Unconventional quantized edge transport in the presence of interedge coupling in intercalated graphene

It is generally believed that the interedge coupling destroys the quantum spin Hall (QSH) effect along with the gap opening at the Dirac points. Using first-principles calculations, we find that the quantized edge transport persists in the presence of interedge coupling in Ta intercalated epitaxial graphene on SiC(0001), being a QSH insulator with the nontrivial gap of 81 meV. In this case, the band is characterized by two perfect Dirac cones with different Fermi velocities, yet only one maintains the edge state feature. We attribute such an anomalous behavior to the orbital-dependent decay of edge states into the bulk, which allows the interedge coupling between just one pair of edge states rather than two.

Figure 1

Illustration of the interedge coupling between the Dirac states on two edges. Red and blue represent the directions of spin while the arrows represent the directions of propagation. $R$ and $T$ are two interaction parameters corresponding to the two possible tunneling processes between the same and different spins. Generally speaking, there should exist three cases dependent upon the interedge coupling, i.e., (i) $R=0$ and $T=0$, (ii) $R\ne 0$ and $T\ne 0$, (iii) $R=0$ or $T=0$. The knowledge about the first two situations has been well established, but as yet there is no report on the third case.

Figure 2

(a) Top and (b) side view of the optimized geometries of the $\mathrm{G}/i-\mathrm{Ta}/\mathrm{SiC}$. Green dashed rhombus in (a) represents the surface cell. Note that ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ denote the two suspended carbon atoms not directly bonding to Ta. (c) Band structures of the $\mathrm{G}/i-\mathrm{Ta}/\mathrm{SiC}$ system without (left) and with (right) spin-orbit coupling. The inset is the magnified plot of the parabolic dispersion around the $\mathrm{\Gamma }$ point. (d) Local density of states of the $\mathrm{G}/i-\mathrm{Ta}/\mathrm{SiC}$ system. The Fermi levels are set to zero.

Figure 3

(a) Band structures corresponding to different ribbon widths without (a, c, e) and with (b, d, f) spin-orbit coupling. The ribbons contain 10, 26, and 46 zigzag carbon dimers from left to right, respectively. Fermi levels are at energy zero.

Figure 4

(a) Electronic structure of zigzag $\mathrm{G}/i-\mathrm{Ta}/\mathrm{SiC}$ ribbon with $W=26$ as well as the real-space charge distributions at the top (b) and down (c) Dirac point and the $k=$ 0.05 $\pi /c$ ($c$ is the lattice constant) for the first (d) and second (e) topmost valence band with an isosurface = 0.001 6 e/$\text{Å}{}^3$. Note that only one of the twofold degenerate states is plotted at the $\mathrm{\Gamma }$ point.

Figure 5

Electronic properties of the $\mathrm{G}/i-\mathrm{Ta}/\mathrm{SiC}$ ribbon as a function of width. When the ribbon is rather narrow, the Dirac cone is fully gapped due to the quantum size effect. With the increase of ribbon width, the gap decreases and is finally closed, meaning a topological phase transition. However, there still exists interedge coupling and the Dirac fermions are contributed both from the bulk and the edge. Further increasing the ribbon width, the interedge coupling becomes negligible and the system enters the true time-reversal invariant QSH state.