Complex band structure of topologically protected edge states

One of the great successes of modern condensed matter physics is the discovery of topological insulators (TIs). A thorough investigation of their properties could bring such materials from fundamental research to potential applications. Here, we report on theoretical investigations of the complex band structure (CBS) of two-dimensional (2D) TIs. We utilize the tight-binding form of the Bernevig, Hughes, and Zhang model as a prototype for a generic 2D TI. Based on this model, we outline the conditions that the CBS must satisfy in order to guarantee the presence of topologically protected edge states. Furthermore, we use the Green's function technique to show how these edge states are localized, highlighting the fact that the decay of the edge-state wave functions into the bulk of a TI is not necessarily monotonic and, in fact, can exhibit an oscillatory behavior that is consistent with the predicted CBS of the bulk TI. These results may have implications for electronic and spin transport across a TI when it is used as a tunnel barrier.

Figure 1

(a) Energy-level diagram of the model. (b) Schematic demonstrating the phase factors appearing in the intersite s-p hopping elements in Eq. (1) for the spin-up channel. Circles represent $s$ orbital basis states on sites neighboring the central $p$ orbital.

Figure 2

Schematic geometry of the edge showing the construction of the Dyson equation. The unperturbed system, described by Green's function $g$, consists of an isolated, infinite chain of lattice sites along the $y$ direction and a semi-infinite half plane. Introduction of coupling between the chain and the edge leads to the perturbed system, a lone semi-infinite half plane with $n$>0, described by the Green's function $G$.

Figure 3

The layer-resolved LDOS near the edge of the model TI (a) over the entire range of bands and (b) in the region of the bulk band gap. Model parameters are $a=1$, $t=1.2$, $t_{sp}=0.3$, and $\varepsilon =3.0$. The total DOS in the bulk (filled curves) was integrated using a uniform $500\times 500\mathbf{k}$-mesh. The LDOS for layers near the edge is computed using Eq. (14), with an energy broadening of $\eta =0.01$ and integrated using a mesh of 500 $k_y$ points. (c)–(e) Layer dependence of the LDOS at $E=0.50$, 0.25, and 0, respectively. Curves are calculated using Eq. (19), with parameters derived from the CBS given in the figures.

Figure 4

The $k_y$-resolved spin-up LDOS of the model topological insulator in the region around the bulk band gap for (a) the edge, (b)–(f) layers one through five, respectively, and (g) the bulk. Panels (a)–(f) use a broadening of $\eta =0.01$ to resolve the δ-function-like gap states. The bulk DOS, (g), does not require broadening. The spin-down LDOS is the mirror image of each panel around $k_y=0$. Model parameters are given in the caption of Fig. 3.

Figure 5

Dependence of the CBS and band topology on ε. (a) and (b) The imaginary and real parts of $k_x=-{ia}^{-1}\ln \lambda$ for the CBS state at $k_y=0$ and $E=0$. Solid and dotted curves correspond to ${\mathbf{u}}_{+}$ and ${\mathbf{u}}_{-}$ from Eq. (15), respectively. Criteria for a topologically nontrivial state are satisfied for ε < $4t$, denoted by the filled background. (c) and (d) $k_y$-resolved LDOS at the edge for (c) $\varepsilon =2t$ showing a topologically protected edge state and (d) $\varepsilon =5t$ showing a simple band-insulating state. The color scale is the same as in Fig. 4. Parameters $t$ and $t_{sp}$ are fixed to the values given in the caption of Fig. 3.

Figure 6

The complex bands for ${\mathbf{u}}_{+}$ described by Eq. (16). The real part of $k_x$ is plotted in (a) and the corresponding imaginary part plotted in (b). Solid and dashed curves correspond to ${\lambda }_{+}^{+}$ and ${\lambda }_{+}^{-}$, respectively. The shaded region indicates the range over which the criteria for forming an edge state are satisfied. Model parameters are given in the caption of Fig. 3.