Topologically confined states at corrugations of gated bilayer graphene

We investigate the electronic and transport properties of gated bilayer graphene with one corrugated layer, which results in a stacking AB/BA boundary. When a gate voltage is applied to one layer, topologically protected gap states appear at the corrugation, which reveal as robust transport channels along the stacking boundary. With increasing size of the corrugation, more localized, quantum-well-like states emerge. These finite-size states are likewise conductive along the fold, but in contrast to the stacking boundary states, which are gapless, they present a gap. Additionally, we have studied periodic corrugations in bilayer graphene; our findings show that such corrugations between AB- and BA-stacked regions also behave as conducting channels in the direction of the folds. These topological states could be easily identified due to the spatial shape of the corrugations.

Figure 1

Top and side view of bilayer graphene with an AB/BA stacking boundary due to a corrugation of the upper layer. The stacking change from AB to BA is clearly visible in the top view. The side view below shows the geometry chosen for the corrugation or fold, namely, half a nanotube. $N_B$ and $N_T$ are the lengths given in the number of unit cells of the bottom and top layer at the fold.

Figure 2

Conductance (black line) and LDOS (blue line) of two bilayer ribbons of $W=4$ and two different fold lengths: (a) $N_B=6\text{and}{N}_T=10$ and (b) $N_B=14\text{and}{N}_T=23$. The red line shows the conductance of the infinite bilayer ribbons acting as leads. (c) shows the band structure close to the Fermi level of a bilayer ribbon with Bernal stacking. Open circles indicate that the corresponding states are localized on disconnected sublattices, i.e., B1 and A2 sites for AB stacking and A1 and B2 sites in the BA case.

Figure 3

(a) LDOS and (b) conductance of a bilayer corrugated ribbon with a $V=0.3$ V gate voltage applied to the bottom layer. The width of the ribbon is $W=4$ and the folded region lengths are $N_B=6$ and $N_T=10$. The red line in (b) shows the conductance of the infinite bilayer leads with the same bias. (c) Band structure of the infinite bilayer ribbons that constitute the leads. Open circles indicate that the corresponding states are localized on disconnected sublattices, whereas filled circles indicate states with weight in any sublattice.

Figure 4

Spatial distribution of the localized state with energy $E_1=0.081$ eV found in the corrugated bilayer ribbon of width $W=4$ and corrugation given by $N_B=6$ and $N_T=10$. (a) Atom-resolved LDOS; the circles plotted on each node have radii proportional to the corresponding LDOS values. Red and blue colors indicate that the node belongs to the bottom and top layers, respectively. (b) Unit-cell averaged LDOS at the bottom (red) and the top layer (blue); the zero labels mark the limits of the stacking boundary. (c) and (d) Unit-cell-averaged LDOS at both layers (red - bottom; blue - top), plotted separately for (c) A and (d) B sublattices; in these two latter panels, open circles indicate that the LDOS is located at unconnected nodes.

Figure 5

Conductance between leads (left) and LDOS at the corrugation (right) for the $N_B=6$ and $N_T=10$ case as functions of the energy $E$ and wave vector $k$. (a) and (b) $V=0$; (c) and (d) $V=0.3\mathrm{V}$ applied to the bottom layer. Notice that the LDOS scale is logarithmic.

Figure 6

LDOS as a function of the energy $E$ and wave vector $k$ at the corrugation, summed over all its nodes, for two large corrugations: (a) $N_B=14$ and $N_T=23$ and (b) $N_B=30$ and $N_T=48$. Because of the large values of the LDOS at the localized states, the LDOS scale is logarithmic. (c) Zoom of the gap states presented in (b), with two types of symmetries labeled as S1 and S2; the topologically protected states are marked with an asterisk. (d) Layer-resolved LDOS of the same system, $ln$(${\mathrm{LDOS}}_B/{\mathrm{LDOS}}_T$), plotted only when ${\mathrm{LDOS}}_B+{\mathrm{LDOS}}_T$ is larger than a threshold value ${10}^{-3}$.

Figure 7

Conductances and band structures $E\left(k_y\right)$ calculated for $k_x=0$ of graphene bilayers with periodic corrugations ($N_B=6, N_T=10$). The separation between folds is $d=10$ unit cells and the applied gate voltage is $V=0.3$ V. (a) and (b) correspond to the case where the corrugations constitute stacking boundaries between AB and BA regions. In (c) and (d), the corrugations separate bilayer areas with the same stacking.