collaborators: ji-hoon shim, s.savrasov, g.kotliar kristjan haule, physics department and center for...
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Collaborators: Ji-Hoon Shim, S.Savrasov, G.Kotliar
Kristjan Haule, Physics Department and
Center for Materials TheoryRutgers University
Electronic structure of Strongly Correlated
Materials: A Dynamical Mean Field Perspective.
ES 07 - Raleigh
Standard theory of solidsStandard theory of solids
Band Theory: electrons as waves: Rigid non-dipersive band
picture: En(k) versus k
Landau Fermi Liquid Theory applicable
Very powerful quantitative tools: LDA,LSDA,GWVery powerful quantitative tools: LDA,LSDA,GW
Predictions:
•total energies,
•stability of crystal phases
•optical transitions
•……
• Fermi Liquid Theory does NOT work . Need new concepts to replace rigid bands picture!
• Breakdown of the wave picture. Need to incorporate a real space perspective (Mott).
• Non perturbative problem.
Strong correlation – Strong correlation –
Standard theory failsStandard theory fails
V2O3Ni2-xSex organics
Universality of the Mott transitionUniversality of the Mott transition
First order MITCritical point
Crossover: bad insulator to bad metal
1B HB model 1B HB model (DMFT):(DMFT):
Delocalization Localization
Basic questionsBasic questions
• How to describe the physics of strong correlations close to the Mott boundary?
• How to computed spectroscopic quantities (single particle spectra, optical conductivity phonon dispersion…) from first principles?
• New concepts, new techniques….. DMFT maybe simplest approach to meet this challenge
DMFT + electronic structure methodDMFT + electronic structure method
Effective (DFT-like) single particle spectrum consists of delta like peaks
DMFT stectral function contains renormalized quasiparticles and Hubbard bands
Basic idea of DMFT: reduce the quantum many body problem to a problemof an atom in a conduction band, which obeys DMFT self-consistency condition (A. Georges et al., RMP 68, 13 (1996)). DMFT in the language of functionals: DMFT sums up all local diagrams in BK functional
Basic idea of DMFT+electronic structure method (LDA or GW): For less correlated bands (s,p): use LDA or GWFor correlated bands (f or d): with DMFT add all local diagrams(G. Kotliar S. Savrasov K.H., V. Oudovenko O. Parcollet and C. Marianetti, RMP 2006).
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:
LDA+DMFT
(G. Kotliar et.al., RMP 2006).
Variation gives st. eq.:
LDA functional ALL local diagrams
Generalized Q. impurity problem!
Exact Exact functionalfunctional of the of the local Green’s functionlocal Green’s function exists, its form exists, its form unknown!unknown!
General impurity problem
Diagrammatic expansion in terms of hybridization +Metropolis sampling over the diagrams
•Exact method: samples all diagrams!•Allows correct treatment of multiplets
k
K.H. Phys. Rev. B 75, 155113 (2007)
Exact “QMC” impurity solver, expansion in terms of hybridization
P. Werner, Phys. Rev. Lett. 97, 076405 (2006)
Trivalent metals with nonbonding f shell
f’s participate in bonding
Partly localized, partly delocalized
Volume of actinides
Anomalous Resistivity
Maximum metallic resistivity:
=e2 kF/h
Fournier & Troc (1985)
Dramatic increase of specific heat
Heavy-fermion behavior in an element
NO Magnetic moments!
Pauli-like from melting to lowest T
No curie Weiss up to 600K
Curium versus Plutonium
nf=6 -> J=0 closed shell
(j-j: 6 e- in 5/2 shell)(LS: L=3,S=3,J=0)
One hole in the f shell One more electron in the f shell
No magnetic moments,large massLarge specific heat, Many phases, small or large volume
Magnetic moments! (Curie-Weiss law at high T, Orders antiferromagnetically at low T) Small effective mass (small specific heat coefficient)Large volume
Standard theory of solids:DFT:
All Cm, Am, Pu are magnetic in LSDA/GGA LDA: Pu(m~5), Am (m~6) Cm (m~4)
Exp: Pu (m=0), Am (m=0) Cm (m~7.6)Non magnetic LDA/GGA predicts volume up to 30% off.In atomic limit, Am non-magnetic, but Pu magnetic with spin ~5B
Can LDA+DMFT account for anomalous properties of actinides?
Can it predict which material is magnetic and which is not?
Many proposals to explain why Pu is non magnetic: Mixed level model (O. Eriksson, A.V. Balatsky, and J.M. Wills) (5f)4 conf. +1itt. LDA+U, LDA+U+FLEX (Shick, Anisimov, Purovskii) (5f)6 conf.
