Transcript E Cappelluti

Workshop on Quantum Fielt Theory aspects of Condensed Matter Physics,
LNF, Frascati, 7 September 2011
Infrared phonon activity and quantum
Fano interference in multilayer graphenes
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
(intensity and Fano asymmetry modulation)
theoretical approach
unified theory for phonon intensity (charged phonon)
and Fano asymmetry
tunable phonon switching effect
comparison with experiments
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
electronic states
optical conductivity
ZQ Li et al., Nat. Phys. 4, 532 (2008)
- doping dependence
- electronic interband features
- possible to extract bandgap 
KF Mak et al, PRL 102, 256405 (2009)
Probing interactions (and characterization) in graphenes
lattice dynamics
optical transitions
single layer
in-plane
in-plane
E2g (G)
out-of-plane
bilayer
Eg
Raman
Eu
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
integrated area
W'
VG Baonza, SSC 130, 383 (2004)
 d '()  ' 
BG
W'  Z 2
IR phonon spectroscopy
bilayer graphene
one allowed in-plane IR mode: antisymmetric (A) Eu
homo-atomic compound
first approximation: all the C atoms equal
charge equally distributed
q
no net dipole
q
q
no IR activity
q
IR phonon spectroscopy
taking into account the slight difference
between atomic sites
small charge disproportion
finite dipole Z ≈ (q1-q2)
q1
q2
q2
however
q 1, q 2 < n
limited by the total amount
of doped charge n
Z ≈ 10-3
q1
(static dipole)
no hope, thus..... but.....
Exp. results: Geneve group
tunable phonon
peak intensity
Zmax ~ 1.2!! huge!
as large as 1 electron over N=4 (sp3) !!
AB Kuzmenko et al, PRL 103, 116804 (2009)
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)
q2 -1 - 2qz
A = ABG + A' 2 2
q z +1


non coupled phonon
z=
|q| ≈ 

weakly coupled
q =
asymmetry
Fano parameter
strongly coupled


symmetric lineshape
 - 0
asymmetric lineshape
|q| ≈ 1
negative peak
|q| ≈ 0
Exp. results: Geneve group
four independent parameter fit
p2 q2 -1 - 2qz
 '( )   'BG ( ) =
4  q2 z2 +1
  - 0 
z=





p : related to intensity
q
: Fano asymmetry
0 : phonon frequency
 : phonon linewidth
p2 
1 
W'= 1  2 
8  q 
W=
p2
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
Eg (S) mode
Eu (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??
Eu (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 C60 systems
KxC60
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
 el ( )  i ( )
: el. polarizability (interband transitions)
 ( ) 
electronical background
of optical conductivity
direct light-phonon coupling
but these no polar materials:....
Charge-phonon effect
 el ( )  i ( )
: el. polarizability (interband transitions)
 ( ) 
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
 el ( )  i ( )
: el. polarizability (interband transitions)
 ( ) 
irreducible diagrams
electronical background
of optical conductivity
no phonon resonance
phonon mediated contribution
giving rise to resonance at phonon energy
Rice (Michael) theory
fundamental ingredients:
 tot ( )  i  ( )   x ( )  ( )Dph ( )
phonon resonance

Rice (Michael) theory
fundamental ingredients:
current/
electron-phonon
response function
 tot ( )  i  ( )   x ( )  ( )Dph ( )
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
: 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
: complex quantity
tunable charged-phonon effects controlled by external
voltage biases (doping and gap)
Microscopic Rice theory in bilayer graphene
three different response functions:
jj (el.background)
AA (ph. self-energy)
jA (charged-phonon effect)
we can compute microscopically each of them
Fano-Rice theory in bilayer graphene
interband transitions at low energy:
jA = RejA +iImjA
DAA ( ) 
 'ep ( )
jA complex quantity!!!
(in gapped systems: ImjA = 0)
1
  A  iA
 A



2  'jA (A )
AA
2
q  1  2zqA
q2A (1  z2 )
2
A
 'jA (A )
qA  
"jA (A )
Fano formula!
Fano and charged-phonon effects same origin!

it permits a microscopical identification
Peak parameters in Fano systems
Fano fit
 'ep ( )
2WA q 1  2zqA

A q2A (1  z2 )
2
A
 A
-integrated area
W'A 
  'jA (A )
2
WA 

A
  'jA (A )  "jA (A )
2

2

A
|qA| ≈ 0 (RejA=0)  negative peak but WA=0
not good
|qA| ≈ 1 (RejA = ImjA)  asymmetric peak but W’A=0
not good
pA 


2

  'jA (A )  "jA (A )
 
A
2
phonon
strength
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
system like a gapped semiconductor
Im = 0
4

