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Electron phonon renormalization of electronic band structure

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Electron phonon renormalization of
electronic band structure

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Electron phonon renormalization of electronic band structure

  1. 1. Elena Cannuccia Institut Laue-Langevin, Grenoble (France) Electron phonon renormalization ofElectron phonon renormalization of electronic band structureelectronic band structure
  2. 2. The N particleThe N particle world:world: ionsions andand electronelectronss all togetherall together Electron phonon renormalizationElectron phonon renormalization of electronic band structureof electronic band structure
  3. 3. Born–Oppenheimer approximationBorn–Oppenheimer approximation a perturbative approacha perturbative approach Electron phonon at workElectron phonon at work beyond thebeyond the Born–Oppenheimer approximationBorn–Oppenheimer approximation
  4. 4. The separated worlds ofThe separated worlds of phononsphonons and electronand electronss Electrons live in the bands generated by the ionic potential Phonons are the quantized ionic vibrations on the potential generated by the electrons
  5. 5. Born–Oppenheimer approximationBorn–Oppenheimer approximation a perturbative approacha perturbative approach Electron phonon at workElectron phonon at work beyond thebeyond the Born–Oppenheimer approximationBorn–Oppenheimer approximation
  6. 6. Coupling electrons and phonons …Coupling electrons and phonons … Superconductivity Joule's heating Electron relaxation (luminescence) Polaronic transport Coherent Phonons Peierls instability Raman Spectroscopy etc......
  7. 7. EPC on the electronic structureEPC on the electronic structure Kink in the band structure Mass Enhancement Temperature dependence of band gaps A. Marini, PRL 101,106405 (2008)
  8. 8. Energy levels renormalizationEnergy levels renormalization ThermalThermal expansionexpansion Electron-PhononElectron-Phonon interactioninteraction P.B. Allen and M. Cardona Phys. Rev. B 27 4760 (1983) >> Where the coupling comes from?
  9. 9. Born–Oppenheimer approximationBorn–Oppenheimer approximation a perturbative approacha perturbative approach Electron phonon at workElectron phonon at work beyond thebeyond the Born–Oppenheimer approximationBorn–Oppenheimer approximation
  10. 10. A perturbative approach:A perturbative approach: Heine-Allen-Cardona 1/2Heine-Allen-Cardona 1/2 For a review see M. Cardona, Solid State Commun. 133, 3 (2005). H (x+u)=H (x) + ∂V scf ∂ x u + 1 2 ∂ 2 V scf ∂ x2 u2 +... Using Perturbation TheoryPerturbation Theory, we get the correction to the energy δ Ei=〈Ψi (0) ∣ ∣Ψi (0) 〉 + 〈Ψi (0) ∣ ∣Ψi (0) 〉 + 〈Ψi (0) ∣ ∣Ψi (1) 〉 +... First order PT Second order PT V scf (x+u)=V scf (x) + ∂V scf ∂ x u + 1 2 ∂ 2 V scf ∂ x 2 u2 +....
  11. 11. A perturbative approach:A perturbative approach: Heine-Allen-Cardona 2/2Heine-Allen-Cardona 2/2 Debye-Waller Fan δ Ei(β) = [ 1 2 〈 ∂ 2 V scf ∂ x2 〉 + ∑j (Ei−Ej) −1 〈 ∂V scf ∂ x ∣j〉〈 j∣ ∂V scf ∂ x 〉] 〈u 2 〉 Clear dependence on the Temperature B(w) = Bose function δ En k (β)=∑q λ n' [ ∣gn n' k qλ ∣ En k−En' k+q − Λnn' k q λ En k−En' k ](2B(ωq λ)+1) Thermal average Average on the electronic wavefunction FINAL FORMULA
  12. 12. All the previous theory can be reformulated in term  of Green's function including non­adiabatic effects  Beyond the staticBeyond the static perturbation theoryperturbation theory
  13. 13. Born–Oppenheimer approximationBorn–Oppenheimer approximation a perturbative approacha perturbative approach Electron phonon coupling at workElectron phonon coupling at work beyond thebeyond the Born–Oppenheimer approximationBorn–Oppenheimer approximation
  14. 14. Spectroscopy: theoretical point of view What really theoreticians calculate!!
  15. 15. Finite temperature electronicFinite temperature electronic and opticaland optical properties of zb-GaNproperties of zb-GaN H. Kawai, K. Yamashita, E. Cannuccia, A. Marini Phys. Rev. B. 89, 085202 (2014) What we can do now!!! BroadeningBroadening induced by electron-phonon scattering and temperature dependence
  16. 16. The gap ofThe gap of diamonddiamond F. Giustino, et al.  PRL, 105, 265501 (2010) E. Cannuccia, Phys. Rev. Lett. 107, 255501 (2011) Logothedis et al. PRB 46, 4483 (1992)  Electronic Gap: 7.715 eV Renormalization: 615 meV Classicalions
  17. 17. It's time to revise previous electronic structure calculations?
  18. 18. What about dynamical effects?What about dynamical effects?
  19. 19. Dynamical effects in diamondDynamical effects in diamond Logothedis et al. PRB 46, 4483 (1992)  -670 meV E. Cannuccia, Phys. Rev. Lett. 107, 255501 (2011) Signature ofSignature of the dynamicalthe dynamical effectseffects
  20. 20. Breakdown of the QP pictureBreakdown of the QP picture E. Cannuccia and A. MariniE. Cannuccia and A. Marini Europ. Phys. J. B.Europ. Phys. J. B. 8585, 320 (2012), 320 (2012)
  21. 21. ConclusionsConclusions  Perturbative approach to the electron­phonon coupling     Finite temperature optical spectra  Band gap renormalization induced by the EPC  Dynamical effects on the electronic properties AcknowledgmentAcknowledgment Andrea Marini CNR Rome, Italy Hiroki Kawai Tokyo University Japano
  22. 22. Thank you for  your attention
  23. 23. S. Ponce, G. Antonius, P. Boulanger, E. Cannuccia, et al. Comp. Mat. Science, 83, 341, (2014) Implementation of large formula: source of infinite errors but ...
  24. 24. Carbon contributionSi contrib. Total Renorm. Temp. Dep. of gap: SiC, path integral molecular dynamics Hernández, Herrero, Ramírez, Cardona, PRB 77 045210 2008
  25. 25. Electron-phonon coupling from a MBPT perspective ϵnk En k (T )+i Γn k (T ) Electron scatters with  1 phonon at a time Electron­Phonon Self Energy Temperature dependenceSpectral Functions Enk Γnk
  26. 26. A. Eiguren and C. Ambrosch-Draxl, PRL 101 036402 (2008) Quasi-particle Band Structure Induced by the Electron-phonon interaction on a surface

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