Search for Second-Order Josephson tunneling in SFS Josephson junctions

SFS (Superconductor-Ferromagnet-Superconductor) Josephson junctions can exhibit transitions between ordinary Josephson (0-junction) tunneling and pi-junction behavior as a function of barrier thickness or temperature. Close to the 0-$\pi $ crossover at which the first-order Josephson component vanishes, it has been predicted that second-order Josephson tunneling, characterized by a \textit{sin(}2$\phi )$ component in the supercurrent, can dominate. If present, this component can be detected directly by measurements of the current-phase relation and can induce period doubling in the critical current diffraction patterns and generate half-integer Shapiro steps. However, such effects can also arise near the 0-$\pi $ transition from a distribution of 0-junction and $\pi $-junction regions due to a non-uniform ferromagnetic barrier. We compare the results of measurements on junctions with uniform and non-uniform ferromagnetic barriers to determine whether observed second harmonics arise from a microscopic \textit{sin(}2$\phi )$ component or from junction non-uniformity.