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Electronic Structure of Strongly Correlated Electron Materials: A Dynamical Mean Field Perspective. Collaborators: G.Kotliar, S. Savrasov, V. Oudovenko. Kristjan Haule, Physics Department and Center for Materials Theory Rutgers University Rutgers University. Overview. - PowerPoint PPT Presentation
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Collaborators: G.Kotliar, S. Savrasov, V. Oudovenko
Kristjan Haule, Physics Department and
Center for Materials TheoryRutgers University Rutgers University
Electronic Structure of Strongly CorreElectronic Structure of Strongly Correlated Electron Materials: A Dynamicalated Electron Materials: A Dynamica
l Mean Field Perspective.l Mean Field Perspective.
• Application of DMFT to real materials (Spectral density functional approach). Examples: – alpha to gamma transition in Ce, optics near
the temperature driven Mott transition. – Mott transition in Americium under pressure
• Extensions of DMFT to clusters. Examples:– Superconducting state in t-J the model– Optical conductivity of the t-J model
OverviewOverview
V2O3Ni2-xSex organics
Universality of the Mott transitionUniversality of the Mott transition
First order MIT
Critical point
Crossover: bad insulator to bad metal
1B HB model 1B HB model (DMFT):(DMFT):
Coherence incoherence crossover in the Coherence incoherence crossover in the
1B HB model (DMFT)1B HB model (DMFT)
Phase diagram of the HM with partial frustration at half-fillingPhase diagram of the HM with partial frustration at half-filling
M. Rozenberg et.al., Phys. Rev. Lett. M. Rozenberg et.al., Phys. Rev. Lett. 7575, 105 (1995)., 105 (1995).
mappingmapping
fermionic bathfermionic bath
Basic idea of DMFT: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.Basic idea of Spectral density functional approach: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission]
Dynamical Mean Field TheoryDynamical Mean Field Theory
Density functional theory
Dynamical mean field theory:
observable of interest is the electron densityobservable of interest is the electron density
observable of interest is the local Green's function observable of interest is the local Green's function
(on the lattice(on the lattice uniquely defined uniquely defined))
fermionic bathfermionic bath
mappingmapping
exact BK
functional
DMFT
approximation
DFT & DMFT from the unifying point of viewDFT & DMFT from the unifying point of view
LDA+U corresponds to LDA+DMFT when impurity is solved in the Hartree Fock approximation
observable of interestobservable of interest is the "local“is the "local“ Green's functionsGreen's functions (spectral (spectral function)function)
Currently feasible approximations: LDA+DMFT and GW+DMFT:
G. Kotliar et.al., cond-mat/0511085
Spectral density functional theory
basic idea: sum-up all local diagrams for electrons in correlated orbitals
f1
L=3,S=1/2 J=5/2
f6
L=3,S=3 J=0
Periodic table
Cerium
Ce overview
volumes exp. LDA LDA+U 28Å3 24.7Å3
34.4Å3 35.2Å3
•Transition is 1.order•ends with CP
isostructural phase transition ends in a critical point at (T=600K, P=2GPa)
(fcc) phase
[ magnetic moment
(Curie-Wiess law),
large volume,
stable high-T, low-p]
(fcc) phase
[ loss of magnetic
moment (Pauli-para),
smaller volume,
stable low-T, high-p]
with large
volume collapse
v/v 15
LDA and LDA+U
f DOStotal DOSvolumes exp. LDA LDA+U
28Å3 24.7Å3
34.4Å3 35.2Å3
ferromagnetic
LDA+DMFT alpha DOS
TK(exp)=1000-2000K
LDA+DMFT gamma DOS
TK(exp)=60-80K
Photoemission&experiment
Fenomenological Landau approach:Kondo volume colapse (J.W. Allen, R.M. Martin, 1982)Kondo volume colapse (J.W. Allen, R.M. Martin, 1982)
•A. Mc Mahan K Held and R. Scalettar (2002)
•K. Haule V. Udovenko and GK. (2003)
Optical conductivity
*
+
+ K. Haule, et.al., Phys. Rev. Lett. 94, 036401 (2005)
* J.W. van der Eb, A.B. Ku’zmenko, and D. van der Marel, Phys. Rev. Lett. 86, 3407 (2001)
Americium
Americium
"soft" phase
f localized
"hard" phase
f bonding
Mott Transition?
f6 -> L=3, S=3, J=0
A.Lindbaum, S. Heathman, K. Litfin, and Y. Méresse, Phys. Rev. B 63, 214101 (2001)
J.-C. Griveau, J. Rebizant, G. H. Lander, and G.KotliarPhys. Rev. Lett. 94, 097002 (2005)
Am within LDA+DMFT
S. Y. Savrasov, K. Haule, and G. KotliarPhys. Rev. Lett. 96, 036404 (2006)
F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV
Large multiple effects:
Am within LDA+DMFT
nf=6
Comparisson with experiment
from J=0 to J=7/2
•“Soft” phase very different from Cenot in local moment regime since J=0 (no entropy)
•"Hard" phase similar to Ce,
Kondo physics due to hybridization, however, nf still far from Kondo regime
nf=6.2
Different from Sm!
