Bonn, 2008

Band structure of strongly correlated materials from the Dynamical Mean Field perspective

K HauleRutgers University

Collaborators : J.H. Shim & Gabriel Kotliar, S. Savrasov

Outline

Dynamical Mean Field Theory in combination with band structure

LDA+DMFT results for 115 materials (CeIrIn5) Local Ce 4f - spectra and comparison to AIPES) Momentum resolved spectra and comparison to ARPES Optical conductivity Two hybridization gaps and its connection to optics Fermi surface in DMFT

Actinides Absence of magnetism in Pu and magnetic ordering in Cm

explained by DMFT Valence of correlates solids, example of Pu

References:•J.H. Shim, KH, and G. Kotliar, Science 318, 1618 (2007).•J.H. Shim, KH, and G. Kotliar, Nature 446, 513 (2007).

Standard theory of solidsStandard theory of solids

Band Theory: electrons as waves: Rigid 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

M. Van SchilfgardeM. Van Schilfgarde

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): B

ad in

sula

tor

Bad metal1B HB model 1B HB model (plaquette):(plaquette):

Basic questions to addressBasic questions to address

How to computed spectroscopic quantities (single particle spectra, optical conductivity phonon dispersion…) from first principles?

How to relate various experiments into a unifying picture.

New concepts, new techniques….. DMFT maybe simplest approach to meet this challenge

atom solidHund’s rule, SO coupling, CFS

DMFT + electronic structure methodDMFT + electronic structure method

(G. Kotliar S. Savrasov K.H., V. Oudovenko O. Parcollet and C. Marianetti, RMP 2006).

Basic idea of DMFT+electronic structure method (LDA or GW): For less correlated orbitals (s,p): use LDA or GWFor correlated orbitals (f or d): add all local diagrams by solving QIM

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!

DMFT + electronic structure methodDMFT + electronic structure method

obtained by DFT

Ce(4f) obtained by “impurity solution”Includes the collective excitations of the system

Self-energy is local in localized basis,in eigenbasis it is momentum dependent!

all bands are affected: have lifetimefractional weight

correlated orbitals

other “light” orbitals

hybridization

Dyson equation

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.H. Phys. Rev. B 75, 155113 (2007) ; P Werner, PRL (2007); N. Rubtsov PRB 72, 35122 (2005).

An exact impurity solver, continuous time QMC - expansion in terms of hybridization

Analytic impurity solvers (summing certain types of diagrams), expansion in terms of hybridization

K.H. Phys. Rev. B 64, 155111 (2001)

Fully dressed atomic propagators

hybridization

•Allows correct treatment of multiplets•Very precise at high and intermediate

frequencies and high to intermediate temperatures

Complementary to CTQMC (imaginary axis -> low energy)

“Bands” are not a good concept in DMFT!

Frequency dependent complex object instead of “bands”

lifetime effectsquasiparticle “band” does not carry weight 1

DMFTDMFT

Spectral function is a good concept

DMFT is not a single impurity calculation

Auxiliary impurity problem:

High-temperature given mostly by LDA

low T: Impurity hybridization affected by the emerging coherence of the lattice

(collective phenomena)

Weiss field temperature dependent:

Feedback effect on makes the crossover from incoherent to coherent state very slow!

high T

low T

DMFT SCC:

CeIn3 CeCoIn5 CeRhIn5 CeIrIn5 PuCoG5 Na

Tc[K] 0.2K 2.3K 2.1K 0.4K 18.3K n/a

Tcrossover ~50K ~50K ~50K ~370K

Cv/T[mJ/molK^2] 1000 300 400 750 100 1

Phase diagram of CeIn3 and 115’s

N.D. Mathur et al., Nature (1998)

CeIn3

0

1

2

3

4

?SC

SCSC

X

0.50.50.5 IrRh CoCo

AFM

T* (

K)

