Transmission spectra and valley processing of graphene and carbon nanotube superlattices with inter-valley coupling

We numerically investigate the electronic transport properties of graphene nanoribbons and carbon nanotubes with inter-valley coupling, e.g., in $\sqrt{3}N\times \sqrt{3}N$ and $3N\times 3N$ superlattices. By taking the $\sqrt{3}\times \sqrt{3}$ graphene superlattice as an example, we show that tailoring the bulk graphene superlattice results in rich structural configurations of nanoribbons and nanotubes. After studying the electronic characteristics of the corresponding armchair and zigzag nanoribbon geometries, we find that the linear bands of carbon nanotubes can lead to the Klein tunnelling-like phenomenon, i.e., electrons propagate along tubes without backscattering even in the presence of a barrier. Due to the coupling between K and $K^{\prime }$ valleys of pristine graphene by $\sqrt{3}\times \sqrt{3}$ supercells, we propose a valley-field-effect transistor based on the armchair carbon nanotube, where the valley polarization of the current can be tuned by applying a gate voltage or varying the length of the armchair carbon nanotubes. (paper)