Title: Electronic%20Structure%20of%20Strongly%20Correlated%20Materials%20:%20a%20DMFT%20Perspective
1Electronic Structure of Strongly Correlated
Materials a DMFT Perspective
- Gabriel Kotliar
- Physics Department and
- Center for Materials Theory
- Rutgers University
Boston March 2002
2Outline
- Introduction to strongly correlated electrons
- Dynamical Mean Field Theory
- Model Hamiltonian Studies. Universal aspects
insights from DMFT - System specific studies LDADMFT
- 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.
- High Temperature Superconductivity in doped
filled Bucky Balls (J. H. Schon et.al Science
Express 1064773 (2001)) CHBr3 C60 Tc117K . - 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.
9Localization 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 b
and Hubbard bands.
10Failure of the standard model Mott transition in
V2O3 under pressure or chemical substitution on
V-site
11Mott transition in layered organic conductors
S Lefebvre et al. cond-mat/0004455, Phys. Rev.
Lett. 85, 5420 (2000)
12Failure of the Standard Model NiSe2-xSx
Miyasaka and Takagi (2000)
13Failure of the standard model Anomalous
ResistivityLiV2O4
Takagi et.al. PRL 2000
14Failure of the StandardModel Anomalous Spectral
Weight Transfer
Optical Conductivity o of FeSi for T,20,20,250
200 and 250 K from Schlesinger et.al (1993)
Neff depends on T
15Strong Correlation Problem
- Large number of compounds (d,f,p.).Qualitative
and quantitive failures of the standard model. - Treat the itinerant and localized aspect of the
electron - The Mott transition, head on confrontation with
this issue - Dynamical Mean Field Theory simplest approach
interpolating between bands and atoms with open
shells.
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
Phys. Rev. B 45, 6497,1992
19Mean-Field Classical vs Quantum
A. Georges, G. Kotliar (1992)
Phys. Rev. B 45, 6497
Quantum case
Classical case
20Solving the DMFT equations
- Wide variety of computational tools
(QMC,NRG,ED.)Analytical Methods - Extension to ordered states.
- Review A. Georges, G. Kotliar, W. Krauth and
M. Rozenberg Rev. Mod. Phys. 68,13 (1996)
21Single site DMFT, functional formulation.
Construct a functional of the local Greens
function
- Expressed in terms of Weiss field
(semicircularDOS) G. Kotliar EBJB 99
22Insights from DMFT
- Low temperatures several competing phases .
Their relative stability depends on chemistry
and crystal structure - High temperature behavior around Mott endpoint,
more universal regime, captured by simple models
treated within DMFT
23Schematic DMFT phase diagram Hubbard model
(partial frustration)
M. Rozenberg G. Kotliar H. Kajueter G Tahomas D.
Rapkikne J Honig and P Metcalf Phys. Rev. Lett.
75, 105 (1995)
24Kuwamoto Honig and AppellPRB (1980)
25Phase Diag Ni Se2-x Sx
26Insights 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
27Evolution 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)
28Anomalous transfer of optical spectral weight,
NiSeS. Miyasaka and Takagi
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)
30ARPES measurements on NiS2-xSexMatsuura et. Al
Phys. Rev B 58 (1998) 3690
.
31Insights from DMFT think in term of spectral
functions (branch cuts) instead of well defined
QP (poles )
Resistivity near the metal insulator endpoint (
Rozenberg et.al 1995) exceeds the Mott limit
32Anomalous Resistivity and Mott transition Ni
Se2-x Sx
33Anomalous 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)
34Insights from DMFT
- Mott transition as a bifurcation of an effective
action - 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.
35Realistic 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. Beyond
LDAU approach (Anisimov, Andersen and Zaanen) - Anisimov Poteryaev Korotin Anhokin and Kotliar J.
Phys. Cond. Mat. 35, 7359 (1997)
36Combining 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
37Spectral 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))
38LDADMFT Self-Consistency loop
E
U
DMFT
39 Case study in f electrons, Mott transition in
the actinide series. B. Johanssen 1974 Smith and
Kmetko Phase Diagram 1984.
