Understanding f-electron materials using Understanding f-electron materials using Dynamical Mean Field Theory Dynamical Mean Field Theory

Gabriel KotliarGabriel Kotliar

and Center for Materials Theory

$upport : NSF -DMR DOE-Basic Energy Sciences

Collaborators: K. Haule and J. Shim

Solid State Seminar U. Oregon January 15Solid State Seminar U. Oregon January 15thth 2010 2010

11

OutlineOutline

• Introduction to Correlated Materials

• Introduction to Dynamical Mean Field Theory

• Applications to f electrons:

• CeIrIn5

• URu2Si2

• Pu-Am-Cm

• PuSe PuTe

• Conclusions

Landau Fermi Liquid Excitation spectrum of a FermiExcitation spectrum of a Fermi system system has the has the

same structure as the excitation spectrum of a same structure as the excitation spectrum of a perfect Fermi gasperfect Fermi gas..

Electrons in a Solid:the Standard Model Electrons in a Solid:the Standard Model

Kohn Sham Density Functional Theory

2 / 2 ( )[ ] [ ]KS kn n knV r kr y e y- Ñ + =

Rigid bands , optical transitions , thermodynamics, transport………

Static Mean Field Theory.

22

[ ]totE r

Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965)

[ ]nk ken band index, e.g. s, p, d,,fn band index, e.g. s, p, d,,f

Bloch waves in a periodic potential Bloch waves in a periodic potential

• Quantum mechanical description of the states in metals and semiconductors. Bloch waves. En(k).

• Inhomogenous systems. Doping. Theory of donors and acceptors . Interfaces. p-n junctions. Transistors. Integrated circuits computers.

Physical Insights into Materials -> Technology Physical Insights into Materials -> Technology

GW= First order PT in screened Coulomb GW= First order PT in screened Coulomb interactions around LDAinteractions around LDA

33

Correlated materials: simple Correlated materials: simple recipe recipe

Transition metal oxidesTransition metal oxides

OxygenOxygen

transition metal iontransition metal ion

Cage : e.g 6 oxygen atoms (octahedra) Cage : e.g 6 oxygen atoms (octahedra)

or other ligands/geometryor other ligands/geometry

Build a microscopic crystal with this building blockBuild a microscopic crystal with this building block

Layer the structure Layer the structure

Transition metal insideTransition metal insideTransition metal ionsTransition metal ions

Rare earth ionsRare earth ions

ActinidesActinides

44

LixLixCoOCoO22 NaNaxx CoO CoO22

YBa2Cu3O7 YBa2Cu3O7

VO2VO2

55

How do we know that the electrons How do we know that the electrons are heavy ?are heavy ?

Heavy Fermions: intermetallics containing 4f elements Heavy Fermions: intermetallics containing 4f elements Cerium, and 5f elements Uranium. Broad spd bands + Cerium, and 5f elements Uranium. Broad spd bands + atomic f open shells.atomic f open shells.

Heavy Fermion Metals Heavy Fermion Metals

100100

200200

300300

100100 200200T(K)T(K)

CeAlCeAl33

UBeUBe1313

-1-1 (

emu

/mol

(em

u/m

ol)-

1)-

1

0000

Coherence Incoherence Coherence Incoherence

CrossoverCrossover

Magnetic Oscillations Magnetic Oscillations

A Very Selected Class of HF A Very Selected Class of HF

URu2Si2URu2Si2

UU

SiSiRuRu

A signature A signature problem ?problem ?

Correlated Electron Systems Pose Basic Correlated Electron Systems Pose Basic Questions in CMT: from atoms to solidsQuestions in CMT: from atoms to solids

• How to describe electron from localized to itinerant ?

• How do the physical properties evolve ?

• Non perturbative techniques

Needed!! (Dynamical) mean field theory for this Needed!! (Dynamical) mean field theory for this problem problem ,, 88

Classical case Quantum case

A. Georges, G. Kotliar (1992)

Mean-Field : Classical vs QuantumMean-Field : Classical vs Quantum

†

0 0 0

( )[ ( ' ] ( '))o o o oc c U n nb b b

s st m tt

t t ¯

¶+ D-

¶- +òò ò

( )wD

†( )( ( ) )) (

MFo n oo n n Sc i c iG i s ss ww w D=- á ñ

( )

(()

)

11

([ ]

)[ ]n

n

kn

G i

G it ki m

w

wwD

D

=- - +

å

,ij i j i

i j i

J S S h S- -å å

eMF offhH S=-

effh

00 ( )MF effH hm S=á ñ

ijff jj

e mh J h= +å

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

Easy!!!

0 [ ]S th heffbá ñ=

Hard!!!but doable QMC, PT , ED , DMRG…….