Exp: J. R. Naegele, L. Manes, J. C. Spirlet, and W. MüllerPhys. Rev. Lett. 52, 1834-1837 (1984)
Theory: S. Y. Savrasov, K. Haule, and G. KotliarPhys. Rev. Lett. 96, 036404 (2006)
V=V0 Am IV=0.76V0 Am IIIV=0.63V0 Am IV
Beyond single site DMFT
What is missing in DMFT?
•Momentum dependence of the self-energy m*/m=1/Z
•Various orders: d-waveSC,…
•Variation of Z, m*, on the Fermi surface
•Non trivial insulator (frustrated magnets)
•Non-local interactions (spin-spin, long range Columb,correlated hopping..)
Present in DMFT:•Quantum time fluctuations
Present in cluster DMFT:•Quantum time fluctuations•Spatially short range quantum fluctuations
The simplest model of high Tc’s
t-J, PW Anderson
Hubbard-Stratonovich ->(to keep some out-of-cluster quantum fluctuations)
BK Functional, Exact
cluster in k space cluster in real space
t’=0
Phase diagram
What can we learn from “small” Cluster-DMFT?
Insights into superconducting state (BCS/non-BCS)?
J. E. Hirsch, Science, 295, 5563 (2226)
BCS: upon pairing potential energy of electrons decreases, kinetic energy increases(cooper pairs propagate slower)Condensation energy is the difference
non-BCS: kinetic energy decreases upon pairing(holes propagate easier in superconductor)
D van der Marel, Nature 425, 271-274 (2003)
cond-mat/0601478
Optical conductivity
overdoped
optimally doped
Bi2212
~1eVWeight bigger in SC,
K decreases (non-BCS)
Weight smaller in SC, K increases (BCS-like)
Optical weight, plasma frequency
D. van der Marel et.al., in preparation
Hubbard model
DrudeU
t2/U
t-J model
J
Drude
no-U
Experiments
intraband interband transitions
~1eV
Excitations into upper Hubbard band
Kinetic energy in Hubbard model:•Moving of holes•Excitations between Hubbard bands
Kinetic energy in t-J model•Only moving of holes
Hubbard versus t-J model
Phys Rev. B 72, 092504 (2005)
cluster-DMFT, cond-mat/0601478
Kinetic energy change
Kinetic energy decreases
Kinetic energy increases
Kinetic energy increases
Exchange energy decreases and gives
largest contribution to condensation energy
cond-mat/0503073
electrons gain energy due to exchange energyholes gain kinetic energy (move faster)
underdoped
electrons gain energy due to exchange energyhole loose kinetic energy (move slower)
overdoped
BCS likesame as RVB (see P.W. Anderson Physica C, 341, 9 (2000),
or slave boson mean field (P. Lee, Physica C, 317, 194 (1999)
Kinetic energy upon condensation
J
J J
J
Optics mass and plasma frequency
Extended Drude model
•Within DMFT, optics mass is m*/m=1/Z and diverges at the Mott transition •Plasma frequency vanishes as 1/Z (Drude shrinks as Kondo peak shrinks)
•In cluster-DMFT optics mass constant at low doping doping
•Plasma frequency vanishes because the active (coherent) part of
the Fermi surface shrinks
line: cluster DMFT (cond-mat 0601478),symbols: Bi2212, Van der Marel (in preparation)
Optimal doping: Powerlaws
cond-mat/0605149D. van der Marel et. al., Nature 425, 271 (2003).
