finite temperature calculations of the electronic and optical

Finite temperature calculations of the

electronic and optical properties of solids and nano-structures

E. Cannuccia

Institut Laue Langevin BP 156 38042 Grenoble,

France

793 K = 68 meV QP gap is 1200 meV

YES ! We do need phonons.

““... unfortunately theorists do ... unfortunately theorists do not even bother to compare not even bother to compare their calculations with low-their calculations with low-

temperature measurements, temperature measurements, using more easily accessible using more easily accessible room temperature spectra."room temperature spectra."

Real life is at finite temperatureReal life is at finite temperature

M. Cardona Solid State Comm. 133, 3 (2005)

⟨u2T ⟩≈ ℏ

4M⟨12NBose T ⟩

The quantistic The quantistic zero-point zero-point

motion effectmotion effect

Outline... Outline...

Finite temperature excitons

Ab-Initio Polarons

Spectral functions and the QP-approximation

Components of the energy renormalizationComponents of the energy renormalization

Thermal Thermal expansionexpansion

Electron-Phonon interaction

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

>>

The Heine-Allen-Cardona Approach (I)The Heine-Allen-Cardona Approach (I)

For a review see M. Cardona, Solid State Commun. 133, 3 (2005).

I

RI

H (R+u)=H (0,e )+H (0, p)

+∑I∇ IV scf ({R})(r )uI+

12∑I∑J

∇ I∇ JV scf ({R})(r)u IuJ

H (0)δH e−p

(1)δH e−p

(2)

Ei(0)=ϵe+ϵpnp

∣Ψi(0) ⟩=∣e ⟩∣np ⟩

∣Ψi ⟩ , E i

δ Ei

Using the Rayleigh-Schrödinger Perturbation Theory, we get the

correction to the energy

δ Ei=⟨Ψi(0)∣δH e− p

(1) ∣Ψi(0)⟩+⟨Ψ i

(0)∣δH e−p(2) ∣Ψi

(0)⟩+⟨Ψi

(0)∣δH e−p(1) ∣Ψi

(1)⟩+...

First order PT Second order PT

V scf ({R})(r)=V ion(R)+V H ({R})(r )+V xc({R})(r )

The Heine-Allen-Cardona Approach (II)The Heine-Allen-Cardona Approach (II)

δ Ei=⟨Ψi(0)∣δH e− p

(2) ∣Ψi(0)⟩+⟨Ψ i

(0)∣δH e− p(1) ∣Ψi

(1)⟩

12∑IJ

⟨e∣⟨np∣∇ IJ2 V scf uI uJ∣np⟩∣e ⟩

We assume that [Ei(0)−H (0)

](−1)

≈(ϵe−H (0,e))(−1)

Phonons as a thermal bath !!!

δ Ee(β)=∑IJ[12⟨∇ IJ

2 V scf ⟩e+⟨∇ IV scf

∇ JV scf

ϵe−H(0, e)⟩e

] ⟨np∣u I uJ∣np⟩β

e

∑IJ⟨e∣⟨np∣∇ IV scf uI

∇ JV scf

Ei(0)−H (0)

uJ∣np ⟩∣e⟩

e→n k

β=1KT

Debye-Waller Fan

The Heine-Allen-Cardona Approach (III)The Heine-Allen-Cardona Approach (III)

δ En k (β)=∑IJ[12⟨∂2V scf

∂RI ∂RJ

⟩+∑m p(En k−Em p)

−1⟨∂V scf

∂RI

∣m p⟩ ⟨m p∣∂V scf

∂RI

⟩]⟨uI uJ ⟩

δ En k(β)=∑IJ{[≈ ]

∑n p

e−βn p ϵp ⟨np∣uI uJ∣np⟩

Z}

Thermal average

Bose Function 1+2B (ϵp)=1+2

eβϵp−1

uI∝∑p(b−p

+ +b p)

Clear dependence on the temperature

Polaron damping neglected

The Heine-Allen-Cardona Approach (IV)The Heine-Allen-Cardona Approach (IV)

The theory satisfies the “translational invariance condition”

∑I⟨n k∣

∂V scf

∂RI

∣m k ' ⟩u I=∑q λgnm k

qλ δk ' ,k−q(b−qλ+ +bqλ)

∑IJ⟨n k∣

∂2V scf

∂RI∂RJ

∣n k ⟩u IuJ=∑q λΛnn ' k

qλ (b−q λ+ +bq λ)(bqλ

+ +b−q λ)

Electron-Phonon Matrix Elements

p→qλ ,ϵp→ωq λk k⃗−q⃗

q

k kq

q

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)

Generalized Eliashberg Function Eliashberg function allows to visualize which phonon modes contribute to the electronic energy renormalization

Top Valence Band “Bottom” Conduction Band

to the role played by the FAN and DW in order to satisfy the translational invariance of the theory

g2FFAN (ω)=∑q λ[∑n '

gnn ' kq λ N q

−1

ϵn k−ϵn ' k ']δ(ω−ωqλ)

g2FDW (ω)=∑qλ[∑n '

