the self-energies represent the broadening of these energy levels. Since this broadening is much larger than the thermal broadening, we restrict our all cal- culations only at absolute zero temperature. All the information about the ribbon-to-electrode coupling are included into these two self-energies. they correspond to the width and length of the rib- bon respectively. Thus, for example, a nanorib- bon with N = 1 and M = 3 represents a linear chain of three hexagons. Hence the parameter M determines the total number of hexagons in a single chain. Following this rule, a nanoribbon with N = 3 and M = 4 corresponds to three linear chains at- The current passing across the ribbon can be de- picted as a single-electron scattering process be- tween the two reservoirs of charge carriers. The current I can be computed as a function of the ap- plied bias voltage V through the relation, 34 Calculation of the current: I ( V ) = 2 e h E$_{F}$ + eV/ 2 ∫ E$_{F}$ − eV/ 2 T ( E ) dE (8) where E$_{F}$ is the equilibrium Fermi energy. Here we make a realistic assumption that the entire voltage is dropped across the ribbon-electrode interfaces, and it is examined that under such an assumption the I - V characteristics do not change their qualita- tive features. This assumption is based on the fact that, the electric field inside the ribbon especially for narrow ribbons seems to have a minimal effect on the conductance-voltage characteristics. On the other hand, for quite larger ribbons and high bias voltages the electric field inside the ribbon may play a more significant role depending on the internal structure and size of the ribbon, 36 but the effect becomes too small. To have a better insight in the present problem i.e., the dependence of the electron transport on the lengths and widths of ribbons, we focus our at- tention only on the perfect systems rather than any disordered one. To get these perfect systems we fix the on-site energies ϵ$_{i}$ = 0 for all the sites i of the honeycomb lattice ribbons. The values of the other parameters are taken as follow. The nearest- neighbor hopping integral t in the ribbon is set to 1, the on-site energy ϵ$_{0}$ and the hopping integral t$_{0}$ for the two electrodes are fixed to 0 and 1 respec- tively. The parameters τ$_{S}$ and τ$_{D}$ are set as 0 . 75, where they correspond to the hopping strengths of the ribbon to the source and drain respectively. In addition to these, to describe the size of a ribbon we introduce two other parameters N and M where 3 Numerical results and dis- cussion tached side by side (see Fig. 1) where each chain contains four hexagons. For simplicity, throughout our study we set the Fermi energy E$_{F}$ = 0 and choose the units where c = e = h = 1. Let us begin our discussion with the variation of the conductance g as a function of the energy E . In Fig. 2 we plot the conductance-energy charac- teristics for some typical lattice ribbons with fixed width N = 3 and varying lengths where (a) and (b) correspond to the lengths M = 6 and 8 respectively. The sharp resonant peaks in the conductance spec- tra are observed almost for all energies, while for some other energies either the conductance g gets much small value or drops to zero. At the reso- nances the conductance gets the value 2, and there- fore, the transmission probability T becomes unity 4 Figure 2: (Color online). Conductance g as a func- tion of the energy E for some lattice ribbons with fixed width N = 3 and varying lengths where (a) M = 6 and (b) M = 8. GLYPH<Minus> 2 GLYPH<Minus> 1 0 1 2 E 0.0 1.0 GLYPH<LParen1> b GLYPH<RParen1> 2.0 g GLYPH<LParen1> E GLYPH<RParen1> GLYPH<Minus> 2 GLYPH<Minus> 1 0 1 2 E 0.0 1.0 GLYPH<LParen1> a GLYPH<RParen1> 2.0 g GLYPH<LParen1> E GLYPH<RParen1>