• Prushke T. et. al Adv. Phys. (1995) • Georges Kotliar Krauth Rosenberg RMP

(1996) Kotliar et. al. RMP (2006)

,……………………………………...

Dynamical Mean Field Theory Dynamical Mean Field Theory • Describes the electron both in the itinerant

(wave-like) and localized (particle-like) regimes and everything in between!.

• Follow different mean field states (phases)

Compare free energies.

• Non Gaussian reference frame for correlated materials.

• Reference frame can be cluster of sites CDMFT

1111

, ,

,

[ ] [ ]( )

[ ] [ ]spd sps spd f

f spd ff

H k H kH k

H k H k

æ ö÷ç ÷ç ÷ç ÷çè ø®

| 0 ,| , | , | | ... JLSJM g> > ¯> ¯> >® Determine energy and andDetermine energy and andself consistently from extremizing a self consistently from extremizing a functional . Savrasov and Kotliar PRB 69, 245101, (2001) Full self functional . Savrasov and Kotliar PRB 69, 245101, (2001) Full self

consistent implementation consistent implementation

1212

1( , )

( ) ( )G k i

i H k i

Spectra=- Im G(k,Spectra=- Im G(k,))

LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997).

0 0

0 ff

æ ö÷ç ÷S ç ÷ç ÷ç Sè ø®

,[ ] [ , ]dft lda dmft locG Ur r+G ¾¾®G

abcdU U®

DMFT ConceptsDMFT Concepts

Valence Histograms. Describes the history of the “atom” in the solid, multiplets!

*

( )a b

ab V Va a

a a

ww e

D =-å

,[ ] [ , ]dft lda dmf loct G Ur r+G ¾¾®G

Weiss Weiss field, collective Weiss Weiss field, collective hybridizationfunction, quantifies the hybridizationfunction, quantifies the degree of localization degree of localization

Functionals of density and Functionals of density and spectra give total energiesspectra give total energies

Photoemission Spectral functions and the Photoemission Spectral functions and the

State of the ElectronState of the Electron Probability of removing

an electron and transfering energy =Ei-Ef, and momentum k

f() A() M2

e

Angle integrated spectraAngle integrated spectra

( , ) ( )dkA k A 99

a)a) Weak correlationsWeak correlations

b)b) Strong correlation: FL parameters can’t be Strong correlation: FL parameters can’t be evaluated in PT or FLT does not work.evaluated in PT or FLT does not work.

A(k,A(k, A(k,A(k,

Qualitative Phase diagram :frustrated Hubbard model, Qualitative Phase diagram :frustrated Hubbard model, integer filling integer filling M. Rozenberg et.al. PRL,75, 105 (1995)M. Rozenberg et.al. PRL,75, 105 (1995)

T/WT/W

1313

CONCEPT:CONCEPT:

(orbitally (orbitally resolved) resolved) spectral spectral function. function. Transfer of Transfer of spectral weight. spectral weight.

CONCEPT:CONCEPT:

(orbital (orbital selective) selective) Mott Mott transitiontransition. .

CONCEPT: CONCEPT: Quasiparticle bands, Quasiparticle bands, T*, and Hubbard T*, and Hubbard bandsbands

OutlineOutline

• Introduction to Correlated Materials

• Introduction to Dynamical Mean Field Theory

• Applications to f electrons:

• CeIrIn5

• Pu-Am-Cm

• PuSe PuTe

• URu2Si2

• Conclusions

CeRhIn5CeRhIn5: : TNTN=3.8 K; =3.8 K; 450 mJ/molK2 450 mJ/molK2 CeCoIn5CeCoIn5: : TcTc=2.3 K; =2.3 K; 1000 mJ/molK2; 1000 mJ/molK2; CeIrIn5CeIrIn5: : TcTc=0.4 K; =0.4 K; 750 mJ/molK2 750 mJ/molK2

4f’s heavy fermions, 115’s, CeMIn5 M=Co, Ir, Rh

out of plane

in-plane

Ce

In

Ir

2121

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

•At 10K: At 10K:

•very narrow Drude peakvery narrow Drude peak

•First MI peak at 0.03eV~250cmFirst MI peak at 0.03eV~250cm-1-1

•Second MI peak at 0.07eV~600cmSecond MI peak at 0.07eV~600cm-1-1

Optical conductivity in Optical conductivity in LDA+DMFT Shim, HK Gotliar LDA+DMFT Shim, HK Gotliar Science (2007)Science (2007)

K. Burch et.al.K. Burch et.al.

D. Basov et.al.D. Basov et.al.

CeCe InIn

InIn

Structure Property Relation: Ce115’s Structure Property Relation: Ce115’s Optics and Multiple hybridization Optics and Multiple hybridization gapsgaps

