Josephson effect in spin-singlet superconductor/ferromagnetic insulator/spin-triplet superconductor junctions with helical $p$-wave states

We study the Josephson effect in spin-singlet superconductor/helical $p$-wave superconductor junctions with a ferromagnetic barrier using the quasiclassical Green function method. It is found that both $\sin{\phi}$-type and $\cos{\phi}$-type current-phase relations always exist, irrespective of the gap symmetries in superconductors. The indispensable condition for the $\sin{\phi}$-type and $\cos{\phi}$-type current is that the magnetization must have a component parallel to the crystallographic $a$ or $b$ axis, which is distinct from the case of $p$-wave superconductor described by a $\vect{d}$-vector with a uniform direction. The relation between the condition and the symmetries of the gap functions is analysed. We investigate in detail the symmetries and the sign reversal of the Josephson current when the magnetization is rotated.