Correlations Magnetism and Structure across the actinide series : a Dynamical Mean Field Theory Perspective Plutonium Futures Asilomar July 9-13 (2006). G.Kotliar Physics Department and Center for Materials Theory Rutgers University.. Collaborators E. Abrahams K. Haule Ji-Hoon Shim A. Toropova (Rutgers) S. Savrasov (UCDavis ) L. Pourovski (E. Polytechnique). Expts. : M. Fluss J. C Griveaux G Lander A. Lawson A. Migliori J.Singleton J.Smith J Thompson J. Tobin Support: DOE- BES DOE-NNSA.

Outline Experimental motivation Brief introduction to DMFT ideas. Valence changes across the late actinides a SUNCA-DMFT study [ K. Haule J. Shim] Interplay of electronic and structural properties in phase transformations. Conclusion

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

Pu phases: A. Lawson Los Alamos Science 26, (2000) Experimentally Pu is not magnetic. No trace of ordered or low frequency fluctuating local moments. [Lashley et. al. cond-matt ] PRB (2005). Approach the Mott transition from the left. (delocalized side).

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

J. Tobin et.al. PRB 72, (2005) XAS and EELS

DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). Happy marriage of atomic and band physics. 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 (to appear in RMP). Extremize a functional of the local spectra. Local self energy.

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).. Mott transition in one band model. Review Georges et.al. RMP 96

Applications to actinides S. Savrasov G.K and E. Abrahams [neglect multiplets] energy and spectra IPT imp. Solver Phonons [Dai et.al.] [Hubbard I imp solver] Could not address the existence of magentically ordered states. Recent review see G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti to appear in RMP. cond-mat Recent progress : K. Haule SUNCA imp. solver, full multiplet structure and Kondo physics compete on equal footing. Can consider DMFT eqs. in different ordered phases.

Curie-Weiss Tc Trends in Actinides alpa->delta volume collapse transition 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

in the late actinides [DMFT results: K. Haule and J. Shim ]

The “DMFT- valence” in the late actinides

Outline Experimental motivation Brief introduction to DMFT ideas. Valence changes across the late actinides a SUNCA-DMFT study [ K. Haule J. Shim] Interplay of electronic and structural properties in phase transformations. Conclusion

Minimum in melting curve and divergence of the compressibility at the Mott endpoint

DMFT Phonons in fcc  -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory Experiment ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

Why is Epsilon Pu (which is smaller than delta Pu) stabilized at higher temperatures ??Compute phonons in bcc structure.

Phonon entropy drives the epsilon delta phase transition Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta. At the phase transition the volume shrinks but the phonon entropy increases. Estimates of the phase transition following Drumont and G. Ackland et. al. PRB.65, (2002); (and neglecting electronic entropy). TC ~ 600 K.

Total Energy as a function of volume for Pu Total Energy as a function of volume for Pu W (ev) vs (a.u ev) (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Moment is first reduced by orbital spin moment compensation. The remaining moment is screened by the spd and f electrons

Double well structure and  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. Phys vol.126, (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 +Finvar

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

Conclusion DMFT studies of electrons and lattice displacements. Valence changes and transfers of spectral weight. [ Consistent picture of Pu-Am-Cm]. Alpha and delta Pu, screened (5f)^5 configuration. Differ in the degree of screening. Different views [ Pu non magnetic (5f)^6, Pu magnetic ] Magnetism and defects. Important role of phonon entropy in phase transformations.

LS vs jj coupling in Am and Cm

Other views on Pu non magnetic 5f 6. Shorikov Lukoyanov Korotin and Anisimov. PRB (2006). LDA+U with around the localized limit double counting. (5f)^6 configuration stabilized by a) small Hunds rule J H =.48 ev and small U=2.5 ev. Strong sensitivity to the value of J H. J H =.5 critical value instability to magnetic state.

Other views on Pu: Pu non mangetic 5f 6 Shick A. Drachl V. Havela L. Europhysics Letters 69, 588 (2005). Pourovskii Katsnelson Lichtenstein L Havela T Gouder F. Wastin A. Shick V. Drachl and G. Lander (2005) LDA+U with Edc around mean field. +DMFT Flex.

