Josephson effect in mesoscopic graphene strips with finite width

Abstract

We study Josephson effect in a ballistic graphene strip of length $L$ smaller than the superconducting coherence length and arbitrary width $W$. We find that the dependence of the critical supercurrent ${I}_{c}$ on $W$ is drastically different for different types of the edges. For smooth and armchair edges at low concentration of the carriers ${I}_{c}$ decreases monotonically with decreasing $W∕L$ and tends to a constant minimum for a narrow strip $W∕L\ensuremath{\lesssim}1$. The minimum supercurrent is zero for smooth edges but has a finite value $e{\ensuremath{\Delta}}_{0}∕\ensuremath{\hbar}$ for the armchair edges. At higher concentration of the carriers, in addition to this overall monotonic variation, the critical current undergoes a series of peaks with varying $W$. On the other hand in a strip with zigzag edges the supercurrent is half-integer quantized to $(n+1∕2)4e{\ensuremath{\Delta}}_{0}∕\ensuremath{\hbar}$, showing a stepwise variation with $W$.