We use the self-consistent mean field approach to study the dispersion relation of graphene on a substrate with hexagonal lattice structure, similar to that of graphene, taking into account on-site Hubbard interactions in the substrate sites. Different Hubbard interactions in the sublattices that constitute the substrate breaks the electron hole symmetry. The interaction between the substrate and the graphene sheet opens two gaps in the dispersion relation of graphene out of the Dirac K point. The position of the gaps depends on the ratio between the first neighbor interaction in graphene and the Hubbard on-site interaction in the substrate lattice.

我们使用自洽平均场方法来研究石墨烯在具有六方晶格结构的基底（类似于石墨烯）上的色散关系，同时考虑了基底位点的现场Hubbard相互作用。构成衬底的亚晶格中不同的Hubbard相互作用破坏了电子空穴的对称性。基板和石墨烯片之间的相互作用在Dirac K点之外打开了石墨烯色散关系中的两个间隙。间隙的位置取决于石墨烯中的第一个相邻相互作用与衬底晶格中的Hubbard现场相互作用之间的比率。