Title: Electronic Structure of Strongly Correlated Materials : a DMFT Perspective
1Electronic Structure of Strongly Correlated
Materials a DMFT Perspective
- Gabriel Kotliar
- Physics Department and
- Center for Materials Theory
- Rutgers University
Palo Alto May 2002
2Outline
- Correlated electrons he Mott transition problem
and electronic structure. - Dynamical Mean Field Theory
- Model Hamiltonian Studies of the Mott
transition. Universal aspects. - System specific studies LDADMFT, d Pu (with S
. Savrasov and E. Abrahams) - Outlook
3 Strongly Correlated Materials
- Copper Oxides. .(La2-x Bax) CuO4 High Temperature
Superconductivity.150 K in the Ca2Ba2Cu3HgO8 . - Uranium and Cerium Based Compounds. Heavy
Fermion Systems,CeCu6,m/m1000 - (La1-xSrx)MnO3 Colossal Magneto-resistance.
4 Strongly Correlated Materials.
- Battery Materials LiCoO2. Large stability for
changing carrier density. - Large thermoelectric response in CeFe4 P12 (H.
Sato et al. cond-mat 0010017). Ando et.al. - NaCo2-xCuxO4 Phys. Rev. B 60, 10580 (1999).
- Large and ultrafast optical nonlinearities
Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85,
2204 (2000) )
5The electron in a solid wave picture
Momentum Space (Sommerfeld)
Maximum metallic resistivity 200 mohm cm
Standard model of solids (Bloch, Landau)
Periodic potential, waves form bands , k in
Brillouin zone . Interactions renormalize away.
6Standard Model of Solids
- Qualitative predictions low temperature
dependence of thermodynamics and transport. - Optical response, transition between the bands.
- Qualitative predictions filled bands give rise
to insulting behavior. Compounds with odd number
of electrons are metals. - Quantitative tools Density Functional Theory
with approximations suggested by the Kohn Sham
formulation, (LDA GGA) is a successful
computational tool for the total energy. Good
starting point for perturbative calculation of
spectra,eg. GW. Kinetic equations yield transport
coefficients.
7The electron in a solid particle picture.
- Array of hydrogen atoms is insulating if agtgtaB.
Mott correlations localize the electron - e_ e_ e_
e_ - Superexchange
Think in real space , solid collection of
atoms High T local moments, Low T spin-orbital
order
8Mott Correlations localize the electron
Low densities, electron behaves as a particle,use
atomic physics, real space One particle
excitations Hubbard Atoms sharp excitation
lines corresponding to adding or removing
electrons. In solids they broaden by their
incoherent motion, Hubbard bands (eg. bandsNiO,
CoO MnO.) H H H H H H
motion of H forms the lower Hubbard band H
H H H- H H motion of H_
forms the upper Hubbard band Quantitative
calculations of Hubbard bands and exchange
constants, LDA U, Hartree Fock. Atomic Physics.
9Photoemission spectroscopy.
Measures density ofstates for (BIS) adding and
(PES) removing electrons
10Localization vs Delocalization Strong Correlation
Problem
- A large number of compounds with electrons in
partially filled shells, are not close to the
well understood limits (localized or itinerant).
Non perturbative problem. - These systems display anomalous behavior
(departure from the standard model of solids). - Neither LDA or LDAU or Hartree Fock work well.
- Dynamical Mean Field Theory Simplest approach to
electronic structure, which interpolates
correctly between atoms and bands. treats QP
and Hubbard bands.
11Failure of the standard model Mott transition in
V2O3 under pressure or chemical substitution on
V-site
12Localization vs Delocalization Strong Correlation
Problem
- A large number of compounds with electrons in
partially filled shells, are not close to the
well understood limits (localized or itinerant).
Non perturbative problem. - These systems display anomalous behavior
(departure from the standard model of solids). - Neither LDA or LDAU or Hartree Fock work well.
- Dynamical Mean Field Theory Simplest approach to
electronic structure, which interpolates
correctly between atoms and bands. Treats QP
bands and Hubbard bands.
13 Mott transition in V2O3 under pressure or
chemical substitution on V-site
14Mott transition in layered organic conductors
S Lefebvre et al. cond-mat/0004455, Phys. Rev.
Lett. 85, 5420 (2000)
15Failure of the Standard Model NiSe2-xSx
Miyasaka and Takagi (2000)
16Hubbard model
- U/t
- Doping d or chemical potential
- Frustration (t/t)
- T temperature
Mott transition as a function of doping, pressure
temperature etc.
17Limit of large lattice coordination
Metzner Vollhardt, 89
Muller-Hartmann 89
18Dynamical Mean Field Theory, cavity construction
A. Georges G. Kotliar 92
19Mean-Field Classical vs Quantum
Classical case
Quantum case
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
20Solving the DMFT equations
- Wide variety of computational tools
(QMC,ED.)Analytical Methods - Extension to ordered states.
