sults, and the summary of this work is available in Section 4. to the source and drain respectively, and they can be defined in terms of their self-energies. For the complete system i.e., the ribbon, source and drain, the Green’s function is defined as, Let us start with Fig. 1, where a honeycomb lat- tice ribbon with zigzag edges is attached to two semi-infinite one-dimensional metallic electrodes, viz, source and drain. As the electron transport properties are significantly influenced by the quan- where E = E + iδ . E is the injecting energy of the source electron and δ gives an infinitesimal imagi- nary part to E . To Evaluate this Green’s function, the inversion of an infinite matrix is needed since the complete system consists of the finite ribbon and the two semi-infinite electrodes. However, the entire system can be partitioned into sub-matrices corresponding to the individual sub-systems and the Green’s function for the ribbon can be effec- tively written as, where H$_{rib}$ is the Hamiltonian of the ribbon which can be written in the tight-binding model within the non-interacting picture like, In the above Hamiltonian ( H$_{rib}$ ), ϵ$_{i}$ ’s are the site energies, c † i ( c$_{i}$ ) is the creation (annihilation) oper- ator of an electron at the site i and t is the nearest- neighbor hopping integral. A similar kind of tight- binding Hamiltonian is also used to describe the two semi-infinite one-dimensional perfect electrodes where the Hamiltonian is parametrized by constant on-site potential ϵ$_{0}$ and nearest-neighbor hopping integral t$_{0}$ . In Eq. 4, Σ$_{S}$ = h † S − $_{rib}$g$_{S}$ h$_{S}$$_{−}$$_{rib}$ and Σ$_{D}$ = h$_{D}$$_{−}$$_{rib}$g$_{D}$h † D − rib are the self-energy opera- tors due to the two electrodes, where g$_{S}$ and g$_{D}$ correspond to the Green’s functions of the source and drain respectively. h$_{S}$$_{−}$$_{rib}$ and h$_{D}$$_{−}$$_{rib}$ are the coupling matrices and they will be non-zero only for the adjacent points of the ribbon, and the elec- trodes respectively. The matrices Γ$_{S}$ and Γ$_{D}$ can be calculated through the expression, G = ( E − H ) − 1 (3) 2 Model and synopsis of the theoretical background Source Drain Honeycomb lattice ribbon G$_{rib}$ = ( E − H$_{rib}$ − Σ$_{S}$ − Σ$_{D}$) − 1 (4) H$_{rib}$ = ∑ i ϵ$_{i}$c † $_{i}$c$_{i}$ + ∑ <ij> t ( c † $_{i}$c$_{j}$ + c † $_{j}$c$_{i}$ ) (5) Figure 1: (Color online). Schematic view of a hon- eycomb lattice ribbon with zigzag edges attached to two semi-infinite one-dimensional metallic elec- trodes, viz, source and drain. Filled circles corre- spond to the position of the atomic sites. tum interference effects, throughout the study, we contact the electrodes at the two extreme ends of nanoribbons (see Fig. 1) to make these interference effects uniform. We use the Landauer conductance formula 34 − 35 to calculate the conductance g of the ribbon. At very low temperature and bias voltage it ( g ) can be presented in the form, Calculation of the conductance: g = 2 e 2 h T (1) Γ$_{S}$$_{(}$$_{D}$$_{)}$ = i [ Σ r S ( D ) − Σ a S ( D ) ] (6) Σ r S ( D ) = Λ$_{S}$$_{(}$$_{D}$$_{)}$ − i Δ$_{S}$$_{(}$$_{D}$$_{)}$ (7) T = Tr [Γ$_{S}$ G r $_{rib}$Γ$_{D}$ G a $_{rib}$] (2) where T gives the transmission probability of an electron in the ribbon which can be expressed in terms of the Green’s function of the ribbon and its coupling to the two electrodes by the relation, 34 − 35 where Σ r S ( D ) and Σ a S ( D ) are the retarded and ad- vanced self-energies respectively, and they are con- jugate with each other. These self-energies can be written as, 36 where G r rib and G a rib are respectively the retarded and advanced Green’s functions of the ribbon in- cluding the effects of the electrodes. The parame- ters Γ$_{S}$ and Γ$_{D}$ describe the coupling of the ribbon where Λ$_{S}$$_{(}$$_{D}$$_{)}$ are the real parts of the self-energies which correspond to the shift of the energy eigenval- ues of the ribbon and the imaginary parts Δ$_{S}$$_{(}$$_{D}$$_{)}$ of 3