Infrared phonon activity and quantum Fano interference in multilayer graphenes

Infrared phonon activity and Infrared phonon activity and quantum Fano interference in quantum Fano interference in multilayer graphenesmultilayer graphenes

Emmanuele Cappelluti

Workshop on Quantum Fielt Theory aspects of Condensed Matter Physics, LNF, Frascati, 7 September 2011

Instituto de Ciencia de Materiales de Madrid (ICMM) , CSIC, Madrid, SpainInstitute 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

(intensity and Fano asymmetry modulation)

theoretical approach

unified theory for phonon intensity (charged phonon)

and Fano asymmetry

comparison with experiments

conclusions

tunable phonon switching effect

Probing interactions (and characterization) in graphenes

electronic states

ARPES- dispersion anomalies- renormalization

A Bostwick et al., NJP 9, 385 (2007)

- linewidth

DC Elias et al., Nat Phys 7, 701 (2011)


electronic states

optical conductivity

KF Mak et al, PRL 102, 256405 (2009)- possible to extract bandgap - electronic interband features

ZQ Li et al., Nat. Phys. 4, 532 (2008)

- doping dependence


lattice dynamics

in-plane

out-of-plane E2g (G)

in-plane

Eg

RamanEu

IR

single layer

bilayer

optical transitions

Raman spectroscopy

C Casiraghi, PRB 80, 233407 (2009)

phonon intensity

I Calizo et al, JAP 106, 043509 (2009)

difficult access to absolute phonon intensityrelative intensity between different peaks instead used

Raman spectroscopy

J Yan et al, PRL 98, 166802 (2007)

ph. frequency

ph. linewidth

focus on:

- not only characterization, also fundamental physics

S Pisana et al, Nat Mat 6, 198 (2007)

doping dependenceof phonon frequency and linewidth:

evidence of nonadiabaticbreakdown of Born-Oppenheimer

Raman spectroscopy

Raman spectroscopy

investigation tools:

peak frequency

peak linewidthrelative (non absolute) peak intensity

J Yan et al, PRL 98, 166802 (2007)

but

no modulation of intensityno asymmetric peak lineshape

IR phonon spectroscopy

suitable tool???


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'= dω σ '(ω) − σ 'BG[ ]∫integrated area

€

W' ∝ Z 2


bilayer graphene

homo-atomic compound

qq

one allowed in-plane IR mode: antisymmetric (A) Eu

first approximation: all the C atoms equal

qq

no net dipole

no IR activity

charge equally distributed


small charge disproportion

q2q1

taking into account the slight differencebetween atomic sites

q1q2

finite dipole Z ≈ (q1-q2)however

q1, q2 < nlimited by the total amount

of doped charge nZ ≈ 10-

3 (static dipole)

no hope, thus..... but.....

Exp. results: Geneve group

AB Kuzmenko et al, PRL 103, 116804 (2009)

Zmax ~ 1.2!! huge!as large as 1 electron over N=4 (sp3) !!

tunable phononpeak intensity



tunable phononpeak intensity

also problem: negative peak area…Z not defined…?

neutrality point (NP) n=0

Negative peak: Fano effect and quantum interference

non coupled phonon weakly coupled

arising from quantum interference (coupling)between a discrete state (phonon) with continuum spectrum (electronic)

€

A=ABG +A'q2 - 1- 2qz

q2 z2 +1( )

€

z =ω - ω0

Γ

|q| ≈ symmetric lineshape asymmetric lineshape

|q| ≈ 1

negative peak

|q| ≈ 0

strongly coupled

asymmetryFano parameter

€

q =


four independent parameter fit

€

σ '(ω) − σ 'BG (ω) =ωp

2

4 πΓ

q2 - 1- 2qz

q2 z 2 +1( )

€

z =ω - ω0

Γ ⎡ ⎣ ⎢

⎤ ⎦ ⎥


ωp : related to intensityq : Fano asymmetryω0 : phonon frequency

Γ : phonon linewidth

€

W =ωp

2

8

€

W'=ωp

2

81−

1

q2

⎛

⎝ ⎜

⎞

⎠ ⎟

“bare” intensity (in the absence of Fano)



phonon softening with doping:ok with LDA and TB theory

T Ando, JPSJ 76, 104711 (2007)

Eu (A) mode

Eg (S) mode



phonon linewidth: strong increase at NP: why??

