Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 6 / e / Solution

Let $L$ be the junction length perpendicular to the width and $J_0$ be uniform. Put $\kappa =2\pi \Phi /(W{\Phi }_0)$. Integrating the local current-phase relation of a superconducting weak link over the junction givesThe odd-in-$z$ term integrates to zero. At fixed central phase ${\phi }_c=\phi (0)$, define $I(0)=J_{0}LW\sin {\phi }_c$. ThenThe value at zero flux is the continuous limit, and zeros occur at nonzero integer multiples of ${\Phi }_0$. This signed formula is the source's expression for a fixed phase. If the measured quantity is the maximum sustainable current, maximize over the central phase instead:For example, at $\Phi =3{\Phi }_0/2$ the signed coefficient is $-2/(3\pi )$, whereas the critical current is positive. The absolute-value envelope is the Fraunhofer pattern of a Josephson junction, shown in the right panel of the original tunnelling figure. Its simple form relies on uniform $J_0$ and the linear phase profile; it is not the two-junction cosine pattern of a DC SQUID.