Sub-Sharvin Conductance and Josephson Effect in Graphene

Titov and Beenakker Phys. Rev. B 74, 041401 (R) (2006) found, by solving the Dirac–Bogoliubov–De-Gennes equation, that the product of critical current and normal-state resistance for superconductor–graphene–superconductor (S–g–S) Josephson junction takes values (for a short junction and zero temperature) between \ (I_ cR_ N 2. 1\) and \ (I_ cR_ N 2. 4\) in units of \ (e/ ₀\), where \ (₀\) is the superconducting gap. These values are notably higher than the tunnelling bound (\ (/2\) ), but lower than the ballistic bound (\ (\) ). Here, we analyze the tunneling of Cooper pairs numerically through S–g–S junctions in which the longitudinal electrostatic potential profile is tuned, within gates electrodes, from a rectangular to a parabolic one. In the unipolar regime (i. e. , when the chemical potential is above the top of a barrier, \ (0\) ), it is found that \ (I_ cR_ N\) gradually evolves from the graphene-specific to the ballistic value. At the same time, the normal-state conductance increases from the sub-Sharvin value of \ (1/R_ N (/4) \, G ₒ₇₀ₑₕ₈₍\) towards to the Sharvin value \ (G ₒ₇₀ₑₕ₈₍=g₀| |W/ (v ₅) \), with the conductance quantum \ (g₀=4e²/h\), the junction width \ (W\), and the Fermi velocity in graphene \ (v ₅\). In contrast, in the tripolar regime (\ (0\) ), both normal-state conductance and the critical current are suppressed when smoothing the potential; however, \ (I_ cR_ N\) remains close to the graphene-specific range, even for a parabolic potential. The skewness of the current-phase relation is also discussed. Abstract Published by the Jagiellonian University 2026 authors