Introduction to the electron-phonon renormalization of electronic band structure
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
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
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
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?
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
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
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)
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.
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.