Optical phonon properties, Fano interference and charged-phonon ...

1° Workshop on Nanoscience: Graphene, NCKU, Tainan City, Taiwan, 16 December 2011
Emmanuele Cappelluti
Instituto de Ciencia de Materiales de Madrid (ICMM) , CSIC, Madrid, Spain
Institute of Complex Systems (ISC), CNR, Rome, Italy,
Lara Benfatto
ISC, CNR, Rome, Italy
Alexey B. Kuzmenko
Dept. Physics Uni. Geneve, Switzerland
and: Z.Q. Li, C.H. Lui, T. Heinz (Columbia, NY, USA)

Outline
motivations (limits of Raman spectroscopy)
experimental measurements in bilayer graphene
(intensity and Fano asymmetry of ph peak in optical conductivity)
theoretical approach
unified theory for phonon intensity (charged phonon)
and Fano asymmetry
tunable phonon switching effect
multilayer graphenes
conclusions

Probing interactions (and characterization) in graphenes
electronic states
ARPES
- dispersion anomalies
- renormalization
- linewidth
A Bostwick et al., NJP 9, 385 (2007)
DC Elias et al., Nat Phys 7, 701 (2011)

Probing interactions (and characterization) in graphenes
optical conductivity
electronic states
ZQ Li et al., Nat. Phys. 4, 532 (2008)
- doping dependence
- electronic interband features
- possible to extract bandgap Δ
KF Mak et al, PRL 102, 256405 (2009)

Probing interactions (and characterization) in graphenes
optical transitions
lattice dynamics
single layer
in-plane
in-plane
out-of-plane
E 2g (G)
bilayer
E g
Raman
E u
IR

Raman spectroscopy
phonon intensity
C Casiraghi, PRB 80, 233407 (2009)
I Calizo et al, JAP 106, 043509 (2009)
difficult access to absolute phonon intensity
relative intensity between different peaks instead used

Raman spectroscopy
focus on:
ph. frequency
ph. linewidth
J Yan et al, PRL 98, 166802 (2007)

Raman spectroscopy
- not only characterization, also fundamental physics
doping dependence
of phonon frequency and linewidth:
evidence of nonadiabatic
breakdown of Born-Oppenheimer
S Pisana et al, Nat Mat 6, 198 (2007)

Raman spectroscopy
investigation tools:
peak frequency
peak linewidth
relative (non absolute) peak intensity
but
no modulation of intensity
no asymmetric peak lineshape
J Yan et al, PRL 98, 166802 (2007)

IR phonon spectroscopy
suitable tool???

IR phonon spectroscopy
IR phonon peak best resolved in ionic systems
-Z +Z
Z: dipole effective charge
(related to oscillator strength S, f)
ex. Na + Cl - → Z = 1
VG Baonza, SSC 130, 383 (2004)
W'=
integrated area
∫
[ ]
dω σ'(ω) − σ' BG
W' ∝ Z 2

IR phonon spectroscopy
bilayer graphene
one allowed in-plane IR mode: antisymmetric (A) E u
homo-atomic compound
first approximation: all the C atoms equal
charge equally distributed
no net dipole
q q
no IR activity
q
q

IR phonon spectroscopy
taking into account the slight difference
between atomic sites
small charge disproportion
finite dipole Z ≈ (q 1 -q 2 )
q 1
q 2
however
q 1 , q 2 < n
limited by the total amount
of doped charge n
Z ≈ 10 -3
(static dipole)
q 2
q 1
no hope, thus..... but.....

Exp. results: Geneve group
tunable phonon
peak intensity
W’ (integrated area)
Z ∝
W'
AB Kuzmenko et al, PRL 103, 116804 (2009)
effective Born charge: Z max ~ 1.2!! huge!
as large as 1 electron over N=4 (sp 3 ) !!

Exp. results: Geneve group
tunable phonon
peak intensity
neutrality point (NP) n=0
also problem: negative peak area…
Z not defined…?
AB Kuzmenko et al, PRL 103, 116804 (2009)

Negative peak: Fano effect and quantum interference
arising from quantum interference (coupling)
between a discrete state (phonon) with continuum spectrum (electronic)
A=A BG
+A' q2 -1-2qz
q 2 ( z 2 +1) z=ω -ω 0
Γ
q =
asymmetry
Fano parameter
non coupled phonon
weakly coupled
strongly coupled
symmetric lineshape
|q| ≈ ∞
asymmetric lineshape
|q| ≈ 1
negative peak
|q| ≈ 0

Exp. results: Geneve group
four independent parameter fit
σ'(ω) − σ' BG
(ω) = ω 2
p
4πΓ
⎡
z= ω -ω ⎤
0
⎣
⎢
Γ ⎦
⎥
q 2 -1-2qz
q 2 z 2 +1
( )
ω p : related to intensity
q : Fano asymmetry
ω 0 : phonon frequency
Γ : phonon linewidth
W'= ω 2
⎛
p
⎜
8 1 − 1 ⎝ q 2
⎞
⎟
⎠
W= ω 2
p
8
“bare” intensity (in the absence of Fano)
AB Kuzmenko et al, PRL 103, 116804 (2009)

