Microscopic theory of in-plane critical field in two-dimensional Ising superconducting systems

We study the in-plane critical magnetic field of two-dimensional Ising superconducting systems, and propose the microscopic theory for these systems with or without inversion symmetry. Protected by certain specific spin-orbit interaction which polarizes the electron spin to the out-of-plane direction, the in-plane critical fields largely surpass the Pauli limit and show remarkable upturn in the zero temperature limit. The impurity scattering and Rashba spin-orbit coupling, treated on equal-footing in the microscopic framework, both weaken the critical field but in qualitatively different manners. The microscopic theory is consistent with recent experimental results in stanene and Pb superconducting ultra-thin films.