Electronic structure of single-wall silicon nanotubes and silicon nanorribons: Helical symmetry treatment and effect of dimensionality

Helical method of tube formation for band structure calculations and Hartree-Fock self-consistent field method (HF-SCF) modified for periodic solids have been applied in study of electronic properties of single-wall silicon nanotubes (SWSiNT), graphene-like parent 2D-hP silicone sheet and nanoribbons (SiNR). The results obtained for nanotubes of the length of $\approx$ 358 \AA in diameter range $\approx$ 3.7 \AA – 116 \AA of different helicity-types have shown that only small-diameter SWSiNTs up to} $\phi${6.3 \AA ~ are metallic due to the effect of curvature which induces coupling of}{$\sigma$}{and}{$\pi$}{orbitals. From the calculated band structures follow that irrespective of helicity, the SWSiNTs of larger diameter are all small-gap semiconductors with direct gap between the Dirac-like cones of}($\pi$\textsuperscript{*}, $\pi$) bands{.}Gap of SWSiNTs exhibits, however, an oscillatory-decreasing character with increase of the tube diameter. In the oscillatory series, minima of the gap in {\textquotedblleft}saw-teeth{\textquotedblright} pattern are reached for helicity numbers m \textsubscript{a} that are an integer multiple of 3, whilst m\textsubscript{a} value itself directly determine the fold-number of particular tubular rotational axis symmetry. Oscillations are damped and gap decreases toward {$\approx$ 0.33 eV} for tube diameter {$\approx$ 116 \AA .} {Irrespective of the width, the SiNRs are all small-gap semiconductors, characteristic by oscillatory decreasing gap with increasing ribbon widths. The gap of SWSiNTs and SiNRs is tuneable through modulation of tube diameter or ribbon width, respectively.