Λn n' kqλ (−2N q)

−1

ϵn k−ϵn ' k]δ(ω−ωq λ)

δ En k (β)=∫dω(g2 FFAN (ω)+g2FDW (ω))(2B(ωq λ)+1)

Temperature dependence

Theory of Polarons: the MBPT perspective

Electron-phonon Self Energy Electron-electron Self Energy

Σ=iGD Σ=i GW

http://arxiv.org/abs/1304.0072E. Cannuccia and A. Marini (submitted PRB)

Many Body

States are not accessible All interactions between electron and phonons are included in the

Electron-Phonon Self Energy Operator

Heine Cardona Allen approach

“Standard” Rayleigh-Schrodinger 2nd order perturbation theory

Solid St. Comm. 133, 3 (2005)

PRB 33, 5501 (1986)PRB 23, 1495 (1981)

∣ϵnk−ϵn ' k−q∣≫ωq λ

STATIC & ADIABATIC

limitω≈ϵnk

Zn k=1

Enk=ϵnk+Zn k(ΣnkFan(ϵnk)+Σnk

DW )

Γnk=ℑΣnk (ϵn k)

Theory of Polarons: from MBPT to HAC approach



Ab-Initio Polarons


Excitons: the polaronic pictureExcitons: the polaronic picture

The poles of Lare the eigenstates of the Bethe-Salpeter Hamiltonian

HK , K 'el

=(ϵe−ϵh)δK ,K '+(v−W )K ,K '

Quasiparticle energies are real in the optical

range

The BS Hamiltonian is Hermitian

+ -=

L=L0−i L0[v−W ]LQuasihole and quasielectron

K=(e ,h)

Excitons: the polaronic pictureExcitons: the polaronic picture

The BS Hamiltonian is NOT Hermitian

=Quasihole and quasielectron

2 ,T ∝∑S T −E T

−1

τλ(T )∝[ℑ(Eλ (T ))]

−1

polaronspolarons

HK ,K '(T )=(Ee(T )−Eh (T ))+i [Γe(T )−Γh(T )]−i (v−W )K , K'

AM, AM, Phys. Rev. Lett.Phys. Rev. Lett. 101101, 106405 (2008), 106405 (2008)

Finite T excitonsFinite T excitons

Bright to dark (and vice versa) transitions

...gradual worsening of optical efficiency



Ab-Initio Polarons


Quasi particle SF

Green's functions: an (over)simplified pictureGreen's functions: an (over)simplified picture

Real particle SF

Spectral Function

Enk=ϵnk+Zn k(ΣnkFan(ϵnk)+Σnk

DW )

QP approximation

Enk=ϵnk

Enk=ℜ(Enk)+ iℑ(Enk)

Gn k(ω)=1

ω−ϵnk−ℜΣn kFan

(ω)−Σn kDW

−iℑΣn kFan

(ω)

An k(ω)=ℑΣn k

(ω−ϵn k−ℜΣn k)2+(ℑΣnk)

2

An kqp(ω)=

ℑEn k

(ω−ℜEn k)2+(ℑEn k)

2

The case of The case of DiamondDiamond

Logothedis et al. PRB 46, 4483 (1992)

-670 meV

“… a disagreement concerning the energy position of the first direct gap and its origin...”

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

Trans-polyacetylene

C-based nanostructures: polymersC-based nanostructures: polymers

Polyethylene

Zero-Point Motion

√ ⟨u2 ⟩≈0.4 a.u.√ ⟨u2 ⟩≈0.1a.u.√ ⟨uC

2 ⟩≈0.2a.u.

√ ⟨uH2 ⟩≈0.3a.u.

Breakdown of the QP pictureBreakdown of the QP picture

E. Cannuccia, E. Cannuccia, Phys. Rev. Phys. Rev. Lett.Lett. 107107, 255501 (2011), 255501 (2011)

Conclusions... Conclusions...


Ab-Initio Polarons


HAC approach = Static Perturbation Theory and as an adiabatic and static limit of the MBPT approachEvaluation of the renormalization of the electronic

energy as a function of T

The coupling with the lattice vibrations modifies the state-of-the-art picture of the excitonic states based on a frozen-atom approximation.

Polaronic-induced effect can be HUGE. They can even lead to the breakdown of the electronic

picture.

Thank you for your attention

2008

A. Marini PRL 101, 106405 (2008)

E. CannucciaPRL 107, 255501 (2011)

2005

R. B. Capaz et al. PRL. 94, 36801 (2005)F. Giustino, et al.

PRL, 105, 265501 (2010)

Electronic Gap: 7.715 eVRenormalization: 615 meV

2010 2011

ReferencesReferences

S. Zollner et al. PRB 45, 3376 (1992)

1992

finite temperature calculations of the electronic and optical

Documents