P. Delplace 1, D. Ullmo 2, G. Montambaux 3
Physical Review B 84 (2011) 195452
We develop a method to predict the existence of edge states in graphene ribbons for a large class of boundaries. This approach is based on the bulk-edge correspondence between the quantized value of the Zak phase Z(k), which is a Berry phase across an appropriately chosen one-dimensional Brillouin zone, and the existence of a localized state of momentum k at the boundary of the ribbon. This bulk-edge correspondence is rigorously demonstrated for a one dimensional toy model as well as for graphene ribbons with zigzag edges. The range of k for which edge states exist in a graphene ribbon is then calculated for arbitrary orientations of the edges. Finally, we show that the introduction of an anisotropy leads to a topological transition in terms of the Zak phase, which modifies the localization properties at the edges. Our approach gives a new geometrical understanding of edge states, it con?firms and generalizes the results of several previous works.
- 1. Département de Physique,
Université de Genève - 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI – Paris Sud - 3. Laboratoire de Physique des Solides (LPS),
CNRS : UMR8502 – Université Paris XI – Paris Sud