Configuration-Independent Low-Energy Gaps in Graphene Nanoribbons: A $δ$-Scatterer Transfer-Matrix Approach

Abstract

Transport measurements on lithographically patterned graphene nanoribbons suggest that the gap depends primarily on ribbon width rather than on crystallographic edge orientation, in contrast to predictions of ideal tight-binding models. To address this, we develop a transfer-matrix approach, based on the theory of finite periodic systems, in which each carbon atom of the honeycomb lattice is represented by an individual $δ$-scatterer with interaction strength $γ^{}_δ$ and the nanoribbon is treated as a confined multichannel finite periodic system. We derive the unit-cell transfer matrix for both arm-chair and zig-zag edge configurations and obtain the transmission and channel-coupling coefficients in closed form. The transmission spectra display the resonant band structure characteristic of finite periodic systems, with the number of resonances per allowed band increasing with the number of unit cells. For $γ_δ=0$ the ribbons are fully transparent; for $γ_δ>0$ a low-energy transport gap opens and grows linearly with the scatterer strength, $Δ=0.364265\,γ_δ$, identically for arm-chair and zig-zag edges. These results are consistent with the reported absence of a systematic dependence on orientation, while the analytical results show that interchannel coupling vanishes exactly for arm-chair ribbons and decreases with increasing width for zig-zag ribbons, revealing the role of edge geometry. Together, these findings indicate that atomic-scale scattering, represented by the $δ$-potential formalism, provides a common orientation-independent contribution to the low-energy transport gap in the pristine ribbons considered here.