1. Dynamical Mean Field Theory (DMFT) of correlated solids. G.Kotliar Physics Department Center for Materials Theory Rutgers University. Collaborators: K. Haule (Rutgers), S. Savrasov (UC Davis) September 2-7 (2007) Magdalen College Oxford United Kingdom Gordon Research Conference on Solid State Chemistry II

2. Outline 1]Introduction to DMFT ideas. 2]Application to elemental actinides, what is valence in a correlated solid ? 3]Application to cobaltates, why are correlation stronger near a band insulator than near a Mott insulator? [C. Delmas talk A.Maignan talk] Central theme, localization-delocalization ! 4]Application to 115’s and the tale of multiple hybridization gaps. [F. Steglich talk]

3. 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 ?

4. DMFT Spectral Function Photoemission and correlations Probability of removing an electron and transfering energy  =Ei-Ef, and momentum k f(  ) A(  ) M 2 e Angle integrated spectral function 8

5. D MFT approximate quantum solid as atom in a medium10

6. (GW) DFT+DMFT: determine H[k] and density and  self consitently from a functional and obtain total energies. 12 Spectra=- Im G(k,  ) Self consistency for V and 

7. Summary: part 1 Gabriel Kotliar and Dieter Vollhardt, Physics Today 57, 53 (2004). A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. of Mod. Phys. 68, 13-125 (1996). G. Kotliar, S. Savrasov, K. Haule, V. Oudovenko, O. Parcollet, and C. Marianetti, Rev. of Mod. Phys. 78, 000865 (2006). Spectral function in DMFT analogous to density in DFT Self consistent Impurity problem, natural language to describe localization/delocalization phenomena. combines atomic physics and band theory Systematically improvable, cluster DMFT Recent progress in implementation

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

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

10. DMFT Phonons in fcc  -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

11. Curie-Weiss Tc Photoemission of Actinides alpa->delta volume collapse transition Curium has large magnetic moment and orders antif Pu does is non magnetic. F0=4,F2=6.1 F0=4.5,F2=7.15 F0=4.5,F2=8.11

12. What is the valence in the late actinides ?

13. Summary part 2 Modern understanding of the Mott transition across the actinde series (B. Johanssen) sheds light on the physics of actinides. Important role of multiplets. Pu is non magnetic and mixed valent element mixture of f 6 and f 5 f electrons are localized in Cm f 7 K. Haule and J. Shim Ref: Nature 446, 513, (2007)

14. DMFT study of Na x CoO 2

15. Foo et.al. PRL 247001

16. CoO 2 NaCoO 2

17. Assume Na patterns of Zandbergen et. al.PRB 70 024101 C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98, 176405 (2007). A

18. DMFT calculations with and without disorder U=3 ev. C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98, 176405 (2007)

19. x=.33 QP dispersion DMFT LDA

20. References: part 3 C. Marianetti, G. Kotliar, and G. Ceder, Nature Materials 3, 627 - 631 (2004). C. A. Marianetti and G. Kotliar Phys. Rev. Lett. 98,176405 (2007) C. Marianetti, K. Haule and O Parcollet cond-mat (2007) Alternative theory : low spin to high spin Khaliullin Phys. Rev. Lett. 96, 216404 (2006)

21. Summary part 3 What is the minimal model of the cobaltes ? t2g orbitals + binary potential a see which results of the Li /Na vacancy. Why are correlations stronger near a band insulator than near a Mott insulator ? U < Uc2, hole moves in a restricted space (where potential is low) and is strongly correlated. DMFT calculations account for the Curie Weiss phase and the Fermi liquid phase

22. Conclusion DMFT as a technique, makes contact with experiments, total energies, phonons, photoemission, ARPES,optics,…thermopower…..neutron scattering ….. Concepts, atom in a quantum medium, Weiss field, local spectral function, A(  ), three peak structure,transfer of spectral weight, valence histogram, [bridges between atomic physics and band theory ] Under constant development, but already gives some exciting results.

23. Actinides, phonons, role of multiplets, spectral signatures, Pu as mixed valent metal. Cobaltates, key role of inhomogeneities bringing correlations near a (correlated) insulator. DMFT treatment of an alloy. Conclusions :chemistry brings out different aspects of localization delocalization physics. 115’s delocalization transition as a function of T. Spectral function as a coherence order parameters. Multiple hybridization gaps.

24. Thanks! Acknowldegment. NSF-DMR. DOE-BES. Collaborators:K. Haule, C. Marianetti, J. Shim, and S. Savrasov Conclusions :chemistry brings out different aspects of localization delocalization physics.