Exp. results: Geneve group
phonon softening with doping:
ok with LDA and TB theory
E g (S) mode
E u (A) mode
T Ando, JPSJ 76, 104711 (2007)
AB Kuzmenko et al, PRL 103, 116804 (2009)

Exp. results: Geneve group
phonon linewidth: strong
increase at NP: why??
E u (A) mode?
T Ando, JPSJ 76, 104711 (2007)
AB Kuzmenko et al, PRL 103, 116804 (2009)

Exp. results: Geneve group
linear dependence of bare
intensity with doping:
where from? why so huge Z?
NB: tight-binding
calculations
AB Kuzmenko et al, PRL 103, 116804 (2009)

Exp. results: Geneve group
linear dependence of bare
intensity with doping:
where from? why so huge Z?
Fano asymmetry: where from?
related to el. optical background?
points out finite intensity at n=0…!
AB Kuzmenko et al, PRL 103, 116804 (2009)

Charge-phonon effect
doped insulators: organic and C 60 systems
K x C 60
huge intensity increase
of selected IR modes
upon electron doping x
doping
SC Erwin, in Backminsterfullerenes (1993)
K-J Fu et al, PRB 46, 1937 (1992)

Charge-phonon effect
σ el
(ω) = −iωχ(ω)
χ: el. polarizability (interband transitions)
χ(ω) =
electronical background
of optical conductivity
direct light-phonon coupling
but these no polar materials:....

Charge-phonon effect
σ el
(ω) = −iωχ(ω)
χ: el. polarizability (interband transitions)
χ(ω) =
electronical background
of optical conductivity
direct light-phonon coupling
but these no polar materials:....
no intrinsic dipole
further channels to be considered

Rice (Michael) theory
electronic polarizability provides finite IR intensity to
phonon modes allowed but otherwise not active
σ el
(ω) = −iωχ(ω)
χ: el. polarizability (interband transitions)
χ(ω) =
irreducible diagrams
electronical background
of optical conductivity
phonon mediated contribution
giving rise to resonance at phonon energy
no phonon resonance

Rice (Michael) theory
fundamental ingredients:
σ tot
(ω) = −iω[ χ(ω) + λ ν
xχ(ω)χ(ω)D ph
(ω)]
phonon resonance

Rice (Michael) theory
fundamental ingredients:
current/
electron-phonon
response function
σ tot
(ω) = −iω[ χ(ω) + λ ν
xχ(ω)χ(ω)D ph
(ω)]
intensity ruled by the current/electron-phonon response function

Rice theory in bilayer graphene
χ: real function (α doping) tuning the phonon intensity

Rice theory in bilayer graphene
χ: real function (α doping) tuning the phonon intensity
Rice theory: in its original application: semiconductors
effective theory:
interesting peculiarities of bilayer graphene:
zero gap semiconductor:
low energy interband transitions
Fano asymmetry
χ: complex quantity
tunable charged-phonon effects controlled by external
voltage biases (doping and gap)

Microscopic Rice theory in bilayer graphene
three different response functions:
χ jj (el.background)
χ AA (ph. self-energy)
χ jA (charged-phonon effect)
we can compute microscopically each of them

Fano-Rice theory in bilayer graphene
interband transitions at low energy:
D AA
(ω) =
χ jA = Reχ jA +iImχ jA
1
ω − ω A
+ iΓ A
σ' ep
(ω) = 2 [ χ' (ω ) ] 2
jA A
ω ≈ωA
ω A
Γ A
χ jA complex quantity!!!
(in gapped systems: Imχ jA = 0)
q 2 A
−1 + 2zq A q A
= − χ' (ω ) jA A
q 2 A
(1 + z 2 )
χ" jA
(ω A
)
Fano formula!
Fano and charged-phonon effects same origin!
it permits a microscopical identification

Peak parameters in Fano systems
Fano fit
σ' ep
(ω)
ω ≈ωA
= 2W A
πΓ A
q A 2 −1 + 2zq A
q A 2 (1 + z 2 )
W A
= π [ χ' (ω ) ] 2
jA A
ω A
ω-integrated area
W' A
=
{[ ] 2 − [ χ" jA
(ω A
)] 2
}
π χ' jA
(ω A
)
ω A
|q A | ≈ 0 (Reχ jA =0) ⇒ negative peak but W A =0
not good
|q A | ≈ 1 (Reχ jA = Imχ jA ) ⇒ asymmetric peak but W’ A =0 not good
p A
=
{[ ] 2 + [ χ" jA
(ω A
)] 2
}
π χ' jA
(ω A
)
ω A
phonon
strength

