Indirect band gap in graphene from modulation of the Fermi velocity

In this work we study theoretically the electronic properties of a sheet of graphene grown on a periodic heterostructure substrate. We write an effective Dirac equation, which includes a dependence of both the band gap and the Fermi velocity on the position, due to the influence of the substrate. This way, both bandgap and Fermi velocity enter the Dirac equation as operators. The Dirac equation is solved exactly and we find the superlattice minibands with gaps due to the breaking of translational symmetry induced by the underlying heterostructure. The spatial dependence of the Fermi velocity makes the band gap be indirect, bringing about interesting possibilities for applications in the design of nanoelectronic devices. In the limit of constant Fermi velocity we obtain a band structure, with direct band gap, very close to the one previously found in the literature, obtained using the transfer matrix method.