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Kronig-Penney model on bilayer graphene: spectrum and transmission periodic in the strength of the barriers

We show that the transmission through single and double δ-function potential barriers of strength P in bilayer graphene is periodic in P with period π. For a certain range of P values we find states that are bound to the potential barrier and that run along the potential barrier. Similar periodic behaviour is found for the conductance. The spectrum of a periodic succession of δ-function barriers (Kronig-Penney model) in bilayer graphene is periodic in P with period 2π. For P smaller than a critical value, the spectrum exhibits two Dirac points while for P larger than this value an energy gap opens. These results are extended to the case of a superlattice of δ-function barriers with P alternating in sign between successive barriers; the corresponding spectrum is periodic in P with period π.