Josephson effect in ballistic graphene

We solve the Dirac–Bogoliubov–de Gennes equation in an impurity-free superconductor–normal-metal–superconductor junction, to determine the maximal supercurrent $I_c$ that can flow through an undoped strip of graphene with heavily doped superconducting electrodes. The result $I_c\simeq (W∕L)e{\Delta }_0∕\hslash$ is determined by the superconducting gap ${\Delta }_0$ and by the aspect ratio of the junction (length $L$ small relative to the width $W$ and to the superconducting coherence length). Moving away from the Dirac point of zero doping, we recover the usual ballistic result $I_c\simeq (W∕{\lambda }_F)e{\Delta }_0∕\hslash$, in which the Fermi wavelength ${\lambda }_F$ takes over from $L$. The product $I_c{R}_{\mathrm{N}}\simeq {\Delta }_0∕e$ of the critical current and normal-state resistance retains its universal value (up to a numerical prefactor) on approaching the Dirac point.

Figure 1

Schematic of a graphene layer, partially covered by two superconducting electrodes (S). A dissipationless supercurrent flows in equilibrium through the normal region (N), depending on the phase difference between the two superconductors. Separate gate electrodes (not shown) make it possible to vary independently the carrier concentration in the normal and superconducting regions of the graphene layer.

Figure 2

Critical current $I_c$ and $I_c{R}_{\mathrm{N}}$ product of a ballistic Josephson junction (length $L$ short compared to the width $W$ and superconducting coherence length $\xi$), as a function of the Fermi energy $\mu$ in the normal region. The asymptotes (21, 22) are indicated by dashed lines.