Ising superconductors: Interplay of magnetic field, triplet channels, and disorder

We study the superconducting instability in disordered noncentrosymmetric monolayers with intrinsic Ising spin-orbit coupling (SOC) subjected to an in-plane Zeeman magnetic field. The pairing interaction contains the channels allowed by crystal symmetry, such that, in general, the pairing state is a mixture of singlet and triplet Cooper pairs. The joint action of the SOC and Zeeman field selects a specific in-plane $\mathbf{d}$-vector triplet component to couple with the singlets, which gains robustness against disorder through the coupling. The out-of-plane $\mathbf{d}$-vector component, that in the clean case is immune to both the Zeeman field and SOC, is obliterated by a very small impurity scattering rate. We formulate the quasiclassical theory of Ising superconductors and solve the linearized Eilenberger equations to obtain the pair-breaking equations that determine the Zeeman field: temperature dependence of the continuous superconducting transition. Our discussion emphasizes how the Zeeman field, SOC, and disorder affect the different superconducting order parameters, and we show how the spin fields inevitably induce odd-frequency pairing correlations.