Controlling edge states in the Kane-Mele model via edge chirality
We investigate the dependence of band dispersion of the quantum spin Hall effect (QSHE) edge states in the Kane-Mele model on crystallographic orientation of the edges. Band structures of the one-dimensional honeycomb lattice ribbons show the presence of the QSHE edge states at all orientations of the edges given sufficiently strong spin-orbit interactions. We find that the Fermi velocities of the QSHE edge-state bands increase monotonically when the edge orientation changes from zigzag (chirality angle ) to armchair (). We propose a simple analytical model to explain the numerical results.
In their seminal paper kan05-1 (), Kane and Mele proposed a simple two-dimensional model which realizes the quantum spin Hall effect (QSHE). The model essentially considers a tight-binding model on a honeycomb lattice akin to graphene with added spin-orbit interactions. Realistic graphene is characterized by only very weak intrinsic spin-orbit coupling of the order of 10 eV min06 (); yao07 (); boe07 (); gmi09 (); kon10 (). While no QSHE has been experimentally observed in graphene, the Kane-Mele construction has become a popular model of topological insulators Hasan10 (); Qi11 (). Significant attention is currently devoted towards understanding the relationship between the crystallographic orientation of edges and surfaces in topological insulators and the resulting properties of topologically non-trivial boundary states Zhang12 (); Silvestrov12 ().
In this work, we establish a dependence between the band dispersion of the QSHE edge states in the Kane-Mele model and the crystallographic orientation of the edges. In particular, we show that Fermi velocities of the topological edge states strongly depend on edge orientation.
We investigate the electronic band structures of 1D periodic honeycomb lattice ribbons within the Kane-Mele model Hamiltonian kan05-1 ()
where and indicate first and second nearest neighbors, respectively, and is the spin index. In this expression, the first term corresponds to an ordinary nearest-neighbor tight-binding model with hopping energy . The second term introduces spin-orbit coupling of strength . is the Haldane factor hal88 () defined as for a pair of second nearest neighbor sites connected via a common neighbor . is a Pauli matrix describing electron spin. In graphene-like systems, the spin-orbit term opens a gap at the Dirac points. Following Kane and Mele kan05-1 (), we choose the spin-orbit second neighbor hopping . We stress that the value used overestimates the intrinsic spin-orbit coupling present in realistic graphene, which was predicted to be between 1 and 50 eV according to first-principles calculations min06 (); yao07 (); boe07 (); gmi09 (); kon10 ().
The configurations of investigated 1D periodic honeycomb lattice ribbons are defined by two parameters: (i) the crystallographic orientation of the edges and (ii) the width of the ribbon. The edge direction is described by a translation vector of the graphene lattice (see Fig. 1a). The high symmetry directions, armchair and zigzag, correspond to vectors and , respectively. Equivalently, the edge orientation can be described in terms of chirality angle defined as the angle between the edge and the zigzag direction yaz11 (); tao11 (). The edge translation vectors and chirality angles are related to each other by the following relation:
Figure 2 shows the band structures of honeycomb ribbons with chirality angles ranging from ( zigzag edge) to ( armchair edge) via a series of intermediate edge orientations (chiral edges). All considered models have comparable width defined by . In Figure 2, one can immediately notice that all band structures feature linear band crossings occurring either at or at the Brillouin-zone boundary . The crossings display a clear increase of the Fermi velocity upon increasing . This relationship will be discussed in detail below. Analysis of the electronic states at the band crossings reveals that the channels of opposite spins are localized at the opposite edges of 1D ribbon structures. That is, all investigated 1D honeycomb ribbons are in quantum spin Hall phase and exhibit spin-filtered edge states topologically protected against backscattering.
The effects of spin-orbit term are clearly illustrated for the case of a zigzag edge shown in Fig. 2. In the absence of spin-orbit coupling (; red dashed line) the band structure of the ribbon model exhibits a dispersionless band at . This band is four times degenerate (2 spins 2 edges); it corresponds to edge-localized states originating from the lifted compensation between the two sublattices of the honeycomb lattice nak96 (). The flat band connects and ( is the lattice constant of the honeycomb lattice). These momenta correspond to the projections of points and of the hexagonal Brillouin zone (the locations of the Dirac cones in the band structure of graphene) onto the momentum space of the 1D ribbon structures (points and in Fig. 1b). The introduction of spin-orbit term opens a band gap at and lifting the degeneracy of edge states and leading to a non-zero value of . In the quantum spin Hall phase the edge states connect the valence band at with the conduction band , and vice versa (Fig. 1c).
The increase of edge chirality angle has a distinct effect on the electronic structure of honeycomb ribbons as it reduces the separation between points and (see Figs. 1b,c). More precisely, the distance between points and is given by yaz11 (); Akhmerov08 ()
This allows us to provide an estimate of the Fermi velocity as a function of spin-orbit interaction strength and chirality angle :
Figure 3 compares the magnitudes of obtained from band structure calculations performed on honeycomb ribbons, with the estimates provided by analytic expression (4). For , the analytic formula (4) shows very good agreement with the numerical results. As the chirality angle approaches the armchair edge limit, the computed values of deviate from analytic estimates eventually resulting in a finite Fermi velocity at . The case of armchair edges is special as both and are projected onto in the 1D ribbon band structure. The edge state dispersion in this situation is illustrated in Fig.1d as well as in the calculated band structure in Fig. 2. It follows that the Fermi velocity of the linear edge-state bands recovers the Fermi velocity of the massless Dirac fermions in the bulk when spin-orbit interactions are absent (the case of graphene), (Fig. 3), confirming the recent result of Gosálbez-Martínez et al. gos12 (). Interestingly, this result does not depend on the strength of spin-orbit interactions, contrary to the low- regime. On the other hand, armchair ribbons as well as high- chiral ribbons of finite width are semiconducting in the absence of spin-orbit interactions yaz11 (); son06 (). Thus, spin-orbit coupling above certain critical strength is required in order to bring these systems into the quantum spin Hall regime.
In summary, we investigated the dependence of the band dispersion of the topologically non-trivial edge states in the Kane-Mele model on the crystallographic orientation of the edges. It was shown that the Fermi velocity of the quantum spin Hall edge states increases monotonically upon varying the edge chirality angle from (zigzag edge) to (armchair edge). A simple analytical model estimates the minimum Fermi velocity as . The maximum value achieved for armchair edges recovers the Fermi velocity of the Dirac fermions on honeycomb lattice in the absence of spin-orbit interactions. The relations established for this prototypical topological insulator provide an important insight into tailoring the properties of topologically protected boundary states in realistic materials.
We acknowledge support by the Swiss National Science Foundation (grant No. PP00P2_133552).
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