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 {\delta}-function potential barriers of strength P in bilayer graphene is periodic in P with period {\pi}. 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 {\delta}-function barriers (Kronig-Penney model) in bilayer graphene is periodic in P with period 2{\pi}. 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 {\delta}-function barriers with P alternating in sign between successive barriers; the corresponding spectrum is periodic in P with period {\pi}.