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Introduction to theIntroduction to the
electron-phononelectron-phonon
renormalization ofrenormalization of
electronic band...
Electron phonon renormalizationElectron phonon renormalization
of electronic band structureof electronic band structure
Th...
Born–OppenheimerBorn–Oppenheimer
approximationapproximation
a perturbative approacha perturbative approach
Electron phonon...
The separated worlds ofThe separated worlds of
phonons and electronphonons and electronss
Electrons live in the bands
gene...
How to take a picture of the
electrons
ARPES
ARPES: direct method toARPES: direct method to
photograph the electronicphotograph the electronic
structure of surfaces 1/...
ARPES: direct method toARPES: direct method to
photograph the electronicphotograph the electronic
structure of surfaces 2/...
ARPES: direct method toARPES: direct method to
photograph the electronicphotograph the electronic
structure of surfaces 3/...
ContemporaryContemporary
example: grapheneexample: graphene
Electrons cannot stay put
Born–Oppenheimer approximationBorn–Oppenheimer approximation
a perturbative approacha perturbative approach
Electron phono...
Coupling electrons and phononsCoupling electrons and phonons
……
Superconductivity
Joule's heating
Electron relaxation
(lum...
EPC on the electronicEPC on the electronic
structurestructure
Kink in the band structure
Mass Enhancement
Temperature depe...
Energy levels renormalizationEnergy levels renormalization
ThermalThermal
expansionexpansion
Electron-PhononElectron-Phono...
Born–Oppenheimer approximationBorn–Oppenheimer approximation
A perturbativeA perturbative
approachapproach
Electron phonon...
A perturbative approach:A perturbative approach:
Heine-Allen-Cardona 1/2Heine-Allen-Cardona 1/2
For a review see M. Cardon...
A perturbative approach:A perturbative approach:
Heine-Allen-Cardona 2/2Heine-Allen-Cardona 2/2
Debye-Waller Fan
δ Ei(β) =...
Born–Oppenheimer approximationBorn–Oppenheimer approximation
a perturbative approacha perturbative approach
Electron phono...
The gap of diamondThe gap of diamond
(1/2)(1/2)
F. Giustino, et al.  PRL, 105, 265501 (2010)
E. Cannuccia, Phys. Rev. Lett...
The gap of diamondThe gap of diamond
(2/2)(2/2)
Exp: Logothetidis et al.
PRB 46, 4483 (1992)
Quantum (PI)
MD calculations
...
Isotopic EffectsIsotopic Effects
〈u
2
〉=〈
h
4Mω
{2[e
−hω/KT
−1]
−1
+1}〉
At high T,
independent of M (classical effect)
At ...
Spectroscopy:
theoretical point of
view
What really theoreticiansWhat really theoreticians
calculate!!calculate!!
Finite temperature electronic and opticalFinite temperature electronic and optical
properties of zb-GaNproperties of zb-Ga...
Results: electronic band structureResults: electronic band structure
Breakdown ofBreakdown of
the QP picturethe QP picture...
ConclusionsConclusions
 Perturbative approach to the electron­phonon coupling
 Band gap renormalization induced by the EPC...
Thank you for 
your attention
Shrinking of the gapShrinking of the gap
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Introduction to the electron-phonon renormalization of electronic band structure

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March 2016 - Seminar at CINaM

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Introduction to the electron-phonon renormalization of electronic band structure

  1. 1. Introduction to theIntroduction to the electron-phononelectron-phonon renormalization ofrenormalization of electronic band structureelectronic band structure
  2. 2. Electron phonon renormalizationElectron phonon renormalization of electronic band structureof electronic band structure The N particlesThe N particles world:world: ionsions andand electronelectronss all togetherall together
  3. 3. Born–OppenheimerBorn–Oppenheimer approximationapproximation a perturbative approacha perturbative approach Electron phonon at workElectron phonon at work beyond thebeyond the rigid ions approximationrigid ions approximation
  4. 4. The separated worlds ofThe separated worlds of phonons and electronphonons and 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. How to take a picture of the electrons ARPES
  6. 6. ARPES: direct method toARPES: direct method to photograph the electronicphotograph the electronic structure of surfaces 1/3structure of surfaces 1/3
  7. 7. ARPES: direct method toARPES: direct method to photograph the electronicphotograph the electronic structure of surfaces 2/3structure of surfaces 2/3
  8. 8. ARPES: direct method toARPES: direct method to photograph the electronicphotograph the electronic structure of surfaces 3/3structure of surfaces 3/3
  9. 9. ContemporaryContemporary example: grapheneexample: graphene
  10. 10. Electrons cannot stay put
  11. 11. Born–Oppenheimer approximationBorn–Oppenheimer approximation a perturbative approacha perturbative approach Electron phonon at workElectron phonon at work beyond thebeyond the rigid ionsrigid ions approximationapproximation
  12. 12. Coupling electrons and phononsCoupling electrons and phonons …… Superconductivity Joule's heating Electron relaxation (luminescence) Polaronic transport Coherent Phonons Peierls instability Raman Spectroscopy etc......
  13. 13. EPC on the electronicEPC on the electronic structurestructure Kink in the band structure Mass Enhancement Temperature dependence of band gaps A. Marini, PRL 101,106405 (2008)
  14. 14. Energy levels renormalizationEnergy 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?
  15. 15. Born–Oppenheimer approximationBorn–Oppenheimer approximation A perturbativeA perturbative approachapproach Electron phonon at workElectron phonon at work beyond the rigid ions approximation
  16. 16. 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 ∂ x 2 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 ∂ x2 u 2 +....
  17. 17. 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' [ |gnn' k q λ | 2 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
  18. 18. Born–Oppenheimer approximationBorn–Oppenheimer approximation a perturbative approacha perturbative approach Electron phonon couplingElectron phonon coupling at workat work beyond the rigid ions approximation
  19. 19. The gap of diamondThe gap of diamond (1/2)(1/2) 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: ~700 meV Classicalions
  20. 20. The gap of diamondThe gap of diamond (2/2)(2/2) Exp: Logothetidis et al. PRB 46, 4483 (1992) Quantum (PI) MD calculations Ramirez et al. PRB 73, 245202 (2006)
  21. 21. Isotopic EffectsIsotopic Effects 〈u 2 〉=〈 h 4Mω {2[e −hω/KT −1] −1 +1}〉 At high T, independent of M (classical effect) At low T, zero point vibrations (quantum) 〈u 2 〉∝KT 〈u 2 〉∝M −1/2 The quantisticThe quantistic zero-pointzero-point motion effectmotion effect Parks et al. PRB 49,14244 (1994) Eg M M→∞Eg electronic
  22. 22. Spectroscopy: theoretical point of view What really theoreticiansWhat really theoreticians calculate!!calculate!!
  23. 23. Finite temperature electronic and opticalFinite temperature electronic and optical properties of zb-GaNproperties of zb-GaN H. Kawai, K. Yamashita, E. Cannuccia, A. Marini Phys. Rev. B. 89, 085202 (2014) BroadeningBroadening induced by electron-phonon scattering and temperature dependence
  24. 24. Results: electronic band structureResults: electronic band structure Breakdown ofBreakdown of the QP picturethe QP picture E. Cannuccia and A. MariniE. Cannuccia and A. Marini Europ. Phys. J. B.Europ. Phys. J. B. 8585, 320 (2012), 320 (2012)
  25. 25. ConclusionsConclusions  Perturbative approach to the electron­phonon coupling  Band gap renormalization induced by the EPC    Finite temperature optical spectra
  26. 26. Thank you for  your attention
  27. 27. Shrinking of the gapShrinking of the gap

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