Correlations Magnetism and Structure across the actinide series IWOSMA-3 Lyon June 2-3 (2006). G.Kotliar Physics Department and Center for Materials Theory Rutgers University. CPHT Ecole Polytechnique, France and CPHT CEA Saclay. Support: -DOE- BES Chaire International de Recherche Blaise Pascal de l”Etat et de la Region d’Ille de France geree par la Fondation de l’Ecole Normale. Collaborators S. Savrasov (UCDavis ) K. Haule (Rutgers) Ji-Hoon Shim (Rutgers) L. Pourovski (E. Polytechnique). Discussions : M. Fluss J. C Griveaux G Lander A. Lawson A. Migliori J.Singleton J. Thompson J. Tobin

Outline Brief introduction to the Mott transition across the Actinides series and to DMFT. The Mott transition from the left.DMFT results for Pu. f  or f 6 The Mott transition from the right. The closed shell case. DMFT results for Am. S. Savrasov K. Haule and GK PRL(2005). The Mott transition from the right. Cm.

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.

Pu phases: A. Lawson Los Alamos Science 26, (2000) GGA LSDA predicts  Pu to be magnetic with a large moment ( ~ 5 Bohr). Experimentally Pu is not magnetic. [Lashley et. al. cond- matt ] PRB (2005). Approach the Mott transition from the left. (delocalized side).

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”

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

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

Towards ab-initio DMFT.  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.

Mott transition in open (right) and closed (left) shell systems. Superconductivity ? S S U U  T Log[2J+1] Uc  ~1/(Uc-U) J=0 ??? Tc

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

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. Nick Zein Following Aryasetiwan et. al. PRB (2004) Moment is first reduced by orbital spin moment compensation. The remaining moment is screened by the spd and f electrons

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.

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 only if the excited state has zero stiffness.

Alpha and delta Pu Photoemission Spectra DMFT(Savrasov et.al.) EXP ( Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000))

Photoemission studies of Pu. [Havela Gouder. Joyce and Arko. J. Tobin et. al. PHYSICAL REVIEW B 68, ,2003

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).

Conclusion Pu I still bet on the (5f) 5 Kondo screened magnetic configuration. But….. More work is needed to understand the correct way to compute nf in DMFT in order to interpret the high energy spectroscopy. How to measure, compute, valence in Pu ?

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)

Mott transition in open (right) and closed (left) shell systems. Superconductivity ? Localized (5f) 6 in L.S coupling or jj coupling ? S S U U  T Log[2J+1] Uc  ~1/(Uc-U) J=0 ??? Tc

Photoemission spectra using Hubbard I solver and Sunca. [Savrasov Haule and Kotliar cond-mat PRL (2006)] Hubbard bands width is determined by multiplet splittings.

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

Photomission Spectra of Am under pressure. Sunca. Onset of mixed valence. Savrasov Haule Kotliar (2005) PRL (2006)

Conclusion Am Americium undergoes Mott transition under pressure. [AmIII-AmIV] boundary. Unusual superconductivity and resistivities. Theoretical clue mixed valent due to admixture of (5f) 7. Unlike Sm…..

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

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.

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.

The schematic phase diagram, the Mott (Johansen ) and the Kondo collapse (Allen- Martin) two scenarios: how to tell between the two ? J.W. Allen and L.Z. Liu, Phys. Rev. B 46, 5047 (1992). Kondo impurity model + elastic terms. DMFT phase diagram of a Hubbard model at integer filling, has a region between Uc1(T) and Uc2(T) where two solutions coexist. A. Georges G. Kotliar W. Krauth and M Rozenberg RMP 68,13,(1996). Coupling the two solutions to the lattice gives a phase diagram akin to alpha gamma cerium. Majumdar and Krishnamurthy PRL 73 (1994).

Photoemission&experiment A. Mc Mahan K Held and R. Scalettar (2002) Zoffl et. al (2002) K. Haule V. Udovenko S. Savrasov and GK. (2004) B. Amadon S. Biermann A. Georges F. Aryastiawan cond-mat

To resolve the conflict between the Mott transition and the Kondo volume collapse picture : Turn to Optics! Haule et.al. Qualitative idea. The spd electrons have much larger velocities, so optics will be much more senstive to their behavior. See if they are simple spectators (Mott transition picture ) or wether a Kondo binding unbinding takes pace (Kondo collapse picture). General method, bulk probe.

Theory: Haule et. al. cond-matt 04 Expt: J.W. vanderEb PRL 886,3407 (2001)

Temperature dependence of the optical conductivity.

Origin of the features.

Conclusions 4f materials Single site DMFT describes well the photoemission, total energy, and optical spectra of alpha and gamma cerium. Analysis of the DMFT results favors (and provides a moder reformulation of) the volume collapse transition. Combining experimental and theoretical spectroscopies, we get new understanding.

