Surface solitons at the edges of graphene nanoribbons

We demonstrate numerically that armchair graphene nanoribbons can support vibrational localized states in the form of surface solitons. Such localized states appear through self-localization of the vibrational energy along the edge of the graphene nanoribbon, and they decay rapidly inside the structure. We find five types of such solitary waves including in-plane and out-of-plane edge breathers and moving envelope solitons. Copyright c © EPLA, 2010 Introduction. – Remarkable properties of graphite structures make their study one of the hot topics of nanoscience [1]. Graphene nanoribbons are effectively low-dimensional structures similar to carbon nanotubes, but their main feature is the presence of edges. Due to the edges, graphene nanoribbons can demonstrate many novel geometry-driven properties, depending on their width and helicity. A majority of the current studies of graphene nanoribbons are devoted to the analysis of their electronic and magnetic properties modified by the presence of edges, including the existence of the edge modes [2,3], which are an analog of surface states in the two-dimensional geometry. In general, surface states are spatially localized modes which can be generated at the surfaces [4], and such surface states have been studied in many branches of physics including electrons in crystals [5,6], surface phonons [7], surface polaritons [8], and optical surface modes in modulated waveguide arrays [9]. Vibrational surface modes localized at the edges of graphene nanoribbons [10] can be considered as a phonon analog of Tamm states which is well known in the electronic theory. Edge phonon modes can be localized only in one direction, transversally to the ribbon length, while they remain propagating modes along the edges. However, taking into account anharmonicity may lead to the nonlinearityinduced localization of the vibrational energy along the edges, so that the vibrations become self-trapped in the second dimension creating both in-plane and out-of-plane (a)E-mail: [email protected] surface solitons. Such solitons differ substantially from the twisting solitons found earlier for carbon nanotubes [11], and they are supported by the edges. The main aim of this letter is to study surface solitons of this type for the case of armchair nanoribbons. In particular, we find numerically five types of such solitary waves including in-plane and out-of-plane edge breathers and moving envelope solitons. Model of graphene nanoribbons. – We model a graphene nanoribbon as a planar stripe of graphite, with properties depending on the stripe width and chirality. Here we consider the armchair nanoribbon (see fig. 1(a)). The structure of the armchair nanoribbon can be presented as a longitudinal repetition of the elementary cell composed of K atoms (the number K is a multiple of 4), as shown in fig. 1(a). We use the atom numbering shown in fig. 1(a). In this case, each carbon atom has a two-component index α= (n, k), where n= 0,±1,±2, . . . stands for the number of the elementary cells, and k= 1, 2, . . . ,K stands for the number of atoms in the cell. Each elementary cell of the armchair nanoribbon has four edge atoms. In fig. 1(a), we show these edge atoms as filled circles. We consider a hydrogen-terminated nanoribbon, where edge atoms correspond to the molecular group CH. We consider such a group as a single effective particle at the location of the carbon atom. Therefore, in our model of graphene nanoribbons we take the mass of atoms inside the stripe as M0 = 12mp, and for the edge atoms we consider a larger mass M1 = 13mp (where mp = 1.6603 · 10 kg is the proton mass).