Abstract
We will present a brief overview of the electronic and transport properties of graphene nanoribbons focusing on the effect of edge shapes and impurity scattering. The low-energy electronic states of graphene have two non-equivalent massless Dirac spectra. The relative distance between these two Dirac points in the momentum space and edge states due to the existence of zigzag-type graphene edges is a deciding factor in the electronic and transport properties of graphene nanoribbons. In graphene nanoribbons with zigzag edges (zigzag nanoribbons), two valleys related to each Dirac spectrum are well separated in momentum space. The propagating modes in each valley contain a single chiral mode originating from a partially flat band at the band center. This feature gives rise to a perfectly conducting channel in the disordered system, if impurity scattering does not connect the two valleys, i.e. for long-range impurity (LRI) potentials. Ribbons with short-range impurity potentials, however, display ordinary localization behavior through inter-valley scattering. On the other hand, the low-energy spectrum of graphene nanoribbons with armchair edges (armchair nanoribbons) is described as the superposition of two non-equivalent Dirac points of graphene. In spite of the lack of two well separated valley structures, the single-channel transport subjected to LRIs is nearly perfectly conducting, where the backward scattering matrix elements in the lowest order vanish as a manifestation of internal phase structures of the wave function. For the multi-channel energy regime, however, conventional exponential decay of the averaged conductance occurs. Symmetry considerations lead to the classification of disordered zigzag ribbons into the unitary class for LRIs, and the orthogonal class for short-range impurities. Since inter-valley scattering is not completely absent, armchair nanoribbons can be classified into the orthogonal universality class irrespective of the range of impurities.
Highlights
Ribbons with zigzag edges possess localized edge states with energies close to the Fermi level [7]–[10]
In spite of the lack of two well separated valley structures, the single-channel transport subjected to long-range impurity (LRI) is nearly perfectly conducting, where the backward scattering matrix elements in the lowest order vanish as a manifestation of internal phase structures of the wave function [67]
We have presented a brief overview of the electronic and transport properties of graphene nanoribbons focusing on the effect of edge shapes and impurity scattering
Summary
There are two typical shapes of a graphene edge, called armchair and zigzag. The two edges have 30◦ difference in their cutting direction. (c) The π band structure and (d) the density of states of the graphene sheet. In figures 2(b)–(d), the first Brillouin zone (BZ) of the graphene lattice, the energy dispersion of π -bands in the first BZ and the corresponding density of states are depicted, respectively. Near the point, both valence and conduction bands have the quadratic form of kx and ky, i.e. Ek = ±(3 − 3|k|2/4). The energy band structures of armchair ribbons are shown in figures 3(a)–(c), for three different ribbon widths, together with the density of states. This convergence condition defines the region 2π/3 |k| π where the flat band appears
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