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

Abstract

We show that the transmission through single and double $\ensuremath{\delta}$-function potential barriers of strength $P=V{W}_{b}/\ensuremath{\hbar}{v}_{F}$ in bilayer graphene is periodic in $P$ with period $\ensuremath{\pi}$. 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 behavior is found for the conductance. The spectrum of a periodic succession of $\ensuremath{\delta}$-function barriers (Kronig-Penney model) in bilayer graphene is periodic in $P$ with period $2\ensuremath{\pi}$. For $P$ smaller than a critical value ${P}_{c}$, the spectrum exhibits two Dirac points while for $P$ larger than ${P}_{c}$ an energy gap opens. These results are extended to the case of a superlattice of $\ensuremath{\delta}$-function barriers with $P$ alternating in sign between successive barriers; the corresponding spectrum is periodic in P with period $\ensuremath{\pi}$.