T Ando, JPSJ 76, 104711 (2007)

Eu (A) mode?



linear dependence of bare intensity with doping:where from? why so huge Z?

NB: tight-bindingcalculations



Fano asymmetry: where from?related to el. optical background?points out finite intensity at n=0…!

linear dependence of bare intensity with doping:where from? why so huge Z?

Charge-phonon effect

K-J Fu et al, PRB 46, 1937 (1992)

KxC60

dopinghuge intensity increase of selected IR modes upon electron doping x

SC Erwin, in Backminsterfullerenes (1993)

doped insulators: organic and C60 systems

€

σel(ω) = −iωχ (ω) : el. polarizability (interband transitions)

€

(ω) =

electronical backgroundof optical conductivity

direct light-phonon coupling

but these no polar materials:....


€


€

(ω) =


direct light-phonon coupling

but these no polar materials:....

further channels to be considered

no intrinsic dipole


Rice (Michael) theory

€


irreducible diagrams

electronic polarizability provides finite IR intensity tophonon modes allowed but otherwise not active

€

(ω) =


no phonon resonance

phonon mediated contribution

giving rise to resonance at phonon energy


€

σtot (ω) = −iω χ (ω) + λ ν xχ (ω)χ (ω)Dph (ω)[ ]

fundamental ingredients:

phonon resonance


€

σtot (ω) = −iω χ (ω) + λ ν xχ (ω)χ (ω)Dph (ω)[ ]

fundamental ingredients:

current/electron-phononresponse function

intensity ruled by the current/electron-phonon response function

Rice theory in bilayer graphene

: real function (α doping) tuning the phonon intensity


interesting peculiarities of bilayer graphene:

zero gap semiconductor:low energy interband transitions : complex quantity Fano asymmetry

tunable charged-phonon effects controlled by externalvoltage biases (doping and gap)

effective theory:

: real function (α doping) tuning the phonon intensity

Rice theory: in its original application: semiconductors

three different response functions:

we can compute microscopically each of them

Microscopic Rice theory in bilayer graphene

jj (el.background) AA (ph. self-energy) jA (charged-phonon effect)

Fano-Rice theory in bilayer graphene

interband transitions at low energy:

jA = RejA +iImjA jA complex quantity!!!

€

DAA (ω) =1

ω − ωA + iΓA

€

σ 'ep (ω)ω ≈ωA

=2 χ 'jA (ωA)[ ]

2

ωAΓA

qA2 −1+ 2zq A

qA2 (1+ z 2 )

Fano formula!

Fano and charged-phonon effects same origin!

€

qA =− 'jA (ωA)

χ "jA (ωA)

it permits a microscopical identification

(in gapped systems: ImjA = 0)

Peak parameters in Fano systems

€


=2W A

πΓA

qA2 −1+ 2zq A

qA2 (1+ z 2 )

Fano fit

|qA| ≈ 0 (RejA=0) negative peak but WA=0

€

WA =π 'jA (ωA)[ ]

2

ωA

€

W'A =π 'jA (ωA)[ ]

2− χ "jA (ωA)[ ]

2

{ }

ωA

ω-integrated area

|qA| ≈ 1 (RejA = ImjA) asymmetric peak but W’A=0 not good

not good

€

pA =π 'jA (ωA)[ ]

2+ χ "jA (ωA)[ ]

