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Tunable Electronic Band Structures and Zero-Energy Modes of Heterosubstrate-induced Graphene Superlattices

We propose a tunable electronic band gap and zero-energy modes in periodic heterosubstrate-induced graphene superlattices. Interestingly, there is an approximate linear relation between the band gap and the proportion of inhomogeneous substrate (i.e., percentages of different components) in the proposed superlattice, and the effect of structural disorder on the relation is discussed. In inhomogeneous substrate with equal widths, zero-energy states emerge in the form of Dirac points by using asymmetric potentials, and the positions of Dirac points are addressed analytically. Further, the Dirac point exists at $\mathbf{k}=\mathbf{0}$ only for specific potentials; every time it appears, the group velocity vanishes in $k_y$ direction and the resonance occurs. For general cases that inhomogeneous substrate with unequal widths, a part of zero-energy states are described analytically, and differently, they are not always Dirac points. Our prediction may be realized on the heterosubstrate such as SiO$_2$/BN type.