  Using highly efficient simulations of the tight-binding Bogoliubov-de Gennes
model we solved self-consistently for the pair correlation and the Josephson
current in a Superconducting-Bilayer graphene-Superconducting Josephson
junction. Different doping levels for the non-superconducting link are
considered in the short and long junction regime. Self-consistent results for
the pair correlation and superconducting current resemble those reported
previously for single layer graphene except in the Dirac point where remarkable
differences in the proximity effect are found as well as a suppression of the
superconducting current in long junction regime. Inversion symmetry is broken
by considering a potential difference between the layers and we found that the
supercurrent can be switched if junction length is larger than the Fermi
length.
