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Impact of electronic correlation on the electron-phonon coupling

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Impact of electronic correlation on the electron-phonon coupling

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Impact of electronic correlation on the electron-phonon coupling

  1. 1. Impact of electronic correlation on electron-phonon couplingelectron-phonon coupling Claudio Attaccalite Institute Neel, Grenoble (France) 22 April 2016 KAIST
  2. 2. The separated worlds of phonons and electrons Electrons live in the bands generated by the ionic potential Phonons are the quantized ionic vibrations on the potential generated by the electrons
  3. 3. Electron-phonon coupling: why? Superconductivity Joule's heating Electron relaxation (luminescence) Raman Spectroscopy
  4. 4. Electron-phonon coupling: definition For every single particle Hamiltonian (Kohn-Sham, Tight-binding, etc..) Second order expansion H (x+u)=H (x) + ∂ H ∂ x u + 1 2 ∂2 H ∂ x2 u 2 +... (3-fold) LUMO Changes in electronic structureStructural deformation Stepwise deformation EPC ~ |slope|2
  5. 5. Electron-phonon coupling: how? DFT is an exact theoryDFT is an exact theory for the ground statefor the ground state (atomic structure,(atomic structure, phonons, etc..)phonons, etc..) EPC matrix elementsEPC matrix elements can be obtained from thecan be obtained from the Kohn-Sham eigenstates, butKohn-Sham eigenstates, but there is no guaranteethere is no guarantee that they are goodthat they are good
  6. 6. RAMAN on graphene 1/2
  7. 7. RAMAN in graphene 2/2 K K ' ℏ q { q → D-line dispersion → Double resonant RAMAN I. Pócsik et al., J. Non­Cryst. Solids 227, 1083 (1998) P. H. Tan et al., Phys. Rev. B 66, 245410 (2002). J. Maultzsch et al., Phys. Rev. B 70, 155403 (2004).
  8. 8. Correlation? ...an old story
  9. 9. Electron-phonon coupling: definition For every single particle Hamiltonian (Kohn-Sham, Tight-binding, etc..) Second order expansion H (x+u)=H (x) + ∂ H ∂ x u + 1 2 ∂2 H ∂ x2 u 2 +... (3-fold) LUMO Changes in electronic structureStructural deformation Stepwise deformation EPC ~ |slope|2
  10. 10. Correction to the band structure ϵQP≃ϵKS+∫G(ω−ω, )W (ω, )+... Using Many-body Perturbation Theory GW to correct the Kohn-Sham band structure and its derivatives W (ω)= V ee ϵ(ω) GW is the first order correction in terms of screened interaction It works very well!!! Aryasetiawan, F., & Gunnarsson, O. The GW method.  Reports on Progress in Physics, 61(3), 237.(1998). 
  11. 11. GW correction to the EPC Pq= 4 N k ∑k ∣D kq  , k∣2 k , −kq ,  Impact of the electron­electron correlation on phonon dispersion: Failure of LDA and GGA DFT  functionals in graphene and graphite Michele Lazzeri, Claudio Attaccalite, Ludger Wirtz, and Francesco Mauri Phys. Rev. B 78, 081406(R) (2008)
  12. 12. A. Grüneis et al . Phys. Rev. B 80, 085423 (2009) Phonon dispersions around the K point  of graphite by inelastic x­ray scattering 〈 DK 2 〉F=166eV / ˚A 〈 DK 2 〉F=164eV / ˚A GW inelastic x­ray Phonons dispersion of graphene revisited
  13. 13. EPC in fullerenes LDA 73 meV evGW 101 meV Exp. 107 meV DFT: J. Laflamme-Janssen, PRB 2010; GW: C. Faber, J. of Mat. S., 47, 7472(2012)  Exp: Wang, JCP 2005; Hands, PRB 2008; (3-fold) LUMO ev-GW DFT-LDA Electron-phonon potential Electron–phonon coupling and charge­transfer excitations in organic systems from  many­body perturbation theory C. Faber et al.,  J. of Material Science, 47, 7472(2012) 
  14. 14. New physics appears... EPC for the K­A  phonon aquires a strong doping dependence′ Doped graphene as tunable electron­phonon coupling material C. Attaccalite et al.,  Nanoletters, 10, 1172(2010)  W (ω)= V ee ϵ(ω)
  15. 15. Experimental confirmation Raman signature of electron-electron correlation in chemically doped few-layer graphene Phys. Rev. B 83, 241401(R) (2011) Matteo Bruna and Stefano Borini Pressure-mediated doping in graphene Nanoletters, 11, 3564 (2011) J. Nicolle,D. Machon, P. Poncharal,O. Pierre-Louis and Alfonso San-Miguel
