Nontrivial superconductivity in two-dimensional superconductors with both magnetic field and spin-orbit coupling

Symmetry broken in a superconducting (SC) system is believed to affect the superconductivity, especially in the case of both time-reversal and inversion symmetry breaking. Accordingly, a two-dimensional (2D) SC system coexistence with both magnetic field and spin-orbit coupling (SOC) is investigated in the framework of mean-field theory. We find that SOC will introduce a mixture of spin-singlet and triplet pairings independent of magnetic field. However, SOC effects on each pairings are dependent on the presence of magnetic field both for its magnitude and direction. For both magnetic field out and in the plane, there seem to be some critical magnetic fields, below and above which SOC and magnetic fields show different effects on the two pairings. Moreover, these critical magnetic fields are determined and enhanced by SOC. The composite effects of magnetic field and SOC on SC pairings might be helpful for understanding pairing mechanism in nontrivial 2D superconductors.