2

{ }

ωA

phononstrength

Phonon intensity in bilayer graphene

Step by step analysis: gating induces doping but not Ez

in this case low-energy transitions between 2 and 3

1

4

2

3

doping depedence of ω-integrated area W’perfectly reproduced

what about WA?negative area?

system like a gapped semiconductor

Im = 0 no Fano effect

E Cappelluti et al, PRB 82, 041402 (2010)

Exp. results: Berkeley group

T-Ta Tang et al, Nat Nanotechn 5, 32 (2010)

double-gated devicepossible tuning doping and

in independent way

n = 0

n = 0 and 0: negative peak like us

Fano effect as a function of

they attribute originof negative peak at n = 0to Eg (S) (Raman-active) mode

(S allowed by symmetry in IR when 0)

Different phonon channels in optical conductivity

> 0gating induces z-axis asymmetry Ez

two main IR channels present

probes DAA ph. propagator

probes DSS ph. propagator

relative “intensity” ruled by pA and pS

total spectra dependent on the relative dominanceof one channel vs. the other one

Eg (S) mode also IR active!

Optical channels and phonon switching in optical conductivity

- phase diagram

Eu-A and Eg-S modesdominant in different regions

of phase diagram:possible switching of intensityfrom one mode to other one

Geneve

Berkeley

E Cappelluti et al, PRB 82, 041402 (2010)

Phonon switching in optical conductivity

Geneve group


experimental integrated area and Fano asymmetryinterpolates and switches from A to S mode

E Cappelluti et al, PRB 82, 041402 (2010)Eu (A)

Eg (S)

Eu (A)

Trilayer graphenes and stacking order

ABA and ABC deeplydifferent

stacking revealed

phonon intensityand phonon frequency

strongly doping dependentin ABC but not in ABA

good agreementwith theory

CH Lui et al, submitted to PRL (2011)

Trilayer graphenes and stacking order

fundamental ingredient: electronic band structure

reminder: phonon activity is triggered by electronic particle-hole excitations

CH Lui et al, submitted to PRL (2011)

upon doping, el. transitionsat ω = √2 γ1 ≈ 0.55 eV in ABA,

at ω ≤ γ1 ≈ 0.39 eV in ABC

ABC closer to ω0 ≈ 0.2 eV

phonon activity amplified

Raman spectroscopy in bilayer graphene

remarkable features:

intensity does not depend on doping !!!

|q| ≈ no Fano asymmetry !!! (in IR S mode had q ≈ 0)

C Casiraghi, PRB 80, 233407 (2009)

unlikeIR probes!

why?

J Yan et al,PRL 98, 166802 (2007)

Fano-Rice theory for Raman spectroscopy

€

ˆ γ xy ∝d ˆ H k

dkxdky

€

(τ ) = − Tτ γ(τ )γ electronicRaman background

effective mass approximation

Raman vertex

€

tot (ω) = χ γγ

irr (ω) + χ γSirr (ω)DSS (ω)χ S γ

irr (ω)

Rice theory

Raman activeS mode


ReS scaling with UV dispersion cut-off Ec

€

qS =−ReS (ωA)

Im χ γS (ωA)≈ − ∞ no Fano profile

W’S ≈ WS Ec2 weakly dependent on band-structure

details (doping, )

ECReS ~ EC

ImS ~ const.

RejA ~ const.

ImjA ~ const.

IR Raman

ReS >> ImS

Conclusions

unified theory of IR intensity and Fano profile

phonon mode switching predicted (and observed)

alternative and powerful tool to characterize ML graphenes

more information encoded in phonon intensity and Fano factor

differences between IR and Raman spectroscopy accounted for

source of microscopic IR phonon intensity

Additional slides


focus on Eg symmetric mode Raman active

J Yan et al, PRL 101, 136804 (2008) T Ando, JPSJ 76, 104711 (2007)

frequency and linewidth OK with theoretical calculations

present also in single-layer graphene


two main quantities: S, A

ex.: isotropic Raman scattering

scaling with UV dispersion cut-off Ec

ReS ~ EC, ImS ~ const.

