Introduction to the electron phonon renormalization of the electronic band structure
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![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](https://image.slidesharecdn.com/epctalkroma-150426161225-conversion-gate01/75/introduction-to-the-electron-phonon-renormalization-of-the-electronic-band-structure-11-2048.jpg?cb=1669651841)
![Isotopic EffectsIsotopic Effects
〈u2
〉=〈
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) This renormalization should be taken
into account when state of the art
ab initio calculations of the gap are
compared
with experimental results.
Eg
M
M→∞Eg electronic](https://image.slidesharecdn.com/epctalkroma-150426161225-conversion-gate01/75/introduction-to-the-electron-phonon-renormalization-of-the-electronic-band-structure-19-2048.jpg?cb=1669651841)
Introduction to the electron phonon renormalization of the electronic band structure
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Introduction to the electron phonon renormalization of the electronic band structure
- 1. Elena Cannuccia Institut Laue-Langevin, Grenoble (France) Introduction to the electronIntroduction to the electron phonon renormalization ofphonon renormalization of electronic band structureelectronic band structure
- 2. The N particlesThe N particles 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 does the coupling come 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. 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
- 14. 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
- 15. Spectroscopy: theoretical point of view What really theoreticians calculate!!
- 16. 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
- 17. 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: ~400 meV Classicalions
- 18. 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)
- 19. Isotopic EffectsIsotopic Effects 〈u2 〉=〈 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) This renormalization should be taken into account when state of the art ab initio calculations of the gap are compared with experimental results. Eg M M→∞Eg electronic
- 20. Shrinking of the gapShrinking of the gap
- 21. It's time to revise previous electronic structure calculations?
- 22. What about dynamical effects?What about dynamical effects?
- 23. Dynamical effects in diamondDynamical effects in diamond Logothedis et al. PRB 46, 4483 (1992) E. Cannuccia, Phys. Rev. Lett. 107, 255501 (2011) Signature ofSignature of the dynamicalthe dynamical effectseffects
- 24. 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)
- 25. What if LDA fails to describe the electron-phonon coupling? What if the electron-phonon coupling causes degeneracy and crossing of quasi-particle levels? In the next talks......In the next talks......
- 28. 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 ...
- 29. Carbon contributionSi contrib. Total Renorm. Temp. Dep. of gap: SiC, path integral molecular dynamics Hernández, Herrero, Ramírez, Cardona, PRB 77 045210 2008
- 32. A. Eiguren and C. Ambrosch-Draxl, PRL 101 036402 (2008) Quasi-particle Band Structure Induced by the Electron-phonon interaction on a surface
Editor's Notes
- H e' quella elettronica della DFT. Fermiarmi al 2 ordine → expansione armonica significa assumere che le frquenze fononiche non dipendono dal volume del cristallo, quindi non sto tenendo conto di effetti anarmonici che sono legati all'expansione termica.
- Cambio di base: cartesiane → fonone displacement k-q
- Cambio di base: cartesiane → fonone displacement k-q
- C, N, O .. have no p-electrons in the core and the p valence electrons, as the atoms vibrate, can get much closer to the core than in cases where p-electrons are present in the core: germanium, silicon, GaAs…. The dipendence of the gap at high temperatures is linear and then it deviates because of quantum effects. Classically the gap correction is equal to zero, than at T=0 the intersection yields the electronic gap.
- Path integral molecular dynamics (PIMD) is a method of incorporating quantum mechanics into molecular dynamics simulations using Feynman path integrals. Such simulations are particularly useful for studying nuclear quantum effects in light atoms and molecules such as hydrogen, helium, neon and argon, as well as in quantum rotators such as methane and hydrogen bonded systems such as water. In PIMD, one uses the Born–Oppenheimer approximation to separate the wavefunction into a nuclear part and an electronic part. The nuclei are treated quantum mechanically by mapping each quantum nuclei onto a classical system of several fictitious particles connected by springs (harmonic potentials) and governed by an effective Hamiltonian