26. Ce In Ir Ce In Crystal structure of 115’s CeMIn5 M=Co, Ir, Rh CeIn 3 layer IrIn 2 layer Tetragonal crystal structure 4 in plane In neighbors 8 out of plane in neighbors 3.27au 3.3 au

27.  CeRhIn5: TN=3.8 K;   450 mJ/molK2  CeCoIn5: Tc=2.3 K;   1000 mJ/molK2;  CeIrIn5: Tc=0.4 K;   750 mJ/molK2 CeMIn 5 M=Co, Ir, Rh out of plane in-plane

28. ARPES Fujimori, 2006 Angle integrated photoemission Experimental resolution ~30meV, theory predicts 3meV broad band Surface sensitive at 122eV

29. Very slow crossover! T*T* Slow crossover more consistent with NP&F T*T* coherent spectral weight T NP&F: Nakatsuji,Pines&Fisk, 2004 Buildup of coherence in single impurity case TKTK coherent spectral weight T scattering rate coherence peak Buildup of coherence Crossover around 50K

30. 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 Optical conductivity in LDA+DMFT

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

32. Summary part 4 115’s model systems to study the evolution of the f electron as a function of temperature Multiple hybridization gaps in optics. Very different Ce-In hybridizations with In out of plane being larger. J. Shim K Haule and G.K. Submitted to Science. (2007).

36. PRL 80, (1998) GPalsson and GK Thermoelectricity near a Mott transition La1-xSrxTiO3

40. ARPES Fujimori, 2006 Angle integrated photoemission vs DMFT Experimental resolution ~30meV, theory predicts 3meV broad band Surface sensitive at 122eV

41. Angle integrated photoemission vs DMFT ARPES Fujimori, 2006 Nice agreement for the Hubbard band position SO split qp peak Hard to see narrow resonance in ARPES since very little weight of q.p. is below Ef Lower Hubbard band

42. . arXiv:0704.1065 [pdf]arXiv:0704.1065pdf –Title: Precise Control of Band Filling in NaxCoO2 –Authors: Daisuke Yoshizumi, Yuji Muraoka, Yoshihiko Okamoto, Yoko Kiuchi, Jun-Ichi Yamaura, Masahito Mochizuki, Masao Ogata, Zenji HiroiDaisuke YoshizumiYuji MuraokaYoshihiko OkamotoYoko KiuchiJun-Ichi YamauraMasahito MochizukiMasao OgataZenji Hiroi –Comments: 4 pages, 5 figures, submitted to J. Phys. Soc. Jpn –Journal-ref: J. Phys. Soc. Jpn. 76 (2007) 063705 –Subjects: Strongly Correlated Electrons (cond- mat.str-el)

44. DMFT Impurity cavity construction

45. Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000) and Phys. Rev.B (2001). Phys. Rev. B 62, 12715 (2000) Ex. Ir>=|R,  > Gloc=G(R , R  ’)  R,R’ ’ Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc. Sum of 2PI graphs One can also view as an approximation to an exact Spetral Density Functional of Gloc and Wloc.

46. Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). After finishing the loop treat the graphs involving Gnonloc Wnonloc in perturbation theory. P.Sun and GK PRL (2004). Related work, Biermann Aersetiwan and Georges PRL 90,086402 (2003). EDMFT loop G. Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated G Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259-301. cond-mat/0208241 S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004)

47. Anomalous Resistivity Maximum metallic resistivity

48. Total Energy as a function of volume for Pu Total Energy as a function of volume for Pu  (ev) vs (a.u. 27.2 ev) (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Zein (2005) Following Aryasetiwan Imada Georges Kotliar Bierman and Lichtenstein. PRB 70 195104. (2004) Pu

49. alpa->delta Photoemission Gouder Havela Lande PRB(2001)r

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

51. Resistivity of Am under pressure. J. C. Griveau Rebizant Lander and Kotliar PRL 94, 097002 (2005).

52. Photoemission spectra using Hubbard I solver [Lichtenstein and Katsnelson, PRB 57, 6884,(1998 ), Svane cond-mat 0508311] and Sunca. [Savrasov Haule and Kotliar cond-mat 0507552] Hubbard bands width is determined by multiplet splittings.

