  Silicon nanotube is constructed by rolling up a silicene, i.e., a monolayer
of silicon atoms forming a two-dimensional honeycomb lattice. It is a
semiconductor or an insulator owing to relatively large spin-orbit interactions
induced by its buckled structure. The key observation is that this buckled
structure allows us to control the band structure by applying electric field
$E_z$. When $E_z$ is larger than a certain critical value $E_{\text{cr}}$, by
analyzing the band structure and also on the basis of the effective Dirac
theory, we demonstrate the emergence of four helical zero-energy modes
propagating along nanotube. Accordingly, a silicon nanotube contains three
regions, namely, a topological insulator, a band insulator and a metallic
region separating these two types of insulators. The wave function of each zero
mode is localized within the metallic region, which may be used as a quantum
wire to transport spin currents in future spintronics. We present an analytic
expression of the wave function for each helical zero mode. These results are
applicable also to germanium nanotube.

Mesoscale and Nanoscale Physics;Materials Science