Abstract
A graphene nano-ribbon in the zigzag edge geometry exhibits a specific type of gapless edge modes with a partly flat band dispersion. We argue that the appearance of such edge modes are naturally understood by regarding graphene as the gapless limit of a Z2 topological insulator. To illustrate this idea, we consider both Kane-Mele (graphene-based) and Bernevig-Hughes-Zhang models: the latter is proposed for HgTe/CdTe 2D quantum well. Much focus is on the role of valley degrees of freedom, especially, on how they are projected onto and determine the 1D edge spectrum in different edge geometries.
Highlights
Graphene has a unique band structure with two Dirac points, K- and K’-valleys–in the first Brillouin zone [1,2]
A direct consequence of this is the perfect transmission in a graphene pn-junction, or Klein tunneling [6,7], whereas its strong tendency not to localize, i. e., the anti-localization [8,9,10], is a clear manifestation of the Berry phase π in the interference of electronic wave functions. Another feature characterizing the electronic property of graphene lies in the appearance of partly flat band edge modes in a ribbon geometry [11,12,13]
We argue that the flat band edge modes of zigzag graphene nano-ribbon can be naturally understood from the viewpoint of underlying Z2 topological order in the KaneMele model
Summary
Graphene has a unique band structure with two Dirac points, K- and K’-valleys–in the first Brillouin zone [1,2]. In the Kane-Mele model (with a finite t2) the existence of a pair of gapless helical edge modes is ensured by bulkedge correspondence [20] They appear both in armchair and zigzag edges. In the sense stated above, we propose that the flat band edge modes of a zigzag graphene ribbon is a precursor of helical edge modes characterizing the Z2 topological insulator Note here that such surface phenomena as fiat and helical edge states are characteristics of a system of a finite size, and the evolution of such gapless surface states is continuous, free from discontinuities characterizing a conventional phase transition as described by the Landau theory of symmetry breaking.
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