Electronic and optical properties of ultra-small single-walled carbon nanotubes: Axial lattice parameter dependence of Metal–Semiconductor transitions

In this work, we address inconsistencies reported in the literature regarding the electronic character and optical response of ultra-small SWCNTs, with particular emphasis on the armchair (2,2) nanotube, and investigate the possible occurrence of a metal–semiconductor transition in zig-zag (3,0) and (4,0) systems as the tube length varies. To this end, density functional theory calculations were performed using the PBE and PBEsol exchange–correlation functionals, as well as the GLLB-SC potential. Structural, electronic, and optical properties were systematically analyzed across different tube-axis lengths. Moreover, the electronic character of the (2,2) nanotube is highly sensitive to both structural variations and the choice of exchange–correlation functional. A small indirect band gap emerges at a tube length of 2.582 Å, with values of 68 meV (PBEsol) and 40 meV (GLLB-SC). In contrast, the (3,0) and (4,0) nanotubes exhibit robust metallic behavior independent of tube length, with no evidence of a metal–semiconductor transition. Even when the band structure is metallic, the optical absorption lacks a Drude tail, resembling a semiconductor in the infrared region. These findings provide a consistent description of ultra-small SWCNTs and establish reliable reference data for future theoretical and experimental studies.