Supergap and subgap enhanced currents in asymmetric {S_1FS_2} Josephson junctions

We have theoretically studied the supercurrent profiles in three-dimensional normal metal and ferromagnetic Josephson configurations, where the magnitude of the superconducting gaps in the superconducting leads are unequal, i.e., Δ_1neq Δ₂, creating asymmetric S_1NS2 and S_1FS2 systems. Our results reveal that by increasing the ratio of the superconducting gaps Δ₂/Δ₁, the critical supercurrent in a ballistic S_1NS2 system can be enhanced by more than 100\%, and reaches a saturation point, or decays away, depending on the junction thickness, magnetization strength, and chemical potential. The total critical current in a diffusive S_1NS2 system was found to be enhanced by more than 50\% parabolically, and reaches saturation by increasing one of the superconducting gaps. In a uniform ferromagnetic junction, the supercurrent undergoes reversal by increasing Δ₂/Δ₁>1. Through decomposing the total supercurrent into its supergap and subgap components, our results illustrate their crucial relative contributions to the Josephson current flow. It was found that the competition of subgap and supergap currents in a S_1FS2 junction results in the emergence of second harmonics in the current-phase relation. In contrast to a diffusive asymmetric Josephson configuration, the behavior of the supercurrent in a ballistic system with Δ₂/Δ₁=1 can be properly described by the subgap current component only, in a wide range of parameter sets, including Fermi level mismatch, magnetization strength, and junction thickness. Interestingly, when Δ₂/Δ₁>1, our results have found multiple parameter sets where the total supercurrent is driven by the supergap component. Therefore, our comprehensive study highlights the importance of subgap and supergap supercurrent components in both the ballistic and diffusive regimes.