300K300K

ee VV

10K10K

•Larger gap due to hybridization with Larger gap due to hybridization with out of plane Inout of plane In

•Smaller gap due to hybridization with Smaller gap due to hybridization with in-plane Inin-plane In

non-f spectranon-f spectra

J. Shim et. J. Shim et. al. Scienceal. Science

Difference between Co,Rh,Ir Difference between Co,Rh,Ir 115’s115’s

more localizedmore localizedmore itinerantmore itinerantIrIr CoCo RhRh

superconductingsuperconducting magnetically orderedmagnetically ordered““goodgood”” Fermi liquid Fermi liquid

Total and f DOSTotal and f DOS f DOSf DOS

Haule Yee and Kim arXiv:0907.0195

URu2Si2URu2Si2

UU

SiSiRuRu

A signature A signature problem ?problem ?

Two Broken Symmetry SolutionsTwo Broken Symmetry Solutions

Hidden OrderHidden Order

LMA LMA

K. Haule and GKK. Haule and GK

*

( )a b

ab V Va a

a a

ww e

D =-åWeiss fieldWeiss field

Order parameter:Order parameter:

Different orientation gives different Different orientation gives different

phases: “adiabatic continuity” explained.phases: “adiabatic continuity” explained.

Hexadecapole order testable by resonant X-rayHexadecapole order testable by resonant X-ray

In the atomic limit:In the atomic limit:

Hidden order parameterHidden order parameter

Paramagnetic phase Paramagnetic phase low lying singlets f^2low lying singlets f^2

Valence HistogramValence Histogram

Mean fieldMean fieldExp. by E. Hassinger et.al. PRL 77, 115117 (2008)Exp. by E. Hassinger et.al. PRL 77, 115117 (2008)

Simplified toy model phase Simplified toy model phase diagram mean field theorydiagram mean field theory

Orbitally resolved DOS Orbitally resolved DOS

DMFT “STM” URu2Si2 T=20 KDMFT “STM” URu2Si2 T=20 K

Fano lineshape:Fano lineshape:

q~1.24, q~1.24, ~6.8meV, very similar to exp Davis~6.8meV, very similar to exp Davis

UU

SiSi

RuRu

SiSi

Lattice responseLattice response

Localization Delocalization in ActinidesLocalization Delocalization in Actinides

Mott Transition

Modern understanding of this phenomenaDMFT. Modern understanding of this phenomenaDMFT.

PuPu

1717

Total Energy as a function of volume for Total Energy as a function of volume for Pu Pu

(Savrasov, Kotliar, Abrahams, Nature ( 2001)(Savrasov, Kotliar, Abrahams, Nature ( 2001)

Non magnetic Non magnetic correlated state of fcc Pu. correlated state of fcc Pu.

Moment is first reduced by orbital spin Moment is first reduced by orbital spin moment compensation. The remaining moment compensation. The remaining moment is screened by the spd and f moment is screened by the spd and f electronselectrons

The f electron in The f electron in --phase is only slightly phase is only slightly more localized than in more localized than in the the -phase which has -phase which has larger spectral weight in larger spectral weight in the quasiparticle peak the quasiparticle peak and smaller weight in the and smaller weight in the Hubbard bandsHubbard bands

Localization Delocalization in ActinidesLocalization Delocalization in Actinides

Mott Transition

Modern understanding of this phenomenaDMFT. Modern understanding of this phenomenaDMFT.

PuPu

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The standard model of solids fails near Pu The standard model of solids fails near Pu

• Spin Density functional theory: Pu , Am, magnetic, large orbital and spin moments.

• Experiments (Lashley et. al. 2005, Heffner et al. (2006)): Pu is non magnetic. No static or fluctuating moments. Susceptibility, specific heat in a field, neutron quasielastic and inelastic scattering, muon spin resonance…

•Paramagnetic LDA underestimates Volume of Paramagnetic LDA underestimates Volume of Pu Pu. .

•Thermodynamic and transport properties similar to Thermodynamic and transport properties similar to strongly correlated materials.strongly correlated materials.

•Plutonium: correlated paramagnetic metal. Plutonium: correlated paramagnetic metal.

DMFT Phonons in fcc DMFT Phonons in fcc -Pu-Pu

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)(experiments from Wong et.al, Science, 22 August 2003)

K.Haule J. Shim and GK K.Haule J. Shim and GK Nature 446, 513 (2007)Nature 446, 513 (2007)

Trends in ActinidesTrends in Actinidesalpa->delta volume collapse transitionalpa->delta volume collapse transition

Curium has large magnetic moment and Curium has large magnetic moment and orders antiferromagnetically Pu does is orders antiferromagnetically Pu does is

non magnetic.non magnetic.

