Post on 30-Dec-2015
description
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)
Probing interactions (and characterization) in graphenes
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
Probing interactions (and characterization) in graphenes
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 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'= dω σ '(ω) − σ 'BG[ ]∫integrated area
€
W' ∝ Z 2
IR phonon spectroscopy
bilayer graphene
homo-atomic compound
one allowed in-plane IR mode: antisymmetric (A) Eu
first approximation: all the C atoms equal
no net dipole
no IR activity
charge equally distributed
IR phonon spectroscopy
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
Exp. results: Geneve group
AB Kuzmenko et al, PRL 103, 116804 (2009)
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 =
Exp. results: Geneve group
four independent parameter fit
€
σ '(ω) − σ 'BG (ω) =ωp
2
4 πΓ
q2 - 1- 2qz
q2 z 2 +1( )
€
z =ω - ω0
Γ ⎡ ⎣ ⎢
⎤ ⎦ ⎥
AB Kuzmenko et al, PRL 103, 116804 (2009)
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)
Exp. results: Geneve group
AB Kuzmenko et al, PRL 103, 116804 (2009)
phonon softening with doping:ok with LDA and TB theory
T Ando, JPSJ 76, 104711 (2007)
Eu (A) mode
Eg (S) mode
Exp. results: Geneve group
AB Kuzmenko et al, PRL 103, 116804 (2009)
phonon linewidth: strong increase at NP: why??
T Ando, JPSJ 76, 104711 (2007)
Eu (A) mode?
Exp. results: Geneve group
AB Kuzmenko et al, PRL 103, 116804 (2009)
linear dependence of bare intensity with doping:where from? why so huge Z?
NB: tight-bindingcalculations
Exp. results: Geneve group
AB Kuzmenko et al, PRL 103, 116804 (2009)
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:....
Charge-phonon effect
€
σel(ω) = −iωχ (ω) : el. polarizability (interband transitions)
€
(ω) =
electronical backgroundof optical conductivity
direct light-phonon coupling
but these no polar materials:....
further channels to be considered
no intrinsic dipole
Charge-phonon effect
Rice (Michael) theory
€
σel(ω) = −iωχ (ω) : el. polarizability (interband transitions)
irreducible diagrams
electronic polarizability provides finite IR intensity tophonon modes allowed but otherwise not active
€
(ω) =
electronical backgroundof optical conductivity
no phonon resonance
phonon mediated contribution
giving rise to resonance at phonon energy
Rice (Michael) theory
€
σtot (ω) = −iω χ (ω) + λ ν xχ (ω)χ (ω)Dph (ω)[ ]
fundamental ingredients:
phonon resonance
Rice (Michael) theory
€
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
Rice theory in bilayer graphene
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
€
σ 'ep (ω)ω ≈ωA
=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
AB Kuzmenko et al, PRL 103, 116804 (2009)
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
Fano-Rice theory for Raman spectroscopy
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
Raman spectroscopy in bilayer graphene
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
Fano-Rice theory for Raman spectroscopy
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
W’S ≈ WS Ec2 weakly dependent on band-structure
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
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
= 0
only S modecoupled
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
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)
+ χ γSirr (ω)DSA (ω)χ Aγ
irr (ω) + h.c.[ ]
Fano-Rice theory for Raman spectroscopy
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
W’S ≈ WS Ec2 weakly dependent on band-structure
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
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
€
σ 'ep (ω)ω ≈ωA
=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
Rice theory in bilayer graphene
€
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
Raman spectroscopy in bilayer graphene
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
Fano-Rice theory for Raman spectroscopy
only direct coupling to S-channel in Raman response
€
tot (ω) = χ γγ
irr (ω) + χ γSirr (ω)DSS (ω)χ S γ
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
Raman spectroscopy in bilayer graphene
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)