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Electronic Structure of Strongly Correlated Materials : a DMFT Perspective

Electronic Structure of Strongly Correlated Materials : a DMFT Perspective. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University. Palo Alto May 2002. Outline. Correlated electrons he Mott transition problem and electronic structure.

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Electronic Structure of Strongly Correlated Materials : a DMFT Perspective

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  1. Electronic Structure of Strongly Correlated Materials : a DMFT Perspective Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Palo Alto May 2002

  2. Outline • Correlated electrons he Mott transition problem and electronic structure. • Dynamical Mean Field Theory • Model Hamiltonian Studies of the Mott transition. Universal aspects. • System specific studies: LDA+DMFT, d Pu (with S . Savrasov and E. Abrahams) • Outlook THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  3. Strongly Correlated Materials • Copper Oxides. .(La2-x Bax) CuO4 High Temperature Superconductivity.150 K in the Ca2Ba2Cu3HgO8 . • Uranium and Cerium Based Compounds. Heavy Fermion Systems,CeCu6,m*/m=1000 • (La1-xSrx)MnO3 Colossal Magneto-resistance. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  4. Strongly Correlated Materials. • Battery Materials LiCoO2. Large stability for changing carrier density. • Large thermoelectric response in CeFe4 P12 (H. Sato et al. cond-mat 0010017). Ando et.al. NaCo2-xCuxO4 Phys. Rev. B 60, 10580 (1999). • Large and ultrafast optical nonlinearities Sr2CuO3 (T Ogasawara et.a Phys. Rev. Lett. 85, 2204 (2000) ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  5. The electron in a solid: wave picture Momentum Space (Sommerfeld) Maximum metallic resistivity 200 mohm cm Standard model of solids (Bloch, Landau) Periodic potential, waves form bands , k in Brillouin zone . Interactions renormalize away. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  6. Standard Model of Solids • Qualitative predictions: low temperature dependence of thermodynamics and transport. • Optical response, transition between the bands. • Qualitative predictions: filled bands give rise to insulting behavior. Compounds with odd number of electrons are metals. • Quantitative tools: Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy. Good starting point for perturbative calculation of spectra,eg. GW. Kinetic equations yield transport coefficients. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  8. Mott : Correlations localize the electron Low densities, electron behaves as a particle,use atomic physics, real space One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bandsNiO, CoO MnO….) H H H+ H H H motion of H+ forms the lower Hubbard band H H H H- H H motion of H_ forms the upper Hubbard band Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  9. Photoemission spectroscopy. Measures density ofstates for (BIS) adding and (PES) removing electrons THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  10. Localization vs Delocalization Strong Correlation Problem • A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem. • These systems display anomalous behavior (departure from the standard model of solids). • Neither LDA or LDA+U or Hartree Fock work well. • Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. treats QP and Hubbard bands. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  11. Failure of the standard model Mott transition in V2O3 under pressure or chemical substitution on V-site THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  12. Localization vs Delocalization Strong Correlation Problem • A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem. • These systems display anomalous behavior (departure from the standard model of solids). • Neither LDA or LDA+U or Hartree Fock work well. • Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  13. Mott transition in V2O3 under pressure or chemical substitution on V-site THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  14. Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  15. Failure of the Standard Model: NiSe2-xSx Miyasaka and Takagi (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  16. Hubbard model • U/t • Doping d or chemical potential • Frustration (t’/t) • T temperature Mott transition as a function of doping, pressure temperature etc. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  17. Limit of large lattice coordination Metzner Vollhardt, 89 Muller-Hartmann 89 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  18. Dynamical Mean Field Theory, cavity construction A. Georges G. Kotliar 92 THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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

