Date Published: December 6, 2019
Author(s): W Jaskólski, M Pelc, Garnett W Bryant, Leonor Chico, A Ayuela.
Experiments in gated bilayer graphene with stacking domain walls present topological gapless states protected by no-valley mixing. Here we research these states under gate voltages using atomistic models, which allow us to elucidate their origin. We find that the gate potential controls the layer localization of the two states, which switches non-trivially between layers depending on the applied gate voltage magnitude. We also show how these bilayer gapless states arise from bands of single-layer graphene by analyzing the formation of carbon bonds between layers. Based on this analysis we provide a model Hamiltonian with analytical solutions, which explains the layer localization as a function of the ratio between the applied potential and interlayer hopping. Our results open a route for the manipulation of gapless states in electronic devices, analogous to the proposed writing and reading memories in topological insulators.
Two-dimensional Dirac materials, like graphene, have attracted remarkable interest for novel nanoelectronic applications due to their reduced dimensionality and extraordinary transport properties [1–6]. Recently, great effort has been devoted to seek and develop methods to control the transport of different degrees of freedom in these new materials. Spin , valley [8, 9], angular momentum  or cone  are being proposed in addition to charge, as a means to convey and store information in future devices. In this field, bilayer graphene (BLG) stands out as a promising candidate for nanoelectronics [12–16]. In its most common form, the so-called Bernal or AB-stacked BLG, an energy gap can be opened and tuned by an applied gate voltage [17–20], which is not possible in single-layer graphene.
We first review the main features of the system. The left panel of figure 1 shows a single domain wall between AB and BA regions in bilayer graphene joined in the zigzag direction. Strained or corrugated graphene presents larger and more realistic domain walls, but for our theoretical analysis we choose an abrupt boundary that allows for an AB/BA stacking change. Previous works [23, 24, 28, 30] have shown the robustness of topological states for smoother boundaries, with their main features preserved. For such a single boundary under a gate there are two topological gapless modes around the K point, as shown in the right panel of figure 1. The plot shows the local density of states (LDOS) resolved in energy E and wave vector k. These calculations employ a Green’s function matching method7 and a pz tight-binding model, with an intralayer hopping parameter γ0 = −2.7 eV and a single interlayer hopping γ1 = 0.1γ0 [12, 13]. There is another valley with another couple of gapless states with negative wave vectors with respect to figure 1 due to time-reversal symmetry. Note that the valley separation has motivated the proposal to employ such topologically protected modes for graphene valleytronics [31, 32].
We report on the effect of layer localization exchange of topological states at a single stacking domain wall for small and large gate voltages. We use the same Green’s function matching method and graphene model when there is an applied V. The LDOS around the K valley projected in the boundary nodes is shown in figure 2 for different voltages. The localization in the top and bottom layers is presented by the color scale from blue to red8. We start with the LDOS for ungated bilayer in panel (a). The LDOS localizes differently at the top and bottom layers because the symmetry between them is broken by the stacking domain wall. For a small gate voltage applied to the bottom layer, e.g. V = 0.1 eV in panel (b), the gap states appear, as it is well known, but they turn out to be separated in the two layers. The state on the left side of the cone is more localized at the bottom layer, while the state on the right is at the top layer. This localization can be explained as a perturbation of the LDOS of the ungated bilayer for small voltages applied. For increasing voltages, around V = γ1 in panel (c), the states become fully mixed between layers. Next, for a large voltage, e.g. V = 0.5 eV in panel (d), the states are again separated in the top and bottom layer. This time, however, the state on the left side of the cone is more localized at the top layer, while the state on the right is at the bottom layer. Therefore, the topological states at the boundary atoms reverse localization for small and large voltages of the same sign.
We have investigated the gapless states with topological character that appear in gated bilayer graphene with stacking domain walls. By employing atomistic models, we find that each of the two topological states in a valley is layer-resolved; furthermore, their localization is switched between the top and bottom layer by varying the magnitude, but not the sign, of the gate voltage. Therefore, besides the valley and sublattice degrees of freedom, these states can also be labeled by a layer index.