Abstract
Planar carbonbased electronic devices, including metal/semiconductor junctions, transistors and interconnects, can now be formed from patterned sheets of graphene. Most simulations of charge transport within graphenebased electronic devices assume an energy band structure based on a nearestneighbour tight binding analysis. In this paper, the energy band structure and conductance of graphene nanoribbons and metal/semiconductor junctions are obtained using a third nearestneighbour 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.
Keywords:
Graphene nanoribbon junction; Tight binding; Conductance; Band structureIntroduction
Since the report of the preparation of graphene by Novoselov et al. [1] 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 [47]. To minimise computation time, the nearestneighbour 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 carbonbased materials such as carbon nanotubes [8] and carbon nanocones [9]. Recently, Reich et al. [10] have demonstrated that this approximation is only valid close to the K points, and a tight binding approach including up to third nearestneighbours 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 nearestneighbour tight binding energy dispersion. For transport studies in nanoribbons and junctions, the formulation of the problem differs from that required for bulk graphene. Third nearestneighbour interactions introduce additional exchange and overlap integrals significantly modifying the Green's function. Calculation of device characteristics is facilitated by the inclusion of a SanchoRubio [11] iterative scheme, modified by the inclusion of third nearestneighbour interactions, for the calculation of the selfenergies. We find that the conductance is significantly altered compared with that obtained based on the nearestneighbour tight binding dispersion even in an isolated nanoribbon. Hong et al. [12] 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.
Theory
The basis for our analysis is the hexagonal graphene lattice shown in Figure 1. a_{1} and b_{1} are the principal vectors of the unit cell containing two carbon atoms belonging to the two sublattices. Atoms on the concentric circles of increasing radius correspond to the nearestneighbours, second nearestneighbours and third nearestneighbours, respectively.
Figure 1. Armchairedge graphene metal (index N = 23)/semiconductor (index N = 13) junction. The rectangle shows the semiconductor unit cell, and the concentric circles of increasing radius show first, second and third nearestneighbours, respectively.
Saito et al. [8] derived the dispersion relation below using a nearestneighbour tight binding analysis including the overlap integral s_{0}.
Here, f(k) = 3 + 2 cos k · a_{1} +2 cos k · b_{1} + 2 cos k · (a_{1}  b_{1}) and the parameters, ε_{2p}, γ_{0} and s_{0} are obtained by fitting to experimental results or ab initio calculations.
Most analyses of charge transport in graphenebased structures simplify the result further by ignoring s_{0}. Reich et al. [10] derived the dispersion relation for graphene based on third nearestneighbours. In this work, the energy band structure of a graphene nanoribbon including third nearestneighbour interactions is obtained from the block Hamiltonian and overlap matrices given below for the unit cell defined by the rectangle in Figure 1.
For the nth row of the above equation, we have
Considering the energy dispersion in the direction of charge transport, the Bloch form of the wavefunction ensures that φ_{n}~e^{ikn}. Substitution of φ_{n} into the above equation yields the secular equation
In the case of first nearest approximation without orbital overlap, S_{n,n1} and S_{n,n+1} are empty matrices. To facilitate comparison with published results, we use an armchairedge with index [13]N = 13 as our model nanoribbon. In the paper by Reich, tight binding parameters were obtained by fitting the band structure to that obtained by ab initio calculations. Recently, Kundo [14] has reported a set of tight biding parameters based on fitting to a first principle calculation but more directly related to the physical quantities of interest. These parameters have been utilised in our calculation and are presented below for third nearestneighbour interactions (Table 1).
Figure 2 compares the energy band structure of the modelled armchairedge graphene nanoribbon obtained from the first nearestneighbour tight binding method with that obtained by including up to third nearestneighbours. Agreement is reasonable close to the K point but significant discrepancies occur at higher energies.
Figure 2. Energy band structure of an N = 13 armchair graphene nanoribbon, (a) obtained from the first nearestneighbour tight binding method and (b) including third nearestneighbours.
Conductance of Graphene Nanoribbons and Junctions
Conductance in graphene nanoribbons and metal/semiconductor junctions is determined using an efficient nonequilibrium Green's function formalism described by Li and Lu [15]. The retarded Green's function is given by
Here, E^{+} = E + iη and η is a small positive energy value (10^{5} eV in this simulation) which circumvents the singular point of the matrix inversion [16]. H is a tight binding Hamiltonian matrix including up to third nearestneighbours, and S is the overlap matrix. Open boundary conditions are included through the left and right selfenergy matrix elements, σ^{L.R}. The selfenergies are independently evaluated through an iterative scheme described by Sancho et al. [11], modified to include third nearestneighbour interactions. Determination of the retarded Green's function through equation 5 is facilitated by the inclusion of the body of the device in the righthand contact through the recursive scheme described in ref. [15]. We will now outline the numerical procedure for deriving the conductance with third nearestneighbour interactions included. Figure 3 shows a schematic of the unit cell labelling used to formulate the Green's function.
Figure 3. Schematic showing the unit cell labelling used to formulate the Green's function.
We calculate the surface retarded Green's functions of the left and right leads by
where θ and are the appropriate transfer matrices calculated from the following iterative procedure.
where t_{i} and are defined by
and
The process is repeated until with δ arbitrarily small. The nonzero elements of the selfenergies and can be then obtained by
The conductance is obtained from the relation
where the transmission coefficient is obtained from
with Γ^{L,R} = i[Σ^{L,R}  (Σ^{L,R})^{†}.
Figure 4a, b compares the conductance of a graphene armchairedge nanoribbon of index N = 13 and metal/semiconductor junction formed with the nanoribbon assuming first and third nearestneighbour interactions, respectively. For graphene nanoribbons, differences are observed in the steplike structure, reflecting differences in the calculated band structure. When only first nearestneighbour interactions are considered, the conductance of the conduction and valence bands is always symmetrical as determined by the formulation of the energy dispersion relation, equation 1. In the case of graphene nanoribbons, the conductance within a few electron volts of the Fermi energy is symmetrical for both first and third nearestneighbour interactions. However, it is notable that at higher energies, overlap integrals introduced by third nearestneighbour interactions result in asymmetry between the conductance in the conduction and valence bands. For metal/semiconductor junctions, significant differences in conductivity occur even at low energies due to mismatches of the subbands. Asymmetry in the conduction and valence band conductance (absent for first nearestneighbour interactions) is also apparent when third nearestneighbour interactions are included in the Green's function. Differences are also seen when defects are incorporated within a metal/semiconductor junction, an interesting system explored by Hong et.al. [12]. In this work, vacancies are introduced in the lattice at the positions marked by the solid rectangle and triangle in Figure 1 and the conductance obtained in each case. Hong et al. derive a coupling term associated with differences in band structure. For third nearestneighbour, the solution to the coupling strength must be derived numerically.
Figure 4. Conductance vs Energy for the junction shown in Figure 1, a using first nearestneighbour parameters and b using third nearestneighbours parameters. Dotted lines are for N = 13 armchair nanoribbon, solid lines are for ideal metal/semiconductor junctions, dot–dash lines and dash lines are for junctions with a single defect type A (triangle in Figure 1) and type B (rectangle in Figure 1) respectively.
Conclusions
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 nearestneighbour and third nearestneighbour. Significant differences are observed, suggesting the commonly used first nearestneighbour 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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