Order parameters and current-phase relations in 3He-B Josephson junctions through a porous layer

Recent discoveries of the $\ensuremath{\pi}$ states in ${}^{3}\mathrm{He}\ensuremath{-}\mathrm{B}$ Josephson junctions through an array of apertures and a single aperture have aroused much theoretical interest on the mechanism of $\ensuremath{\pi}$ states. Both tunneling junction and single orifice junction models were successfully applied to explain the occurrence of $\ensuremath{\pi}$ states and their relationship with the texture orientations of $\mathrm{n\ifmmode \hat{}\else \^{}\fi{}}$ vectors in two ${}^{3}\mathrm{He}\ensuremath{-}\mathrm{B}$ reservoirs. In this paper, we study a model ${}^{3}\mathrm{He}\ensuremath{-}\mathrm{B}$ Josephson junction through a porous layer. The order parameters and current-phase relations are calculated self-consistently using the quasiclassical theory. In agreement with previous theories, the $\ensuremath{\pi}$ state is also observed when the $\mathrm{n\ifmmode \hat{}\else \^{}\fi{}}$'s are aligned antiparallel and normal to the porous layer. In this model, however, the $\ensuremath{\pi}$ state exists only when the coupling between two ${}^{3}\mathrm{He}\ensuremath{-}\mathrm{B}$ reservoirs is strong, and the usual 0 state is present when the coupling diminishes. Being contrary to the single aperture case, the $\ensuremath{\pi}$ state in our model is robust only when the magnetic field is aligned either nearly normal to or within the porous layer.