Cylindrical Josephson tunneling

Abstract

Josephson-tunneling experiments between small cylindrical superconductors are proposed. The essential idea is that the wave functions in the superconductors must be of the form ${\ensuremath{\psi}}_{1}=|{\ensuremath{\psi}}_{\ensuremath{\infty}}|f(r){e}^{i{n}_{1}\ensuremath{\theta}}$ and ${\ensuremath{\psi}}_{2}=|{\ensuremath{\psi}}_{\ensuremath{\infty}}|f(r){e}^{i({n}_{2}\ensuremath{\theta}+{\ensuremath{\delta}}_{0})}$ to guarantee that they are single valued. This, together with the Josephson relation $J={J}_{0}sin\ensuremath{\delta}$, where ${J}_{0}$ is the maximum Josephson current and $\ensuremath{\delta}$ is the phase difference between the superconductors, leads to the conclusion that the Josephson current will be zero for ${n}_{1}\ensuremath{\ne}{n}_{2}$. For ${n}_{1}={n}_{2}$ the current will be proportional to ${f}_{1}^{2}{f}_{2}^{2}$ in the thin-film approximation. If the inner cylinder is solid and of higher transition temperature then ${J}_{0}\ensuremath{\propto}{f}_{2}^{2}$ providing a means for determining the order parameter as a function of magnetic field. The ratio of cylinder radius to coherence length determines, in large measure, the variation of ${J}_{0}$ with field. Specific examples are given.