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Strongly Correlated Electron Materials : A Dynamical Mean Field Theory (DMFT) Perspective.

This presentation provides an introduction to strongly correlated materials and an overview of Dynamical Mean Field Theory (DMFT) with a case study on CeIrIn5. It discusses the challenges of describing electron behavior in solids and the applications of DMFT in understanding correlated electron systems. The Hubbard model, Mott-Hubbard physics, and the evolution of the density of states are also discussed. The presentation concludes with a summary of DMFT and its implementation in studying materials like CeIrIn5.

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Strongly Correlated Electron Materials : A Dynamical Mean Field Theory (DMFT) Perspective.

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  1. Strongly Correlated Electron Materials : A Dynamical Mean Field Theory (DMFT) Perspective. Gabriel Kotliar and Center for Materials Theory Cornell Ithaca NY November 27 2007 $upport : NSF -DMR DOE-Basic Energy Sciences 1

  2. Outline • Introduction to strongly correlated materials. • Brief overview of Dynamical Mean Field Theory. • Application to heavy fermions: a case study of CeIrIn5 [with K. Haule and J. Shim, Science Express Nov 1st (2007) ] • Conclusions – and some thought about the 5f elemental metals.

  3. Electrons in a Solid:the Standard Model .Interactions renormalize away (Landau) . Band Theory: electrons as waves Rigid bands , optical transitions , thermodynamics, transport……… • Quantitative Tools. Density Functional Theory Kohn Sham (1964) Static Mean Field Theory. 2

  4. Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) Self Energy M VanShilfgaarde et. al. PRL 96, 226402 (2006) 3

  5. Correlated Electron Systems Pose Basic Questions in CMT • FROM ATOMS TO SOLIDS • How to describe electron from localized to itinerant ? • How do the physical properties evolve ?

  6. Strong Correlation Problem:where the standard model fails • Fermi Liquid Theory works but parameters can’t be computed in perturbative 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

  7. Strongly correlated materials do “big” things • Huge volume collapses Pu ……. • Masses as large as 1000 me (heavy fermions UPt3, CeIrIn5….. • High Temperature Superconductivity. 150 K Ca2Ba2Cu3HgO8 . • Large thermoelectric response in NaxCo2O4 • Large change in resistivity. MIT in TM oxides (V2O3, VO2, LaSrMnO3……..) • ………………….. 5

  8. Hubbard model • U/t • Doping d or chemical potential • Frustration tij • T temperature 6

  9. Mott-Hubbard Physics H H H+ H H Real space picture High T : local moments Low T: spin orbital order Excitations: Excitations: adding (removing ) e, Upper Hubbard band. 7

  10. Dynamical Mean Field Theory. Cavity Construction.A. Georges and G. Kotliar PRB 45, 6479 (1992). A(w) 10

  11. A(w) A. Georges, G. Kotliar (1992) 11

  12. Dynamical Mean Field Theory • Weiss field is a function. Multiple scales in strongly correlated materials. • Exact in the limit of large coordination (Metzner and Vollhardt 89) , kinetic and interaction energy compete on equal footing. • Immediate extension to real materials DFT+DMFT 12

  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 n n Angle integrated spectral function 8

  14. Evolution of the DOS. Theory and experiments 13

  15. Summary: DMFT Self consistent Impurity problem, natural language to quantify localization/delocalization phenomena. Combines atomic physics and band theory Systematically improvable, cluster DMFT Implementation

  16.  CeRhIn5: TN=3.8 K;   450 mJ/molK2CeCoIn5: Tc=2.3 K;   1000 mJ/molK2; CeIrIn5: Tc=0.4 K;   750 mJ/molK2 Ir In Ce CeMIn5 M=Co, Ir, Rh out of plane in-plane

  17. Phase diagram of 115’s Why CeIrIn5? • Ir atom is less correlated than • Co or Rh (5d / 3d or 4d) • CeIrIn5 is more itinerant(coherent) • than Co (further away from QCP)

  18. Generalized Anderson Lattice Model • High temperature • Ce-4f local moments C+ff+ • Low temperature – • Itinerant heavy bands 6

