First-principles calculations for the electronic band structures of small diameter single-wall carbon nanotubes

We present first-principles calculations on the band structures of 40 different small diameter $(d)$ single-wall carbon nanotubes (SWCNTs), including 14 chiral ones, employing density functional theory (DFT) within the local density approximation (LDA), using the Vienna ab initio simulation package (VASP). The band gaps are calculated and discussed for all of the tubes. From small to large diameters, the gap of semiconducting zigzag tubes first increases, then reaches a maximum of about $1\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ for (11,0), after which it decreases, approximately as $1∕d$, showing a ``buckling'' around this average behavior. The smallest diameter zigzag tubes are all metallic, due to $\ensuremath{\sigma}\ensuremath{-}\ensuremath{\pi}$ mixing caused by high curvature. The Fermi wave-vector of armchair tubes shows a downshift from its ideal, zone folding expected value; this shift is proportional to $1∕{d}^{\phantom{\rule{0.1em}{0ex}}2}$. Eight of the ZF-metallic tubes (4 zigzags, 4 chirals) show a small gap in the band structure. For the zigzag tubes, this band gap is roughly given by the formula $\ensuremath{\Delta}=1.99∕{d}^{\phantom{\rule{0.1em}{0ex}}2}+140.9∕{d}^{4}$ ($\ensuremath{\Delta}$ in eV, $d$ in \AA{}). The appearance of the $1∕{d}^{4}$ higher order correction term is due to high curvature at small diameters, however it's apparently overestimated in our calculations. This overestimation can likely be eliminated by considering many-electron effects.