- Review A. Georges, G. Kotliar, W. Krauth and
M. Rozenberg Rev. Mod. Phys. 68,13 (1996)
21Insights from DMFT
- Low temperature Ordered phases . Stability
depends on chemistry and crystal structure - High temperature behavior around Mott endpoint,
more universal regime, captured by simple models
treated within DMFT. Role of magnetic frustration.
22Kuwamoto Honig and Appell PRB (1980)M.
Rozenberg G. Kotliar H. Kajueter G Thomas D.
Rapkikne J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
23Insights from DMFT
- The Mott transition is driven by transfer of
spectral weight from low to high energy as we
approach the localized phase - Control parameters doping, temperature,pressure
24Spectral Evolution at T0 half filling full
frustration
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
25Parallel development Fujimori et.al
26Evolution of the Spectral Function with
Temperature
Anomalous transfer of spectral weight connected
to the proximity to the Ising Mott endpoint
(Kotliar Lange and Rozenberg Phys. Rev. Lett. 84,
5180 (2000)
27ARPES measurements on NiS2-xSexMatsuura et. al
Phys. Rev B 58 (1998) 3690. Doniach and Watanabe
Phys. Rev. B 57, 3829 (1998)
.
28Insights from DMFT
- The Mott transition is driven by transfer of
spectral weight from low to high energy as we
approach the localized phase - Control parameters doping, temperature,pressure
29Anomalous transfer of optical spectral weight V2O3
- M Rozenberg G. Kotliar and H. Kajuter Phys. Rev.
B 54, 8452 (1996). - M. Rozenberg G. Kotliar H. Kajueter G Tahomas D.
Rapkikne J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
30Anomalous transfer of optical spectral weight,
NiSeS. Miyasaka and Takagi
31Anomalous Resistivity and Mott transition Ni
Se2-x Sx
Insights from DMFT think in term of spectral
functions (branch cuts) instead of well defined
QP (poles )
32Insights from DMFT
- Important role of the incoherent part of the
spectral function at finite temperature - Physics is governed by the transfer of spectral
weight from the coherent to the incoherent part
of the spectra. Real and momentum space. - Mott transitions bifurcation points of an
effective action functional. Simple behavior.
33Two roads for ab-initio calculation of electronic
structure of strongly correlated materials
Crystal structure Atomic positions
Model Hamiltonian
Correlation Functions Total Energies etc.
34Realistic Calculationsof the Electronic
Structure of Correlated materials
- Combinining DMFT with state of the art electronic
structure methods to construct a first principles
framework to describe complex materials. - Anisimov Poteryaev Korotin Anhokin and Kotliar J.
Phys. Cond. Mat. 35, 7359 (1997) - Savrasov Kotliar and Abrahams Nature 410, 793
(2001))
35Combining LDA and DMFT
- The light, SP (or SPD) electrons are extended,
well described by LDA - The heavy, D (or F) electrons are localized,treat
by DMFT. - LDA already contains an average interaction of
the heavy electrons, subtract this out by
shifting the heavy level (double counting term) - The U matrix can be estimated from first
principles or viewed as parameters
36Spectral Density Functional effective action
construction (Fukuda, Valiev and Fernando ,
Chitra and Kotliar).
- DFT, consider the exact free energy as a
functional of an external potential. Express the
free energy as a functional of the density by
Legendre transformation. GDFTr(r) - Introduce local orbitals, caR(r-R)orbitals, and
local GF - G(R,R)(i w)
- The exact free energy can be expressed as a
functional of the local Greens function and of
the density by introducing Gr(r),G(R,R)(iw) - A useful approximation to the exact functional
can be constructed, the DMFT LDA functional.
Savrasov Kotliar and Abrahams Nature 410, 793
(2001))
37LDADMFT Self-Consistency loop
E
U
DMFT
38 Case study in f electrons, Mott transition in
the actinide series. B. Johanssen 1974 Smith and
Kmetko Phase Diagram 1984.
39Pu Anomalous thermal expansion (J. Smith LANL)
40Problems with LDA treatements of d Pu
- DFT in the LDA or GGA is a well established tool
for calculation of ground state properties. Many
studies (APW Freeman, Koelling 1972, ASA and
FP-LMTO, Soderlind et. al 1990, Kollar et.al
1997, Boettger et.al 1998, Wills et.al. 1999)
show - an equilibrium volume of the d phase Is 35
lower than experiment - Largest discrepancy ever known in DFT based
calculations. - LSDA predicts magnetic long range order which is
not observed experimentally (Solovyev et.al.) - If one treats the f electrons as part of the core
LDA overestimates the volume by 30 - Weak correlation picture for alpha phase.