  19. e- S. Doniach, 1978. DONIACH PHASE DIAGRAM Jk= V2/ef Kondo Exchange Kondo scale RKKY scale

  20. Angle integrated photoemission Expt Fujimori et al., PRB 73, 224517 (2006) P.R B 67, 144507 (2003). Experimental resolution ~30meV Surface sensitivity at 122 ev , theory predicts 3meV broad band Theory: LDA+DMFT, impurity solvers SUNCA and CTQMC Shim Haule and GK (2007)

  21. Buildup of coherence in single impurity case Very slow crossover! coherent spectral weight TK T T* Buildup of coherence coherence peak scattering rate Slow crossover pointed out by NPF 2004 Crossover around 50K

  22. Consistency with the phenomenological approach of NPF +C Remarkable agreement with Y. Yang & D. Pines cond-mat/0711.0789!

  23. DMFT is not a single impurity calculation high T low T Auxiliary impurity problem: temperature dependent: Weiss field High-temperature D given mostly by LDA low T: Impurity hybridization affected by the emerging coherence of the lattice (collective phenomena) DMFT SCC: Feedback effect on D makes the crossover from incoherent to coherent state very slow!

  24. Momentum resolved total spectra trA(w,k) Most of weight transferred into the UHB LDA f-bands [-0.5eV, 0.8eV] almost disappear, only In-p bands remain Very heavy qp at Ef, hard to see in total spectra Below -0.5eV: almost rigid downshift Unlike in LDA+U, no new band at -2.5eV ARPES, HE I, 15K LDA+DMFT at 10K Fujimori, PRB Short lifetime of HBs -> similar to LDA(f-core) rather than LDA or LDA+U

  25. Optical conductivity in LDA+DMFT Expts: F. P. Mena, D. van der Marel, J. L. Sarrao, PRB 72, 045119 (2005). 16. K. S. Burch et al., PRB 75, 054523 (2007). 17. E. J. Singley, D. N. Basov, E. D. Bauer, M. B. Maple, PRB 65, 161101(R) (2002). • At 300K very broad Drude peak (e-e scattering, spd lifetime~0.1eV) • At 10K: • very narrow Drude peak • First MI peak at 0.03eV~250cm-1 • Second MI peak at 0.07eV~600cm-1

  26. 10K In eV Ce In Multiple hybridization gaps non-f spectra 300K • Larger gap due to hybridization with out of plane In • Smaller gap due to hybridization with in-plane In

  27. DMFT -Momentum resolved Ce-4f spectra Af(w,k) Hybridization gap q.p. band Fingerprint of spd’s due to hybridization scattering rate~100meV SO Not much weight T=10K T=300K

  28. DMFT qp bands LDA bands LDA bands DMFT qp bands Quasiparticle bands three bands, Zj=5/2~1/200

  29. Quantum Phase Transition: Kondo Breakdown vs SDW. V> Vc Neglect Magnetic order V < Vc . Magnetic Order • SDW picture. Focus on order parameters. Neglect changes in the electronic structure.[Hertz, Morya] • Kondo breakdown scenario. Drastic changes in the electronic structure. [Doniach] [ Coleman, Pepin, Paul, Senthil, Sachdev, Vojta, Si ]

  30. DMFT: consider the underlying paramagnetic solution. Study finite T.

  31. Kondo Breakdown as an Orbitally selective Mott Transition. [L. DeMedici, A. Georges GK and S. Biermann PRL (2005), C. Pepin (2006) , L. DeLeo M. Civelli and GK ] • Analogous situation to the Mott transition. Mott / Slater. • f localization - Jump in the Fermi volume-Jump in DeHaas VanAlven frequencies. • f Localized and f Itinerant phases have different compressibilities. • Low but finite temperature aspects of the transition governed by a two impurity model.