CeCoIn5 CeRhIn5CeIrIn5 CeCoIn5

CeXIn5

layering

Tcrossover α Tc

Ce

In

Ir

CeIn

In

Crystal structure of 115’s

CeIn3 layer

IrIn2 layer

IrIn2 layer

Tetragonal crystal structure

4 in plane In neighbors

8 out of plane in neighbors

3.27au

3.3 au

Crossover scale ~50K

in-plane

out of plane

•Low temperature – Itinerant heavy bands

•High temperature Ce-4f local moments

ALM in DMFTSchweitzer&Czycholl,1991

Coherence crossover in experiment

•How does the crossover from localized moments to itinerant q.p. happen?

•How does the spectral

weight redistribute?

•How does the hybridization gap look like in momentum space?

?

k

A()

•Where in momentum space q.p. appear?

•What is the momentum dispersion of q.p.?

Issues for the system specific study

(e

Temperature dependence of the local Ce-4f spectra

•At low T, very narrow q.p. peak (width ~3meV)

•SO coupling splits q.p.: +-0.28eV

•Redistribution of weight up to very high frequency

SO

•At 300K, only Hubbard bands

J. H. Shim, KH, and G. Kotliar Science 318, 1618 (2007).

Broken symmetry (neglecting strong correlations) can give Hubbard bands, but not both Hubbard bands

And quasiparticles!

Very slow crossover!

T*

Buildup of coherence in single impurity case

TK

cohere

nt

spect

ral

weig

ht

T scattering rate

coherence peak

Buildup of coherence

Crossover around 50K

Slow crossover pointed out byS. Nakatsuji, D. Pines, and Z. FiskPhys. Rev. Lett. 92, 016401 (2004)

Consistency with the phenomenological approach of

NPF

Remarkable agreement with Y. Yang & D. Pines cond-mat/0711.0789!

Anom

alo

us

Hall

coeffi

cient

Fraction of itinerant heavy fluid

m* of the heavy fluid

ARPESFujimori, 2006 (T=10K)

Angle integrated photoemission vs DMFT

Very good agreement, but hard to see resonancein experiment: resonance very asymmetric in Ce ARPES is surface sensitive at 122eV

Angle integrated photoemission vs DMFT

ARPESFujimori, 2006

Nice agreement for the• Hubbard band position•SO split qp peak

Hard to see narrow resonance

in ARPES since very little weight

of q.p. is below Ef

Lower Hubbard band

T=10K T=300Kscattering rate~100meV

Fingerprint of spd’s due to hybridization

Not much weight

q.p. bandSO

Momentum resolved Ce-4f spectraAf(,k)

Hybridization gap

DMFT qp bands

LDA bands LDA bands DMFT qp bands

Quasiparticle bands

three bands, Zj=5/2~1/200

Momentum resolved total spectra A(,k)

Fujimori, 2003

LDA+DMFT at 10K ARPES, HE I, 15K

LDA f-bands [-0.5eV, 0.8eV] almostdisappear, only In-p bands remain

Most of weight transferred intothe UHB

Very heavy qp at Ef,hard to see in total spectra

Below -0.5eV: almost rigid downshift

Unlike in LDA+U, no new band at -2.5eV

Large lifetime of HBs -> similar to LDA(f-core)rather than LDA or LDA+U

Optical conductivity

Typical heavy fermion at low T:

Narrow Drude peak (narrow q.p. band)

Hybridization gap

k

Interband transitions across hybridization gap -> mid IR peak

CeCoIn5

no visible Drude peak

no sharp hybridization gap

F.P. Mena & D.Van der Marel, 2005

E.J. Singley & D.N Basov, 2002

second mid IR peakat 600 cm-1

first mid-IR peakat 250 cm-1

•At 300K very broad Drude peak (e-e scattering, spd lifetime~0.1eV) •At 10K:

