- The document discusses electron-phonon coupling in graphene using density functional theory (DFT) and many-body perturbation theory (MBPT). - DFT underestimates the phonon dispersion near the K point compared to experiments. MBPT within the GW approximation improves upon DFT by increasing the bandwidth and Fermi velocity. - Frozen phonon calculations within GW can accurately reproduce the electron-phonon coupling and explain experimental results like the Raman D-line dispersion.

1. o
Electron-Phonon Coupling
in graphene
Claudio Attaccalite
Trieste 10/01/2009

2. Outline
●
DFT: ground state properties
Many­Body Perturbation Theory: excited state properties
●
Quasi­Particle band structure
●
Phonons and Electron­Phonon Coupling (EPC) in DFT (LDA
and GGA)
●
It is possible to go beyond DFT and obtain an accurate
description of the EPC
●
Recent Raman experiments can be reproduced completely
ab­initio

3. Density Functional Theory
Usually Exc
in the local density approximation (LDA) or the general
gradient approximation (GGA) successfully describes the ground state
properties of solids.
E[n]=T 0[n]−
e2
2
∫
nrnr ' 
∣r−r '∣
dr dr 'Exc[n]−∫nrvxcrdr
The ground state energy is expressed in terms of the density and
an unknown functional Exc
Kohn­Sham eigenfunctions are obtained from
−ℏ
2
2m
∂
2
∂ r
2
V SCF r
 nr=n nr

4. Beyond DFT:
Many­Body Perturbation Theory
[T V hVext ] n, k r∫dr'  r ,r' ;En, k  n, k r' =En, k n, k r
Starting from the LDA Hamiltonian we construct the
Quasi­Particle Dyson equation:
: Self­Energy Operator; En, k
: Quasi­particle energies;
... following Hedin(1965): the self­energy operator
is written as a perturbation series
of the screened Coulomb interaction
=i G W ⋯
G: dressed Green Function
W: in the screened interaction
W =−1
v

5. Quasi­Particles Band Structure
Comparison of GW with ARPES experiments
for graphite A. Gruenis, C.Attaccalite et al. PRL 100, 189701 (2008)
In GW the bandwidth is increased and
consequently the Fermi velocity vF
is enhanced

6. what about the phonons?

7. Motivations
●
Kohn anomalies
●
Phonon­mediated superconductivity
●
Jahn­Teller distortions
●
Electrical resistivity
Electron-Phonon Coupling (EPC)
determines:

8. Phonon dispersion of Graphite
(IXS measurements) and DFT calculations
● M. Mohr et al. Phys. Rev. B 76, 035439 (2007)
● J. Maultzusch et al. Phys. Rev. Lett. 92, 075501 (2004)
● L. Wirtz and A. Rubio, Solid State Communications 131, 141 (2004)

9. Phonon dispersion close to K
In spite of the general good agreement the
situation is not clear close to KK

10. Phonon dispersion close to K
In spite of the
general good
agreement the
situation is not clear
close to KK

11. Raman Spectroscopy of graphene

12. k
K
ωphonon
at Γ point, k~0
→ G-line
Single-resonant
G-line
Raman G­line

13. K K'
{
ℏ q {
q
Raman D­line
dispersive

14. .... but also from Raman....

15. How to calculate phonons in GW?
Ideal solution: calculate total energy and its derivatives in GW
Problem: how to calculate total energy in GW
(questions of self-consistence and of numerical feasibility)
Phonon frequencies (squared) are
eigenvalues of the dynamical matrix
Dst
 
q=
∂
2
E
∂ us
 
q∂t

q

16. Electron­Phonon Coupling
and dynamical matrix
q=Dq/ m
Dq=BqPq
Pq= 4
N k
∑k
∣D kq 
, k∣2
k , −kq ,

-bands self-energy-bands self-energy
Dkq 
,k 
=〈kq ,
∣ V∣k ,〉Electron-Phonon Coupling

17. Phonon dispersion without
dynamical matrix of the  bands
=Bq/m

18. ..but using GW band structure
provides a worse result
Pq= 4
N k
∑k
∣D kq 
, k∣2
k , −kq ,

q=BqPq/ m
because
where
In fact the GW correction to the electronic bands alone results
in a larger denominator providing a smaller phonon slope and
a worse agreement with experiments.