40Total energy vs Volume (Savrasov Kotliar and
Abrahams Nature 410, 793 (2001))
41Small amounts of Ga stabilize the d phase
42Problems with density functional treatements of d
Pu
- DFT in the LDA or GGA is a well established tool
for the 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 - This is the 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.
43Pu DMFT total energy vs Volume (Savrasov Kotliar
and Abrahams Nature 410, 793 (2001)
44Lda vs Exp Spectra (Joyce et.al.)
45Pu Spectra DMFT(Savrasov) EXP (Joyce , Arko et.al)
46Case 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
-
47Iron and Nickel crossover to a real space
picture at high T (Lichtenstein, Katsnelson and
Kotliar Phys Rev. Lett 87, 67205 , 2001)
48Iron and Nickelmagnetic properties
(Lichtenstein, Katsenelson,GK PRL 01)
49Ni and Fe theory vs exp
- m/ mB ordered moment
- Fe 2.5 ( theory) 2.2(expt)
- Ni .6 (theory) .6(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)
50Fe 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.
51Photoemission Spectra and Spin Autocorrelation
Fe (U2, J.9ev,T/Tc.8) (Lichtenstein,
Katsenelson,Kotliar Phys Rev. Lett 87, 67205 ,
2001)
52Photoemission and T/Tc.8 Spin Autocorrelation
Ni (U3, J.9 ev)
53Summary
- Introduction to strongly correlated electrons
- Dynamical Mean Field Theory
- Model Hamiltonian Studies. Universal aspects
insights from DMFT - System specific studies LDADMFT
- Outlook
54Outlook
- 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
55Challenges
- Short Range Magnetic Correlations without
magnetic order. - Single Site DMFT does not capture these effects
56Outlook
- Extensions to take into account longer range
correlations and interactions Cellular DMFT G.
Kotliar S. Savrasov G. Palsson and G. Biroli
Phys. Rev. Lett. 87, 186401, 2001 - Mott transition magnetic correlations and
momentum space differentiation. RVB, multipatch
models of transport A. Perali M. Sindel and G.
Kotliar Eur. Phys. J. B 24, 487 (2001). - Exploration of materials.
-
57Acknowledgements
Collaborators V. Anisimov, R. Chitra, V.
Dobrosavlevic, D. Fisher, A. Georges, H.
Kajueter, W.Krauth, E. Lange, G. Moeller, Y.
Motome, G. Palsson, M. Rozenberg, S.
Savrasov, Q. Si, V. Udovenko, X.Y. Zhang
Support National Science Foundation. Work on Fe
and Ni Office of Naval Research Work on Pu
Departament of Energy and LANL.
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61LDADMFT functional
F Sum of local 2PI graphs with local U matrix and
local G
62Comments 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.
63Anomalous Resistivities DopedHubbard Model (QMC)
Prushke and Jarrell 1993
64Anomalous ResistivitiesDoped Hubbard ModelG.
Palsson 1998
NCA
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65DMFT Methods of Solution
66LDA functional
Conjugate field, VKS(r)
67Minimize LDA functional
Kohn Sham eigenvalues, auxiliary quantities.
68Anomalous transfer of spectral weight heavy
fermions
69Anomalous transfer of spectral weight
70Anomalous transfer of spectral weigth heavy
fermions
71V2O3 resistivity
72LDADMFT Self-Consistency loop
E
U
DMFT
73DMFT Impurity cavity construction A. Georges,
G. Kotliar, PRB, (1992)
Weiss field
74Spectral Evolution at T0 half filling full
frustration
X.Zhang M. Rozenberg G. Kotliar (PRL 1993)
75Parallel development Fujimori et.al
76Landau Functional
G. Kotliar EPJB (1999)
77Ni 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)
78Problems with LDA
- DFT in the LDA or GGA is a well established tool
for the calculation of ground state properties. - Many studies (Freeman, Koelling 1972)APW methods
- ASA and FP-LMTO Soderlind et. Al 1990, Kollar
et.al 1997, Boettger et.al 1998, Wills et.al.
1999) give - an equilibrium volume of the d phase Is 35
lower than experiment - This is the largest discrepancy ever known in DFT
based calculations.
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80Spectral 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))
81LDA functional
Conjugate field, VKS(r)