Phonon dispersion in graphene

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LECTRONIC PROPERTIES OF SOLIDS 1

Phonon Dispersion in Graphene L. A. Falkovsky

Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow, 117334 Russia Institute of High Pressure Physics, Russian Academy of Sciences, Troitsk, Moscow oblast, 142190 Russia e-mail: [email protected] Received February 19, 2007

Abstract—Taking into account the constraints imposed by the lattice symmetry, we calculate the phonon dispersion for graphene with interactions between the first and second nearest neighbors. We show that only five force constants give a very good fitting to the elastic constants and phonon frequencies observed in graphite. PACS numbers: 63.20.Dj, 81.05.Uw, 71.15.Mb DOI: 10.1134/S1063776107080122 1

1. INTRODUCTION

Since the discovery of graphene (a single atomic layer of graphite) [1,2], much attention has been devoted to its electronic properties. Now, Raman spectroscopy [3] extends to investigations of graphene. For interpretations of Raman scattering and transport phenomena, detailed knowledge of the lattice dynamics and electron–phonon interactions is needed [4]. Several models [5–12] have been proposed to calculate the phonon dispersion in bulk graphite. The most improved ones [9, 10] involve many (up to twenty) parameters. Recently, detailed measurements and firstprinciple calculations of optical phonon frequencies were performed for graphite [13]. They show qualitative disagreement with models [5, 12], which employ central and angular atomic forces between the first and second neighbors in the graphite lattice. The passage in the lattice dynamics from graphite to graphene and then to nanotubes was examined in [14] using the model in [5]. Numerical calculations of the dynamical matrix in terms of the electron energy for graphene were performed in [15]. The first-principle calculations [16] of the dynamical properties of graphite and graphene (as well as of diamond) show that differences between the phonon frequencies in graphene and the related ones in graphite are negligible in comparison with the experimental errors for these frequencies in graphite. This could be intuitively expected for the highest frequencies because interactions between the layers in graphite are weak. Our aim here is to find an analytic description of the phonon dispersion in graphene. This can be done in the framework of the Born–von Karman model for the honeycomb graphene lattice with interactions only between the first and second nearest neighbors, but with 1 The

the constraints imposed by the lattice symmetry taken into account. We find that the out-plane (bending) modes are described by two force constants, one of which is determined by the corresponding Raman frequency and the other by the smallest elastic constant C44. For the in-plane modes, the lattice stability condition with respect to rotation of the layer as a whole around the z axis allows reducing the number of force constants to three. These constants are extracted from comparison with experimental data for graphite. We do not pay close attention

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