Van der Waals heterostructures of graphene and hexagonal boron nitride feature a moire superlattice for graphenes Dirac electrons. Here, we review the effects generated by this superlattice, including a specific miniband structure featuring gaps and
secondary Dirac points, and a fractal spectrum of magnetic minibands known as Hofstadters butterfly.
We consider the role of deformations in graphene heterostructures with hexagonal crystals (including strain, wrinkles and dislocations) on the geometrical properties of moire patterns characteristic for a pair of two incommensurate misaligned isostru
ctural crystals. By, employing a phenomenological theory to describe generic moire perturbation in van der Waals heterostructures of graphene and hexagonal crystals we investigate the electronic properties of such heterostructure.
We find a systematic reappearance of massive Dirac features at the edges of consecutive minibands formed at magnetic fields B_{p/q}= pphi_0/(qS) providing rational magnetic flux through a unit cell of the moire superlattice created by a hexagonal sub
strate for electrons in graphene. The Dirac-type features in the minibands at B=B_{p/q} determine a hierarchy of gaps in the surrounding fractal spectrum, and show that these minibands have topological insulator properties. Using the additional $q$-fold degeneracy of magnetic minibands at B_{p/q}, we trace the hierarchy of the gaps to their manifestation in the form of incompressible states upon variation of the carrier density and magnetic field.
We present a phenomenological theory of the low energy moire minibands of Dirac electrons in graphene placed on an almost commensurate hexagonal underlay with a unit cell pproximately three times larger than that of graphene.A slight incommensurabili
ty results in a periodically modulated intervalley scattering for electrons in graphene. In contrast to the perfectly commensurate Kekule distortion of graphene, such supperlattice perturbation leaves the zero energy Dirac cones intact, but is able to open a band gap at the edge of the first moire subbband, asymmetrically in the conduction and valence bands.
Using a general symmetry-based approach, we provide a classification of generic miniband structures for electrons in graphene placed on substrates with the hexagonal Bravais symmetry. In particular, we identify conditions at which the first moire min
iband is separated from the rest of the spectrum by either one or a group of three isolated mini Dirac points and is not obscured by dispersion surfaces coming from other minibands. In such cases the Hall coefficient exhibits two distinct alternations of its sign as a function of charge carrier density.
