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Understanding Heavy Fermion Systems: a DMFT perspective

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**1. ** Understanding Heavy Fermion Systems: a DMFT perspective Gabriel Kotliar
and Center for Materials Theory

**2. **Electrons in a Solid:the Standard Model
Framework is reiigorous and predictive.
BANDS AT THE FERMI LEVEL--? Metal
No bands at the fermi level insulator.
Wave metals are NOT very resistive.
Thermodynamics, cv chi hall coefficient etc.. As free electrons.
KEY
Framework is reiigorous and predictive.
BANDS AT THE FERMI LEVEL--? Metal
No bands at the fermi level insulator.
Wave metals are NOT very resistive.
Thermodynamics, cv chi hall coefficient etc.. As free electrons.
KEY

**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. **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),

**6. **Localization Delocalization in Actinides

**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) The main characters alpha, ground state monoclinic. Delta fcc negative thermal expansion. Epsilon bcc smaller than delta. Contracts in the liquid.
Gigantic volume changes. < Volume collapse, bigger than cerium>The main characters alpha, ground state monoclinic. Delta fcc negative thermal expansion. Epsilon bcc smaller than delta. Contracts in the liquid.
Gigantic volume changes. < Volume collapse, bigger than cerium>

**9. **Small amounts of Ga stabilize the d phase (A. Lawson LANL) Extraordinary sensitivity to impurities. Focus on delta mostly. Important for applications, ductile, stable. Any element (except Np stabilizes it) bigger or smaller.Extraordinary sensitivity to impurities. Focus on delta mostly. Important for applications, ductile, stable. Any element (except Np stabilizes it) bigger or smaller.

**10. **Anomalous Resistivity Apply the standard model to Pu, runs into problems. Certainly for delta.Apply the standard model to Pu, runs into problems. Certainly for delta.

**11. **Specific heat and susceptibility. Pu is non magnetic Localized picture, starting point for f electrons in UBe 13. Heavy fermion behavior. Combining the two models.
High temperature localized. Low temperature band like. Difficult crossover. Pu is heavy but NOT magnetic. Discuss susceptiblity puzzle.
Polarized neutron scattering. No magnetic scattering. Localized picture, starting point for f electrons in UBe 13. Heavy fermion behavior. Combining the two models.
High temperature localized. Low temperature band like. Difficult crossover. Pu is heavy but NOT magnetic. Discuss susceptiblity puzzle.
Polarized neutron scattering. No magnetic scattering.

**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

**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)

**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)

**16. **Bistability of a material near the Mott transition. Model realization of the Johanssen ideas.
Central for understanding the physics of Pu.. New paradigm for thinking, about materials.Bistability of a material near the Mott transition. Model realization of the Johanssen ideas.
Central for understanding the physics of Pu.. New paradigm for thinking, about materials.

**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)]

**18. **Notice the agreement. Usefulness of theory. Notice the discreapancy. Scientfici opportunity. Notice the agreement. Usefulness of theory. Notice the discreapancy. Scientfici opportunity.

**19. **The “DMFT-valence” in the late actinides.

**21. **Photoemission Spectra[ Shim. Haule,GK Nature (2007)]

**22. **Photoemission and Mixed valence in Pu

**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 ?

**28. **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.

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

**30. **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.

**31. **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.

**32. **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.

**33. **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..

**38. **K.Haule and J. Shim Trends in Actinides

**39. **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).

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

**42. **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, ……..

**43. **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, ….

**44. **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.

**45. **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.

**46. **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 ]

**47. ** 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.

**48. **Cluster DMFT: removes limitations of single site DMFT

**49. **Two Site Cellular DMFT (G.. Kotliar et.al. PRL (2001)) in the 1D Hubbard model M.Capone M.Civelli V. Kancharla C.Castellani and GK PRB 69,195105 (2004)T. D Stanescu and GK PRB (2006) Edit. LISA.Edit. LISA.

**50. **
Framework is reiigorous and predictive.
BANDS AT THE FERMI LEVEL--? Metal
No bands at the fermi level insulator.
Wave metals are NOT very resistive.
Thermodynamics, cv chi hall coefficient etc.. As free electrons.
KEY
Framework is reiigorous and predictive.
BANDS AT THE FERMI LEVEL--? Metal
No bands at the fermi level insulator.
Wave metals are NOT very resistive.
Thermodynamics, cv chi hall coefficient etc.. As free electrons.
KEY

**54. **Smith Kmeko Phase diagram. Minimum in melting curve and divergence of the compressibility at the Mott endpoint

**55. **The enhancement of the specific heat, further evidence for an open shell configuration, presence of electronic entropy.

**56. **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)]

**57. **“Invar model “ for Pu-Ga. (Data fits if the excited state has zero stiffness.

**58. **Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992).

**59. **Animate, and expand.Animate, and expand.

**60. **Expt. Wong et. al.

**61. **Elastic Deformations 1 Pa= 10 dyn/cm^2 , others gamma Ce c44/c’=2.8, La = 4.1, Th 3.6 find out about alpha cerium.
From Kittel U=1/2 C11 ( exx^2 + eyy^2 +ezz^2)+ ½ c44 (eyz^2 +ezx^2+exy^2)+
C12(eyy ezz+ ezz exx +exx eyy).
And exx=dux/dx and exy=duy/dx+dux/dy B= 1/3(c11+2 c12)
C’= c11-c121 Pa= 10 dyn/cm^2 , others gamma Ce c44/c’=2.8, La = 4.1, Th 3.6 find out about alpha cerium.
From Kittel U=1/2 C11 ( exx^2 + eyy^2 +ezz^2)+ ½ c44 (eyz^2 +ezx^2+exy^2)+
C12(eyy ezz+ ezz exx +exx eyy).
And exx=dux/dx and exy=duy/dx+dux/dy B= 1/3(c11+2 c12)
C’= c11-c12

**63. **Localization Delocalization in Actinides

**65. **Spectral Function and Photoemission Probability of removing an electron and transfering energy w=Ei-Ef, and momentum k
f(w) A(w, K) M2

**66. **
Framework is reiigorous and predictive.
BANDS AT THE FERMI LEVEL--? Metal
No bands at the fermi level insulator.
Wave metals are NOT very resistive.
Thermodynamics, cv chi hall coefficient etc.. As free electrons.
KEY
Framework is reiigorous and predictive.
BANDS AT THE FERMI LEVEL--? Metal
No bands at the fermi level insulator.
Wave metals are NOT very resistive.
Thermodynamics, cv chi hall coefficient etc.. As free electrons.
KEY

**67. **W110 =2/3<l.s> and banching ratio

**68. **Different way of thinking was generated by the study of the Mott transition at integer filling.
Universality and system specificity. . Bridge atomic physic and band physics. Crossovers with changing degrees of freedom. Different way of thinking was generated by the study of the Mott transition at integer filling.
Universality and system specificity. . Bridge atomic physic and band physics. Crossovers with changing degrees of freedom.

**69. **OUTLINE The challenge of strongly correlated electron systems. Heavy Fermions and Late actinides: experimental overview
Introduction to Dynamical Mean Field Theory
(DMFT).
Theory of delta Pu
Theory of Am and Cm
Conclusions

**70. **Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.

**72. **2/3<l.s> in the late actinides [DMFT results: K. Haule and J. Shim ]

**73. **Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003