ReA ~ const., ImA ~ const.

€

qS =−ReS (ωA)

Im χ γS (ωA)≈ − ∞

dominant DSS channelpS » pA

no Fano profile


details (doping, )

€

ep(ω) = χ γS

irr (ω)DSS (ω)χ S γirr (ω)

+ χ γAirr (ω)DAA(ω)χ Aγ

irr (ω)

+ χ γSirr (ω)DSA (ω)χ Aγ

irr (ω) + h.c.[ ]

EC


€


dkxdky

€



Raman vertex

€

tot (ω) = χ γγ


irr (ω)

Rice theory

= 0

only S modecoupled


€


dkxdky

€



Raman vertex

Rice theory

0phonon switching

possible(in principle)

€

ep(ω) = χ γS

irr (ω)DSS (ω)χ S γirr (ω) + χ γA

irr (ω)DAA(ω)χ Aγirr (ω)


irr (ω) + h.c.[ ]


two main quantities: S, A

ex.: isotropic Raman scattering

scaling with UV dispersion cut-off Ec

ReS ~ EC, ImS ~ const.

ReA ~ const., ImA ~ const.

€

qS =−ReS (ωA)

Im χ γS (ωA)≈ − ∞

dominant DSS channelpS » pA

no Fano profile


details (doping, )

€

ep(ω) = χ γS

irr (ω)DSS (ω)χ S γirr (ω)

+ χ γAirr (ω)DAA(ω)χ Aγ

irr (ω)


irr (ω) + h.c.[ ]

EC

Probing electronic spectrum: optical conductivity

bilayer (BL)

KF Mak et al, PRL 102, 256405 (2009)

AB Kuzmenko et al, PRB 80, 165406 (2009)

possible to extract gap and doping n vs. gate voltage Vg

Effective charge in IR spectroscopy

€

W'= dω σ '(ω) − σ 'BG[ ]∫integrated areaW’

VG Baonza, SSC 130, 383 (2004)

€

Z =2VW ' MC

CπNe2effective charge

V: volume unit cell, MC: carbon mass, C constant, N: # atoms/cell

Z: effective charge put on ion positions to producethe same exp. dipole upon lattice distortion as an ionic crystal +Z-Z

ex. Na+ Cl- Z = 1

(related to oscillatorstrength S, f)

Phonon intensities in Fano systems??

€

σ =σBG +A'q2 - 1- 2qz

q2 z2 +1( )

€

A'=ωp2 /4 πΓ

ωp: phonon oscillator strength

however for q 0, A’ 0 σ -1/(z2+1)

negative peak, but ωp = 0 no good parameter

€

W = dω σ(ω) - σBG[ ]∫integrated spectral area

however for q 1, W 0 σ -2z/(z2+1)

€

W'=ωp2 /8[ ]

negative and positive areascancel out no good parameter

for q W and W’ coincidephonon intensity well defined

two main popular choices:

Peak parameters in Fano systems

€


=2W A

πΓA

qA2 −1+ 2zq A

qA2 (1+ z 2 )

Fano fit

|qA| ≈ 0 (RejA=0) negative peak but WA=0

€

WA =π 'jA (ωA)[ ]

2

ωA

€

W'A =π 'jA (ωA)[ ]

2− χ "jA (ωA)[ ]

2

{ }

ωA

ω-integrated area

|qA| ≈ 1 (RejA = ImjA) asymmetric peak but W’A=0 not good

not good

€

pA =π 'jA (ωA)[ ]

2+ χ "jA (ωA)[ ]