41Conclusion
- The character of the localization delocalization
in simple( Hubbard) models within DMFT is now
fully understood, nice qualitative insights. - This has lead to extensions to more realistic
models, and a beginning of a first principles
approach interpolating between atoms and band,
encouraging results for simple elements
42Outlook
- The Strong Correlation ProblemHow to deal with
a multiplicity of competing low temperature
phases and infrared trajectories which diverge in
the IR - Strategy advancing our understanding scale by
scale - Generalized cluster methods to capture longer
range magnetic correlations - New structures in k space. Cellular DMFT
43Outlook
- Systematic improvements, short range
correlations. - Take a cluster of sites, include the effect of
the rest in a G0 (renormalization of the
quadratic part of the effective action). What
to take for G0 or D - DCA (M. Jarrell et.al) , CDMFT (Kotliar
Savrasov Palsson and Biroli ) - include the effects of the electrons to
renormalize the quartic part of the action (spin
spin , charge charge correlations) E. DMFT
(Kajueter and GK, Si et.al) - Exploration of materials.
44Acknowledgements Development of DMFT
Collaborators V. Anisimov, R. Chitra, V.
Dobrosavlevic, D. Fisher, A. Georges, H.
Kajueter, W.Krauth, E. Lange, A. Lichtenstein,
G. Moeller, Y. Motome, G. Palsson, M.
Rozenberg, S. Savrasov, Q. Si, V. Udovenko,
X.Y. Zhang
Support National Science Foundation. Work on
Pu Departament of Energy and LANL.
45Case study Fe and Ni
- Archetypical itinerant ferromagnets
- LSDA predicts correct low T moment
- Band picture holds at low T
- Main puzzle at high temperatures c has a Curie
Weiss law with a moment much larger than the
ordered moment. - Magnetic anisotropy
-
46Iron and Nickel crossover to a real space
picture at high T (Lichtenstein, Katsnelson and
Kotliar Phys Rev. Lett 87, 67205 , 2001)
47Iron and Nickelmagnetic properties
(Lichtenstein, Katsenelson,GK PRL 01)
48Ni and Fe theory vs exp
- m( T.9 Tc)/ mB ordered moment
- Fe 1.5 ( theory) 1.55 (expt)
- Ni .3 (theory) .35 (expt)
- meff / mB high T moment
- Fe 3.1 (theory) 3.12 (expt)
- Ni 1.5 (theory) 1.62 (expt)
- Curie Temperature Tc
- Fe 1900 ( theory) 1043(expt)
- Ni 700 (theory) 631 (expt)
49Fe and Ni
- Consistent picture of Fe (more localized) and Ni
- (more correlated)
- Satellite in minority band at 6 ev, 30
reduction of bandwidth, exchange splitting
reduction .3 ev - Spin wave stiffness controls the effects of
spatial flucuations, it is about twice as large
in Ni and in Fe - Mean field calculations using measured exchange
constants(Kudrnovski Drachl PRB 2001) right Tc
for Ni but overestimates Fe , RPA corrections
reduce Tc of Ni by 10 and Tc of Fe by 50.
50Photoemission Spectra and Spin Autocorrelation
Fe (U2, J.9ev,T/Tc.8) (Lichtenstein,
Katsenelson,Kotliar Phys Rev. Lett 87, 67205 ,
2001)
51Photoemission and T/Tc.8 Spin Autocorrelation
Ni (U3, J.9 ev)
52LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
53Landau Functional
G. Kotliar EPJB (1999)
54Comments on LDADMFT
- Static limit of the LDADMFT functional , with F
FHF reduces to LDAU - Removes inconsistencies and shortcomings of this
approach. DMFT retain correlations effects in
the absence of orbital ordering. - Only in the orbitally ordered Hartree Fock limit,
the Greens function of the heavy electrons is
fully coherent - Gives the local spectra and the total energy
simultaneously, treating QP and H bands on the
same footing.
55LDA functional
Conjugate field, VKS(r)
56Minimize LDA functional
Kohn Sham eigenvalues, auxiliary quantities.
57Anomalous transfer of spectral weight
58Anomalous Resistivities DopedHubbard Model (QMC)
Prushke and Jarrell 1993
59Anomalous ResistivitiesDoped Hubbard ModelG.
Palsson 1998
NCA
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60Failure of the standard model Anomalous
ResistivityLiV2O4
Takagi et.al. PRL 2000
61Anomalous Spectral Weight Transfer Optics
Below energy
ApreciableT dependence found.
Schlesinger et.al (FeSi) PRL 71 ,1748 , (1993) B
Bucher et.al. Ce2Bi4Pt3PRL 72, 522 (1994),
Rozenberg et.al. PRB 54, 8452, (1996).
62Schematic DMFT phase diagram Hubbard model
(partial frustration)
M. Rozenberg G. Kotliar H. Kajueter G Thomas D.
Rapkikne J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
63ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe
Phys. Rev. B 57, 3829 (1998)
.
64Spectral Density Functional
- The exact functional can be built in perturbation
theory in the interaction (well defined
diagrammatic rules )The functional can also be
constructed expanding around the the atomic
limit. No explicit expression exists. - DFT is useful because good approximations to the
exact density functional GDFTr(r) exist, e.g.
LDA, GGA - A useful approximation to the exact functional
can be constructed, the DMFT LDA functional.
Savrasov Kotliar and Abrahams Nature 410, 793
(2001))
65LDA functional
Conjugate field, VKS(r)
66Phase Diag Ni Se2-x Sx
67V2O3 resistivity