Elemental Plutonium Electrons at the Edge The Mott

Elemental Plutonium: Electrons at the Edge The Mott transition across the actinide series. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Santa Fe November 2003

Outline , Collaborators, References Physical properties of plutonium. n. Dynamical Mean Field Theory (DMFT) n. DMFT study of elemental plutonium. n. Conclusions n Los Alamos Science, 26, (2000). S. Savrasov and G. Kotliar PRL 84 3670 (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, Vol 300, 954 (2003).

Pu in the periodic table actinides THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Pu is famous because of its nucleus. Fission: Pu 239 absorbs a neutron and breaks apart into pieces releasing energy and more neutrons. Pu 239 is an alpha emitter, making it into a most toxic substance. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition in the actinide series (Smith-Kmetko phase diagram) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phases of Pu (A. Lawson LANL) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Small amounts of Ga stabilize the d phase (A. Lawson LANL) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Elastic Deformations Uniform compression: Dp=-B Volume conserving DV/Vdeformations: F/A=c 44 Dx/L F/A=c’ Dx/L In most cubic materials the shear does not depend strongly on crystal orientation, fcc Al, c 44/c’=1. 2, in Pu C 44/C’ ~ 7 largest shear anisotropy of any element. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

The electron in a solid: wave picture Sommerfeld Bloch, Landau: Periodic potential, waves form bands , k in Brillouin zone. [Density functional theory ] Landau: Interactions renormalize parameters. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Anomalous Resistivity Maximum metallic resistivity THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Pu Specific Heat THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Electronic specific heat(J Lashley et. al. LANL) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Localized model of electron in solids. (Peierls Mott)particle picture. Solid=Collection of atoms L, S, J • Think in real space , solid collection of atoms • High T : local moments, Low T spin-orbital order THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Specific heat and susceptibility. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Density Functional Theory and K n n n Total energy is minimizes a functional of the density (spin density). Exact form of the functional is unknown but good approximations exist. (LDA, GGA) In practice, one solves a one particle shrodinger equation in a potential that depends on the density. A band structure is generated (Kohn Sham system). and in many systems this is a good starting point for perturbative computations of the spectra (GW). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Delta phase of Plutonium: Problems with LDA o n n Many studies and implementations. (Freeman, Koelling 1972)APW methods, ASA and FP-LMTO Soderlind et. Al 1990, Kollar et. al 1997, Boettger et. al 1998, Wills et. al. 1999). all give an equilibrium volume of the d phase Is 35% lower than experiment this is the largest discrepancy ever known in DFT based calculations. LSDA predicts magnetic long range (Solovyev et. al. ) Experimentally d Pu is not magnetic. If one treats the f electrons as part of the core LDA overestimates the volume by 30% THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DFT Studies of Pu n n DFT in GGA predicts correctly the volume of the a phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that a Pu is a weakly correlated system Alternative models: 1) For the delta phase a model with 4 5 f electrons localized and 1 electron as itinerant was proposed by Wills et. al, in the spirit of SIC corrected LDA. This model produces correct volume of delta. 2) Strong random potential. (B. Cooper). . THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Dynamical Mean Field Theory n n Basic idea: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition. [A. Georges and GK 1992] Basic idea: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission] [GK R. Chitra and S. Savrasov 2000, 2002] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phys. Rev. B 45, 6497 DMFT Referen ce System THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A. Georges, G. Kotliar (1992)

One Particle Local Spectral Fun e Probability of removing an electron and transfering energy w=Ei-Ef, f(w) A(w) M 2 n Probability of absorbing an electron and transfering energy w=Ei-Ef, (1 -f(w)) A(w) M 2 n Theory. Compute one particle greens function and use spectral function. n n n e THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

n n Simple interface with electronic structure. Treat the spd electrons within LDA (static self energy approximated by xc potential). Treat the f electrons with DMFT. LDA+DMFT. Extensions. Treat the electric field and the electronic fields using DMFT. [E-DMFT] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT functional formulation. n n n Focus on the local spectral function A(w) of the solid. Write a functional of the local spectral function such that its stationary point, give the energy of the solid. No explicit expression for the exact functional exists, but good approximations are available, by making systematic truncations in the range of the BK functional. The spectral function is computed by solving a local impurity model. Which is a new reference system to think about correlated electrons. Ref: A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68, 1 (1996). Generalizations to realistic electronic structure. (G. Kotliar and S. Savrasov 2001 -2002 ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Canonical Phase Diagram of the Localization Delocalization Transition. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Pressure Driven Mott transition THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

More recent work, organics, Limelette et. al. PRL 91, 061401 (2003) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

DMFT has bridged the gap be n Delocalized picture, it should resemble the density of states, (perhaps with some additional shifts and satellites). Localized picture. Two peaks at the ionization and affinity energy of the atom. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

One electron spectra near the Mott transition. Transfer of Spectral Weight. [Zhang Rozenberg and Kotliar 93] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

n DMFT studies of elemental Plutonium THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

What is the dominant atomic configuration? Local moment? Snapshots of the f electron n Dominant configuration: (5 f)5 n Naïve view Lz=-3, -2, -1, 0, 1 n ML=-5 m. B S=5/2 Ms=5 m. B Mtot=0 L=5, S=5/2, J=5/2, n Mtot=Ms=m. B g. J =. 7 m. B n Crystal fields G 7 +G 8 n n GGA+U estimate ML=-3. 9 Mtot=1. 1 (Savrasov GK 2000) This bit is quenches by the f and spd electrons Neutron Scattering in a field (Lander) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Pu: DMFT total energy vs Volume (Savrasov Kotliar and Abrahams 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Double well structure and d Pu Qualitative explanation of negative thermal expansion Sensitivity to impurities which easily raise the energy of the a -like minimum. RUTGERS THE STATE UNIVERSITY OF NEW JERSEY

