Spin and charge transport in hybrid triplet Josephson junctions

We report a theoretical study on the spin and charge transport in hybrid triplet Josephson junctions, of which the triplet pair potentials could have both different orbital symmetries and spin states. Based on a lattice model and a Hamiltonian method, we find that the spin/charge supercurrent is proportional to sin2 with being the macroscopic superconducting phase, coming from the second-order Josephson effect, when the orbital symmetries of pair potentials in the two triplet superconductors are orthogonal to each other. A dissipationless transverse spin current is also found flowing at the interface of the junction and its polarization points along the cross-product of two d vectors, which arises from the combined effect of the orthogonal orbital symmetries and misalignment of d vectors. In a special hybrid junction, where the zero-energy states are absent at the interface of the junction, there is no net spin supercurrent flowing through the junction although two d vectors can be perpendicular to each other, whereas a mode-resolved spin supercurrent is flowing in the system instead.