introduction to the electron phonon renormalization of the electronic band structure

Elena CannucciaInstitut Laue-Langevin, Grenoble (France)

Introduction to the electron Introduction to the electron phonon renormalization of phonon renormalization of electronic band structureelectronic band structure

The N particles The N particles world: world:

ionsions and and electronelectronss all togetherall together

Electron phonon renormalizationElectron phonon renormalization of electronic band structure of electronic band structure

Born–Oppenheimer approximation Born–Oppenheimer approximation

a perturbative approacha perturbative approach

Electron phonon at workElectron phonon at work

beyond the beyond the Born–Oppenheimer approximationBorn–Oppenheimer approximation

The separated worlds of The separated worlds of phononsphonons and electron and electronss

Electrons live in the bands generated by the ionic potential

Phonons are the quantizedionic vibrations on the potential

generated by the electrons

Coupling electrons and phonons …Coupling electrons and phonons …Superconductivity Joule's heating

Electron relaxation(luminescence)

Polaronic transport Coherent PhononsPeierls instability

Raman Spectroscopyetc......

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)

Energy levels renormalizationEnergy levels renormalization

Thermal Thermal expansionexpansion

Electron-Phonon Electron-Phonon interactioninteraction

P.B. Allen and M. Cardona Phys. Rev. B 27 4760 (1983)

>>

Where does the coupling come from?

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 +

12

∂

2V scf

∂ x2 u2 +...

Using Perturbation TheoryPerturbation Theory, we get the correction

to the energy

δ Ei=⟨Ψi(0)∣ ∣Ψi

(0)⟩ + ⟨Ψi

(0)∣ ∣Ψ i

(0)⟩ + ⟨Ψi

(0)∣ ∣Ψi

(1)⟩ +...

First order PT Second order PT

V scf (x+u)=V scf (x) + ∂V scf

∂ x u +

12

∂

2V scf

∂ x2 u2 +....

A perturbative approach:A perturbative approach:Heine-Allen-Cardona 2/2Heine-Allen-Cardona 2/2

Debye-Waller Fan

δ Ei(β) = [12

⟨∂

2V scf

∂ x2⟩ + ∑ j

(Ei−E j)−1

⟨∂V scf

∂ x∣j ⟩ ⟨ j∣

∂V scf

∂ x⟩] ⟨u2

⟩

Clear dependence on the Temperature

B(w) = Bose function

δ En k (β)=∑q λ n '[

∣gn n' kqλ ∣

En k−En ' k+q

−Λnn ' k

q λ

En k−En ' k

](2B (ωq λ)+1)

Thermal average

Average on theelectronic

wavefunction

FINAL FORMULA

All the previous theory can be reformulated in term of Green's function including nonadiabatic effects

Beyond the static Beyond the static perturbation theoryperturbation theory

Electron-phonon coupling from a MBPT perspective

ϵn k En k(T )+iΓn k(T )

Electron scatters with 1 phonon at a time

ElectronPhonon Self Energy

Temperature dependenceSpectral Functions

Enk

Γnk



Electron phonon coupling at workElectron phonon coupling at work


Spectroscopy: theoretical point of

viewWhat really theoreticians

calculate!!

Finite temperature electronic Finite temperature electronic and optical and optical

properties of zb-GaNproperties of zb-GaN

H. Kawai, K. Yamashita, E. Cannuccia, A. MariniPhys. Rev. B. 89, 085202 (2014)

What we can

do now!!!BroadeningBroadening induced

by electron-phonon scattering andtemperature dependence

The gap of diamond The 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

Classical ions

The gap of diamond The 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)

Isotopic EffectsIsotopic Effects

⟨u2⟩=⟨

h4Mω

{2[e−hω/KT−1]

−1+1}⟩

At high T, independent of M (classical effect)

At low T, zero point vibrations (quantum)

⟨u2⟩∝KT

⟨u2⟩∝M−1/2

The quantistic The quantistic zero-point zero-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

comparedwith experimental results.

Eg

MM→∞Eg electronic

Shrinking of the gapShrinking of the gap

It's time to revise previous electronic structure

calculations?

What about dynamical effects?What about dynamical effects?

Dynamical effects in diamondDynamical effects in diamond

Logothedis et al. PRB 46, 4483 (1992)

E. Cannuccia, Phys. Rev. Lett. 107, 255501 (2011)

Signature of Signature of the dynamical the dynamical

effectseffects

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)

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......

ConclusionsConclusions Perturbative approach to the electronphonon coupling

Finite temperature optical spectra

Band gap renormalization induced by the EPC

Dynamical effects on the electronic properties

Thank you for your attention

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 ...

Carbon contributionSi contrib.

TotalRenorm.

Temp. Dep. of gap: SiC, path integral molecular dynamics

Hernández, Herrero, Ramírez, Cardona, PRB 77 045210 2008

Polarons in an Hamiltonian representation (I)

k kq

k kq

q

q

Phonon population:

Basis setT = 0 °K

H

Many Body

GF's evaluated from the matrix diagonalization are equivalent to the

MB ones

Two bands model

Polaronic State:

More polarons than electrons!

H

T = 0 °KBasis Set

0

E

Spectral Function

Polarons in an Hamiltonian representation (II)

A. Eiguren and C. Ambrosch-Draxl, PRL 101 036402 (2008)

Quasi-particle Band Structure Induced by the Electron-phonon interaction on a surface