  32. Fermi surface changes under pressure in CeRhIn5 • Fermi surface reconstruction at 2.34GPa . Sudden jump of dHva frequencies • Delocalization. Increase of electron FS frequencies . Localization decreases them. localized itinerant Shishido, (2005) We can not yet address FS change with pressure  We can study FS change with Temperature - At high T, Ce-4f electrons are excluded from the FS At low T, they are included in the FS

  33. No c in DMFT! No c in Experiment! Slight decrease of the electron FS with T A R A Z R R A A R Electron fermi surfaces at (z=p) LDA+DMFT (400 K) LDA+DMFT (10 K) LDA b2 b2 c

  34. M X M G X X M M X Hole fermi surfaces at z=0 Big change-> from small hole like to large electron like LDA+DMFT (400 K) LDA+DMFT (10 K) LDA e1 g h h g

  35. Conclusion- future directions • Long wavelength vs short distance [ mean field ] physics in correlated materials. • Further improvements and developments of DMFT [ CDMFT, electronic structure] • Other systems. […..] System specific studies. Variety and universality in the localization delocalization phenomena. • Towards a (Dynamical ) Mean Field Theory based theoretical spectroscopy.

  36. Conclusions [115’s] • DMFT in action: collective behavior of the hybridization field. Very slow crossover. Spectral evolution. Valence histograms. • Theory/Experiment Spectroscopy. Multiple hybridization gaps in optics. • Very different Ce-In hybridizations with In out of plane being larger. • Kondo breakdown as an orbitally selective Mott transition. dhv orbits. • Lessons for the 5f’s. Elemental actinides.

  37. Thanks! • $upport NSF-DMR. • Collaborators: K. Haule, L. DeLeo, J. Shim, M. Civelli. K. Haule and J. Shim and GK, Science Express Nov 1st (2007). To appear in science.

  38. Mott transition across the actinides. B. Johansson Phil Mag. 30,469 (1974)] Mott Transition d Pu a d a after G. Lander, Science (2003) and Lashley et. al. PRB (2006).

  39. DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer filling T/W 14

  40. Fermi surfaces Increasing temperature from 10K to 300K: • Gradual decrease of electron FS • Most of FS parts show similar trend • Big change might be expected in the G plane – small hole like FS pockets (g,h) merge into electron FS e1 (present in LDA-f-core but not in LDA) • Fermi surface a and c do not appear in DMFT results

  41. Summary part 2 • Modern understanding (DMFT) of the (orbitally selective) Mott transition across the actinde series (B. Johanssen 1970 ) sheds light on 5f physics. • Important role of multiplets. Pu is non magnetic and mixed valent element mixture of f6 and f5 • f electrons are localized in Cm f7 • Physics of 5f’s and 4f’s is similar but different. Main difference, the coherence scale in 5f’s much larger, resulting in a much larger coupling to the lattice. K. Haule and J. Shim Ref: Nature 446, 513, (2007)

  42. Pu phases: A. Lawson Los Alamos Science 26, (2000) GGA LSDA predicts d Pu to be magnetic with a large moment ( ~5 Bohr). Experimentally Pu is not magnetic. [PRB 054416(2005). Valence of Pu is controversial.

  43. 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) (experiments from Wong et.al, Science, 22 August 2003)

  44. Photoemission of Actinides Curie-Weiss Tc 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.

  45. What is the valence in the late actinides ?

  46. Electron fermi surfaces at (z=0) Slight decrease of the electron FS with T M X M G X X M M X LDA+DMFT (400 K) LDA LDA+DMFT (10 K) a2 a2

  47. Electron fermi surfaces at (z=p) Slight decrease of the electron FS with T No a in DMFT! No a in Experiment! A R A Z R R A A R LDA+DMFT (400 K) LDA LDA+DMFT (10 K) a3 a3 a

  48. Slight decrease of the electron FS with T M X M G X X M M X Electron fermi surfaces at (z=0) LDA+DMFT (400 K) LDA+DMFT (10 K) LDA b1 b1 b2 b2 c

  49. No c in DMFT! No c in Experiment! Slight decrease of the electron FS with T A R A Z R R A A R Electron fermi surfaces at (z=p) LDA+DMFT (400 K) LDA+DMFT (10 K) LDA b2 b2 c

  50. M X M G X X M M X Hole fermi surfaces at z=0 Big change-> from small hole like to large electron like LDA+DMFT (400 K) LDA+DMFT (10 K) LDA e1 g h h g

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