F0=4,F2=6.1F0=4,F2=6.1

F0=4.5,F2=7.15F0=4.5,F2=7.15

F0=4.5,F2=8.11F0=4.5,F2=8.11

PhotoemissioPhotoemissio

nn

Havela et. al. Phys. Rev. Havela et. al. Phys. Rev. B 68, 085101 (2003)B 68, 085101 (2003)

What is the valence in the late actinides ?What is the valence in the late actinides ?

Plutonium has an unusual form of MIXED VALENCEPlutonium has an unusual form of MIXED VALENCE

LDA resultsLDA results

Finding the f Finding the f occupancyTobin et. al. PRB 72, 085109 2005 occupancyTobin et. al. PRB 72, 085109 2005 K. Moore and G. VanDerLaan K. Moore and G. VanDerLaan RMP (2009). Shim et. al. Europhysics Lett (2009) RMP (2009). Shim et. al. Europhysics Lett (2009)

DMFT DMFT resultsresults

Localization delocalization of f Localization delocalization of f electrons in compounds. electrons in compounds.

• Pu Chalcogenides [PuSe, PuS, PuTe]: Pauli susceptibility, small gap in transport.

• Pu Pnictides [PuP, PuAs, PuSb], order magnetically.

• Simple cubic NaCl structure• Going from pnictides to

chalcogenides tunes the

degree of localization of the

f electron. Earlier work Shick et. al. Pourovski Earlier work Shick et. al. Pourovski et. al. et. al.

LDA+DMFT C. Yee Expts. T. Rurakiewicz et. al. PRB 70, 205103LDA+DMFT C. Yee Expts. T. Rurakiewicz et. al. PRB 70, 205103

PuTe: a 5f mixed valent semi-PuTe: a 5f mixed valent semi-conductor conductor

PuSb: a local moment metalPuSb: a local moment metal

Summary Summary

• Correlated Electron Systems. Huge Phase Space. Fundamental questions. Promising applications.

• DMFT reference frame to think about electrons in solids. Quasiparticles Hubbard bands. Compare with the standard model.

• Many succesful applications, some examples illustrating a) the concepts, b) the role of realistic modelling, and c) the connection between theory and experiment and the role of theoretical spectroscopy.

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Conclusion:Conclusion:

• DMFT provides a surprisingly accurate description of f electron systems.

• It’s physical content at very low temperatures is that of a heavy Fermi liquid in common with other methods but asymptotia is hardly reached (and relevant). Complete description of the crossover.

• Variety and Universality.

Outlook Outlook • “Locality “ as an alternative to Perturbation Theory.• Needed: progress in implementation. e.g. full solution

of DMFT equations on a plaquette, robust GW+DMFT ………….

• Fluctuation around DMFT. • Interfaces, junctions, heterostructures………..• Motterials, Materials,…….• Towards rational material design with correlated

electrons systems http://www.kitp.ucsb.edu/activities/auto/?id=970

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Looking for moments. Pu under (negative ) pressure. C Looking for moments. Pu under (negative ) pressure. C Marianetti, K Haule GK and M. FlussMarianetti, K Haule GK and M. Fluss

Phys. Rev. Lett. 101, 056403 (2008)Phys. Rev. Lett. 101, 056403 (2008)

Conclusion: some general comments. Conclusion: some general comments. •DMFT approach. Can now start from the material.

•Can start from high energies, high temperatures, where the method (I believe ) is essentially exact, far from critical points, provided that one starts from the right “reference frame”.

•Spectral “fingerprints” and their chemical origin.

•Still need better tools to analyze and solve the DMFT equations.

•Still need simpler approaches to rationalize simpler limit.

•Validates some aspects of slave boson mean field theories, modifies quantitatively and sometimes qualitatively the answers.

•At lower temperatures, one has to study different

broken symmetry states.

•At lower temperatures, one has to study

different broken symmetry states.

•Compare free energies, draw phase diagram

•Beyond DMFT: Write effective low energy theories that match the different regions of the phase diagram.

•Close contact with experiments.

•Many materials are being tried, methods are being refined

•Contemplating material design using correlated electron

systems.

Very slow crossover!Very slow crossover!

TT**

Buildup of coherence in single impurity caseBuildup of coherence in single impurity case

TTKK

cohe

rent

spe

ctra

l co

here

nt s

pect

ral

wei

ght

wei

ght

TT scattering ratescattering rate

coherence peakcoherence peak

Buildup of coherenceBuildup of coherence

Crossover around 50KCrossover around 50KSlow crossover compared to AIMSlow crossover compared to AIM

Plutonium Plutonium