Understanding Heavy Fermion Understanding Heavy Fermion Systems: a DMFT perspective Systems: a DMFT perspective

Gabriel KotliarGabriel Kotliar

and Center for Materials Theory

$upport : NSF -DMR DOE-Basic Energy Sciences

Collaborators: K. Haule and J. Shim Ref: Nature 446, 513, (2007)

Colloquium : MPI Dresden (2007)Colloquium : MPI Dresden (2007)

11

Understanding Strongly Correlated Electron Understanding Strongly Correlated Electron Systems: a DMFT perspectiveSystems: a DMFT perspective

Towards and Understanding of Elemental Late Towards and Understanding of Elemental Late Actinides: a DMFT perspective Actinides: a DMFT perspective

Band Theory: electrons as waves.

Landau Fermi Liquid Theory.

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

•Quantitative Tools. Density Functional Theory

•Kohn Sham (1964)2 / 2 ( )[ ] KS kj kj kjV r r 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 E kn band index, e.g. s, p, d,,fn band index, e.g. s, p, d,,f

Strong Correlation Problem:where Strong Correlation Problem:where the standard model failsthe standard model fails

• Fermi Liquid Theory works but parameters can’t be computed in perturbation theory.

• Fermi Liquid Theory does NOT work . Need new concepts to replace of rigid bands !

• Partially filled d and f shells. Competition between kinetic and Coulomb interactions.

• Breakdown of the wave picture. Need to incorporate a real space perspective (Mott).

• Non perturbative problem.

44

Heavy-fermion system G. R. Stewart RMP 56 (1984)

Since the discovery by Steglich et al. (1979) of superconductivity in the high-effective-mass (-200melectrons) in CeCu2Si2, the search for and characterization of such "heavy-fermion" systems has been a rapidly growing field of study. They include superconductors(CeCu2Si2, UBe13, UPt3), magnets (NpBe13, U,Zn17,, UCdll), and materials in which no ordering is observed (CeA13, CeCu6). These f-electron materials have, in comparison to normal metals, enormous specifIC heat values (450-1600 mJ / m o l K2 large values of the low-temperature magnetic susceptibility 8-50 10-3 emu/molG), with large room temperature, values of the resistivity (100-

200Ohmcm), Ingredients: Band spd + local f electrons Ingredients: Band spd + local f electrons

5f elements: actinide series5f elements: actinide series

s/cs/c AFAF FMFM

Localisation Delocalization Localisation Delocalization

1.4K1.4K 0.40.4KK

0.9K0.9K 0.8K0.8K 52K52K 25K25K 52K52K

Localization Delocalization in ActinidesLocalization Delocalization in Actinides

after G. Lander, Science (2003).

Mott Transition

PuPu

Basic Questions Basic Questions

• 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

Phases of Pu (A. Lawson LANL)Phases of Pu (A. Lawson LANL)

Small amounts of Ga stabilize the Small amounts of Ga stabilize the phase (A. Lawson LANL)phase (A. Lawson LANL)

Anomalous ResistivityAnomalous Resistivity

2 ( )F Fe k k l

h

Maximum metallic resistivity 2

Fe k

h

Specific heat and susceptibility. Specific heat and susceptibility. Pu is non magnetic Pu is non magnetic

Standard model FAILS in the late actinidesStandard model FAILS in the late actinides

• Predicts Pu and Am to be magnetic, with a large moment. (about 5 B)

• Paramagnetic DFT understimates volume of delta Pu by 25 %

• Many modfications have been attempted, to explain why Pu is non magnetic. Mixed level model Zwicknagl and Fulde (Erickson Balatzki and Wills et. al. ) (5f)4 conf.

