Beyond the effective length: How to analyze magnetic interference patterns of thin-film planar Josephson junctions with finite lateral dimensions

The magnetic field dependent critical current $I_{\text{c}}(B)$ of a Josephson junction is determined by the screening currents in its electrodes. In macroscopic junctions, a local vector potential drives the currents, however, in thin film planar junctions, with electrodes of finite size and various shapes, they are governed by non-local electrodynamics. This complicates the extraction of parameters such as the geometry of the effective junction area, the effective junction length and, the critical current density distribution from the $I_{\text{c}}(B)$ interference patterns. Here we provide a method to tackle this problem by simulating the phase differences that drive the shielding currents and use those to find $I_{\text{c}}(B)$. To this end, we extend the technique proposed by John Clem [Phys. Rev. B, \textbf{81}, 144515 (2010)] to find $I_{\text{c}}(B)$ for Josephson junctions separating a superconducting strip of length $L$ and width $W$ with rectangular, ellipsoid and rhomboid geometries. We find the periodicity of the interference pattern ($\Delta B$) to have geometry independent limits for $L \gg W$ and $L \ll W$. By fabricating elliptically shaped S$-$N$-$S junctions with various aspect ratios, we experimentally verify the $L/W$ dependence of $\Delta B$. Finally, we incorporate these results to correctly extract the distribution of critical currents in the junction by the Fourier analysis of $I_{\text{c}}(B)$, which makes these results essential for the correct analysis of topological channels in thin film planar Josephson junctions.