19. Frozen Phonons Calculation of the EPC
The electronic Hamiltonian for
the  and * bands
can be written as 2x2 matrix:
H k ,0=
ℏ vf k
0
0
−ℏ v f k
H k ,u=H k ,0
∂ H k ,0
∂u
uO u2

a distortion of the lattice according to the ­E2G
∂ H k ,0
∂ u
=2〈 D
2
〉F a
b
b
−a
If we diagonalize H(k,u) at the K­point
where:
〈 D
2
〉F =limd 0
1
16  E
d 
2
!!!!!
We can get the EPC for the  bands with a frozen phonon calculation!!!

20. EPC and  in different approximations
q=〈 Dq
2
〉F / g
To study the changes on the phonon slope we recall that Pq
is
the ratio of the square EPC and band energies
Pq= 4
N k
∑k
∣D kq 
, k∣2
k , −kq ,

Thus we studied:
Hartree­Fock equilibrium structure

21. ...the same result ...
but with another approach
Electron and phonon
renormalization
of the EPC vertex
D. M. Basko et I. L. Aleiner
Phys. Rev. B 77, 0414099(R) 2008
Dirac massless fermions
+
Renormalization Group

22. How to model the phonon dispersion
to determine the GW phonon dispersion we assume
q
GW
≃Bq
GW
rGW
Pq
DFT
/m
rGW
=
K
GW
K
DFT
≃
PK
GW
PK
DFT
Bq
GW
≃BK
GW
We assume Bq
constant because it is expected
to have a small dependence from q and fit it from
the experimental measures
where
The resulting K A phonon frequency is ′ 1192 cm−1
which is our best
estimation and is almost 100 cm−1
smaller than in DFT.

23. Results: phonons around K
Phys. Rev. B 78, 081406(R) (2008) M. Lazzeri, C. Attaccalite, L. Wirtz and F. Mauri

24. Results: the Raman D­line dispersion

25. Conclusions
●
The GW approach can be used to calculate EPC
●
In graphene and graphite DFT(LDA and GGA)
underestimates the phonon dispersion of the highest
optical branch at the zone­boundaries
●
B3LYP gives a results similar to GW but overestimates
the EPC
●
It is possible to reproduce completely ab­initio the
Raman D­line shift

26. Acknowledgment
http://www.pwscf.org
http://www.yambo-code.org
2) My collaborators
M. Lazzeri, L. Wirtz, A. Rubio and F. Mauri
Codes I used:Codes I used:
1) The support from:
http://www.crystal.unito.it

27. The case of Doped Graphene
Therefore the effective interaction felt by the
electrons starts to be weaker due
the stronger screening of the Coulomb potential.
With doping graphene evolves from a semi­metal to a real metal.
http://arxiv.org/abs/0808.0786 C. Attaccalite et al.
Solid State Commun. 143, 58 (2007) M. Polini et al.

28. Electron­phonon Coupling at K
LDA results is recovered in doping graphene
WORK IN PROGRESS . . .

29. Tuning the B3LYP
The B3LYP hybrid-functional has the from:
Exc=1−A
˙
Ex
LDA
B ˙Ex
BECKE
A ˙Ex
HF
1−C ˙Ec
VWN
C ˙Ec
LYP
B3LYP consist of a mixture of Vosko-Wilk-Nusair and LYP correlation part Ec
and a mixture of LDA/Becke exchange with Hartree-Fock exchange
The parameter A controls the admixture
of HF exchange in the standard B3LYP is 20%
A(%) M gap
12% 176.96 5.547 31.93
13% 185.50 5.662 32.99
14% 194.39 5.695 34.13
15% 203.65 5.769 35.30
20% 256.03 6.140 41.70
GW 193 4.89 39.5
<DK
2
> K
It is possible to reproduce GW results
tuning the non-local exchange in B3LYP !!!!!

30. Many­Body perturbation Theory 2
Approximations for G and W (Hybertsen and Louie, 1986):
• Random phase approximation (RPA) for the dielectric function.
• General plasmon­pole model for dynamical screening.
G≈GLDA
we use the LDA results as starting point
and so the Dyson equation becomes
TVhV extV xcn, k ∫' r ,r ' ;En, k  n, k=En, k n, k r
' r ,r' ; En , k =' r ,r' ;En, k − r ,r ' V xcr

31. Raman spectroscopy of graphene
k
K
ωphonon
at Γ point, k~0
→ G-line
Single-resonant
G-line
Ref.: S. Reich, C. Thomsen, J. Maultzsch, Carbon Nanotubes, Wiley-VCH (2004)
D-line
0 1500 3000
Raman shift (cm-1)
Intensity(a.u)
graphite 2.33 eV
D
G
D‘ G‘
TO mode between K and M
dispersive

32. Electron­Hole Coupling