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!