1. 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

2. Outline, Collaborators, References 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, Vol300, 954 (2003). Physical properties of plutonium. Dynamical Mean Field Theory (DMFT) DMFT study of elemental plutonium. Conclusions

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

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu is famous because of its nucleus. Fission: Pu239 absorbs a neutron and breaks apart into pieces releasing energy and more neutrons. Pu239 is an alpha emitter, making it into a most toxic substance.

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

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

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

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Elastic Deformations In most cubic materials the shear does not depend strongly on crystal orientation,fcc Al, c 44 /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=c 44  x/L F/A=c’  x/L

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

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

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu Specific Heat

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

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

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

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Density Functional Theory and Kohn Sham Reference System. 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).

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Delta phase of Plutonium: Problems with LDA an equilibrium volume of the  phase  Is 35% lower than experiment o 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  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  Pu is not magnetic. If one treats the f electrons as part of the core LDA overestimates the volume by 30%

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT Studies of Pu DFT in GGA predicts correctly the volume of the  phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that  Pu is a weakly correlated system Alternative models:1) For the delta phase a model with 4 5f 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)..

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory 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]

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

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

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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]

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT functional formulation. Focus on the local spectral function A(  ) 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 )

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

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pressure Driven Mott transition

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

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT has bridged the gap between band theory and atomic physics. 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.

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

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

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 L=5, S=5/2, J=5/2, Mtot=Ms=  B gJ =.7  B Crystal fields     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)

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

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

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Generalized phase diagram T U/W Structure, bands, orbitals

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

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cerium

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

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lda vs Exp Spectra

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

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha and delta Pu

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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.

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon Spectra 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.

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

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev. E = E i - E f Q = k i - k f

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

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

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Shear anisotropy. Expt. vs Theory C’=(C11-C12)/2 = 4.78 GPa C’=3.9 GPa C44= 33.59 GPa C44=33.0 GPa C44/C’ ~ 7 Largest shear anisotropy in any element! C44/C’ ~ 8.4

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The delta –epsilon transition 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)

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Epsilon Plutonium.

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

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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 Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonons epsilon

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

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Superconductivity among 5f elements s/cAFFM Localisatio n 1.4K 0.4 K 0.9K0.8K52K25K52K

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

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Interesting fundamental questions. 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 ?)

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Insights into the anomalous properties 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. 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.

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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. Interplay of correlations and electron phonon interactions (delta-epsilon). Calculations can be refined in many ways, ….

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Quantitative 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?

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions 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 ……..]

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions 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.

60. 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: ONR4-2650 Work on Pu: DOE DE-FG02-99ER45761 and LANL subcontract No. 03737-001-02

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

62. 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

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

64. Summary LDA LDA+U DMFT Spectra Method E vs V

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

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

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

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS 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).

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS What do we want from 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.

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