Cannot account for anomalous transport and thermodynamics
Incre
asin
g F’s a
n
SO
C
N Atom F2 F4 F6 92 U 8.513 5.502 4.017 0.226
93 Np 9.008 5.838 4.268 0.262
94 Pu 8.859 5.714 4.169 0.276
95 Am 9.313 6.021 4.398 0.315
96 Cm 10.27 6.692 4.906 0.380
Very strong multiplet splitting
Atomic multiplet splitting crucial -> splits Kondo peak
Used as input to DMFT calculation - code of R.D. Cowan
-Plutonium
0
1
2
3
4
-6 -4 -2 0 2 4 6
DO
S (
stat
es/e
V)
Total DOS
f DOS
Curium
0
1
2
3
4
-6 -4 -2 0 2 4 6ENERGY (eV)
DO
S (
stat
es/e
V)
Total DOS f, J=5/2,jz>0f, J=5/2,jz<0 f, J=7/2,jz>0f, J=7/2,jz<0
Starting from magnetic solution, Curium develops antiferromagnetic long range order below Tc above Tc has large moment (~7.9 close to LS coupling)Plutonium dynamically restores symmetry -> becomes paramagnetic
J.H. Shim, K.H., G. Kotliar, Nature 446, 513 (2007).
-Plutonium
0
1
2
3
4
-6 -4 -2 0 2 4 6
DO
S (
stat
es/e
V)
Total DOS
f DOS
Curium
0
1
2
3
4
-6 -4 -2 0 2 4 6ENERGY (eV)
DO
S (
stat
es/e
V)
Total DOS f, J=5/2,jz>0f, J=5/2,jz<0 f, J=7/2,jz>0f, J=7/2,jz<0
Multiplet structure crucial for correct Tk in Pu (~800K)and reasonable Tc in Cm (~100K)
Without F2,F4,F6: Curium comes out paramagnetic heavy fermion Plutonium weakly correlated metal
Magnetization of Cm:
Curium
0.0
0.3
0.6
0.9
-6 -4 -2 0 2 4 6ENERGY (eV)
Pro
bab
ility
N =8
N =7
N =6
J=7/
2,g =
0
J=5,
g =0
J=6,
g =0
J=4,
g =0
J=3,
g =0
J=2,
g =0
J=5,
g =0
J=2,
g =0
J=1,
g =0
J=0,
g =0
J=6,
g =0
J=4,
g =0
J=3,
g =0
f
f
f
-Plutonium
0.0
0.3
0.6
Pro
bab
ility
N =6
N =5
N =4
JJ=
0,g =
0J=
1,g =
0J=
2,g =
0J=
3,g =
0J=
4,g =
0J=
5,g =
0
J=6,
g =1
J=4,
g =0
J=5,
g =0
J=2,
g =0
J=1,
g =0
J=2,
g =1
J=3,
g =1
J=5/
2, g
=0
J=7/
2,g =
0J=
9/2,
g =0
f
f
f
Valence histograms
Density matrix projected to the atomic eigenstates of the f-shell(Probability for atomic configurations)
f electron fluctuates
between theseatomic states on the time scale t~h/Tk
(femtoseconds)
One dominant atomic state – ground state of the atom
Pu partly f5 partly f6
Probabilities:
•5 electrons 80%
•6 electrons 20%
•4 electrons <1%
J.H. Shim, K. Haule, G. Kotliar, Nature 446, 513 (2007).
Gouder , Havela PRB
2002, 2003
Fingerprint of atomic multiplets - splitting of Kondo peak
Photoemission and valence in Pu
|ground state > = |a f5(spd)3>+ |b f6 (spd)2>
f5<->f6
f5->f4
f6->f7
Af(
)
approximate decomposition
core
vale
nce
4d3/2
4d5/2
5f5/2
5f7/2
Exci
tati
ons
from
4d c
ore
to 5
f vale
nce
Electron energy loss spectroscopy (EELS) orX-ray absorption spectroscopy (XAS)
Energy loss [eV]
Core splitting~50eV
4d5/2->5f7/2 &
4d5/2->5f5/2
4d3/2->5f5/2
Measures unoccupied valence 5f statesProbes high energy Hubbard bands!
hv
Core
split
ting~
50
eV
Probe for Valence and Multiplet structure: EELS&XAS
A plot of the X-ray absorption as a function of energy
B=B0 - 4/15<l.s>/(14-nf)
Branching ration B=A5/2/(A5/2+A3/2)
LD
A+
DM
FT
2/3<l.s>=-5/2(B-B0) (14-nf)
One measured quantity B, two unknownsClose to atom (IC regime)
Itinerancy tends to decrease <l.s>
[a] G. Van der Laan et al., PRL 93, 97401 (2004).[b] G. Kalkowski et al., PRB 35, 2667 (1987)[c] K.T. Moore et al., PRB 73, 33109 (2006).[d] K.T. Moore et al., PRL in press
Specific heat
Purovskii et.al. cond-mat/0702342:
f6 configuration gives smaller gin Pu than Pu
(Shick, Anisimov, Purovskii) (5f)6 conf
Could Pu be close to f6 like Am?