LDA+U (Shick, Anisimov) (5f)6 conf• Cannot account for anomalou transport and

thermodynamics

DMFT Spectral Function Photoemission and DMFT Spectral Function Photoemission and correlationscorrelations

• Probability of removing an electron and transfering energy =Ei-Ef, and momentum k

f() A() M2

e

Angle integrated spectral Angle integrated spectral function function

( , ) ( )dkA k A 88

a)a) Weak CorrelationWeak Correlation

b)b) Strong CorrelationStrong Correlation

DMFT cavity construction. A. Georges and G. Kotliar PRB 45, 6479 (1992).DMFT cavity construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). H Happy appy marriage of atomic and band physics. Extremize functional of A(marriage of atomic and band physics. Extremize functional of A())

Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004). G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti (RMP 2006).

1( , )

( )k

G k ii i

Extremize a functional of the local spectra. Local self energy.

Dynamical Mean Field TheoryDynamical Mean Field Theory• Weiss field is a function. Multiple scales in strongly

correlated materials.• Exact large coordination (Metzner and Vollhardt 89) .• Not restricted to single site-CDMFT.• Immediate extension to real materials DFT+DMFT Functionals of density and spectra. Review Kotliar et. al.

RMP (2006)

, ,

, 22

[ ] [ ]( )

[ ] [ ]spd sps spd f

f spd ff

H k H kk

H k H ke

æ ö÷ç ÷ç ÷ç ÷çè ø®

| 0 ,| , | , | | ... JLSJM g> ­> ¯> ­ ¯> >®

1212

Total­Energy­as­a­function­of­volume­for­Total­Energy­as­a­function­of­volume­for­Pu­Pu­W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Nick Zein Following Aryasetiwan et. al. PRB 70 195104. (2004)

Moment is first reduced by orbital spin moment compensation. The

remaining moment is screened by the spd and f electrons

Double well structure and Double well structure and Pu Pu Qualitative explanation of negative thermal expansion[Lawson, A. C., Roberts J. A., Martinez, B., and Richardson, J. W., Jr. Phil. Mag. B, 82, 1837,(2002). G. Kotliar J.Low Temp. Physvol.126, 1009 27. (2002)]

Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the

volume expands the insulator and contract the metal.

F(T,V)=Fphonons+F(T,V)=Fphonons+FinvarFinvar

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-Pu-Pu

  C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

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

(experiments from Wong et.al, Science, 22 August 2003)2121

The “DMFT-The “DMFT-valence” in the valence” in the late actinides.late actinides.

Time scale of the fluctuations. Ef*2222

­­Photoemission­Gouder , Havela PRB

2002, 2003

Photoemission­Spectra[­Shim.­Photoemission­Spectra[­Shim.­Haule,GK­Nature­(2007)]Haule,GK­Nature­(2007)]

alpa->delta­volume­collapse­transition

F0=4,F2=6.1

F0=4.5,F2=7.15

2020

Photoemission and Mixed valence in Pu Photoemission and Mixed valence in Pu

-6 -4 -2 0 2 4 6

ENERGY(eV)

0.0

0.1

0.2

0.3

0.4

0.5

DOS

Approach the Mott point from the right Am under Approach the Mott point from the right Am under pressurepressure

Density­functional­based­electronic­structure­calculations: Non­magnetic­LDA/GGA­predicts­volume­50%­off.­ Magnetic­GGA­corrects­most­of­error­in­volume­but­gives­m~6B

(Soderlind et.al., PRB 2000). Experimentally,­Am­has non­magnetic­f6­ground­state­with­J=0­(7F0)

Experimental­Equation­of­State­(after Heathman et.al, PRL 2000)

Mott Transition?“Soft”

“Hard”

Am equation of state. LDA+DMFT.New acceleration Am equation of state. LDA+DMFT.New acceleration technique for solving DMFT equations S. Savrasov K. technique for solving DMFT equations S. Savrasov K. Haule G. Kotliar cond-mat. 0507552 (2005)Haule G. Kotliar cond-mat. 0507552 (2005)

Mott transition in open (right) and closed (left) shell systems. Mott transition in open (right) and closed (left) shell systems. Superconductivity ? Application to Am ?Superconductivity ? Application to Am ?

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

J=0

???

Tc

Resistivity of Am under pressure. J. C. Griveau et.al. PRL 94, 097002 (2005).