Generalized phase diagram T Structure, bands, orbital s U/W THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Minimum in melting curve and divergence of the compressibility at the Mott endpoint THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Cerium THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Photoemission Technique Density of states for removing (adding ) a particle to the sample. n Delocalized picture, it should resemble the density of states, (perhaps with some satellites). n Localized picture. Two peaks at the ionization and affinity energy of the atom. n THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Lda vs Exp Spectra THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Pu Spectra DMFT(Savrasov) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Alpha and delta Pu THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

n n n Alpha phase is also a correlated metal. It differs from delta in the relative weight of the resonance and the Hubbard band. Consistent with resistivity and specific heat measurements. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phonon Spectra n n Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure. Phonon spectra reveals instablities, via soft modes. Phonon spectrum of Pu had not been measured. Short distance behavior of the elastic moduli. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = Ei - Ef Q =ki - kf THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Expt. Wong et. al. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Expts’ Wong et. al. Science 301. 1078 (2003) Theory Dai et. al. Science 300, 953, (2003) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Shear anisotropy. Expt. vs Theory n C’=(C 11 -C 12)/2 = 4. 78 GPa n C 44= 33. 59 GPa n n C’=3. 9 GPa C 44=33. 0 GPa C 44/C’ ~ 7 Largest shear anisotropy in any element! C 44/C’ ~ 8. 4 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

The delta –epsilon transition n The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase. What drives this phase transition? Having a functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Epsilon Plutonium. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phonon frequency (Thz ) vs q in epsilon Pu. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phonon entropy drives the epsilon delta phase transition n 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 Ackland et. al. PRB. 65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Phonons epsilon THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

n Approaching the Mott transition from the “localized side”. Americium under pressure. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Superconductivity among 5 f elements Localisatio n 1. 4 K 0. 4 K s/c 0. 9 K 0. 8 K 52 K 25 K AF 52 K FM

Phase diagram (Lindbaum et. al. PRB 2003) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Interesting fundamental questions. n n n Closed shell system. Mott transition? Where does it occur? Interplay of spin orbit coupling and Coulomb interactions. Superconductivity (how does it depend on pressure ? Is it in the f or the spd system ? Does it correlated with the Mott transition ? ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Insights into the anomalous properties of Pu n 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. The Mott transition across the actinide series [ B. Johansson Phil Mag. 30, 469 (1974)] concept has finally been worked out!. We learned how to think about this unusual situation using spectral functions. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Conclusions n n n DMFT produces non magnetic state, around a fluctuating (5 f)^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, …. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Quantitative calculations n n n 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? THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Conclusions n n Pu is a unique ELEMENT, but by no means unique material. It is one among many strongly correlated electron system, materials for which neither the standard model of solids, either for itinerant or localized electrons works well. They require, new concepts, new computational methods, new algorithms. System specific methods, DMFT and is being used in many other problems. International multidisciplinary effort [ Dresden, Trieste, Leiden, Trieste, Santa Barbara, Trieste ……. . ] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Conclusions n n Methodology applicable to a large number of other problems, involving correlated electrons, thermoelectrics, batteries, optical devices, high temperature dilute magnetic semiconductors…………. Calculations can be refined in many ways, electronic structure calculations for correlated electrons research program, MINDLAB. Bring the method to the point, that we can start focusing in deviations from DMFT, isolate short and long wavelength physics. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Acknowledgements: Development of DMFT Collaborators: E. Abrahams, V. Anisimov, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, K. Haule H. Kajueter, W. Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, A. Poteryaev, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X. Y. Zhang Support: NSF DMR 0096462 Support: Instrumentation. NSF DMR-0116068 Work on Fe and Ni: ONR 4 -2650 Work on Pu: DOE DE-FG 02 -99 ER 45761 and LANL subcontract No. 03737 -001 -02 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Acknowledgements: Development of DMFT Collaborators: E. Abrahams, V. Anisimov, G. Biroli, C Bolech, M. Capone, R. Chitra, M. Civelli, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, K. Haule, V Kancharla, H. Kajueter, W. Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, O. Parcollet, A. Poteryaev, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X. Y. Zhang THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

LDA+DMFT functional F Sum of local 2 PI graphs with local U matrix and local G THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Summary Spectra Method LDA+U DMFT E vs V

More recent work, organics, Limelette et. al. PRL 91, 061401 (2003) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Ising critical endpoint! In V 2 O 3 P. Limelette et. al. Science Vol 302, 89 (2003). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Mott transition in layered organic conductors al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS S Lefebvre et

Magnetic moment n n n L=5, S=5/2, J=5/2, Mtot=Ms=m. B g. J =. 7 m. B Crystal fields G 7 +G 8 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). THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

What do we want from materials theory? New concepts , qualitative ideas n Understanding, explanation of existent experiments, and predictions of new ones. n Quantitative capabilities with predictive power. n Notoriously difficult to achieve in strongly correlated materials. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

Canonical Phase Diagram of the Localization Delocalization Transition. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS