Group velocity in finite graphene superlattices

We have investigated the effects of finite crystal size on the electronic properties of infinite graphene superlattices. After deriving analytical expressions for the dispersion relation and group velocity vg in finite N-period graphene superlattices, we demonstrated that these properties can be dramatically modified by such effects, especially those directly associated with the band structure, such as the Dirac points and the wavevector dependent minigap situated in between them. In particular, it was shown that this minigap disappears and becomes a null-gap curve. It was also shown that the properties of vg differ substantially from those of the infinite superlattice. An interesting difference is that one of the components of vg may attain superluminal values. Our results provide a systematical procedure for describing and understanding the electronic transport and optical characteristics of any finite N-period graphene superlattices, such as the density of states, conductivity and absorption coefficient.