Overview  Various phases : isostructural phase transition (T=298K, P=0.7GPa)  (fcc) phase [ magnetic moment (Curie-Wiess law) ]   (fcc) phase [ loss of magnetic moment (Pauli-para) ] with large volume collapse  v/v  15  (  -phase a  5.16 Å  -phase a  4.8 Å) volume sexp.LDALDA+U  28Å Å 3  34.4Å Å 3   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (localized) : High T phase  Curie-Weiss law (localized magnetic moment),  Large lattice constant  Tk around 60-80K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  Sm  aller lattice constant  Tk around K   -phase (delocalized:Kondo-physics) : Low T phase  Loss of Magnetism (Fermi liquid Pauli susceptibility) - completely screened magnetic moment  Sm  aller lattice constant  Tk around K

H.Q. Yuan et. al. CeCu2(Si 2-x Ge x ). Am under pressure Griveau et. al. Superconductivity due to valence fluctuations ?

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)

Double well structure and  Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low Temp. Phys vol.126, (2002)]See also A. Lawson et.al.Phil. Mag. B 82, 1837 ]

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)

Next work Cm I : 4 atoms in unit cell –Find stable magnetic ground state Calculation of magnetic susceptibility. Finding proper parameters : U, dE dc –Predict experimental photoemission, optical spectrum. Change of transition temperature –U, volume, dE dc etc.

Not fully converged result!!

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) ……….

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. Zein Savrasov and Kotliar (2005) Following Aryasetiwan et. al. PRB (2004)

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

Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = E i - E f Q = k i - k f

Expt. Wong et. al. IXRS at Grenoble.

Conclusion Pu Realistic DMFT calculations provide an overall good description of phonon spectra of delta Pu. Deviations along the (111) direction. Many possibilities, fruitful area of research. Interplay of theory and experiment. DMFT can enhance joint theoretical- experimental advances in the field of correlated electron materials.

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)

Resistivity of Am under pressure. J. C. Griveau Rebizant Lander and Kotliar PRL 94, (2005).

Photoemission spectra using Hubbard I solver [Lichtenstein and Katsnelson, PRB 57, 6884,(1998 ), Svane cond-mat ] and Sunca. [Savrasov Haule and Kotliar cond-mat ] Hubbard bands width is determined by multiplet splittings.

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

Conclusion Am Americium undergoes Mott transition under pressure. [AmIII-AmIV] boundary. Unusual superconductivity and resistivities. Theoretical clue mixed valent due to admixture of (5f)7. Unlike Sm…..

The schematic phase diagram, the Mott (Johansen ) and the Kondo collapse (Allen-Martin) two scenarios: how to tell between the two ? J.W. Allen and L.Z. Liu, Phys. Rev. B 46, 5047 (1992). Kondo impurity model + elastic terms. DMFT phase diagram of a Hubbard model at integer filling, has a region between Uc1(T) and Uc2(T) where two solutions coexist. A. Georges G. Kotliar W. Krauth and M Rozenberg RMP 68,13,(1996). Coupling the two solutions to the lattice gives a phase diagram akin to alpha gamma cerium. Majumdar and Krishnamurthy PRL 73 (1994).

How good is the local approximation ??? Exact in infinite dimensions, very good also in one dimension! Cellular DMFT [Kotliar et. al. PRL (2001) ] Test in 1d Hubbard model Capone Civelli Sarma Castellani and Kotliar PRB 69, (2004) ]

Pu HAS SEVEN STABLE PHASES AT LOW PRESSURE  monoclinic  distorted diamond  fcc  ’ distorted bcc  bcc L liquid  monoclinic T(K) Non magnetic LDA-GGA understimates the volume of  Pu by about 25 %.DFT-GGA succesfully predicts the volume of all phases of plutonium. However it also predicts that all phases of plutonium are magnetic in dsiagreement with experiment.

. LDA+DMFT, Hubbard U matrix and Double Counting Correction Matrix Edc. Functionals of Wloc and Gloc, simultaneous determination of U and Edc. [Chitra and Kotliar PRB 2001] Recent test on semiconductors, agree well with experiments, with clusters as small as 3 coordination spheres. [Zein Savrasov Kotliar cond-mat 2005] Approximate Impurity solvers for : Hubbard I, PT in hybridization, SUNCA, rational interpolative solvers, QMC. Compromise between speed and accuracy.

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

Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = E i - E f Q = k i - k f

 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).

Conclusion Pu Realistic DMFT calculations provide an overall good description of phonon spectra of delta Pu. Deviations along the (111) direction. Many possibilities, fruitful area of research. Interplay of theory and experiment. DMFT can enhance joint theoretical- experimental advances in the field of correlated electron materials.

Contrasting View. Anisimov. Havela. PlutoniumAm Mixtures.

More recent work. Line of research. Connection with Van Der Laand.

How to define the valence of Plutonium Correctly. New Tools.

High energy spectroscopies, experiment and theory.