The Non-Neighbor Effect in Graphene Ribbons

Abstract

Graphene is the name given to a flat monolayer of carbon atoms tightly packed into a twodimensional honeycomb lattice (Novoselov et al., 2004), and is a rapidly rising star on the horizon of materials science and condensed matter physics. This two-dimensional material exhibits exceptionally high crystal and electronic quality and has already revealed a cornucopia of new physics and potential applications. Charge transport properties in graphene are greatly different from that of conventional two-dimensional electronic systems as a consequence of the linear energy dispersion relation near the charge neutrality point (Dirac point) in the electronic band structure (Geim & Novoselov, 2007; Novoselov et al., 2005; Zhang et al., 2005). Theoretically, the energy band structure of a graphite monolayer had been investigated using the tight-binding approximation (Wallace, 1947). In the work of Wallace, the nearestand next-nearest-neighbor interaction for the 2pz orbitals in graphene were considered, but the wave function overlap between carbon atoms was neglected. Since his aim is to show how the π-electron distribution is related to the electrical conductivity of graphite, he did not attempt to draw the band distribution. In 1952, Coulson & Taylor considered the overlap integrals between atomic orbitals in studying the band structure of the graphite monolayer. Their work suggested that the overlap was important for the electronic density of states and referred mainly to the π states, leading to a description of the conduction band (Coulson & Taylor, 1952). To study the valence bands in graphene, Lomer used the group-threoretical method to deal with the electronic energy bands based on the three atomic orbitals 2s, 2px, and 2py (Lomer, 1955). Because there are two atoms per unit cell, there are six basis functions to be considered, and in general the tight binding model must lead to a 6×6 determinantal secular equation for the energy. The method used group theory is able to solve it easily. Slonczewski and Weiss found that the Lomer’s work can be simplified greatly by a different choice of the location of the origin (Slonczewski & Weiss, 1958). A better tight-binding description of graphene was given by Saito et al. (Saito et al., 1998), which considers the nonfinite overlap between nearest neighbors, but includes only interactions between nearest neighbors. To understand the different levels of approximation, Reich et al. started from the most general form of the secular equation, the tight binding Hamiltonian, and the overlap matrix to calculate the band structure (Reich et al., 2002). But their work did not involve the effect of the non-nearest-neighbor interaction on the band structure. This work will be discussed in details in Section 2. Because there is no energy gap, perfect graphene sheets are metallic. How open the gap of graphene? According to the quantum size effect, graphene nanoribbons maybe achieve this