Phase diagram of a bilayer superconductor under an in-plane magnetic field

We study a double layer superconductor in the presence of a parallel magnetic field $B$ by obtaining self-consistent solutions of the Bogoliubov--de Gennes equations, and also using the Pokrovsky-Talapov model for the free energy expressed in terms of the relative phase, namely the difference in the phases of the superconducting order parameters in the two layers. We find that with increasing $B$ a continuous transition occurs from the BCS state, where the relative phase is constant, into a state which contains stripes of the BCS state separated by localized vortices in the relative phase. This state is predicted to manifest through oscillations in the amplitude of the superconducting gap and an alternating pattern of supercurrents. With increasing $B$, the BCS stripe state continuously evolves into the Fulde-Ferrell-Larkin-Ovchinnikov state with linearly varying relative phase and a constant gap amplitude. These predictions apply to superconductivity in bilayer transition-metal-dichalcogenide systems with Ising spin-orbit coupling, and ought to be testable in a recently studied experimental system.