, 5987
(1996) .Unrestricted Hartree Fock calculations
produce large antiferromagnetic gap. M. Cati,
et. al. Phys. Rev. B. f55 , 16122 (1997).
2site-Cluster DMFT with intersite Coulomb
U = 2, J = 0.5, W = 0.5
β = 20 eV-1, LT structure
U = 2, J = 0.5, W = 0.5
β = 10 eV-1, HT
structure
A. Poteryaev
Pauling and Coulomb Ti2O3[S.
Poteryaev S. Lichtenstein and GK cond-mat
0311319 ]
Dynamical GoodenoughHoning picture
Conclusion Ti2O3
• 2 site cluster DMFT describes the MIT in
Ti2O3.
• Different from V2O3 where single site DMFT
works well, and cluster corrections are small
[A. Poteryaev]
• It requires the Coulomb interactions, and a
frequency dependent enhancement of the a1ga1g hopping, induced by the Coulomb
interactions. [Haldane Ph.D thesis, Q Si and
GK 1993 ].Dynamical Pauling-Goodenough
mechanism is able to trigger the MIT at low
Overview
 Various phases :
isostructural phase transition (T=298K, P=0.7GPa)
 (fcc) phase
[ magnetic moment
(Curie-Wiess law) ]
  (fcc) phase
[ loss of magnetic
moment (Pauli-para) ]
with large
volume collapse
v/v  15
( -phase a  5.16 Å
-phase a  4.8 Å)
volume
s
exp.
28Å3

34.4Å3

  -phase
LDA
24.7Å3
LDA+U
35.2Å3
(localized):
High T phase
Curie-Weiss law (localized
magnetic moment),
Large lattice constant
Tk around 60-80K
 -phase (delocalized:Kondo-physics):
Low T phase
Loss of Magnetism (Fermi
liquid Pauli susceptibility) completely screened magnetic
moment
smaller lattice constant
Tk around 1000-2000K
Qualitative Ideas.
• B. Johansson, Philos. Mag. 30, 469 (1974). Mott
transition of the f electrons as a function of
pressure. Ce alpha gamma transition. spd
electrons are spectators.
• Mathematical implementation, “metallic phase”
treat spdf electrons by LDA, “insulating phase”
put f electron in the core.
 J.W. Allen and R.M. Martin, Phys. Rev. Lett. 49,
1106 (1982); Kondo volume collapse picture.
The dominant effect is the spd-f hybridization.
Qualitative Ideas
• alpha phase Kondo effect between spd
and f takes place. “insulating phase” no
Kondo effect (low Kondo temperature).
• Mathematical implementation, Anderson
impurity model in the suplemented with
elastic terms. (precursor of realistic DMFT
ideas, but without self consistency
condition). J.W. Allen and L.Z. Liu, Phys.
Rev. B 46, 5047 (1992).
LDA+DMFT:Ce spectra
M.B.Z¨olfl,I.A.NekrasovTh.Pruschke,V.I.Anisimov
J. Keller,Phys.Rev. Lett 87, 276403
(2001).
K. Held, A.K. McMahan, and R.T. Scalettar,
Phys. Rev.Lett. 87, 276404 (2001)
A.K.McMahan,K.Held,andR.T.Scalettar,Phys
Rev. B 67, 075108 (2003).
Successful calculations of
thermodynamics.
Unfortunately photoemission cannot decide
between the Kondo collapse picture and the
Mott transition picture.
Evolution of the spectra as a function of U ,
half filling full frustration, Hubbard model!!!!
X.Zhang M. Rozenberg
G. Kotliar (PRL 70,
1666(1993)).
The schematic phase diagram of
cannot distinguish between the two
scenarios.
•
J.W. Allen and L.Z. Liu, Phys. Rev.
B 46, 5047 (1992). Kondo impurity
model + elastic terms.
• DMFT phase diagram of a Hubbard
model at integer filling, has a region
between Uc1(T) and Uc2(T) where
two solutions coexist. A. Georges G.
Kotliar W. Krauth and M Rozenberg
RMP 68,13,(1996).
• Coupling the two solutions to the
lattice gives a phase diagram akin to
alpha gamma cerium. Majumdar and
Krishnamurthy PRL 73 (1994).