Optimal doping: Coherence scale seems
to vanish
Tc
underdoped
overdoped
optimally
scattering at Tc
Local density of states of SC
•V shaped gap (d-wave)•size of gap decreases with doping•CDMFT-optimall doping PH symmetric
STM study Cluster DMFT
1 Larger doping6 Smaller doping
Pengcheng et.al., Science 284, (1999)
YBa2Cu3O6.6 (Tc=62.7K)
41meV resonance
local susceptibility
•Resonance at 0.16t~48meV•Most pronounced at optimal doping•Second peak shifts with doping (at 0.38~120meV opt.d.) and changes below Tc – contribution to condensation energy
Pseudoparticle insight
N=4,S=0,K=0
N=4,S=1,K=()
N=3,S=1/2,K=(,0)
N=2,S=0,K=0
A()
’’()PH symmetry,
Large
• LDA+DMFT can describe interplay of lattice and electronic structure near Mott transition. Gives physical connection between spectra, lattice structure, optics,.... – Allows to study the Mott transition in open and closed shell cases. – In both Ce and Am single site LDA+DMFT gives the zeroth order
picture– Am: Rich physics, mixed valence under pressure.
• 2D models of high-Tc require cluster of sites. Some aspects of optimally doped, overdoped and slightly underdoped regime can be described with cluster DMFT on plaquette:– Evolution from kinetic energy saving to BCS kinetic energy cost
mechanism– Optical mass approaches a constant at the Mott transition and
plasma frequency vanishes– At optimal doping: Physical observables like optical conductivity
and spin susceptibility show powerlaw behavior at intermediate frequencies, very large scattering rate – vanishing of coherence scale, PH symmetry is dynamically restored, 41meV resonance appears in spin response
Conclusions
local in localized LMTO base
Impurity problem (14x14):
LDA
Impurity solver
DMFT SCC *
*
LDA+DMFT implementation
Carbone et.al., in preparation
Comparison of spectral weightcluster DMFT / Bi2212
Spectral weight (kinetic energy) changes faster
with T in overdoped
system – larger coherence scale
Partial DOS
4f
5d
6s
Z=0.33
More complicated f systems
•Hunds coupling is important when more than one electron in the correlated (f) orbital•Spin orbit coupling is very small in Ce, while it become important in heavier elements
The complicated atom embedded into fermionic The complicated atom embedded into fermionic
bath (with crystal fileds) is a serious chalange so solve!bath (with crystal fileds) is a serious chalange so solve!
Coulomb interaction is diagonal in the base of total LSJ -> LS base
while the SO coupling is diagonal in the j-base -> jj base
Eigenbase of the atom depends on the strength of the Hund's couling and
strength of the spin-orbit interaction
Mott transition (B. Johansson, 1974):Mott transition (B. Johansson, 1974):
Kondo volume colapse (J.W. Allen, R.M. Martin, 1982):Kondo volume colapse (J.W. Allen, R.M. Martin, 1982):
Classical theories
Hubbard modelHubbard model
Anderson (impurity) modelAnderson (impurity) model
changes and causes Mott tr.changes and causes Mott tr.
changes changes →→ chnange of T chnange of TKK
bath
either constant or
taken from LDA and rescaled
spd electrons pure spectatorsspd electrons pure spectators
hybridization with spd electrons is crucialhybridization with spd electrons is crucial
f electrons insulating
f electrons in local moment regime
(Lavagna, Lacroix and Cyrot, 1982)(Lavagna, Lacroix and Cyrot, 1982)
Fenomenological Landau approach:
LDA+DMFTab initio calculationab initio calculation
is self-consistently determinedis self-consistently determined
contains tcontains tffff and V and Vfdfd hopping hopping
bath for AIMbath for AIM
Kondo volume colapse model resembles DMFT picture:Kondo volume colapse model resembles DMFT picture:
Solution of the Solution of the Anderson impurity modelAnderson impurity model → → Kondo physicsKondo physics
DifferenceDifference: : with DMFT the lattice problem is solved (and therefore with DMFT the lattice problem is solved (and therefore Δ must self-consistently determined) while in KVC Δ is calculated for a fictious impurity (and needs to be rescaled to fit exp.)
In KVC scheme there is no feedback on spd bans, hence optics is not much affected.
An example
Atomic physics of selected Actinides
Basov, cond-mat/0509307
optics mass and plasma w
Two Site CDMFTTwo Site CDMFT in the 1D Hubbard in the 1D Hubbard modelmodel
M.Capone M.Civelli V. Kancharla C.Castellani and Kotliar, PRB 69,195105 (2004)
Slave particle diagrammatic impurity solvers
NCA
OCA
TCA
Luttinger Ward functional
every atomic state represented with a unique pseudoparticle
atomic eigenbase - full (atomic) base
, where
general AIM:
( )
SUNCA vs QMCtwo band Hubbard model, Bethe lattice, U=4D
three band Hubbard model,
Bethe lattice, U=5D, T=0.0625D
three band Hubbard model,
Bethe lattice, U=5D, T=0.0625D