  20. Solving the DMFT equations • Wide variety of computational tools (QMC,ED….)Analytical Methods • Extension to ordered states. • Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  21. Insights from DMFT • Low temperature Ordered phases . Stability depends on chemistry and crystal structure • High temperature behavior around Mott endpoint, more universal regime, captured by simple models treated within DMFT. Role of magnetic frustration. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  22. Kuwamoto Honig and Appell PRB (1980)M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  23. Insights from DMFT • The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase • Control parameters: doping, temperature,pressure… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  24. Spectral Evolution at T=0 half filling full frustration X.Zhang M. Rozenberg G. Kotliar (PRL 1993) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  25. Parallel development: Fujimori et.al THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  26. Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (2000) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  27. ARPES measurements on NiS2-xSexMatsuura et. al Phys. Rev B 58 (1998) 3690. Doniach and Watanabe Phys. Rev. B 57, 3829 (1998) . THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  28. Insights from DMFT • The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase • Control parameters: doping, temperature,pressure… THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  29. Anomalous transfer of optical spectral weight V2O3 :M Rozenberg G. Kotliar and H. Kajuter Phys. Rev. B 54, 8452 (1996). M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  30. Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi] THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  31. Anomalous Resistivity and Mott transition Ni Se2-x Sx Insights from DMFT: think in term of spectral functions (branch cuts) instead of well defined QP (poles ) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  32. Insights from DMFT • Important role of the incoherent part of the spectral function at finite temperature • Physics is governed by the transfer of spectral weight from the coherent to the incoherent part of the spectra. Real and momentum space. • Mott transitions : bifurcation points of an effective action functional. Simple behavior. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  33. Two roads for ab-initio calculation of electronic structure of strongly correlated materials Crystal structure +Atomic positions Model Hamiltonian Correlation Functions Total Energies etc. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  34. Realistic Calculationsof the Electronic Structure of Correlated materials • Combinining DMFT with state of the art electronic structure methods to construct a first principles framework to describe complex materials. • Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys. Cond. Mat. 35, 7359 (1997) • Savrasov Kotliar and Abrahams Nature 410, 793 (2001)) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  35. Combining LDA and DMFT • The light, SP (or SPD) electrons are extended, well described by LDA • The heavy, D (or F) electrons are localized,treat by DMFT. • LDA already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term) The U matrix can be estimated from first principles or viewed as parameters THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  36. Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and Kotliar). • DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. GDFT[r(r)] • Introduce local orbitals, caR(r-R)orbitals, and local GF G(R,R)(i w) = • The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing G[r(r),G(R,R)(iw)] • A useful approximation to the exact functional can be constructed, the DMFT +LDA functional. Savrasov Kotliar and Abrahams Nature 410, 793 (2001)) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  37. LDA+DMFT Self-Consistency loop E U DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  38. Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  39. Pu: Anomalous thermal expansion (J. Smith LANL) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  40. Problems with LDA treatements of d Pu DFT in the LDA or GGA is a well established tool for calculation of ground state properties. Many studies (APW Freeman, Koelling 1972, ASA and FP-LMTO, Soderlind et. al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) show an equilibrium volume of the d phaseIs 35% lower than experiment Largest discrepancy ever known in DFT based calculations. LSDA predicts magnetic long range order which is not observed experimentally (Solovyev et.al.) If one treats the f electrons as part of the core LDA overestimates the volume by 30% Weak correlation picture for alpha phase.

  41. Conclusion • The character of the localization delocalization in simple( Hubbard) models within DMFT is now fully understood, nice qualitative insights. • This has lead to extensions to more realistic models, and a beginning of a first principles approach interpolating between atoms and band, encouraging results for simple elements THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  42. Outlook • The Strong Correlation Problem:How to deal with a multiplicity of competing low temperature phases and infrared trajectories which diverge in the IR • Strategy: advancing our understanding scale by scale • Generalized cluster methods to capture longer range magnetic correlations • New structures in k space. Cellular DMFT THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  43. Outlook • Systematic improvements, short range correlations. • Take a cluster of sites, include the effect of the rest in a G0 (renormalization of the quadratic part of the effective action). What to take for G0 or D: • DCA (M. Jarrell et.al) , CDMFT (Kotliar Savrasov Palsson and Biroli ) • include the effects of the electrons to renormalize the quartic part of the action (spin spin , charge charge correlations) E. DMFT (Kajueter and GK, Si et.al) • Exploration of materials. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  44. Acknowledgements: Development of DMFT Collaborators: V. Anisimov, R. Chitra, V. Dobrosavlevic, 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, X.Y. Zhang Support: National Science Foundation. Work on Pu: Departament of Energy and LANL. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  45. Case study Fe and Ni • Archetypical itinerant ferromagnets • LSDA predicts correct low T moment • Band picture holds at low T • Main puzzle: at high temperatures c has a Curie Weiss law with a moment much larger than the ordered moment. • Magnetic anisotropy THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  46. Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and Kotliar Phys Rev. Lett 87, 67205 , 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  47. Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  48. Ni and Fe: theory vs exp • m( T=.9 Tc)/ mB ordered moment • Fe 1.5 ( theory) 1.55 (expt) • Ni .3 (theory) .35 (expt) meff / mB high T moment Fe 3.1 (theory) 3.12 (expt) Ni 1.5 (theory) 1.62 (expt) Curie Temperature Tc • Fe 1900 ( theory) 1043(expt) • Ni 700 (theory) 631 (expt) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  49. Fe and Ni • Consistent picture of Fe (more localized) and Ni (more correlated) • Satellite in minority band at 6 ev, 30 % reduction of bandwidth, exchange splitting reduction .3 ev • Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe • Mean field calculations using measured exchange constants(Kudrnovski Drachl PRB 2001) right Tc for Ni but overestimates Fe , RPA corrections reduce Tc of Ni by 10% and Tc of Fe by 50%. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

  50. Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,Kotliar Phys Rev. Lett 87, 67205 , 2001) THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

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