Moiré minibands in graphene heterostructures with almost commensurate $\sqrt{3}\times \sqrt{3}$ hexagonal crystals

We present a phenomenological theory of the low-energy moiré minibands of Dirac electrons in graphene placed on an almost commensurate hexagonal underlay with a unit cell approximately three times larger than that of graphene. A slight incommensurability results in a periodically modulated intervalley scattering for electrons in graphene. In contrast to the perfectly commensurate Kekulé distortion of graphene, such superlattice perturbation leaves the zero-energy Dirac cones intact, but is able to open a band gap at the edge of the first moiré subbands, asymmetrically in the conduction and valence bands.

Figure 1

(a) The moiré pattern formed from graphene (blue) on a underlay (red) with $\theta =0$, $\delta =\frac{1}{9}$. The black hexagons follow Kekulé lattice of graphene. (b) The two sets of reciprocal-lattice vectors, ${\mathbf{b}}_m$ and ${\beta }_m$, with their associated Brillouin zones.

Figure 2

Numerically calculated moiré minibands shown in the smaller mBZ (left) and larger mBZ (center), and the corresponding density of states (right). A Van Hove singularity, originating from the first moiré miniband (in both the conduction and valence bands) is always present for the perturbed spectra.

Figure 3

The regions of parameter space for which either an indirect band gap (within the red volume between the black dashed lines) or direct band gap (outside the black dashed lines) is present in the conduction band. The parameter space for the valence band is obtained by flipping the sign of $U_{E^{\prime }}$.