Emergence of Different Replica Dirac Cones and Intra- and Intervalley Scatterings in Short-Wavelength Graphene Superlattices Modulated by an Atomic-Scale Sharp Potential

Here, we calculate the unfolded band structure of short-wavelength graphene superlattices modulated by the atomic-scale sharp potential of a semiconductor surface, using density functional theory. We show that in the case of the superlattice graphene Brillouin zone (SBZ) center folded into the primitive graphene Brillouin zone (GBZ) corner, the emergence of different replica Dirac cones and their scattering behaviors are driven by the substrate-induced potential or atomic-scale disorders as well as by the correlation between the rotation angle of the superlattice and the trigonal warping orientation of the Dirac cone because of the quantum interference associated with the scattering at the SBZ edge. In particular, a superlattice with a rotational angle of φ = 30° (G–N√3 × N√3–R30°, N > 1) facilitates intervalley scattering, whereas an unrotated superlattice (G–N3 × N3) is favorable to intravalley scattering. For a superlattice with a rotational angle in the range of 0 < φ < 30°, the intervalley and intravalley scatterings may be comparable to each other, leading to the emergence of mixed inequivalent replica cones at the SBZ center (or GBZ corner). Interestingly, such inequivalent replica cones facilitate a stronger intervalley scattering at the GBZ corner, compared to the G–N√3 × N√3–R30° and G–N3 × N3 superlattices, thus opening a larger energy gap at the primitive Dirac point. In addition to the inversion symmetry breaking in graphene, we show that intervalley scattering can also generate an energy gap at the secondary Dirac point