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18. Yan, J., Zhang, Y., Kim, P. & Pinczuk, A. Electric field effect tuning of electron-phonon coupling in
graphene. Preprint at (2006).
19. Tuinstra, F. & Koenig, J. Raman spectrum of graphite. J. Chem. Phys. 53, 1126 1130 (1970).
where c is the speed of light. At T = 0, ³ = 11 cm-1 for F = 0
20. Piscanec, S., Lazzeri, M., Mauri, F., Ferrari, A. C. & Robertson, J. Kohn anomalies and electron
and ³ drops to zero for F > hÉ0/2 because the scattering process
¯
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is forbidden by the Pauli exclusion principle22. Figure 2d shows a
of graphite layer and carbon nanotubes. Phys. Rev. B67, 035401 (2003).
good agreement between the experimental and theoretical ³, oncea
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electron-phonon coupling in graphite and nanotubes. Phys. Rev. B73, 155426 (2006).
constant inhomogeneous gaussian broadening of
23. Zhang, Y. et al. Landau-level splitting in graphene in high magnetic fields. Phys. Rev. Lett. 96,
to the electron phonon contribution of equation (7).
136806 (2006).
In summary, graphene is a remarkable example of ABO 24. Moos, G., Gahl, C., Fasel, R., Wolf, M. & Hertel, T. Anisotropy of quasiparticle lifetimes and the role
of disorder in graphite from ultrafast time-resolved photoemission spectroscopy. Phys. Rev. Lett. 87,
violation. Within ABO, the energy of a zone-centre phonon is
267402 (2001).
determined by two contributions: the distortion of the electronic
25. Kampfrath, T., Perfetti, L., Schapper, F., Frischkorn, C. & Wolf, M. Strongly coupled optical phonons
in the ultrafast dynamics of electronic energy and current relaxation in graphite. Phys. Rev. Lett. 95,
bands, associated with the phonon displacement, and the adiabatic
187403 (2005).
rearrangement of the Fermi surface. In graphene, these two
26. Lazzeri, M. & Mauri, F. Non-adiabatic Kohn-anomaly in a doped graphene monolayer. Phys. Rev.
Lett. 97, 266407 (2006).
contributions cancel out exactly because of the peculiar rigid
27. Ando, T. Anomaly of optical phonon in monolayer graphene. J. Phys. Soc. Jpn. 75, 124701 (2006).
motion of the Dirac cones, associated with the E2g phonon.
28. Castro Neto, A. H. & Guinea, F. Electron-phonon coupling and Raman spectroscopy in graphene.
Phys. Rev. B75, 045404 (2007).
In general, a correct phonon treatment should not include
29. Ashcroft, N. W. & Mermin, N. D. Solid State Physics (Saunders College, London, 1976).
the adiabatic rearrangement of the Fermi surface whenever the
30. Piscanec, S., Lazzeri, M., Robertson, J., Ferrari, A. C. & Mauri, F. Optical phonons in carbon
electron-momentum relaxation time is longer than the inverse nanotubes: Kohn anomalies, peierls distortions and dynamic effects. Phys. Rev. B75, 035427 (2007).
of the phonon pulsation, as occurs in graphene and in several
Acknowledgements
metals (see, for example, Table 1.3 of ref. 29). Note that the
The authors thank P. Kim and A. Pinczuk for useful discussions and for sending us a preprint of
stronger the electron phonon coupling with q = 0 phonons, the
ref. 18. A.C.F. acknowledges funding from the Royal Society and The Leverhulme Trust. The
calculations were carried out at IDRIS (Orsay).
larger the difference between ABO and non-ABO frequencies.
Correspondence and requests for materials should be addressed to A.C.F or F.M.
However, the lattice dynamics is well described by ABO if the
Supplementary Information accompanies this paper on www.nature.com/naturematerials.
electron phonon coupling with q = 0 phonons is so strong that
Competing financial interests
the electron-momentum relaxation is faster than the lattice motion.
The authors declare that they have no competing financial interests.
We anticipate that the ABO breakdown described here will
affect the vibrational properties of carbon nanotubes30 and Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/
nature materials VOL 6 MARCH 2007 www.nature.com/naturematerials 201
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