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Understanding Heavy Fermion Systems: a DMFT perspective

Towards and Understanding of Elemental Late Actinides: a DMFT perspective. Understanding Strongly Correlated Electron Systems: a DMFT perspective. Understanding Heavy Fermion Systems: a DMFT perspective. Gabriel Kotliar and Center for Materials Theory. Colloquium : MPI Dresden (2007).

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Understanding Heavy Fermion Systems: a DMFT perspective

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  1. Towards and Understanding of Elemental Late Actinides: a DMFT perspective Understanding Strongly Correlated Electron Systems: a DMFT perspective Understanding Heavy Fermion Systems: a DMFT perspective Gabriel Kotliar and Center for Materials Theory Colloquium : MPI Dresden (2007) $upport : NSF -DMR DOE-Basic Energy Sciences Collaborators: K. Haule and J. Shim Ref: Nature 446, 513, (2007) 1

  2. Electrons in a Solid:the Standard Model Band Theory: electrons as waves. Landau Fermi Liquid Theory. n band index, e.g. s, p, d,,f Rigid bands , optical transitions , thermodynamics, transport……… • Quantitative Tools. Density Functional Theory • Kohn Sham (1964) Static Mean Field Theory. Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) 2

  3. Strong Correlation Problem:where the standard model fails • Fermi Liquid Theory works but parameters can’t be computed in perturbation theory. • Fermi Liquid Theory does NOT work . Need new concepts to replace of rigid bands ! • Partially filled d and f shells. Competition between kinetic and Coulomb interactions. • Breakdown of the wave picture. Need to incorporate a real space perspective (Mott). • Non perturbative problem. 4

  4. Heavy-fermion system G. R. Stewart RMP 56 (1984)Since the discovery by Steglich et al. (1979) of superconductivity in the high-effective-mass (-200melectrons) in CeCu2Si2, the search for and characterization of such "heavy-fermion" systems has been a rapidly growing field of study. They include superconductors(CeCu2Si2, UBe13, UPt3), magnets (NpBe13, U,Zn17,, UCdll), and materials in which no ordering is observed (CeA13, CeCu6). These f-electron materials have, in comparison to normal metals, enormous specifIC heat g values (450-1600 mJ / m o l K2 large values of the low-temperature magnetic susceptibility c 8-50 10-3 emu/molG), with large room temperature, values of the resistivity (100-200mOhmcm), Ingredients: Band spd + local f electrons

  5. 5f elements: actinide series Localisation Delocalization 1.4K 0.4K 0.9K 0.8K 52K 25K 52K s/c AF FM

  6. Localization Delocalization in Actinides Mott Transition d Pu a a d after G. Lander, Science (2003).

  7. Basic Questions • How does the electron go from being localized to itinerant. • How do the physical properties evolve. • How to bridge between the microscopic information (atomic positions) and experimental measurements. • New concepts, new techniques….. DMFT simplest approach to meet this challenge

  8. Phases of Pu (A. Lawson LANL)

  9. Small amounts of Ga stabilize the d phase (A. Lawson LANL)

  10. Anomalous Resistivity Maximum metallic resistivity

  11. Specific heat and susceptibility. Pu is non magnetic

  12. Standard model FAILS in the late actinides • Predicts Pu and Am to be magnetic, with a large moment. (about 5 mB) • Paramagnetic DFT understimates volume of delta Pu by 25 % • Many modfications have been attempted, to explain why Pu is non magnetic. Mixed level model Zwicknagl and Fulde (Erickson Balatzki and Wills et. al. ) (5f)4 conf. LDA+U (Shick, Anisimov) (5f)6 conf • Cannot account for anomalou transport and thermodynamics

  13. DMFT Spectral Function Photoemission and correlations • Probability of removing an electron and transfering energy w=Ei-Ef, and momentum k f(w) A(w, K) M2 e • Weak Correlation • Strong Correlation n n Angle integrated spectral function 8

  14. DMFT cavity construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). Happy marriage of atomic and band physics. Extremize functional of A(w) Extremize a functional of the local spectra. Local self energy. 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 (RMP 2006).

  15. Dynamical Mean Field Theory • Weiss field is a function. Multiple scales in strongly correlated materials. • Exact large coordination (Metzner and Vollhardt 89) . • Not restricted to single site-CDMFT. • Immediate extension to real materials DFT+DMFT Functionals of density and spectra. Review Kotliar et. al. RMP (2006) 12

  16. Total Energy as a function of volume for Pu W(ev) vs (a.u. 27.2 ev) Moment is first reduced by orbital spin moment compensation. The remaining moment is screened by the spd and f electrons (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Nick Zein Following Aryasetiwan et. al. PRB 70 195104. (2004)

  17. 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.