Photoemission spectra using Hubbard I solver and Sunca . Photoemission spectra using Hubbard I solver and Sunca . [Savrasov Haule and Kotliar cond-mat 0507552 PRL (2006)] [Savrasov Haule and Kotliar cond-mat 0507552 PRL (2006)]

Hubbard bands width is determined by multiplet splittings.Hubbard bands width is determined by multiplet splittings.

Photomission Spectra of Am under pressure. Sunca. Onset of Photomission Spectra of Am under pressure. Sunca. Onset of

mixed valence. Savrasov Haule Kotliar (2005) PRL (2006)mixed valence. Savrasov Haule Kotliar (2005) PRL (2006)

ConclusionsConclusions

• Unique properties of Pu and Am under pressure result from a proximity of a localization delocalization transition. Rare form of mixed valence.

• DMFT provides a good start. Qualitative insights, some quantitative predictions into delta Pu. Other Pu phases.

• Meaningful interplay of theory and experiment. Key in condensed matter physics.

ConclusionsConclusions• Pu and Am are unique strongly correlated

elements. It is one among many strongly correlated electron system, materials for which neither the standard model of solids, works well.

• They require, new concepts, new computational methods, new algorithms, DMFT provides all of the above, and is being used in many other problems.

• Many applications to othe problems exist, others are in progress, research opportunity in correlated materials.

Prospects for Extensions and Applications Prospects for Extensions and Applications to More Complex Heavy Fermion Systems to More Complex Heavy Fermion Systems

• More complicated crystal structures, more atoms per unit cell. 115’s , alpha Pu……

• Non local physics. Heavy fermion quantum criticality. a) Local Quantum Criticality scenario of Q. Si and collaborators. Nature 413 (2001) 804. Single site EDMFT

b) Cluster Quantum Multicriticality. L. DeLeo and GK. Requires 2 impurity Kondo model for its description.

Conclusion AmConclusion Am

• Americium undergoes Mott transition under pressure. [AmIII-AmIV] boundary.

• Unusual superconductivity and resistivities.

• Theoretical clue mixed valent due to admixture of (5f) upon application of pressure.

• Realizes Mott transition from the insulating side, towards a close shell configuration..

. Mott transition in the open shell case. Heathman et. al. Science 309,110 (2006)

Approach the Mott transition from the right.

LS coupling L=0 S=7

jj coupling J=7/2

=2S+L

Expt monent . is closer to L S coupling

Curium is magnetic Hurray et.al. Physica. B (1980) 217

K.Haule­and­J.­Shim­K.Haule­and­J.­Shim­Trends­in­ActinidesTrends­in­Actinides

alpa->delta­volume­collapse­transition

Curium has large magnetic moment and orders antif Pu does is non

magnetic.

F0=4,F2=6.1

F0=4.5,F2=7.15

F0=4.5,F2=8.11

Conclusion Conclusion

• A Few References ……

• A.Georges, G. K., W. Krauth and M. J. Rozenberg, Reviews of . Modern Physics 68, 13 (1996).

• G. K, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, RMP 78, 865-951, (2006).

• G. K and D. Vollhardt Physics Today, Vol 57, 53 (2004).

2929

““Invar model “ for Pu-Ga. Lawson et. al.Phil. Mag. Invar model “ for Pu-Ga. Lawson et. al.Phil. Mag. (2006) Data fits only if the excited state has zero (2006) Data fits only if the excited state has zero

stiffness.stiffness.

ConclusionsConclusions• Constant interplay between theory and

experiment has lead to new advances.• General anomalies of correlated electrons

and anomalous system specific studies, need for a flexible approach. (DMFT).

• New understanding of Pu. Methodology applicable to a large number of other problems, involving correlated electrions, thermoelectrics, batteries, optical devices, memories, high temperature superconductors, ……..

ConclusionsConclusions

• DMFT produces non magnetic state, around a fluctuating (5f)^5 configuraton with correct volume the qualitative features of the photoemission spectra, and a double minima structure in the E vs V curve.

• Correlated view of the alpha and delta phases of Pu. Interplay of correlations and electron phonon interactions (delta-epsilon).