53. CeIrIn5 While DMFT aims at local quantities, one can derive non local information

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

56. Am equation of state. LDA+DMFT.New acceleration technique for solving DMFT equations S. Savrasov K. Haule G. Kotliar PRL (2006)

57. The “DMFT- valence” in the late actinides

58. Pu phases: A. Lawson Los Alamos Science 26, (2000) GGA LSDA predicts  Pu to be magnetic with a large moment ( ~ 5 Bohr). Experimentally Pu is not magnetic. [Lashley et. al. cond- matt 0410634] PRB 054416(2005). Approach the Mott transition from the left. (delocalized side).

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

60. Model Hamiltonians Tight binding form. Eliminate the “irrelevant” high energy degrees of freedom Add effective Coulomb interaction terms

61. Summary: part 1

62. LDA+DMFT Self-Consistency loop DMFT U -E dc

63. DMFT Cavity Construction. Happy marriage of atomic and band physics. 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 Rev. Mod. Phys. 78, 865 (2006). G. Kotliar and D. Vollhardt Physics 53 Today (2004) Extremize a functional of the local spectra. Local self energy.

64. A. Georges, G. Kotliar (1992) A(  ) 11

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

66. in the late actinides [DMFT results: K. Haule and J. Shim ]

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

68. Double well structure and  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. Phys vol.126, 1009 27. (2002)] 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. F(T,V)=Fphonons +Finvar

69. Optical conductivity Typical heavy fermion at low T: Narrow Drude peak (narrow q.p. band) Hybridization gap k  Interband transitions across hybridization gap -> mid IR peak CeCoIn 5 no visible Drude peak no sharp hybridization gap F.P. Mena & D.Van der Marel, 2005 E.J. Singley & D.N Basov, 2002 second mid IR peak at 600 cm -1 first mid-IR peak at 250 cm -1

70. T=10K T=300K scattering rate~100meVFingerprint of spd’s due to hybridization Not much weight q.p. bandSO Momentum resolved Ce-4f spectra Af(,k)Af(,k) Hybridization gap

71. DMFT qp bandsLDA bands DMFT qp bands Quasiparticle bands three bands, Z j=5/2 ~1/200

72. Momentum resolved total spectra A( ,k) Fujimori, 2003 LDA+DMFT at 10K ARPES, HE I, 15K LDA f-bands [-0.5eV, 0.8eV] almost disappear, only In-p bands remain Most of weight transferred into the UHB 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 Large lifetime of HBs -> similar to LDA(f-core) rather than LDA or LDA+U

73. DMFT : Evolution of the DOS. 13

74. Approach Mott point from the right Am under pressure Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition? “Soft” “Hard” J. C. Griveau et.al. PRL 94, 097002 (2005).

75. The “DMFT- valence” in the late actinides

76. From Atoms to Solids Band (e.g. LDA) Atomic Physics (eg. CI) )

77. The “DMFT- valence” in the late actinides

78. Late actinide issues All the spin density functional studies of fcc Pu and Am, using either LDA or GGA, predict magnetic long range order with a large moment. Experimentally Pu and Am are not magnetic. If one treats the f electrons as part of the core LDA overestimates the volume by 30% Valence of Pu controversy. LDA+U Schick, Havela Lichtenstein et.al. Anisimov et.al.(5f)^6. Erickson and Wills (5f)^4.

79. CoO2

80. (e Temperature dependence of the local Ce-4f spectra At low T, very narrow q.p. peak (width ~3meV) SO coupling splits q.p.: +-0.28eV Redistribution of weight between the q.p. and the upper Hubbard band SO At 300K, only Hubbard bands

81. arXiv:0708.2528 [pdf]arXiv:0708.2528pdf –Title: Synthesis and properties of CoO2, the x = 0 end member of the LixCoO2 and NaxCoO2 systems –Authors: T. Motohashi, Y. Katsumata, T. Ono, R. Kanno, M. Karppinen, H. YamauchiT. MotohashiY. KatsumataT. Ono R. KannoM. KarppinenH. Yamauchi –Comments: 18 pages, 5 figures –Journal-ref: Chemistry of Materials, cm0702464 (2007)

82. Photoemission Gouder, Havela PRB 2002, 2003

83. CeCoI n 5 CeRhIn 5 CeIrIn 5 Tc[K]2.3K2.1K@ p>1.5G Pa 0.4K C v /T[mJ/mol K^2] 30050750 Why CeIrIn 5 ? Phase diagram of 115’s

84. 5f elements: actinide series 5f elements: actinide seriess/cAFFM Localisation Delocalization 1.4K 0.4 K 0.9K0.8K52K25K52K

85. The “DMFT- valence” in the late actinides Haule Shim Kotliar (2006)