•very narrow Drude peak•First MI peak at 0.03eV~250cm-1

•Second MI peak at 0.07eV~600cm-1

Optical conductivity in LDA+DMFT

CeIn

In

Multiple hybridization gaps

300K

e V

10K

•Larger gap due to hybridization with out of plane In•Smaller gap due to hybridization with in-plane In

non-f spectra

Fermi surfaces of CeM In5 within LDA

Localized 4f:LaRhIn5, CeRhIn5

Shishido et al. (2002)

Itinerant 4f :CeCoIn5, CeIrIn5

Haga et al. (2001)

de Haas-van Alphen experiments

LDA (with f’s in valence) is reasonable for CeIrIn5

Haga et al. (2001)

Experiment LDA

Fermi surface changes under pressure in CeRhIn5

Fermi surface reconstruction at 2.34GPa Sudden jump of dHva frequencies Fermi surface is very similar on both sides, sl

ight increase of electron FS frequencies Reconstruction happens at the point of maxim

al Tc

Shishido, (2005)localized itinerant

We can not yet address FS change with pressure

We can study FS change with Temperature -

At high T, Ce-4f electrons are excluded from the FSAt low T, they are included in the FS

Electron fermi surfaces at (z=0)

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

X M

X

XX

M

MM

2 2

Slight decrease of the electron

FS with T

R A

R

RR

A

AA

3

a

3

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

Electron fermi surfaces at (z=)No a in DMFT!No a in Experiment!

Slight decrease of the electron

FS with T

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

X M

X

XX

M

MM

c

2 2

11

Electron fermi surfaces at (z=0)Slight decrease of the electron

FS with T

R A

R

RR

A

AA

c

2 2

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

Electron fermi surfaces at (z=)No c in DMFT!No c in Experiment!

Slight decrease of the electron

FS with T

LDA+DMFT (10 K)LDA LDA+DMFT (400 K)

X M

X

XX

M

MM

g h

Hole fermi surfaces at z=0

g h

Big change-> from small hole like to large electron like

1

Localization – delocalization transition

in Lanthanides and Actinides

Delocalized Localized

Electrical resistivity & specific heat

J. C. Lashley et al. PRB 72 054416 (2005)

Heavy ferm. in an element

closed shell Am

Itinerant

NO Magnetic moments in Pu!

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~7.9)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

-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

DMFT can describe crossover from local moment regime to heavy fermion state in heavy fermions. The crossover is very slow.

Width of heavy quasiparticle bands is predicted to be only ~3meV. We predict a set of three heavy bands with their dispersion.

Mid-IR peak of the optical conductivity in 115’s is split due to presence of two type’s of hybridization

Ce moment is more coupled to out-of-plane In then in-plane In which explains the sensitivity of 115’s to

substitution of transition metal ion

DMFT predicts Pu to be nonmagnetic (heavy fermion like) and Cm to be magnetic

ConclusionsConclusions

Thank you!

Gradual decrease of electron FS

Most of FS parts show similar trend

Big change might be expected in the plane – small hole like FS pockets (g,h) merge into electron FS 1 (present in LDA-f-core but not in LDA)

Fermi surface a and c do not appear in DMFT results

Increasing temperature from 10K to 300K:

Fermi surfacesFermi surfaces

ARPES of CeIrIn5

Fujimori et al. (2006)

Ce 4f partial spectral functions

LDA+DMFT (10K) LDA+DMFT (400K)

Blue lines : LDA bands

Hole fermi surface at z=

R A

R

RR

A

AANo Fermi surfaces

LDA+DMFT (400 K)LDA+DMFT (10 K)LDA

dHva freq. and effective mass

Analytic impurity solvers (summing certain types of diagrams), expansion in terms of hybridization

K.H. Phys. Rev. B 64, 155111 (2001)

Fully dressed atomic propagators

hybridization

SUNCA

•Allows correct treatment of multiplets•Very precise at high and intermediate

frequencies and high to intermediate temperatures

Complementary to CTQMC (imaginary axis -> low energy)