L. Pourovski (unpublished) Expt  60 mJ/Mol K 2

L. Pourovski (unpublished) Expt  mJ/Mol K 

Mott Transition in the Actinide Series. J. Lashley et.al.(2005)

K. Haule, Pu- photoemission with DMFT using (vertex corrected )NCA. n f =5.7

High energy spectroscopies theory and expt (5f) 5 Intermediate or jj coupling limit. J. Tobin et.al. PRB 68, (2003) resonant photoemission and X ray absortion. K Moore et.al. PRL 90, (2003). Phil Mag 84,1039 (2004).

J. Tobin et.al.

Cm & Pu Cm –No quasiparticle peak at EF –Hubbard bands at ~4eV and ~-5eV –intermediate between jj and LS Pu –Clear quasiparticle peak at EF –Hubbard bands at ~3eV and ~-1eV –closer to jj scheme than Cm

What is the dominant atomic configuration? Local moment? Snapshots of the f electron Dominant configuration:(5f) 5 Naïve view Lz=-3,-2,-1,0,1 ML=-5  B S=5/2 Ms=5  B Mtot=0

Magnetic moment L=5, S=5/2, J=5/2, Mtot=Ms=  B gJ =.7  B Crystal fields     GGA+U estimate (Savrasov and Kotliar 2000) ML=-3.9 Mtot=1.1 This bit is quenched by Kondo effect of spd electrons [ DMFT treatment] Experimental consequence: neutrons large magnetic field induced form factor (G. Lander).

Curie-Weiss Tc Trends in Actinides alpa->delta volume collapse transition Same transition in Am under pressure Curium has large magnetic moment and orders antif. F0=4,F2=6.1 F0=4.5,F2=7.15 F0=4.5,F2=8.11

The “DMFT- valence” in the late actinides

Many theoretical reasons to apply DMFT to Curium. Mott transition from the right from an open shell configuration. (5f) 7 Mott transition or volume collapse ? L.S or jj coupling ? Underscreened Kondo lattice ? Crucial test for DMFT: produce magnetism where there is!

LS coupling L=0 S=7  jj coupling J=7/2   =2s+l Expt. Hurray et. Al. Physica. B (1980) 217

Conclusions Mott transition in Americium and Plutonium. In both cases theory (DMFT) and experiment suggest gradual more subtle evolution than in earlier treatments. DMFT: Physical connection between spectra and structure. Studied the Mott transition open and closed shell cases.. DMFT: method under construction, but it already gives quantitative results and qualitative insights. Interactions between theory and experiments. Pu: simple picture of the phases. alpha delta and epsilon. Interplay of lattice and electronic structure near the Mott transition. Am: Rich physics, mixed valence under pressure. Superconductivity near the Mott transition. Cm -----work in progress.

Summary LDA LDA+U DMFT Spectra Method E vs V

Outline Introduction: some Pu puzzles. DMFT, qualitative aspects of the Mott transition in model Hamiltonians. DMFT as an electronic structure method. DMFT results for delta Pu, and some qualitative insights. Conclusions

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. Calculations can and should be refined and extended.

Conclusions Outsanding question: electronic entropy, lattice dynamics. In the making, new generation of DMFT programs, QMC with multiplets, full potential DMFT, frequency dependent U’s, multiplet effects, combination of DMFT with GW

P63/mmc #194 a=3.490 A c= A Cm(1) 2a (0,0,0) Cm(2) 2c (1/3,2/3,1/4) Fm3m (fcc) a~4.97 A under 30GPa

Antiferromagnetic structures in Cm II (fcc) AFM (I) : AFM ordering along (001) AFM (II) : AFM ordering along (111)

AFM (I) structure of fcc-Cm (II) U=4.5eV,J=0eV dE dc = 3.7 eV No experimental result

Experiments Needed: investigation of the unoccupied states. BIS, Optics, Raman, Inelastic XRay, etc.

