The empirical tightbinding model is a standard, convenient, and accurate method for calculating the electronic structure of semiconductors. It is also referred to as Bloch or linear combination of atomic orbitals (LCAO) method [31]. The Hamiltonian matrix for a simplified 2D system, using the TB method is described as follows: The 2D structure in Fig. 2.1 is composed of chains of lattice points (i.e. atomic sites), each chain with sides. Assuming that each point is represented by one basis orbital, the Hamiltonian matrix will have the size , and is given by:

The only change, here, is in the submatrices and , which now include the self energies of the left and right contacts. In the case of periodic boundary conditions, the bandstructure is calculated by considering the unit cell (index ) of the lattice, connected to the neighboring unit cells (index ) using the matrix elements . For example, as indicated in the structure of Fig. 2.1, once periodic boundary conditions are applied along the axis, then and for all . The bandstructure of the lattice is then obtained by calculating the eigenvalues of the Hamiltonian as:
(2.6) 

In the case of graphene, a third nearest neighbor tightbinding model is used to describe its electronic structure. In this case, the particular atom and its nearest neighbor atoms are shown in Fig. 2.2. The hopping parameter between two nearest atoms separated by distance is . The tightbinding parameter of the thirdnearest neighbor atoms located away from each other is [34]. The hopping parameter for the second nearestneighbor is assumed to be . The bandstructure of graphene along the highsymmetry band line is shown in Fig. 2.3. The tightbinding results are in good agreement with experimental data taken from [33], in particular around the Fermi energy which dominates the electrical properties. As shown in Ref. [34] this method can capture the details of the bandstructure of graphenebased nanostructures. The tightbinding model with calibrated parameters provides bandgap and subbandedge energies in excellent agreement with firstprinciples calculations [34].