  16. 16. Energy levels renormalization ThermalThermal expansionexpansion Electron-PhononElectron-Phonon interactioninteraction P.B. Allen and M. Cardona Phys. Rev. B 27 4760 (1983) >> Where does the coupling come from?
  17. 17. GW renormalization of EPC and diamond gap Many­Body Effects on the Zero­Point Renormalization of the Band Structure G. Antonius, S. Poncé, P. Boulanger, M. Côté, and X. Gonze Phys. Rev. Lett. 112, 215501 (2014)
  18. 18. GW renormalization of EPC is too expensive to calculate. How to solve the problem?
  19. 19. Solution (1) Fit an exchange-correlation functional to reproduce GW
  20. 20. Bismuthates, Chloronitrides and other high-Tc superconductors Correlation­Enhanced Electron­Phonon Coupling: Applications of GW and Screened  Hybrid Functional to Bismuthates, Chloronitrides, and Other High­Tc Superconductors Z. P. Yin, A. Kutepov, and G. Kotliar Phys. Rev. X 3, 021011 (2013)
  21. 21. Bismuthates, Chloronitrides and other high-Tc superconductors Correlation­Enhanced Electron­Phonon Coupling: Applications of GW and Screened  Hybrid Functional to Bismuthates, Chloronitrides, and Other High­Tc Superconductors Z. P. Yin, A. Kutepov, and G. Kotliar Phys. Rev. X 3, 021011 (2013) critical temperature Tc
  22. 22. Solution (2) Use a smart way to make averages
  23. 23. Average on thermal lines Vibrational averages along thermal lines B. Monserrat Phys. Rev. B 93, 014302 (2016) Correlation effects on electron­phonon  coupling in semiconductors: Many­body  theory along thermal lines B. Monserrat Phys. Rev. B 93, 100301 (2016) ^O(qMV ,T )=⟨ ^O(T )⟩ ⟨ ^O(T)⟩=∫d q|Ψ(q ,T )2 | ^O(q)
  24. 24. Solution (3) Use approximate GW formulas
  25. 25. Simplified approach ∂ ΣGW ∂ u q ν ≃∂ ΣCOHSEX ∂ u q ν ∂GW ∂ u qν ≃ ( ∂G ∂ u q ν )W 2) Constant screening respect to the phonon displacement 1) Static screening (COHSEX) Inspired by the Bethe-Salpeter equation W (ω)≃W (0)
  26. 26. The simplified approach works! Approximation to the GW self-energy electron-phonon coupling gradients C. Faber et al. Physical Review B 91 (15), 155109(2015)
  27. 27. Conclusions 1)  Electron­phonon coupling from Kohn­Sham        under/overestimate the real one 2)  Many­Body­Perturbation Theory       can be used to correct the EPC  4)  New physics can appear!  5)  If calculations are too heavy ….                     use a simplified approach  3)  The new EPC matrix elements improve:        Raman, phonons, band­gap renomalization, Tc 
  28. 28. Acknowledgments Xavier Blase Paul Boulanger Elena Cannuccia Ivan Duchemin Carina Faber Alexander Gruneis Francesco Mauri Michele Lazzeri Angel Rubio Ludger Wirtz
  29. 29. Bands renormalization Frozen phonon approach
  30. 30. Electron-phonon coupling (3-fold) LUMO Changes in electronic structure → The frozen-phonon approach: EPC ~ |slope|2 Structural deformation Stepwise deformation
  31. 31. q = 2 ℏq  N f  ∫BZ d k  ∑i, j ∣gki ,k q, j  2 ∣k −k  kq−f  describe the scattering of an electron from the band I to band j due to the phonon ν gki ,kq, j  =〈 kq, j∣ Vq∣k ,i〉 ..where the electron­phonon matrix element.. Has been always considered a constant  with respect to the electron/hole doping Electron-phonon coupling and doping
  32. 32. Diagrammatic approach Electron and phonon renormalization of the EPC vertex Dirac massless fermions + Renormalization Group Interplay of Coulomb and electron­phonon interactions in graphene D. M. Basko and I. L. Aleiner Phys. Rev. B 77, 041409(R) (2008)
  33. 33. Diamond gap? Effect of the Quantum Zero­Point Atomic Motion on the Optical and Electronic Properties of  Diamond and Trans­Polyacetylene Elena Cannuccia and Andrea Marini Phys. Rev. Lett. 107, 255501(2011) Electron­Phonon Renormalization of the Direct Band Gap of Diamond Feliciano Giustino, Steven G. Louie, and Marvin L. Cohen Phys. Rev. Lett. 105, 265501 (2010)

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