• Calculations can be refined in many ways, electronic structure calculations for correlated electrons research program, MINDLAB, ….

What do we want from What do we want from materials theory?materials theory?

• New concepts , qualitative ideas

• Understanding, explanation of existent experiments, and predictions of new ones.

• Quantitative capabilities with predictive

power.

Notoriously difficult to achieve in strongly correlated materials.

Some new insights into the funny Some new insights into the funny properties of Puproperties of Pu

• Physical anomalies, are the result of the unique position of Pu in the periodic table, where the f electrons are near a localization delocalization transition. We learned how to think about this unusual situation using spectral functions.

• Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha). Negative thermal expansion, multitude of phases.

Quantitative calculationsQuantitative calculations• Photoemission spectra,equilibrium volume,

and vibration spectra of delta. Good agreement with experiments given the approximations made.Many systematic improvements are needed.

• Work is at the early stages, only a few quantities in one phase have been considered.

• Other phases? Metastability ? Effects of impurities? What else, do electrons at the edge of a localization localization do ? [ See epsilon Pu spectra ]

Collaborators, Acknowledgements ReferencesCollaborators, Acknowledgements References

Los Alamos Science,26, (2000)

S. Savrasov and G. Kotliar Phys. Rev. Lett. 84, 3670-3673, (2000). S.Savrasov G. Kotliar and E. Abrahams, Nature 410, 793 (2001).

X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science, Vol300, 954 (2003).

Collaborators: S. Savrasov ( Rutgers-NJIT)

X. Dai ( Rutgers), E. Abrahams (Rutgers), A. Migliori (LANL),H Ledbeter(LANL).

Acknowledgements: G Lander (ITU) J Thompson(LANL)

Funding: NSF, DOE, LANL.

Cluster DMFTCluster DMFT: removes limitations of single site DMFTlimitations of single site DMFT

11 23

24

( , ) (cos cos )

cos coslatt k kx ky

kx ky

wS =S +S +

+S

•No k dependence of the self energy.

•No d-wave superconductivity.

•No Peierls dimerization.

•No (R)valence bonds.

Reviews: Reviews: Georges et.al. RMP(1996). Th. Maier et. al. RMP (2005); Kotliar et. .al. RMP (2006).

2323

U/t=4.

Two Site Cellular DMFTTwo Site Cellular DMFT (G.. Kotliar et.al. PRL (2001)) in the 1D in the 1D Hubbard modelHubbard model M.Capone M.Civelli V. Kancharla C.Castellani and GK PRB

69,195105 (2004)T. D Stanescu and GK PRB (2006)

2424

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

Self Energy Self Energy

VanShilfgaarde (2005)VanShilfgaarde (2005)

33

Smith Kmeko Phase diagram. Minimum in melting Smith Kmeko Phase diagram. Minimum in melting curve and divergence of the compressibility at the curve and divergence of the compressibility at the

Mott endpointMott endpoint

( )dT V

dp S

Vsol

Vliq

The enhancement of the specific heat, further evidence for an The enhancement of the specific heat, further evidence for an open shell configuration, presence of electronic entropy. open shell configuration, presence of electronic entropy.

J. Lashley et.al. PRB(2005)

Double well structure and Double well structure and Pu Pu Qualitative explanation of negative thermal expansion[Lawson, A. C., Roberts J. A., Martinez, B., and Richardson, J. W., Jr. Phil. Mag. B, 82, 1837,(2002). G. Kotliar J.Low Temp. Physvol.126, 1009 27. (2002)]

Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the

volume expands the insulator and contract the metal.

F(T,V)=Fphonons+F(T,V)=Fphonons+FinvarFinvar

““Invar model “ for Pu-Ga. Invar model “ for Pu-Ga.

(Data fits if the excited state has zero stiffness.(Data fits if the excited state has zero stiffness.

Dynamical Mean Field Theory. Cavity Construction.Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992).A. Georges and G. Kotliar PRB 45, 6479 (1992).

