Orientation-dependent Josephson effect in spin-singlet superconductor/altermagnet/spin-triplet superconductor junctions

We study the Josephson effect in the spin-singlet superconductor/altermagnet/spin-triplet superconductor junctions using the Green's function method. The current-phase difference relationships in the junctions strongly depend on the orientation of altermagnet and the types of the Cooper pairs. For the orientation angle equal to odd multiples of $\pi/4$, the current-phase difference relationships are of the $\sin{2\phi}$ type, which are irrespective of the pairing wave functions in superconductors. For the other orientation angles, the emergence of the lowest order current becomes possible and its form, $\sin\phi$ or $\cos\phi$, depends on the pairing wave functions in superconductors. The $\phi_{0}$ phase and the $0$-$\pi$ transition can be realized in our junctions due to the appearance of the lowest order current. The selection rules for the lowest order current are presented. The symmetric relations satisfied by the current-phase difference relationships are analyzed through considering the transformations of the junctions under the mirror reflection, the time-reversal and the spin rotation operations. Our results not only provide a method to detect the intrinsic spin-triplet superconductivity but also possess application values in the design of the field-free quantum devices.