Controlled formation of isolated miniband in bilayer graphene on almost commensurate $\sqrt{3}\times \sqrt{3}$ substrate

We investigate theoretically the interplay between the effects of a perpendicular electric field and incommensurability at the interface on the electronic properties of a heterostructure of bilayer graphene and a semiconducting substrate with a unit cell almost three times larger than that of graphene. It is known that the former introduces an asymmetry in the distribution of the electronic wave function between the layers and opens a band gap in the electronic spectrum. The latter generates a long wavelength periodic moiré perturbation of graphene electrons which couples states in inequivalent graphene Brillouin zone corners and leads to the formation of minibands. We show that, depending on the details of the moiré perturbation, the miniband structure can be tuned from that with a single band gap at the neutrality point and overlapping minibands on the conduction/valence band side to a situation where a single narrow miniband is separated by gaps from the rest of the spectrum.

Figure 1

(a) Schematic of the moiré superlattice, with graphene (gray) and an incommensurate $\sqrt{3}\times \sqrt{3}$ substrate (red). Also shown is the Kekulé lattice of graphene (blue). For clarity we choose a large $\delta$, set $\theta =0$, and do not show the top graphene layer of BLG. While the mismatch $\delta$ between the substrate (red) and the graphene Kekulé lattice (blue) results in a superlattice periodicity set by $\mathit{A}$, a shorter periodicity $\tilde{\mathit{A}}$ of the local atomic arrangement surrounding a carbon atom (gray) always exists (see the text and the Appendix for details). (b) The two periodicities set by $\mathit{A}$ and $\tilde{\mathit{A}}$ result in two sets of basic reciprocal vectors, ${\check{\mathbf{\beta }}}_m$ and ${\check{\mathit{b}}}_m,m=0,1,\cdots ,5$, respectively, and two superlattice Brillouin zones that can be used to describe the miniband spectrum. For the larger BZ set by the reciprocal vectors ${\check{\mathit{b}}}_m$, we introduce the high-symmetry points $\gamma ,\mu$, and $\kappa$. (c) Spectrum of bilayer graphene folded onto the sBZ set by vectors ${\check{\mathit{b}}}_m$. States from the ${\mathit{K}}_{+}$ $\left({\mathit{K}}_{-}\right)$ valley are shown in magenta (yellow).

Figure 2

(a)–(c) Moiré miniband spectra and density of states (DoS) corresponding to characteristic behaviors of the miniband spectrum, as discussed in the text.

Figure 3

Regimes of miniband spectra in the $(U_{E^{\prime }},U_G)$ parameter space for the heterostructure of BLG and ${\mathrm{In}}_2{\mathrm{Te}}_2$ $\left(\delta =-0.007\right)$ and (a) $u=10$, (b) $u=0$, and (c) $u=-20$ meV. For the region in blue, a global gap separates the first and second miniband on the valence side. The white region corresponds to the overlapping first and second minibands. The dashed red line represents the boundary between the blue and white regions as obtained using first order perturbation theory analysis of the high-symmetry points in the sBZ. The green dot represents point in the $(U_{E^{\prime }},U_G)$ space for which miniband spectra are shown in Fig. 2.