†

0 0 0

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

s st m tt

t t ­ ¯

¶+ D-

¶- +òò ò

,ij i j i

i j i

J S S h S- -å å eMF offhH S=-† †

, ,

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

t c c c c U n n

*

( )V Va a

a a

ww e

D =-å

† † † † †Anderson Imp 0 0 0 0 0 0 0

, , ,

( +c.c). H c A A A c c UcV c c c

A(A())

1010

A. Georges, G. Kotliar (1992)

( )wDlatt ( ,

1 G [ ]

( ) [( ) ])

[ ]n impn

n

ik ii

ktw m

ww+ + - S

DD

=

latt( ) G ([ [)] ] ,imp n nk

G i i kw wD D=å

[ ]ijij

jm mJth hb= +å

11

( ( )( )

( [))

][ ]

imp n

imp n

kn

G i

Gti

ik

w

ww -D

D

=+-

å

A(A())

1111

Expt. Wong et. al.Expt. Wong et. al.

Elastic DeformationsElastic Deformations

In most cubic materials the shear does not depend strongly on crystal orientation,fcc Al, c44/c’=1.2, in Pu C44/C’ ~ 7 largest shear anisotropy of any element.

Uniform compression:p=-B V/V Volume conserving deformations:

F/A=c44 x/L F/A=c’ x/L

Localization Delocalization in ActinidesLocalization Delocalization in Actinides

after G. Lander, Science (2003).

Mott Transition

Modern understanding of this phenomena using functional Modern understanding of this phenomena using functional approach toDMFT. K Haule S.Savrasov J Shim approach toDMFT. K Haule S.Savrasov J Shim

PuPu

1818

w110

nh

5

2B

3

5

110

h

2 3 + B

5 5

w

n

<w110> = n7/2 – 4/3 n5/2nf = n7/2 + n5/2

Spectral Function and PhotoemissionSpectral Function and Photoemission

• Probability of removing an electron and transfering energy =Ei-Ef, and momentum k

f() A() M2

e

Angle integrated spectral Angle integrated spectral function function

( , ) ( )dkA k A 88

Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in

the screened interactions (Hedin 1965)

Self Energy Self Energy

Succesful description of the total energy and the Succesful description of the total energy and the excitation spectra of a large number of simple metals excitation spectra of a large number of simple metals semiconductors and insulators.semiconductors and insulators.

Succesfully predicts semiconducting gaps, phonon Succesfully predicts semiconducting gaps, phonon frequencies, resistivities, of countless materials. frequencies, resistivities, of countless materials.

33

a)a) Weak CorrelationWeak Correlation

b)b) Strong CorrelationStrong Correlation

WW110110 =2/3<l.s> and banching ratio =2/3<l.s> and banching ratio

See the expt. work of K. Moore G. Van der Laan G. Haire M. Wall and A. Schartz

Am H2

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling. [Rozenberg et. al. PRL 1995] Evolution of the Local Spectra as a function of U,and T. Mott transition driven by transfer of spectral weight Zhang Rozenberg Kotliar PRL (1993)..

.

OUTLINE OUTLINE

• The challenge of strongly correlated electron systems. Heavy Fermions and Late actinides: experimental overview

• Introduction to Dynamical Mean Field Theory

(DMFT).

• Theory of delta Pu

• Theory of Am and Cm

• Conclusions

Inelastic X Ray. Phonon energy 10 Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.mev, photon energy 10 Kev.

E = Ei - EfQ =ki - kf

WW110110 =2/3<l.s> and banching ratio =2/3<l.s> and banching ratio

Moore and van der Laan, Ultramicroscopy 2007.

110

h

2 3 + B

5 5

w

n

2/3<l.s> in the late actinides [DMFT 2/3<l.s> in the late actinides [DMFT results: K. Haule and J. Shim ]results: K. Haule and J. Shim ]

See the expt. work of K. Moore G. Van der Laan G. Haire M. Wall and A. Schartz

Am H2

Phonon freq (THz) vs q in delta Pu X. Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003Dai et. al. Science vol 300, 953, 2003