Sub-Sharvin conductance and Josephson effect in graphene

Titov and Beenakker [Phys. Rev. B 74, 041401® (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 $I_c{R}_N\approx 2.1$ and $I_c{R}_N\approx 2.4$ in units of $e\mathrm{/}{\mathrm{\Delta }}_0$, where ${\mathrm{\Delta }}_0$ is the superconducting gap. These values are notably higher than the tunnelling bound ($\pi \mathrm{/}2$), but lower than the ballistic bound ($\pi$). Here we analyze numerically the tunneling of Cooper pairs 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, $\mu >0$), it is found that $I_c{R}_N$ 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 $1\mathrm{/}{R}_N\approx (\pi \mathrm{/}4)G_{\mathrm{S}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{v}\mathrm{i}\mathrm{n}}$ towards to the Sharvin value $G_{\mathrm{S}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{v}\mathrm{i}\mathrm{n}}=g_0\mathrm{\mid }\mu \mathrm{\mid }W\mathrm{/}(\pi \mathrm{\hslash }{v}_F)$, with the conductance quantum $g_0=4e^2\mathrm{/}h$, the junction width $W$, and the Fermi velocity in graphene $v_F$. In contrast, in the tripolar regime ($\mu <0$), both normal-state conductance and the critical current are suppressed when smoothing the potential; however, $I_c{R}_N$ remains close to the graphene-specific range, even for a parabolic potential. The skewness of the current-phase relation is also discussed.