Zero-energy modes and gate-tunable gap in graphene on hexagonal boron nitride

In this article, we derive an effective theory of graphene on a hexagonal boron nitride (h-BN) substrate. We show that the h-BN substrate generically opens a spectral gap in graphene despite the lattice mismatch. The origin of that gap is particularly intuitive in the regime of strong coupling between graphene and its substrate, when the low-energy physics is determined by the topology of a network of zero-energy modes. For twisted graphene bilayers, where inversion symmetry is present, this network percolates through the system and the spectrum is gapless. The breaking of that symmetry by h-BN causes the zero-energy modes to close into rings. The eigenstates of these rings hybridize into flat bands with gaps in between. The size of this band gap can be tuned by a gate voltage and it can reach the order of magnitude needed to confine electrons at room temperature.