Conductance of Graphene Nanoribbon Junctions and the Tight Binding Model
© Wu and Childs. 2010
Received: 9 July 2010
Accepted: 9 September 2010
Published: 7 October 2010
Planar carbon-based electronic devices, including metal/semiconductor junctions, transistors and interconnects, can now be formed from patterned sheets of graphene. Most simulations of charge transport within graphene-based electronic devices assume an energy band structure based on a nearest-neighbour tight binding analysis. In this paper, the energy band structure and conductance of graphene nanoribbons and metal/semiconductor junctions are obtained using a third nearest-neighbour tight binding analysis in conjunction with an efficient nonequilibrium Green's function formalism. We find significant differences in both the energy band structure and conductance obtained with the two approximations.
Since the report of the preparation of graphene by Novoselov et al.  in 2004, there has been an enormous and rapid growth in interest in the material. Of all the allotropes of carbon, graphene is of particular interest to the semiconductor industry as it is compatible with planar technology. Although graphene is metallic, it can be tailored to form semiconducting nanoribbons, junctions and circuits by lithographic techniques. Simulations of charge transport within devices based on this new technology exploit established techniques for low dimensional structures [2, 3]. The current flowing through a semiconducting nanoribbon formed between two metallic contacts has been established using a nonequilibrium Green's Function (NEGF) formalism based coupled with an energy band structure derived using a tight binding Hamiltonian [4–7]. To minimise computation time, the nearest-neighbour tight binding approximation is commonly used to determine the energy states and overlap is ignored. This assumption has also been used for calculating the energy states of other carbon-based materials such as carbon nanotubes  and carbon nanocones . Recently, Reich et al.  have demonstrated that this approximation is only valid close to the K points, and a tight binding approach including up to third nearest-neighbours gives a better approximation to the energy dispersion over the entire Brillouin zone.
In this paper, we simulate charge transport in a graphene nanoribbon and a nanoribbon junction using a NEGF based on a third nearest-neighbour tight binding energy dispersion. For transport studies in nanoribbons and junctions, the formulation of the problem differs from that required for bulk graphene. Third nearest-neighbour interactions introduce additional exchange and overlap integrals significantly modifying the Green's function. Calculation of device characteristics is facilitated by the inclusion of a Sancho-Rubio  iterative scheme, modified by the inclusion of third nearest-neighbour interactions, for the calculation of the self-energies. We find that the conductance is significantly altered compared with that obtained based on the nearest-neighbour tight binding dispersion even in an isolated nanoribbon. Hong et al.  observed that the conductance is modified (increased as well as decreased) by the presence of defects within the lattice. Our results show that details of the band structure can significantly modify the observed conductivities when defects are included in the structure.
Here, f(k) = 3 + 2 cos k · a 1 +2 cos k · b 1 + 2 cos k · (a 1 - b 1 ) and the parameters, ε2p, γ0 and s0 are obtained by fitting to experimental results or ab initio calculations.
Tight binding parameters 
Conductance of Graphene Nanoribbons and Junctions
with Γ L,R = i[Σ L,R - (Σ L,R )†.
In this paper, we have determined the energy band structure of graphene nanoribbons and conductance of nanoribbons and graphene metal/semiconductor junctions using a NEGF formalism based on the tight binding method approximated to first nearest-neighbour and third nearest-neighbour. Significant differences are observed, suggesting the commonly used first nearest-neighbour approximation may not be sufficiently accurate in some circumstances. The most notable differences are observed when defects are introduced in the metal/semiconductor junctions.
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