Superconducting lens and Josephson effect in AA-stacked bilayer graphene

We study superconducting transport phenomena involving the lensing effect and supercurrent in AA-stacked bilayer graphene, which is characterized by a linear gapless band with two shifted Dirac cones. Our findings indicate that cross Andreev reflection and Josephson current occur exclusively within the intracone process, while intercone scatterings are strictly prohibited. The normal/superconductor/normal junction can act as a superconducting lens for the upper and lower cones. Depending on the cone index, the transmitted electrons and holes can be focused, collimated, or diverged by adjusting the gate voltages. In the superconductor/normal/superconductor junction, due to interlayer coupling, the critical currents of the two cones exhibit distinct oscillation periods with junction width, leading to an irregular oscillation of the total critical current. Furthermore, the oscillations of the critical currents with the exchange field maintain a stable phase difference of one quarter period between the two cones. Consequently, a cone-dependent $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition is achieved in this Josephson junction.