  18. C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 DMFT Phonons in fcc d-Pu ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) 21 (experiments from Wong et.al, Science, 22 August 2003)

  19. The “DMFT-valence” in the late actinides. 22 Time scale of the fluctuations. Ef*

  20. Photoemission Gouder , Havela PRB 2002, 2003

  21. Photoemission Spectra[ Shim. Haule,GK Nature (2007)] alpa->delta volume collapse transition F0=4.5,F2=7.15 20 F0=4,F2=6.1

  22. Photoemission and Mixed valence in Pu

  23. Approach the Mott point from the right Am under pressure Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition? “Soft” “Hard” 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~6mB (Soderlind et.al., PRB 2000). • Experimentally, Am hasnon magnetic f6ground state with J=0(7F0)

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

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

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

  27. Photoemission spectra using Hubbard I solver and Sunca . [Savrasov Haule and Kotliar cond-mat 0507552 PRL (2006)] Hubbard bands width is determined by multiplet splittings.

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

  29. Conclusions • Unique properties of Pu and Am under pressure result from a proximity of a localization delocalization transition. Rare form of mixed valence. • DMFT provides a good start. Qualitative insights, some quantitative predictions into delta Pu. Other Pu phases. • Meaningful interplay of theory and experiment. Key in condensed matter physics.

  30. Conclusions • Pu and Am are unique strongly correlated elements. It is one among many strongly correlated electron system, materials for which neither the standard model of solids, works well. • They require, new concepts, new computational methods, new algorithms, DMFT provides all of the above, and is being used in many other problems. • Many applications to othe problems exist, others are in progress, research opportunity in correlated materials.

  31. Prospects for Extensions and Applications to More Complex Heavy Fermion Systems • More complicated crystal structures, more atoms per unit cell. 115’s , alpha Pu…… • Non local physics. Heavy fermion quantum criticality. a) Local Quantum Criticality scenario of Q. Si and collaborators. Nature 413 (2001) 804. Single site EDMFT b) Cluster Quantum Multicriticality. L. DeLeo and GK. Requires 2 impurity Kondo model for its description.

  32. Conclusion Am • Americium undergoes Mott transition under pressure. [AmIII-AmIV] boundary. • Unusual superconductivity and resistivities. • Theoretical clue mixed valent due to admixture of (5f) upon application of pressure. • Realizes Mott transition from the insulating side, towards a close shell configuration..

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

  34. Curium is magnetic Hurray et.al. Physica. B (1980) 217 m=2S+L LS coupling L=0 S=7 m=7 jj coupling J=7/2 m=3+1=4 Expt monent . is closer to L S coupling

  35. K.Haule and J. Shim Trends in Actinides alpa->delta volume collapse transition F0=4,F2=6.1 F0=4.5,F2=7.15 F0=4.5,F2=8.11 Curium has large magnetic moment and orders antif Pu does is non magnetic.

  36. Conclusion • A Few References …… • A.Georges, G. K., W. Krauth and M. J. Rozenberg, Reviews of . Modern Physics 68, 13 (1996). • G. K, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, C.A. Marianetti, RMP 78, 865-951, (2006). • G. K and D. Vollhardt Physics Today, Vol 57, 53 (2004). 29

  37. “Invar model “ for Pu-Ga. Lawson et. al.Phil. Mag. (2006) Data fits only if the excited state has zero stiffness.

  38. 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, ……..

  39. 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, ….

  40. 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.

  41. 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.

  42. 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 ]

  43. Collaborators, Acknowledgements References 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. 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). X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams  Science,  Vol300, 954 (2003).

  44. Cluster DMFT: removes limitations of single site DMFT • No k dependence of the self energy. • No d-wave superconductivity. • No Peierls dimerization. • No (R)valence bonds. Reviews: Georges et.al. RMP(1996). Th. Maier et. al. RMP (2005); Kotliar et. .al. RMP (2006). 23

  45. Two Site Cellular DMFT (G.. Kotliar et.al. PRL (2001))in the 1D Hubbard modelM.Capone M.Civelli V. Kancharla C.Castellani and GK PRB 69,195105 (2004)T. D Stanescu and GK PRB (2006) U/t=4. 24

  46. Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) Self Energy VanShilfgaarde (2005) 3

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