Graphene based superconducting quantum point contacts

We investigate the Josephson effect in the graphene nanoribbons of length L smaller than the superconducting coherence length and an arbitrary width W . We find that in contrast to an ordinary superconducting quantum point contact (SQPC) the critical supercurrent Ic is not quantized for the nanoribbons with smooth and armchair edges. For a low concentration of the carriers Ic decreases monotonically with lowering W/L and tends to a constant minimum for a narrow nanoribbon with W . L. The minimum Ic is zero for the smooth edges but e∆0/~ for the armchair edges. At higher concentrations of the carriers this monotonic variation acquires a series of peaks. Further analysis of the current-phase relation and the Josephson coupling strength IcRN in terms of W/L and the concentration of carriers revels significant differences with those of an ordinary SQPC. On the other hand for a zigzag nanoribbon we find that, similar to an ordinary SQPC, Ic is quantized but to the half-integer values (n+1/2)4e∆0/~.