1. Interpretation of the Raman spectra of graphene and carbon nanotubes: the effects of Kohn anomalies and non-adiabatic effects
S. Piscanec
Cambridge University Engineering Department: Centre for Advanced Photonics and Electronics, Cambridge, UK

2. G-band in graphite and nanotubes
one single sharp G peak corresponding to q==0, mode E2g
Nanotubes:
Two main bands, G+ and G-.
Modes derived from graphite E2g
Metallic  semiconducting
Calculation of D_q for graphene: trivial and cheap.
Restricting

3. Common interpretation: curvature
Jorio et al. PRB 65, (2002)
G+: no diameter dependence
 LO axial
Calculation of D_q for graphene: trivial and cheap.
Restricting
G- diameter dependence  TO circumferential

4. Common interpretation: Fano resonance
In metallic tubes the G- peak is:
Downshifted
Broader
Depends on diameter
Interpretation
Fano resonance
Phonon-Plasmon interaction
Calculation of D_q for graphene: trivial and cheap.
Restricting
Electron-phonon coupling and Kohn anomalies have to be considered

5. Kohn anomalies Atomic vibrations are screened by electrons
In a metal this screening abruptly changes for vibrations associated to certain q points of the Brillouin zone.
Kink in the phonon dispersions: Kohn anomaly.
Graphite is a semi-metal
Nanotubes are folded graphite
Nanotubes can as well be metallic
Calculation of D_q for graphene: trivial and cheap.
Restricting

6. Kohn anomalies: when?
Everything depends on the geometry of the Fermi surface k1
k2 = k1+ q q
Fermi surface
q = phonon wavevector
k = electron wavevector
Calculation of D_q for graphene: trivial and cheap.
Restricting
k1 & k2= k1+q on the Fermi surface
Tangents to the Fermi surface at k1 and k2= k1+ q are parallel
W. Kohn, Phys. Rev. Lett. 2, 393 (1959) bold

7. Kohn anomalies in graphite
Graphite is a semi metal:
Fermi surface = 2 points: K and K’ = 2 K K K’ p* E G
q = K-K = 0 = G G G
q = K’-K = 2K - K = K K G EF G p
Calculation of D_q for graphene: trivial and cheap.
Restricting
Kohn Anomalies for:

8. Kohn anomalies in graphite
IXS data: J. Maultzsch et al. Phys. Rev. Lett. 92, (2004)
E2g
A’1
E2g
Calculation of D_q for graphene: trivial and cheap.
Restricting
2 sharp kinks for modes E2g at G and A1’ at K
Kohn Anomaly
EPC ≠ 0

9. Kohn anomalies in nanotubes
Metallic tubes: same geometrical conditions as graphite Ef p* p
Calculation of D_q for graphene: trivial and cheap.
Restricting
Metallic tubes: two Giant Kohn anomalies predicted
Semi-conducting tubes: NO Kohn anomalies predicted

10. Metallic tubes: LO-TO splitting
Circumferential
No KA
 G+
LO:
Axial
strong EPC
 G-
Opposite
Interpretation 10

11. Rely on Born-Oppenheimer approximation: electrons see fixed ions
Dynamic Effects
Frozen phonons
Finite differences
Density functional perturbation theory
Rely on Born-Oppenheimer approximation: electrons see fixed ions
Static
approaches
For 3D crystals this is 100% OK
Calculation of D_q for graphene: trivial and cheap.
Restricting
This is no longer true for 1D systems
The dynamic nature of phonons can be taken into account
Beyond Born-Oppenheimer…

12. Dynamic effects in nanotubes
smeared
New
LO: increased
TO: decreased

13. Phonons are not static deformations
Dynamic effects
Phonons are not static deformations
T increases:
weaker
no changes
d increases:
weaker
weaker
smeared
New

14. LO and TO frequencies

15. Th Vs Exp: Room Temperature
Metallic tubes:
G-LO & G+TO
Semiconducting tubes:
G- TO & G+ LO
Fermi golden rule:
EPC FWHM(G-)

16. Interpretation of Raman spectra
TO – circumferential
LO – axial
Semiconducting:
LO-TO splitting  curvature
G+  axial
G-  circumferential
LO – axial
TO – circumferential
Metallic:
LO-TO splitting  Kohn an.
G+  circumferential
G-  axial (KA)
FWHM(G-)  EPC
G- interpretation: EPC and not
Phonon-plasmon resonance
Piscanec et al. PRB (2007)

17. G- band Vs T: experiments
Metallic SWNTs
Dielectrophoresis
HiPCo SWNTs (Houston), d~1.1nm
Vpp = 20 V and f=3MHz
Raman Spectroscopy
 = 514 nm (resonant with semicon.)
 = 633 nm (resonant with metallic)
Linkam stage: 80K < T < 630K
Calculation of D_q for graphene: trivial and cheap.
Restricting
Krupke et al. Science 301, 344 (2003)

18. G- band Vs T: experiments
Calculation of D_q for graphene: trivial and cheap.
Restricting
Semiconducting tubes: G+ - G- constant  Anharmonicity
Metallic tubes: G+ - G- increases with T  ??? (EPC)

19. Th Vs Exp: Temperature Dependence
Metallic tubes from R. Krupke

20. Conclusions Measurement of the Raman G-band Vs T
Metallic tubes from dielecrophoresis
Semiconducting tubes  G+ - G- = constant
Metallic tubes  G+ - G- changes with T
Kohn anomalies and electron phonon coupling and dynamic effects
Interpretation of G-band in SWNTs Raman spectra
Explanation of the T-dependence of the G- in metallic SWNTs