Photoemission&experiment
•A. Mc Mahan K Held and R. Scalettar (2002)
•Zoffl et. al (2002)
•K. Haule V. Udovenko S. Savrasov and GK. (2004)
To resolve the conflict between the
Mott transition and the volume
collapse picture : Turn to Optics!
Haule
et.al.
• Qualitative idea.
The spd
electrons have
much larger velocities, so optics will be
much more senstive to their behavior.
• See if they are simple spectators (Mott
transition picture ) or wether a Kondo
binding unbinding takes pace (Kondo
collapse picture).
• General method, bulk probe.
Temperature dependence of the
optical conductivity.
Theory: Haule et. al. cond-matt 04
Expt: J.W. vanderEb PRL 886,3407 (2001)
Optical conductivity of Ce (expt. Van Der Eb et.al.
theory Haule et.al)
experiment
LDA+DMFT
•K. Haule et.al.
Origin of the features.
Conclusion: Cerium
• Qualitatively good agreement with existing
experiment.
• Some quantitative disagreement, see
however .
• Experiments should study the
temperature dependence of the optics.
• Optics + Theory can provide a simple
resolution of the Mott vs K-Collapse
conundrum.
Conclusion
• DMFT mapping onto “self consistent
impurity models” offer a new “reference
frame”, to think about correlated materials
and compute their physical properties.
• Treats atomic excitations and band-like
quasiparticle on the same footing. Can
treat electrons near a localization
delocalization boundary.
Conclusions
• Essential for many materials. New
physics. Case studies.
• Kappa organics , hot –cold regions.
• Parcollet Biroli and Kotliar PRL 2004.
• Mott transition across the actinide series.
Pu and Am.
• Interplay with dimerization, and Coulomb
interactions. The two impurity model and
Ti2O3.
Conclusions.
• Alpha Gamma Cerium. Optics and Kondo scales. The
mechanism revealed.
• LixCoO3. Mott transition in your cell phone.[ C.
Marianetti, G. Kotliar and G. Ceder Nature materials in
press ]
• Itinerant ferromagnets at high and low temperatures, Ni
and Fe crossover from atomic to band physics.
• Doping driven Mott transition in three dimensional
materials: LaSrTiO3.
• Other work, other materials, ENS, Stuttgart Augsburg,
Ekaterinburg…………….
W (ev) vs
iw
(a.u. 27.2 ev) N.Zein
G. Kotliar and S. Savrasov
Anomalous Resistivity
PRL 91,061401 (2003)
Optical transfer of spectral weight
ET =
k-(ET)2X are across Mott
transition
X
Ground U/t t’/t
-
Insulating
anion layer
X-1
conducting
ET layer
[(ET)2]+1
modeled to triangular lattice
t’
t
State
Cu2(CN)3
Mott
insulator
8.2
1.06
Cu[N(CN)2]Cl
Mott
insulator
7.5
0.75
Cu[N(CN)2]Br
SC
7.2
0.68
Cu(NCS)2
SC
6.8
0.84
Cu(CN)[N(CN)2 SC
]
Ag(CN)2 H2O
SC
6.8
0.68
6.6
0.60
I3
6.5
0.58
SC
Controversy on the unfrustrated case.
Comment on "Absence of a Slater
Transition in the Two-Dimensional
Hubbard Model"
B. Kyung, J.S. Landry, D. Poulin, A.M.S. Tremblay Phys. Rev.
Lett. 90, 099702-1 (2003)
k organics
• ET = BEDT-TTF=Bisethylene dithio
tetrathiafulvalene
 k organics = (ET)2 X
Increasing pressure ----- increasing t’ ----------X0
X1
X3
• (Cu)(2CN)3 Cu(NCN)2 Cl
X2
Cu(NCN2)2Br
Compare with single site results
Rozenberg Chitra Kotliar PRL 2002
Mott transition in cluster (QMC)
Deviations from single site DMFT
W (ev) vs
iw
(a.u. 27.2 ev) N.Zein
G. Kotliar and S. Savrasov
3
Am II
Am I
Am III
Am IV
243
Am - This work
243
Am - Link et al.
241
Am - This work
Tc (K)
2
1
0
0
5
10
15
p (GPa)
20
25
30