Introduction to the electron-phonon renormalization of electronic band structure

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



Contemporary example: graphene

Electrons cannot stay put




beyond the

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?


A perturbative approach


beyond the

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



Electron phonon coupling at work

beyond the

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

Education

Introduction to the electron-phonon renormalization of electronic band structure