Band gaps in incommensurable graphene on hexagonal boron nitride

Devising ways of opening a band gap in graphene to make charge-carrier masses finite is essential for many applications. Recent experiments with graphene on hexagonal boron nitride ($h$-BN) offer tantalizing hints that the weak interaction with the substrate is sufficient to open a gap, in contradiction of earlier findings. Using many-body perturbation theory, we find that the small observed gap is what remains after a much larger underlying quasiparticle gap is suppressed by incommensurability. The sensitivity of this suppression to a small modulation of the distance separating graphene from the substrate suggests ways of exposing the larger underlying gap.

Figure 1

LDA band structures of graphene on $h$-BN for the (a), (b), and (c) commensurable structures, with respect to a common vacuum level. Solid and dashed lines give the DFT results and the TB fits, respectively. The region around the Dirac point is enlarged in the panels below. The formation of an interface dipole is illustrated by the charge displacement $-e\Delta n$ in the $yz$ plane containing B, C, and N atoms. Blue and red indicate regions of negative and positive charge density, respectively, giving a dipole moment p. (d) The interface potential step $\Delta V$ as a function of the position $(x,y)$ of graphene with respect to $h$-BN, at the RPA equilibrium separation $d_{\mathrm{eq}}(x,y)$ [see Fig. 3]. (e) Dependence of $\Delta V$ on $d$ for configurations [(a)–(c)].

Figure 2

Quasiparticle (QP) and LDA gaps opened at the Dirac point as a function of the inverse cell size. The thick solid lines correspond to graphene at the RPA equilibrium distances of configurations (a), (b), and (c) [22]. The dotted, dashed, and dashed-dotted lines correspond to graphene–$h$-BN separations of 3.35, 3.45, and 3.55 Å, respectively.

Figure 3

(a) The RPA equilibrium separation as a function of the lateral displacement of the two lattices for commensurable graphene on $h$-BN. (b) The position of the Dirac point with respect to the center of the $h$-BN band gap, ${\varepsilon }_{\mathrm{D}}$. The plotted values are relative to the constant ${\varepsilon }_{\mathrm{D}}$ obtained for graphene and $h$-BN at an infinite interlayer distance. Schematic representations of (c) commensurable and (d) incommensurable configurations of graphene on $h$-BN where the bonding locally resembles commensurable configurations.