Transmission in graphene–topological insulator heterostructures

We investigate scattering of the topological surface state of a three-dimensional time-reversal invariant topological insulator when graphene is deposited on the topological-insulator surface. Specifically, we consider the (111) surface of a ${\mathrm{Bi}}_2{\mathrm{Se}}_3$-like topological insulator. We present a low-energy model for the graphene–topological insulator heterostructure and we calculate the transmission probability at zigzag and armchair edges of the deposited graphene, and the conductance through graphene nanoribbon barriers, and show that its features can be understood from antiresonances in the transmission probability.

Figure 1

(a) Top view of the most symmetric commensurate $\sqrt{3}\times \sqrt{3}$ $R30$ stackings of the graphene (small red dots) and TIS (large gray dots) heterostructure. The structures differ by the position of the TIS atom in the unit cell: (H) the center of a graphene hexagon, (T) one sublattice on top, and (B) bond on top. (b),(c) Energy spectrum of the (b) H and (c) T structure with $\mu =0$ and $v_t=v_g/2$. In both cases, the dashed curve is the original topological Dirac cone of the TIS. For (b), the spectrum is shown for $t^{\prime \prime }=0.6$ eV, while for (c) $t=t_A=0.3$ eV, $t_B=0$, and the index $n=1,...,5$ labels the scattering channel. The spectrum for B stacking is also given by (c) with $t^{\prime }=t/\sqrt{2}$. (d) Momentum space of the commensurate $\sqrt{3}\times \sqrt{3}$ $R30$ stacking configurations shown in (a) in the extended zone scheme. The small gray hexagons correspond to the TIS, where the dots are reciprocal lattice points, and the large red hexagon is the first BZ of graphene; the $K$ and $K^{\prime }$ point of graphene are folded to the $\overline{\mathrm{\Gamma }}$ point of the surface BZ of the TI.

Figure 2

Representation of the block diagonalization of $h\left(\mathit{k}\right)$ into subspaces that are even $(+$, dashed blue cone) and odd $(-$, red cone) under valley exchange. The spectra are shown for $t=0$ and $2v_t=v_g$. Only the even subspace couples to the topological-insulator surface (green cone).

Figure 3

Low-energy spectrum for T (or B) stacking where the corresponding spin expectation values are shown as arrows. All bands except the two valley-odd Dirac cones that are decoupled in the bulk heterostructure are shown. The explicit expressions of the spectrum are given in Appendix pp3.

Figure 4

Basic edge geometries of graphene (red, small dots) on top of a TIS (gray, large dots) for the T stacking configuration. The Roman numerals indicate different scattering regions. (a) Zigzag edges: two types depending on whether the edge is terminated by the A (ZZ1) or B (ZZ2) sublattice. (b) One of the three physically distinct armchair edges which the continuum model cannot distinguish.

Figure 5

(a)–(e) Transmission probabilities $T_n$ for scattering at the ZZ1 step with $t=0.3$ eV for the scattering channels $n=1,...,5$, respectively, and (f) the total transmission probability $T={\sum }_{n=1}^5{T}_n$.

Figure 6

Transmission probability $T(E,k_y)$ for a zigzag ribbon with $t=0.3$ eV, and (a) $N=10$ and (b) $N=20$. The red lines outside the cone are bound states and the density corresponding to the states marked with an asterisk is shown in Fig. 9.

Figure 7

Transmission probability $T(E,k_y)$ for an armchair ribbon with $t=0.3$ eV, and (a) $N=30$ (insulating) and (b) $N=41$ (metallic). The red lines outside the cone are bound states.

Figure 8

Transmission probability $T(E,k_y)$ for the nanoribbon barrier. (a),(b) Zigzag ribbon with $N=30$ for (a) $t=0.1$ eV and (b) $t=0.2$ eV. (c),(d) Armchair ribbon with $N=30$ for (c) $t=0.1$ eV and (d) $t=0.2$ eV. The red lines are bound states, localized in the barrier, while the orange dashed lines in (a) and (c) are the bound states of a bare graphene nanoribbon.

Figure 9

Projected electron density of the (a) lower and (b) upper branch of edge states for the zigzag ribbon with $N=20$ and $t=0.3$ for $k_y=0.7{\mathrm{nm}}^{-1}$. These states are marked in Fig. 6 with an asterisk.

Figure 10

Energy of the zigzag edge states at $k_y=2{\mathrm{nm}}^{-1}$ as a function of $t$ for $N=10$ and $N=20$. We only show one state for $N=20$, since the other state remains at zero energy for all $t$.

Figure 11

(a),(b) Conductance for the (a) zigzag and (b) armchair barrier for several widths with $t=0.3$ eV as a function of the Fermi energy $E$. The widths of the armchair ribbon are chosen so that it is insulating and matches the corresponding widths in the zigzag case. (c),(d) Conductance for the (c) zigzag barrier with $N=20$ and (d) armchair barrier with $N=34$ for several $t$, whose values are shown in eV.