Localization Delocalization Phenomena across the Mott transition cracking

Localization Delocalization Phenomena across the Mott transition: cracking open the f shell 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 Theory : S. Savrasov (NJIT-- UCDavis ) K. Haule (Jozef Stefan Institute-Rutgers) Xi Dai (Rutgers-Institute of Theoretical Physics Beigng) Experimental Motivation: Am J. C Griveaux Rebizant F Wastin. G Lander Pu: A. Lawson A. Migliori Miniworkshop on Actinides Theory and Experiments Karlsruhe. December 3 -6 (2006).

Outline • Some introductory comments about the Mott transition Actinides and Dynamical Mean Field Theory (DMFT). “Theoretical Spectroscopy”. • The Mott transition across the actinide series, Plutonium and Americium. • DMFT results for Pu. DMFT results for Am under pressure. • Why we need experimental probes of unoccupied states. • Conclusions: advantage of combining theoretical and experimental spectroscopies.

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.

DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). First happy marriage of atomic and band physics. Extremize a functional of the local spectra or the local self energy. Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP 68 , 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).

Mott transition in single site DMFT. Georges Kotliar Krauth and Rozenberg RMP (1996)) 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). .

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). Similar in spirit LDA ++ Lichtentsein and Katsnelson. PRB 57, 6884 (1998). Derive complex Hamiltonians solve them using DMFT. § LDA+DMFT photoemission Allows the computation of realistic photoemission spectra optics etc……

§ Functional formulation (Chitra and Kotliar 2000) Phys. Rev. B 62, 12715 (2000). Self consistent determination of electronic structure. Full implementation S. Savrasov G. Kotliar (2001 -2005). Phys. Rev. B 69, 245101 (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. • linear response phonon spectra [ Savrasov and Kotliar Phys. Rev. Lett. 90, 056401 (2003). ]. Speedup 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. cond-mat. 0507552 (2005).

• 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.

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 1 d Hubbard model Capone Civelli Sarma Castellani and Kotliar PRB 69, 195105 (2004) ]

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

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

Pu phases: A. Lawson Los Alamos Science 26, (2000) o. GGA LSDA predicts d Pu to be magnetic with a large moment ( ~5 Bohr). Experimentally Pu is not magnetic. [Lashley et. al. cond-matt 0410634]

Total Energy as a function of volume for Pu W (ev) vs (a. u. 27. 2 ev) (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Zein Savrasov and Kotliar (2005) Following Aryasetiwan et. al. PRB 70 195104. (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 = Ei - Ef Q =ki - kf

DMFT Phonons in fcc d-Pu C 11 (GPa) C 44 (GPa) C 12 (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)

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, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

Double well structure and d 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)] F(T, V)=Fphonons +Finvar 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.

“Invar model “ for Pu-Ga. Lawson et. al. (2005) 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, 155109 , 2003

K. Haule , Pu- photoemission with DMFT using vertex corrected NCA.

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

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

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

Photoemission spectra using Hubbard I solver and Sunca [Savrasov Haule and Kotliar cond-mat 0507552] Hubbard bands width is determined by multiplet splittings.

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

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

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

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.

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 Uc 1(T) and Uc 2(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 0511085

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. vander. Eb PRL 886, 3407 (2001)

Temperature dependence of the optical conductivity.

Origin of the features.

Conclusions 4 f 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=298 K, P=0. 7 GPa) (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 s exp. 28Å3 34. 4Å3 -phase LDA 24. 7Å3 LDA+U 35. 2Å3 (localized): High T phase ØCurie-Weiss law (localized magnetic moment), ØLarge lattice constant ØTk around 60 -80 K -phase (delocalized: Kondophysics): Low T phase ØLoss of Magnetism (Fermi liquid Pauli susceptibility) completely screened magnetic moment ØSm Øaller lattice constant ØTk around 1000 -2000 K

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

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

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

Double well structure and d Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J. Low Temp. Physvol. 126, 1009 27. (2002)]See also A. Lawson et. al. Phil. Mag. B 82, 1837 ]

A. C. Lawson et. al. LA UR 046008 F(T, V)=Fphonons+Finvar

Invar model A. C. Lawson et. al. LA UR 04 -6008

Prediction of the Invar Model

DMFT and the Invar Model

G. Kotliar and S. Savrasov Strongly Correlated Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic 259 -301. conmat/0208241 S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004) • Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). Semiconductors: Zein Savrasov and Kotliar (2005).

G. Kotliar and S. Savrasov Strongly Correlated Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic 259 -301. conmat/0208241 S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004) • Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). Semiconductors: Zein Savrasov and Kotliar (2005).