Broken symmetry, boundary conditions, and band-gap oscillations in finite single-wall carbon nanotubes

The interplay between the broken symmetry and the boundary conditions alters profoundly the electronic properties of carbon single-wall nanotubes (SWNTs) of finite-lengths. For SWNTs $\mathrm{}(p,q)\mathrm{}$ characterized by $p=k+l,$ $q=k-l,$ $0<~l<~k,$ and $k=1,2,\dots ,$ the energy gaps for finite SWNTs belonging to a given family k exhibit strikingly similar oscillating patterns for even NT sections, but the gap maxima for SWNTs of different lengths exhibit different periodicities with the amplitudes of oscillations diminishing from the armchair, to chiral, and to the zigzag SWNTs.