Electron phonon renormalization of electronic band structure

1. Elena Cannuccia
Institut Laue-Langevin, Grenoble (France)
Electron phonon renormalization ofElectron phonon renormalization of
electronic band structureelectronic band structure

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

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. 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. It's time to revise previous
electronic structure
calculations?

18. What about dynamical effects?What about dynamical effects?

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. 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. 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. Thank you for
your attention

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. Carbon contributionSi contrib.
Total
Renorm.
Temp. Dep. of gap: SiC, path integral molecular dynamics
Hernández, Herrero, Ramírez, Cardona, PRB 77 045210 2008

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. A. Eiguren and C. Ambrosch-Draxl, PRL 101 036402 (2008)
Quasi-particle Band Structure Induced by the
Electron-phonon interaction on a surface