Flat edge modes of graphene and of Z _{2} topological insulator
 KenIchiro Imura^{1, 2}Email author,
 Shijun Mao^{2, 3},
 Ai Yamakage^{2} and
 Yoshio Kuramoto^{2}
DOI: 10.1186/1556276X6358
© Imura et al; licensee Springer. 2011
Received: 5 November 2010
Accepted: 21 April 2011
Published: 21 April 2011
Abstract
A graphene nanoribbon 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 Z _{2} topological insulator. To illustrate this idea, we consider both KaneMele (graphenebased) and BernevigHughesZhang 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.
Introduction
Graphene has a unique band structure with two Dirac points, K and K'valleysin the first Brillouin zone [1, 2]. Its transport characteristics are determined by the interplay of such effective "relativistic" band dispersion and the existence of valleys [3, 4]. The former induces a "Berry phase π," manifesting as the absence of backward scattering [5]. A direct consequence of this is the perfect transmission in a graphene pnjunction, or Klein tunneling [6, 7], whereas its strong tendency not to localize, i.e., the antilocalization [8–10], is also 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 [11–13]. It has been proposed that such flat band edge modes can induce nanomagnetism. The flat band edge modes also show robustness against disorder [14]. The Dirac nature in the electronic properties of graphene is much related to the concept of Z _{2} topological insulator (Z _{2}TI). A Z _{2}TI is known to possess a pair of gapless helical edge modes protected by time reversal symmetry. Similar to the gapless chiral edge mode of quantum Hall systems, responsible for the quantization of (charge) Hall conductance [15], the helical edge modes ensure the quantization of spin Hall conductance. The KaneMele model [16, 17] (= graphene + topological mass term, induced by an intrinsic spinorbit coupling) is a prototype of such Z _{2}TI constructed on a honeycomb lattice. Edge modes of graphene and of the KaneMele model show contrasting behaviors in the zigzag and armchair ribbon geometries [4, 11]. In this article, we argue that the flat band edge modes of zigzag graphene nanoribbon can be naturally understood from the viewpoint of underlying Z _{2} topological order in the KaneMele model. To illustrate this idea and clarify the role of valleys, we deal with the KaneMele and the BernevigHughesZhang (BHZ) models [18] in parallel, the latter being proposed for HgTe/CdTe 2D quantum well [19].
Flat band edge modes in garphene and KaneMele model for Z _{2}topological insulator
Let us consider a minimal tightbinding model for graphene: , where t _{1} is the strength of hopping between nearestneighbor (NN) sites, i and j, on the hexagonal lattice. The tightbinding Hamiltonian H _{1} has two gap closing points, and , in the first Brillouin zone. In the KaneMele model [16], hopping between next NN (NNN) sites (hopping in the same sublattice) is added to H _{1}, the former being also purely imaginary: , where 〈〈...〉〉 represents a summation over NNN sites. s _{ z } is the zcomponent of Pauli matrices associated with the real spin, and ν _{ ij } is a sign factor introduced in [16]. The origin of this NNN imaginary hopping is intrinsic spinorbit coupling consistent with symmetry requirements. it _{2} induces a mass gap of size in the vicinity of and .
Therefore, they appear both in armchair and zigzag edges. In the graphene limit: t _{2} → 0, however, the edge modes survive only in the zigzag edge geometry, as a result of different ways in which  and points are projected onto the edge. In the armchair edge, the helical modes at finite t _{2} are absorbed in the bulk Dirac spectrum in this limit. In the zigzag edge, on the contrary, the helical modes connecting and survive but become completely flat in the limit t _{2} → 0. Notice that and interchange under a timereversal operation. 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 Z _{2} 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. The study of a system of a finite size L can be employed to determine the presence (or absence) of a topological gap with the precision of 1/L. The behavior of such gapless surface states that exist on the topologically nontrivial side is continuous, up to and at the gap closing. They also evolve continuously into gapped surface states on the trivial side. The flat edge modes appear at the gap closing when they do.
Edge modes of BHZ model on square lattice
Four Dirac cones of BHZ model on square lattice
Dirac Points (DP)  Γ  X _{1}  X _{2}  M 
 ∏_{DP} δ _{DP} 

= (k _{ x } , k _{ y }) at the DP  (0, 0)  (0, π/a)  (π/a, 0)  (π/a, π/a)  
Mass gap  Δ  Δ  4B  Δ  4B  Δ  8B  
Chirality χ  +      +  
Δ < 0   (p)  + (s)  + (s)   (p)  0  +1 
0 < Δ < 4B  + (s)  + (s)  + (s)   (p)  2e ^{2}/h  1 
4B < Δ < 8B  + (s)   (p)   (p)   (p)  2e ^{2}/h  1 
8B < Δ  + (s)   (p)   (p)  + (s)  0  +1 
Straight edge
Zigzag edge
In contrast to the straight edge case, deriving an analytic expression for the edge spectrum in the zigzag edge geometry is a much harder task [24].
At Δ/B = 4, the edge spectrum becomes completely flat and covers the entire Brillouin zone. Notice that the horizontal axis is suppressed to make the edge modes legible. Analogous to the flat edge modes in graphene, these edge modes connect the two valleys X _{1} and X _{2} in the bulk, though they reduce to an equivalent point on the edge. As the bulk spectrum is also gapless at Δ/B = 4, the flat edge modes indeed touch the bulk continuum at the zone boundary.
Conclusions
We have studied the edge modes of graphene and of related topological insulator models in 2D. Much focus has been on the comparison between the single versus double valley systems (KaneMele versus BHZ). We have seen that a flat edge spectrum appears in the two cases, whereas in the latter case, the flat band edge modes connect the two valleys that have emerged because of the (square) lattice regularization. The appearance of flat band edge modes in the zigzag graphene nanoribbon was naturally understood from such a point of view.
Abbreviations
 BHZ:

BernevigHughesZhang.
Declarations
Authors’ Affiliations
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