Theory vs. experiments
n c = αV g
Au contacts
modelling field effect:
gating induces doping
but also vertical electric field E z
(inducing gap)
SiO 2
Si
GRAPHENE
step by step analysis:
(1) we first consider only the doping effects induced by gate
(2) we later consider only the electric effects induced by gate

Phonon intensity in bilayer graphene
(1) gating induces doping but not E z
in this case no low-energy transitions between 2 and 3
system like a gapped semiconductor
Imχ = 0 no Fano effect
4
γ
1
3
2
doping depedence of ω-integrated area W’
perfectly reproduced
what about W A ?
negative area?
E Cappelluti et al, PRB 82, 041402 (2010)

Exp. results: Berkeley group
double-gated device
possible tuning doping and
Δ in independent way
n = 0
n = 0 and Δ ≠ 0: negative peak like us
Fano effect as a function of Δ
they attribute origin
of negative peak at n = 0
to E g (S) (Raman-active) mode
(S allowed by symmetry in IR when Δ ≠ 0)
T-Ta Tang et al, Nat Nanotechn 5, 32 (2010)

Different phonon channels in optical conductivity
gating induces z-axis asymmetry E z
two main IR channels present
Δ > 0
E g (S) mode also IR active!
probes D AA ph. propagator
probes D SS ph. propagator
relative “intensity” ruled by p A and p S
total spectra dependent on the relative dominance
of one channel vs. the other one

Optical channels and phonon switching in optical conductivity
Δ-μ phase diagram
Berkeley
E u -A and E g -S modes
dominant in different regions
of phase diagram:
possible switching of intensity
from one mode to other one
Geneve
E Cappelluti et al, PRB 82, 041402 (2010)

Phonon switching in optical conductivity
E u (A)
Geneve group
E g (S)
E u (A)
E Cappelluti et al, PRB 82, 041402 (2010)
AB Kuzmenko et al, PRL 103, 116804 (2009)
experimental integrated area and Fano asymmetry
interpolates and switches from A to S mode

Multilayer graphenes
N=3,..,6 layer with different stacking order on substrate (no gating)
intensity and Fano asymmetry strong depending on N-layer and stacking order
ABA and ABC deeply
different
stacking revealed
ZQ Li, CH Lui, E Cappelluti, L Benfatto, KF Mak, L Carr, J Shan, TF Heinz, arXiv:1109.6367

Multilayer graphenes
N=3,..,6 layer with different stacking order on substrate (no gating)
intensity and Fano asymmetry strong depending on N-layer and stacking order
ABA and ABC deeply
different
stacking revealed
ZQ Li, CH Lui, E Cappelluti, L Benfatto, KF Mak, L Carr, J Shan, TF Heinz, arXiv:1109.6367

Trilayer gated graphenes and stacking order
ABA and ABC deeply
different
stacking revealed
phonon intensity
and phonon frequency
strongly doping dependent
in ABC but not in ABA
good agreement
with theory
CH Lui et al, preprint (2011)

Trilayer gated graphenes and stacking order
fundamental ingredient: electronic band structure
reminder: phonon activity is triggered by electronic particle-hole excitations
upon doping, el. transitions
at ω = √2 γ 1 ≈ 0.55 eV in ABA,
at ω ≤ γ 1 ≈ 0.39 eV in ABC
CH Lui et al, submitted to PRL (2011)
ABC closer to ω 0 ≈ 0.2 eV
phonon activity amplified

Raman spectroscopy in bilayer graphene
remarkable features:
|q| ≈ ∞ no Fano asymmetry !!! (in IR S mode had q ≈ 0)
intensity does not depend on doping !!!
unlike
IR probes!
why?
J Yan et al,
PRL 98, 166802 (2007)
C Casiraghi, PRB 80, 233407 (2009)

Fano-Rice theory for Raman spectroscopy
effective mass approximation
ˆ γ xy
∝
d ˆ H k
dk x
dk y
Raman vertex
χ γγ
(τ) = − T τ
γ(τ)γ
electronic
Raman background
Rice theory
Raman active
S mode
χ tot γγ
(ω) = χ irr γγ
(ω) + χ irr γS
(ω)D SS
(ω)χ irr Sγ
(ω)

Fano-Rice theory for Raman spectroscopy
IR Raman
Reχ jA ~ const.
Reχ γS ~ E C
E C
Imχ jA ~ const.
Imχ γS ~ const.
Reχ γS scaling with UV dispersion cut-off E c
Reχ γS >> Imχ γS
W’ S ≈ W S ∝ E c
2
weakly dependent on band-structure
details (doping, Δ)
q S
= − Reχ γS (ω A )
Imχ γS
(ω A
) ≈ − ∞
no Fano profile

Conclusions
source of microscopic IR phonon intensity
unified theory of IR intensity and Fano profile
more information encoded in phonon intensity and Fano factor
phonon mode switching predicted (and observed)
differences between IR and Raman spectroscopy accounted for
alternative and powerful tool to characterize ML graphenes