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Goos-Hänchen Shifts in AA-Stacked Bilayer Graphene Superlattices

The quantum Goos-Hänchen shifts of the transmitted electron beam through an AA-stacked bilayer graphene superlattices is investigated. We found that the band structures of graphene superlattices can have more than one Dirac point, their locations do not depend on the number of barriers. It was revealed that any $n$-barrier structure is perfectly transparent at normal incidence around the Dirac points created in the superlattices. We showed that the Goos-Hänchen shifts display sharp peaks inside the transmission gap around two Dirac points ($E= V_B + τ$, $E= V_W + τ$), which are equal to those of transmission resonances. The obtained Goos-Hänchen shifts are exhibiting negative as well as positive behaviors and strongly depending on the location of Dirac points. It is observed that the maximum absolute values of the shifts increase as long as the number of barriers is increased. Our analysis is done by considering four cases: single, double barriers, superlattices without and with defect.