Symmetry based analysis of the Kohn anomaly and electron-phonon interaction in graphene and carbon nanotubes I. Milošević, 1, * N. Kepčija, 1 E. Dobardžić, 1 M. Mohr, 2 J. Maultzsch, 2 C. Thomsen, 2 and M. Damnjanović 1 1 NanoLab, Faculty of Physics, University of Belgrade, Studentski Trg 12, 11001 Belgrade, Serbia 2 Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany Received 20 March 2010; revised manuscript received 3 June 2010; published 30 June 2010 Symmetry based analysis of the electron-phonon coupling in graphene and carbon nanotubes is performed and Kohn anomalies, their Brillouin-zone positions together with the complete set of good quantum numbers are predicted. Interestingly, graphene dynamical representation is found to contain only a small portion of quite a large set of inequivalent irreducible representations of the relevant full symmetry group. Besides, vanishing of the electron-phonon interaction for majority of the normal displacements is also shown to be a consequence of the symmetry. The results are further numerically confirmed within full and tight-binding density-functional calculations and force constants model and enhanced coupling to the Fermi level electrons of the Dirac point A 1g mode with respect to the  point E 2g mode is confirmed. Finally, energy dispersion of the kink phonon spectrum is analytically evaluated and compared to the classical phonon spectrum. DOI: 10.1103/PhysRevB.81.233410 PACS numbers: 81.05.ue, 88.30.rh, 31.15.xh, 63.22.Rc Graphene 1,2 being an exciting and unusual material in many respects and having prospects of a new technological marvel attracts tremendous attention of quite a wide spec- trum of researches. Kohn anomaly 3 pertains to the most im- portant fundamental topics of graphene physics due to its relevance for Raman spectroscopy which has an essential role in investigating this, in many ways unique two- dimensional 2D surface being thus easily accessible to Ra- man scattering measurements. 4 Namely, electron-phonon coupling is particularly interesting in graphitic materials due to their specific pointlike Fermi surface. In graphene it is electron-phonon coupling which strongly softens phonon fre- quencies giving rise to Kohn anomalies which occur for phonons having wave number q such that there are two elec- tronic states k 1 and k 2 = k 1 + q at the Fermi surface. In the phononic spectrum of a metal a Kohn anomaly is manifested as discontinuity in the derivative of the dispersion relation that occurs at certain points of the Brillouin zone BZ, pro- duced by the abrupt change in the screening of lattice vibra- tions by conduction electrons. In graphene, the electronic gap vanishes only at the two equivalent K Brillouin-zone points the so called Dirac points because of the linear dis- persion corresponding to the massless Dirac fermions which are connected by the vector K. Thus, Kohn anomalies can occur for central,  point phonons and q = K, Dirac point phonons. Also, in metallic single-wall carbon nanotubes the Fermi surface consists of only two points and Kohn anoma- lies occur only for phonons with zero wave vector or with the wave vector q connecting these two Fermi-surface points. Due to their quasi-one-dimensionality the armchair carbon nanotubes are expected to exhibit stronger Kohn anomaly than graphene. 5,6 In this Brief Report we use full symmetry of graphene 7 and carbon nanotubes 8 in order to discuss electron-phonon interaction and to point out the direct consequences of sym- metry, completely independent on the model of dynamics used. The symmetry-based results are further numerically confirmed within full and tight-binding density-functional calculations and force constants model. Graphene, one-atom-thick allotrope of carbon with a hon- eycomb structure made out of hexagons, is a two- dimensional crystal with diperiodic symmetry 9 group DG80 = TD 6h the symmetry generators are shown in Fig. 1. For considerations restricted to the in-plane modes, horizontal mirror symmetry can be ignored and normal subgroup C 6v of the full symmetry group D 6h can be used 10 . Note that DG80 is not a subgroup of the nonsymmorphic space group P6 3 / mmc of graphite. 11 Graphene is a single orbit system, generated by the subgroup DG3 = TC 2 , with the stabilizer isomorphic to D 3h . Despite this large stabilizer with twelve elements, graphene is not invariant under any Euclidean su- pergroup. However, its dynamical representation D dyn con- tains only a small portion of quite a large set of inequivalent irreducible representations of DG80. It is symmetry which also predicts vanishing of the electron-phonon interaction for many normal displacements. Decomposition of the dynamical representation onto the irreducible components 7 is FIG. 1. Right panel: Brillouin zone and irreducible domain of graphene. Notation of the high symmetry lines and points, as well as of the interior domain, are introduced. Left panel: graphene sym- metry generators: 2D translations, rotation for  / 3, horizontal and vertical mirror reflections. PHYSICAL REVIEW B 81, 233410 2010 1098-0121/2010/8123/2334104 ©2010 The American Physical Society 233410-1