3
2
1
no Fano effect
doping depedence of -integrated area W’
perfectly reproduced
what about WA?
negative area?
E Cappelluti et al, PRB 82, 041402 (2010)
Exp. results: Berkeley group
double-gated device
possible tuning doping and
 in independent way
n=0
n = 0 and   0: negative peak like us
Fano effect as a function of 
they attribute origin
of negative peak at n = 0
to Eg (S) (Raman-active) mode
(S allowed by symmetry in IR when   0)
T-Ta Tang et al, Nat Nanotechn 5, 32 (2010)
Different phonon channels in optical conductivity
gating induces z-axis asymmetry Ez
>0
Eg (S) mode also IR active!
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 dominance
of one channel vs. the other one
Optical channels and phonon switching in optical conductivity
- phase diagram
Berkeley
Eu-A and Eg-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
Geneve group
Eu (A)
Eg (S)
Eu (A)
E Cappelluti et al, PRB 82, 041402 (2010)
experimental integrated area and Fano asymmetry
interpolates and switches from A to S mode
AB Kuzmenko et al, PRL 103, 116804 (2009)
Trilayer 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, submitted to PRL (2011)
Trilayer 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 γ1 ≈ 0.55 eV in ABA,
at ω ≤ γ1 ≈ 0.39 eV in ABC
ABC closer to ω0 ≈ 0.2 eV
CH Lui et al, submitted to PRL (2011)
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 !!!
J Yan et al,
PRL 98, 166802 (2007)
C Casiraghi, PRB 80, 233407 (2009)
unlike
IR probes!
why?
Fano-Rice theory for Raman spectroscopy
effective mass approximation
dHˆ k
ˆ xy 

dkx dky
Raman vertex
electronic
Raman background
 ( )   T  ( )
Rice theory
Raman active
S mode
tot
irr
irr
S
irr
S
 ()   ()   ()DSS () ()
Fano-Rice theory for Raman spectroscopy
IR
Raman
EC
RejA ~ const.
ReS ~ EC
ImjA ~ const.
ImS ~ const.
ReS scaling with UV dispersion cut-off Ec
ReS >> ImS
W’S ≈ WS  Ec2
qS  
ReS (A )
ImS (A )
weakly dependent on band-structure
details (doping, )
 
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
Additional slides
Raman spectroscopy in bilayer graphene
focus on Eg symmetric mode Raman active
present also in single-layer graphene
J Yan et al, PRL 101, 136804 (2008)
T Ando, JPSJ 76, 104711 (2007)
frequency and linewidth OK with theoretical calculations
Fano-Rice theory for Raman spectroscopy
ex.: isotropic Raman scattering
two main quantities: S, A
EC
scaling with UV dispersion cut-off Ec
ReS ~ EC,
ImS ~ const.
ReA ~ const., ImA ~ const.
irr
 ep ( )   irr
S ( )DSS ( )  S ( )
pS » pA
qS  
dominant DSS channel
ReS (A )
ImS (A )
W’S ≈ WS  Ec2

 
irr
  irr
A ( )DAA ( )  A ( )
irr
+  irr
(

)D
(

)

S
SA
A ( )  h.c.
no Fano profile

weakly dependent
on band-structure
details (doping, )
Fano-Rice theory for Raman spectroscopy
effective mass approximation
dHˆ k
ˆ xy 

dkx dky
Raman vertex
electronic
Raman background
 ( )   T  ( )
Rice theory
=0
only S mode
coupled
tot
irr
irr
S
irr
S
 ()   ()   ()DSS () ()
Fano-Rice theory for Raman spectroscopy
effective mass approximation
dHˆ k
ˆ xy 

dkx dky
Raman vertex
electronic
Raman background
 ( )   T  ( )
Rice theory
ep
irr
S
irr
S
irr
A
irr
A
 ( )   ( )DSS ( )  ( )   ( )DAA ( ) ( )
irr
+ irr
(

)D
(

)

S
SA
A ( )  h.c.
0
phonon switching
possible
(in principle)
Fano-Rice theory for Raman spectroscopy
ex.: isotropic Raman scattering
two main quantities: S, A
EC
scaling with UV dispersion cut-off Ec
ReS ~ EC,
ImS ~ const.
ReA ~ const., ImA ~ const.
irr
 ep ( )   irr
S ( )DSS ( )  S ( )
pS » pA
qS  
dominant DSS channel
ReS (A )
ImS (A )
W’S ≈ WS  Ec2

 
irr
  irr
A ( )DAA ( )  A ( )
irr
+  irr
(

)D
(

)

