The document discusses electron-phonon renormalization of electronic band structure. It begins by introducing the concept of electrons and ions interacting together in a material. It then describes the Born-Oppenheimer approximation approach, which treats electron-phonon coupling perturbatively. This approach is used to calculate corrections to electronic energy levels and band structures due to electron-phonon interactions. The summary describes how this coupling affects key properties like band gaps, masses, and spectroscopic measurements in real materials like diamond and GaN.

1. Introduction to theIntroduction to the
electron-phononelectron-phonon
renormalization ofrenormalization of
electronic band structureelectronic band structure

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. 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. 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. How to take a picture of the
electrons
ARPES

6. ARPES: direct method toARPES: direct method to
photograph the electronicphotograph the electronic
structure of surfaces 1/3structure of surfaces 1/3

7. ARPES: direct method toARPES: direct method to
photograph the electronicphotograph the electronic
structure of surfaces 2/3structure of surfaces 2/3

8. ARPES: direct method toARPES: direct method to
photograph the electronicphotograph the electronic
structure of surfaces 3/3structure of surfaces 3/3

9. ContemporaryContemporary
example: grapheneexample: graphene

10. Electrons cannot stay put

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. Coupling electrons and phononsCoupling electrons and phonons
……
Superconductivity
Joule's heating
Electron relaxation
(luminescence)
Polaronic transport
Coherent Phonons
Peierls instability
Raman Spectroscopy
etc......

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. 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. Born–Oppenheimer approximationBorn–Oppenheimer approximation
A perturbativeA perturbative
approachapproach
Electron phonon at workElectron phonon at work
beyond the
rigid ions approximation

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. 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. Born–Oppenheimer approximationBorn–Oppenheimer approximation
a perturbative approacha perturbative approach
Electron phonon couplingElectron phonon coupling
at workat work
beyond the
rigid ions approximation

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. 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. 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. Spectroscopy:
theoretical point of
view
What really theoreticiansWhat really theoreticians
calculate!!calculate!!

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. 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. ConclusionsConclusions
Perturbative approach to the electron­phonon coupling
Band gap renormalization induced by the EPC
Finite temperature optical spectra

26. Thank you for
your attention

27. Shrinking of the gapShrinking of the gap