Approach the Mott point from the right Am under pressure Density functional based electronic structure calculations:  Non magnetic LDA/GGA predicts volume 50% off.  Magnetic GGA corrects most of error in volume but gives m ~6  B (Soderlind et.al., PRB 2000).  Experimentally, Am has non magnetic f 6 ground state with J=0 ( 7 F 0 ) Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition? “Soft” “Hard”

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

A. C. Lawson et. al. LA UR F(T,V)=Fphonons+Finvar

Invar model A. C. Lawson et. al. LA UR

Prediction of the Invar Model

DMFT and the Invar Model

Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, (2002). Semiconductors: Zein Savrasov and Kotliar (2005). G. Kotliar and S. Savrasov Strongly Correlated Systems, A. M. Tsvelik Ed Kluwer Academic conmat/ S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, (2004)

Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, (2002). Semiconductors: Zein Savrasov and Kotliar (2005). G. Kotliar and S. Savrasov Strongly Correlated Systems, A. M. Tsvelik Ed Kluwer Academic conmat/ S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, (2004)

5f’s Mott Transition in the Actinide Series Johansen Phil Mag. 30, 469(1974). J. Lashley et.al.(2004) Revisit with modern DMFT tools. Savrasov and Kotliar PRL 84,3760 (2000) ……….

Photomission Spectra of Am under pressure. Sunca. Onset of mixed valence. Savrasov Haule Kotliar (2005)

 Functional formulation (Chitra and Kotliar 2000) Phys. Rev. B 62, (2000). Self consistent determination of electronic structure. Full implementation S. Savrasov G. Kotliar ( ). Phys. Rev. B 69, (2004). Frequency dependent generalization of the Kohn Sham potential, whose role is to give the exact “local” Greens function. Frequency dependent Kohn-Sham like equations can be derived by extremizing a functional which gives the total energy. Application to Pu. Savrasov et al. Nature (2001)Phys. Rev. B 62, (2000).Phys. Rev. B 69, (2004). Llinear response phonon spectra [ Savrasov and Kotliar Phys. Rev. Lett. 90, (2003). ]. Speed up of the method. “DMFT quality at LDA speed”. Reduction of the DMFT equations, to Kohn Sham equations with additional orbitals. Total energy of complicated structures. Savrasov Haule and Kotliar Am PRL cond-mat (2005). Phys. Rev. Lett. 90, (2003).

A Lindbaum1, SHeathman2, T Le Bihan3, RGHaire4, M Idiri2 and GHLander2 J. Phys.: Condens. Matter 15 (2003) S2297–S2303

Curie-Weiss Tc Trends in Actinides alpa->delta volume collapse transition 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

Spectral Density Functional  Incorporate band structure and orbital degeneracy to achieve a realistic description of materials. LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Related work PRB 57,6884 (1998). Derive complex Hamiltonians solve them using DMFT.  LDA+DMFT photoemission Allows the computation of realistic photoemission spectra optics etc.  Simple concepts. Multiplet structure in the Hubbard bands. K space structure in the resonance.  Difficult technical implementation. Various impurity solvers. Various basis sets. Various orbitals on which correlation are applied. Various double counting corrections.  K. Haule SUNCA as an impurity solver.

DMFT + electronic structure method. Spectral density functional. Review Kotliar Savrasov Haule Parcollet Oudovenko and Marianetti to appear inRMP Spectral density functional. Review Kotliar Savrasov Haule Parcollet Oudovenko and Marianetti to appear inRMP Effective (DFT-like) single particle Spectrum consists of delta like peaks Spectral density usually contains renormalized quasiparticles and Hubbard bands Basic idea of DMFT: reduce the quantum many body problem to a one site or a cluster of sites problem, in a medium of non interacting electrons obeying a self-consistency condition. (review 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 GW For correlated bands (f or d): with DMFT add all local diagrams\ Functionals of the spectra to obtain total energy.

Earlier applications of DMFT to actinides. Savrasov Kotliar and Abrahams (Nature

Approach the Mott point from the right (localized side) Am under pressure Density functional based electronic structure calculations:  Non magnetic LDA/GGA predicts volume 50% off.  Magnetic GGA corrects most of error in volume but gives m ~6  B (Soderlind et.al., PRB 2000).  Experimentally, Am has non magnetic f 6 ground state with J=0 ( 7 F 0 ) Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition? “Soft” “Hard”

The “DMFT- valence” in the late actinides

Mott Transition in the Actinide Series. J. Lashley et.al.(2004)

Smith-Kmetko phase diagram. Mott Transition in the Actinide Series around Pu : Johansen Phil Mag. 30, 469(1974). Early views on the Mott transition. Strongly discontinuous. Implementation with LDA or LDA SIC. Approach to the Mott transition, REDUCTION of the specific heat.