Weak side of strong topological insulators

Strong topological insulators may have nonzero weak indices. The nonzero weak indices allow for the existence of topologically protected helical states along line defects of the lattice. If the lattice admits line defects that connect opposite surfaces of a slab of such a “weak-and-strong” topological insulator, these states effectively connect the surface states at opposite surfaces. Depending on the phases accumulated along the dislocation lines, this connection results in a suppression of in-plane transport and the opening of a spectral gap or in an enhanced density of states and an increased conductivity.

Figure 1

(a) Topological insulator slab of size $L\times W$, with top and bottom surfaces connected via randomly placed dislocation lines with mean distance $d$. Ideal contacts are attached to the left and right, for top and bottom surfaces separately. (b) Zero-angular-momentum ($m=0$) radial waves for nonzero wave number $k$ at the surface of the topological insulator are transmitted perfectly into and out of the one-dimensional helical states along the dislocation line.

Figure 2

Schematic picture of a top view (a) and side view (b) of the topological insulator slab. The calculational scheme involves the computation of the scattering matrix $\tilde{S}$ for a slab of width $\delta L$, followed by the concatenation of scattering matrices of individual slabs to obtain the scattering matrix $S$ of the full structure.

Figure 3

Zero-energy in-plane conductivity ${\sigma }_{\parallel }$ (a) and cross conductance $G_{\perp }$ (b) for a slab of weak-and-strong topological insulator with a concentration $1/d^2$ of randomly placed dislocation lines. The different curves refer to different choices for the phases ${\theta }_{\mathrm{d}}\left({\mathbf{r}}_{\mathrm{d}}\right)$, as shown in the figure. The dashed line in (a) denotes the clean-limit in-plane conductivity ${\sigma }_{\parallel ,0}=2e^2/\pi h$. Data points denote an average over 500 disorder realizations, statistical error bars are typically smaller than the markers.

Figure 4

Density of states $dN/dE$ of a sample with dislocation line density $1/d^2$. The four curves represent the four representative scenarios for the choice of the phase shifts ${\theta }_{\mathrm{d}}\left({\mathbf{r}}_{\mathrm{d}}\right)$, as explained in the text. The vertical dashed lines correspond to energies calculated from Eq. (11). The thin black lines denote the ideal surface-state density of states without dislocation lines. The arrow represents a Dirac delta function at zero energy. Data points denote an average over five disorder realizations and 80 values of the crystal momentum ${\kappa }_x$.