Twisted double-walled nanotubes and bilayer ribbons: Stripe versus island electronic localization in boron nitride systems

In this work, we applied first-principles calculations to address the problem of the electronic localization and the emergence of flat bands in one-dimensional boron nitride systems. We considered bilayer ribbons, hybrid bilayers formed by the stacking of a ribbon on top of an infinite sheet, and double-walled nanotubes. For the ribbon cases, we found results consistent with those reported for extended bilayer systems, with flat bands associated with specific stacking orders in the moir\'e patterns. The influence of the edge states in the moir\'e localizations is also described. For the double-walled nanotubes, we found a distinct phenomenology: The localization is also present in connection with the stacking orders; however, it shows up in the form of extended stripes along the tube axis, and not in the form of islands, as in the ribbon cases. A model to interpret this phenomenology is discussed.