S
SA
A ( )  h.c.
no Fano profile

weakly dependent
on band-structure
details (doping, )
Probing electronic spectrum: optical conductivity
bilayer (BL)
AB Kuzmenko et al, PRB 80, 165406 (2009)
KF Mak et al, PRL 102, 256405 (2009)
possible to extract gap  and doping n
vs. gate voltage Vg
Effective charge in IR spectroscopy
integrated area
effective charge
W'
 d '()  ' 
W’
BG
2VW ' M C
Z
CNe 2
V: volume
 unit cell, MC: carbon mass,
C constant, N: # atoms/cell
Z: effective charge put on ion positions to produce
exp. dipole upon lattice distortion as an
the same
ionic crystal
-Z
+Z
(related to oscillator
strength S, f)
ex. Na+ Cl-  Z = 1
VG Baonza, SSC 130, 383 (2004)


Phonon intensities in Fano systems??
q2 -1 - 2qz
 =  BG + A' 2 2
q z +1
two main popular choices:
A' p2 /4



for q  0, A’ 0    -1/(z2+1)
negative peak,
but p = 0
no good parameter
for q   W and W’ coincide
phonon intensity well defined
W =  d  () - BG 
integrated spectral area
however

W' p2 /8
p: phonon oscillator strength
however

for q  1, W 0    -2z/(z2+1)
negative and positive areas
cancel out
no good parameter
Peak parameters in Fano systems
Fano fit
 'ep ( )
2WA q 1  2zqA

A q2A (1  z2 )
2
A
 A
-integrated area
W'A 
  'jA (A )
2
WA 

A
  'jA (A )  "jA (A )
2

2

A
|qA| ≈ 0 (RejA=0)  negative peak but WA=0
not good
|qA| ≈ 1 (RejA = ImjA)  asymmetric peak but W’A=0
not good
pA 


2

  'jA (A )  "jA (A )
 
A
2
phonon
strength
Rice theory in bilayer graphene
multiband structure
  /2

v(kx  ik y )


v(k x  iky )
 /2



Hˆ k 


 /2
v(kx  ik y )


v(k x  iky )
 /2 

ˆ
d
H
k
ˆj x  e
dkx
 jj ( )   T j( ) j
 jj ( )
 ( )  
i

electronic background
EJ Nicol & JP Carbotte, PRB 77, 155409 (2008)
Microscopic Rice theory in bilayer graphene
el-ph interaction
=0
Eu (antisymmetric) mode
0

i

ˆ
VA  ig
0

0
Hep   kVˆAk A
k
el-ph contribution to ()
irr
irr
irr
tot
(

)


(

)


(

)D
(

)

jj
jj
jA
AA
Aj ()


i 0 0

0 0 0
0 0 i 

0 i 0
Doping dependence of phonon intensity in bilayer graphene
4
 = 0 : analytical calculations
3
jA ()   ()   ()   ()   ()
12
jA
13
jA
24
jA
34
jA
nm
nm
mn
 jA
()   jA
()   jA
()
 ()  ge vNsNv k
nm
jA

1

4 ( vk) 2   2
f (kn  )  f (km  )
kn  km    i
 damping: disorder/impurities/inhomogeneities
 = 0 particle-hole symmetry
jA=0
pA = 0 : no phonon intensity
2
Raman spectroscopy in bilayer graphene
A
A
S
A
S
S
>0
S
A
=0
S
LM Malard et al, PRL 101, 257401 (2008)
double peaks A-S evolve upon 
S mode intensity constant
why no phonon switching?
why no Fano asymmetry in both of them?
Fano-Rice theory for Raman spectroscopy
no Fano effect
only direct coupling to S-channel in Raman response
not two channels making phonon switching possible
irr
tot ()  irr()  irr
(

)D
(

)

S
SS
S ()
double-peak? encoded in DSS,

Optical channels and phonon mixing in optical conductivity
>0
mode mixing in phonon propagators
AA()
but also: current directly coupled to Eg S mode!!!
jS()  0
irr
irr
irr
irr
 ep
(

)


(

)D
(

)

(

)


(

)D
(

)

jj
jA
AA
Aj
jS
SS
Sj ( )
peak at A


irr
+  irr
(

)D
(

)

jA
AS
Sj ( )  h.c.
peak at S
Phonon hybridization self-energy
A and S lattice vibrations eigenmodes only for  = 0
  0 mode mixing through coupling to electronic excitations
DAA

DSA
DAS  D0AA
  
DSS  
1
1 AA
  
0
DSS  SA
AS 

SS 
DAA double peaked: contains a second (weaker) pole at S
 DSS double peaked: contains a second (weaker) pole at A
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 switching
it could never produce a dominant S peak in IR
neither 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
Z
Z
DSS ( ) 

      
conditions to resolve the double-peak structure:

Z- ≈ Z+ (triggered by )
|+ - +|  ph)
Double peaks in Raman spectroscopy
Raman spectroscopy in bilayer graphene
one problem: difficult to obtain absolute intensities
(estimated indirectly by looking at some reference phonon peak)
at =0 only S Eg mode Raman active
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)