  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.
