effect of time reversal symmetry on the current-phase relation in josephson junctions including two-gap superconductors

From a general viewpoint, the josephson effect is a quantum phenomenon that can be described by cooper electron pair tunneling. this junction involves two or more superconductors separated by a thin layer of non-superconductor material. the supercurrent can flow without dissipation through this junction without applying voltage. the dependence of the josephson current on the phase of two superconductors is significantly important to investigate. in this research, we considered a josephson junction consisting of two two-gap superconductors, separated by a thin insulator. also, we assumed a spin-singlet s-wave pair potential in each conduction band. we studied the effect of time-reversal symmetry on the phase difference, and consequently on the josephson current in two cases. the first case is when only one of the superconductors has time-reversal symmetry and the second one is when none have time-reversal symmetry. we compared the mentioned states with the condition where both superconductors have time-reversal symmetry.