We use first-principle density functional theory (DFT) to study the transport properties of single and double barrier heterostructures realized by stacking multilayer h-BN or BC$_{2}$N, and graphene films between graphite leads. The heterostructures
are lattice matched. The considered single barrier systems consist of layers of up to five h-BN or BC$_{2}$N monoatomic layers (Bernal stacking) between graphite electrodes. The transmission probability of an h-BN barrier exhibits two unusual behaviors: it is very low also in a classically allowed energy region, due to a crystal momentum mismatch between states in graphite and in the dielectric layer, and it is only weakly dependent on energy in the h-BN gap, because the imaginary part of the crystal momentum of h-BN is almost independent of energy. The double barrier structures consist of h-BN films separated by up to three graphene layers. We show that already five layers of h-BN strongly suppress the transmission between graphite leads, and that resonant tunneling cannot be observed because the energy dispersion relation cannot be decoupled in a vertical and a transversal component.
