1. Elemental Plutonium: Electrons at the Edge Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Colloquium UT July 2003

2. Outline, Collaborators, References Los Alamos Science,26, (2000) S. Savrasov and G. Kotliar Phys. Rev. Lett. 84, 3670-3673, (2000). S.Savrasov G. Kotliar and E. Abrahams, Nature 410, 793 (2001).Phys. Rev. Lett. 84, 3670-3673, (2000)Nature 410, 793 (2001). X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science, Vol300, 954 (2003). Plutonium Puzzles Solid State Theory, Old and New (DMFT) Results 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 Electronic Physics of Pu

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’ ~ 6 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. 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

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Localized model of electron in solids. (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 (Spin) Density Functional Theory. Focus on the density (spin density ) of the solid. Total energy is obtained by minimizing 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). Works exceedingly well for many systems. W. Kohn, Nobel Prize in Chemistry on October 13, 1998 for its development of the density-functional theory

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Kohn Sham system

17. 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%

18. 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.

19. 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

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Dynamical Mean Field Theory 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. 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 in )

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mean-Field : Classical vs Quantum Classical case Quantum case Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)

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

23. 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.

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS One electron spectra near the Mott transition.

25. 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 More refined estimates ML=-3.9 Mtot=1.1 This bit is quenches by the f and spd electrons

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

27. 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.

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

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

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cerium

31. 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.

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

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

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

35. 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 until recently.

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

37. 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

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

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Expts’ Wong et. al.

40. 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. The Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)] concept has finally been worked out! They require, new concepts, new computational methods, new algorithms, DMFT provides all of the above, and is being used in many other problems.

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusions Constant interplay between theory and experiment has lead to new advances. General anomalies of correlated electrons and anomalous system specific studies, need for a flexible approach. (DMFT). New understanding of Pu. Methodology applicable to a large number of other problems, involving correlated electrions, thermoelectrics, batteries, optical devices, memories, high temperature superconductors, ……..

42. 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, electronic structure calculations for correlated electrons research program, MINDLAB, ….

43. 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.

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Some new insights into the funny 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. We learned how to think about this unusual situation using spectral functions. Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha). Negative thermal expansion, multitude of phases.

45. 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? What else, do electrons at the edge of a localization localization do ? [ See epsilon Pu spectra ]

46. Collaborators, Acknowledgements References Los Alamos Science,26, (2000) S. Savrasov and G. Kotliar Phys. Rev. Lett. 84, 3670-3673, (2000). S.Savrasov G. Kotliar and E. Abrahams, Nature 410, 793 (2001).Phys. Rev. Lett. 84, 3670-3673, (2000)Nature 410, 793 (2001). X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science, Vol300, 954 (2003). Collaborators: S. Savrasov ( Rutgers-NJIT) X. Dai ( Rutgers), E. Abrahams (Rutgers), A. Migliori (LANL),H Ledbeter(LANL). Acknowledgements: G Lander (ITU) J Thompson(LANL) Funding: NSF, DOE, LANL.

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Acknowledgements: Development of DMFT Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, H. Kajueter, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, 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

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

51. 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)

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

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

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonons epsilon

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon entropy drives the epsilon delta phase transition Epsilon is slightly more metallic than delta, 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 neglecting the Electronic entropy: TC 600 K.

56. Results for NiO: Phonons Solid circles – theory, open circles – exp. ( Roy et.al, 1976 ) DMFT Savrasov and GK PRL 2003

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two models of a solid. Itinerant and localized. Mott transition between the two. Spectral function differentiates between the two phases. Insert the phase diagram that I like.

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

59. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The electron in a solid: particle picture. NiO, MnO, …Array of atoms is insulating if a>>a B. Mott: correlations localize the electron e_ e_ e_ e_ Think in real space, solid collection of atoms High T : local moments, Low T spin-orbital order Superexchange

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

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS For future reference.

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Shear anisotropy. C’=(C11-C12)/2 4.78 C44= 33.59 19.70 C44/C’ ~ 6 Largest shear anisotropy in any element!

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Electronic specific heat

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT BOX

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity Maximum metallic resistivity 200 mohm cm

67. 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).