2

{ }

ωA

phononstrength


€

ˆ H k =

Δ /2 v(kx − iky )

v(kx + iky ) Δ /2 γ

γ −Δ /2 v(kx − iky )

v(kx + iky ) −Δ /2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

€

ˆ j x = −ed ˆ H kdkx

multiband structure

€

jj (τ ) = − Tτ j(τ ) j

€

σ(ω) = −χ jj (ω)

iω

EJ Nicol & JP Carbotte, PRB 77, 155409 (2008)electronic background

Microscopic Rice theory in bilayer graphene

= 0 Eu (antisymmetric) mode

€

Hep = ψ k+

k∑ ˆ V Aψ kφA

€

ˆ V A = ig

0 −i 0 0

i 0 0 0

0 0 0 i

0 0 −i 0

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟el-ph contribution to σ(ω)

€

jjtot (ω) = χ jj

irr (ω) + χ jAirr (ω)DAA(ω)χ Aj

irr (ω)

el-ph interaction

Doping dependence of phonon intensity in bilayer graphene

€

jA (ω) = χ jA12 (ω) + χ jA

13 (ω) − χ jA24 (ω) − χ jA

34 (ω)

= 0 : analytical calculations

€

jAnm (ω) = π jA

nm (ω) − π jAmn (ω)

€

π jAnm (ω) = ge hvN sN v

γ

4 (hvk)2 + γ 2k∑ f (εkn − μ) − f (εkm − μ)

εkn − εkm + hω + iη

damping: disorder/impurities/inhomogeneities

= 0 particle-hole symmetry jA=0

pA = 0 : no phonon intensity

1

4

2

3


LM Malard et al, PRL 101, 257401 (2008)

A

A

A

S

=0

>0

S

S

S S

A

double peaks A-S evolve upon

why no phonon switching?why no Fano asymmetry in both of them?

S mode intensity constant


only direct coupling to S-channel in Raman response

€

tot (ω) = χ γγ


irr (ω)

no Fano effect

double-peak? encoded in DSS,

not two channels making phonon switching possible

Optical channels and phonon mixing in optical conductivity

> 0mode mixing in phonon propagators

but also: current directly coupled to Eg S mode!!! jS(ω) 0

€

jjep(ω) = χ jA

irr (ω)DAA(ω)χ Ajirr (ω) + χ jS

irr (ω)DSS (ω)χ Sjirr (ω)

+ χ jAirr (ω)DAS (ω)χ Sj

irr (ω) + h.c.[ ]

AA(ω)

peak at ωA peak at ωS

Phonon hybridization self-energy

A and S lattice vibrations eigenmodes only for = 0

0 mode mixing through coupling to electronic excitations

€

DAA DAS

DSA DSS

⎛

⎝ ⎜

⎞

⎠ ⎟−1

=DAA

0

DSS0

⎛

⎝ ⎜

⎞

⎠ ⎟

−1

−ΠAA ΠAS

ΠSA ΠSS

⎛

⎝ ⎜

⎞

⎠ ⎟

DAA double peaked: contains a second (weaker) pole at ωS

DSS double peaked: contains a second (weaker) pole at ωA

~

LM Malard et al (2008); T Ando M Koshino (2009); P Gava et al (2009)

double peak only at very large origin of double peak deeply different from phonon switchingit could never produce a dominant S peak in IRneither a dominat A peak in Raman

Double peaks in Raman spectroscopy

Raman spectroscopy only probed the direct S-channel

but for > 0: mode mixing in phonon propagators

double paks in DSS

conditions to resolve the double-peak structure:

€

DSS (ω) =Z−

ω −ω−

+Z+

ω −ω+

Z- ≈ Z+ (triggered by )

|ω+ - ω+| Γph)

Double peaks in Raman spectroscopy


one problem: difficult to obtain absolute intensities

at =0 only S Eg mode Raman active

(estimated indirectly by looking at some reference phonon peak)

J Yan et al, PRL 101, 136804 (2008) T Ando, JPSJ 76, 104711 (2007)

frequency and linewidth OK with theoretical calculations

|q| ≈ : while no Fano asymmetry? (in IR S mode had q ≈ 0)

Documents

Infrared phonon activity and quantum Fano interference in multilayer graphenes