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

We study the superconducting instability in disordered non-centrosymmetric 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 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 quasi-classical 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.