We investigate the electronic and transport properties of gated bilayer graphene with one corrugated layer, which results in a stacking AB/BA boundary. When a gate voltage is applied to one layer, topologically protected gap states appear at the corr
ugation, which reveal as robust transport channels along the stacking boundary. With increasing size of the corrugation, more localized, quantum-well-like states emerge. These finite-size states are also conductive along the fold, but in contrast to the stacking boundary states, which are gapless, they present a gap. We have also studied periodic corrugations in bilayer graphene; our findings show that such corrugations between AB- and BA-stacked regions behave as conducting channels that can be easily identified by their shape.
We prescribe general rules to predict the existence of edge states and zero-energy flat bands in graphene nanoribbons and graphene edges of arbitrary shape. No calculations are needed. For the so-called {it{minimal}} edges, the projection of the edge
translation vector into the zigzag direction of graphene uniquely determines the edge bands. By adding extra nodes to minimal edges, arbitrary modified edges can be obtained. The edge bands of modified graphene edges can be found by applying hybridization rules of the extra atoms with the ones belonging to the original edge. Our prescription correctly predicts the localization and degeneracy of the zero-energy bands at one of the graphene sublattices, confirmed by tight-binding and first-principle calculations. It also allows us to qualitatively predict the existence of $E e 0$ bands appearing in the energy gap of certain edges and nanoribbons.
