The Zak phase and the existence of edge states in graphene

P. Delplace (1) , D. Ullmo (2) , G. Montambaux (3)

1 Département de Physique
2 LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
3 LPS - Laboratoire de Physique des Solides

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.

Domaines: Autre [cond-mat.other]; Physique Quantique [quant-ph]

P. Delplace, D. Ullmo, G. Montambaux. The Zak phase and the existence of edge states in graphene. Physical Review B: Condensed Matter and Materials Physics (1998-2015), 2011, 84, pp.195452. ⟨hal-00644295⟩