Electronic band gaps and transport in Cantor graphene superlattices

•The Cantor graphene superlattices are investigated theoretically.•Cantor fractal structure can possess an unusual Dirac point.•The Dirac corresponds to the zero-averaged wave number.•Multi-Dirac-points appear by adjusting the lattice constants and the order.•The electron transport in Cantor fractal structure can be controlled.

The electronic band gap and transport in Cantor graphene superlattices are investigated theoretically. It is found that such fractal structure can possess an unusual Dirac point located at the energy corresponding to the zero-averaged wave number (zero–k‾). The location of the Dirac point shifts to lower energy with the increase of order number. The zero–k‾ gap is robust against the lattice constants and less sensitive to the incidence angle. Moreover, multi-Dirac-points may appear by adjusting the lattice constants and the order, and an expression for their location is derived. The control of electron transport in such fractal structure may lead to potential applications in graphene-based electronic devices.