introduction to the electron-phonon renormalization of electronic band structure
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Diapositiva 1
Introduction to the electron-phonon
renormalization of electronic band structure
Electron phonon renormalization
of electronic band structure
The N particles world:
ions and electrons
all together
BornOppenheimer approximation
a perturbative approach
Electron phonon at work
beyond the
rigid ions approximation
The separated worlds of
phonons and electrons
Electrons live in the bands
generated by the ionic potential
Phonons are the quantized
ionic vibrations on the potential
generated by the electrons
How to take a picture of the electrons
ARPES
ARPES: direct method to photograph the electronic structure of surfaces 1/3
ARPES: direct method to photograph the electronic structure of surfaces 2/3
ARPES: direct method to photograph the electronic structure of surfaces 3/3
Contemporary example: graphene
Electrons cannot stay put
BornOppenheimer approximation
a perturbative approach
Electron phonon at work
beyond the
rigid ions approximation
Coupling electrons and phonons
Superconductivity
Joule's heating
Electron relaxation
(luminescence)
Polaronic transport
Coherent Phonons
Peierls instability
Raman Spectroscopy
etc......
EPC on the electronic structure
Kink in the band structure
Mass Enhancement
Temperature dependence of band gaps
A. Marini, PRL 101,106405 (2008)
Energy levels renormalization
Thermal expansion
Electron-Phonon interaction
P.B. Allen and M. Cardona Phys. Rev. B 27 4760 (1983)
>>
Where does the coupling come from?
BornOppenheimer approximation
A perturbative approach
Electron phonon at work
beyond the
rigid ions approximation
A perturbative approach:Heine-Allen-Cardona 1/2
For a review see M. Cardona,
Solid State Commun. 133, 3 (2005).
Using
Perturbation Theory, we get the correction to the energy
First order PT
Second order PT
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.
A perturbative approach:Heine-Allen-Cardona 2/2
Debye-Waller
Fan
Clear dependence on the Temperature
B(w) = Bose function
Thermal average
Average on the
electronic
wavefunction
FINAL FORMULA
BornOppenheimer approximation
a perturbative approach
Electron phonon coupling at work
beyond the
rigid ions approximation
The gap of diamond (1/2)
The e-h interaction Is not taken into account,It does not modify qualitatively the line shape of the absorptionedge
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
Classical ions
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-electronsare 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.
The gap of diamond (2/2)
Exp: Logothetidis et al.PRB 46, 4483 (1992)
Quantum (PI)MD calculations
Ramirez et al. PRB 73, 245202 (2006)
Isotopic Effects
At high T, independent of M (classical effect)
At low T,
zero point vibrations (quantum)
The quantistic zero-point motion effect
Parks et al. PRB 49,14244 (1994)
Spectroscopy:
theoretical point of view
What really theoreticians calculate!!
Finite temperature electronic and optical properties of zb-GaN
H. Kawai, K. Yamashita, E. Cannuccia, A. MariniPhys. Rev. B. 89, 085202 (2014)
Broadening induced
by electron-phonon scattering and
temperature dependence
Results: electronic band structureBreakdown of the QP picture
E. Cannuccia and A. MariniEurop. Phys. J. B. 85, 320 (2012)
Conclusions
Perturbative approach to the electron-phonon coupling
Band gap renormalization induced by the EPC
Finite temperature optical spectra
Thank you for your attention
Shrinking of the gap
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