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.H., 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 not in local moment regime since J=0 (no entropy)
•"Hard" phase similar to Pu,
Kondo physics due to hybridization, however, nf still far from Kondo regime
nf=6.2
Exp: J. R. Naegele, L. Manes, J. C. Spirlet, and W. MüllerPhys. Rev. Lett. 52, 1834-1837 (1984)
Theory: S. Y. Savrasov, K.H., and G. KotliarPhys. Rev. Lett. 96, 036404 (2006)
V=V0 Am IV=0.76V0 Am IIIV=0.63V0 Am IV
What is captured by single site DMFT?
•Captures volume collapse transition (first order Mott-like transition)•Predicts well photoemission spectra, optics spectra,
total energy at the Mott boundary•Antiferromagnetic ordering of magnetic moments,
magnetism at finite temperature•Branching ratios in XAS experiments, Dynamic valence fluctuations,
Specific heat•Gap in charge transfer insulators like PuO2
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
Plaquette DMFT for the Hubbard modelas relevant for cuprates
Plaquette DMFT for the Hubbard modelas relevant for cuprates
Large onsitecomponent
Small next-nearest neighbor component (except in the underdoped regime)
anomalous SE-SC
Complicated Fermi surface evolution with
temperature
underoped phase“fermi arcs”
“arcs” decreasewith T
Superconducting phase-banana like Fermi surface
• Pu and Am (under pressure) are unique strongly correlated elements. Unique mixed valence.
• They require, new concepts, new computational methods, new algorithms, DMFT!
• Cluster extensions of DMFT can describe many features of cuprates including superconductivity and gapping of fermi surface (pseudogap)
Conclusion
Many strongly correlated compounds await the explanation:
CeCoIn5, CeRhIn5, CeIrIn5
Photoemission of CeIrIn5
LDA+DMFT DOS
Comparisonto experiment
Photoemission of CeIrIn5
Optics of CeIrIn5
LDA+DMFT
K.S. Burch et.al., cond-mat/0604146
Experiment:
Optimal doping: Coherence scale seems
to vanish
Tc
underdoped
overdoped
optimally
scattering at Tc
New continuous time QMC, expansion in terms of hybridization
General impurity problem
Diagrammatic expansion in terms of hybridization +Metropolis sampling over the diagrams
Contains all: “Non-crossing” and all crossing diagrams!Multiplets correctly treated
k
• 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 actinides and their compounds, single site
LDA+DMFT gives the zero-th order picture• 2D models of high-Tc require cluster of sites. Some
aspects of optimally doped regime can be described with cluster DMFT on plaquette:– Large scattering rate in normal state close to optimal
doping
Conclusions
• How does the electron go from being localized to itinerant.
• How do the physical properties evolve.
• How to bridge between the microscopic information (atomic positions) and experimental measurements.
• New concepts, new techniques….. DMFT simplest approach to meet this challenge
Basic questions
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).
Singlet-type Mott state (no entropy) goes mixed valence under pressure-> Tc enhanced (Capone et.al, Science 296, 2364 (2002))
• DMFT in actinides and their compounds (Spectral density functional approach). Examples: – Plutonium, Americium, Curium. – Compounds: PuAmObservables:– Valence, Photoemission, and Optics, X-ray
absorption
OverviewOverview
Why is Plutonium so special?
Heavy-fermion behavior in an element
No curie Weiss up to 600K
Typical heavy fermions (large mass->small TkCurie Weis at T>Tk)
Why is Plutonium so special?
Heavy-fermion behavior in an element
Overview of actinides
Two phases of Ce, and gwith 15% volume difference
25% increase in volume between and phase
Many phases
Current:
Expressed in core valence orbitals:
The f-sumrule: can be expressed as
Branching ration B=A5/2/(A5/2+A3/2)
Energy loss [eV]
Core splitting~50eV
4d5/2->5f7/2
4d3/2->5f5/2
B=B0 - 4/15<l.s>/(14-nf)
A5/2 area under the 5/2 peak
Branching ratio depends on: •average SO coupling in the f-shell <l.s>
•average number of holes in the f-shell nf
B0~3/5
B.T. Tole and G. van de Laan, PRA 38, 1943 (1988)
Similar to optical conductivity:
f-sumrule for core-valence conductivity
2p->5f5f->5f
Pu: similar to heavy fermions (Kondo type conductivity) Scale is large MIR peak at 0.5eVPuO2: typical semiconductor with 2eV gap, charge transfer
Optical conductivity
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:
Spectral density functional theory
(G. Kotliar et.al., RMP 2006).
Variation gives st. eq.:Generalized Q. impurity problem!
Pu-Am mixture, 50%Pu,50%Am
Lattice expands -> Kondo collapse is expected
f6: Shorikov, et al., PRB 72, 024458 (2005); Shick et al, Europhys. Lett. 69, 588 (2005). Pourovskii et al., Europhys. Lett. 74, 479 (2006).
Our calculations suggest charge transfer
Pu phase stabilized by shift tomixed valence nf~5.2->nf~5.4
Hybridization decreases, but nf increases,
Tk does not change significantly!