Theory of diffusive 0 Josephson junctions in the presence of spin-orbit coupling

We present a full microscopic theory to describe the Josephson current through an extended superconductor-normal metal-superconductor (SNS) diffusive junction with an intrinsic spin-orbit coupling (SOC) in the presence of a spin-splitting field h. We demonstrate that the ground state of the junction corresponds to a finite intrinsic phase difference between the superconductor electrodes provided that both h and the SOC-induced SU(2) Lorentz force are finite. The nontrivial is closely related to the appearance of an equilibrium spin current in the normal metal with the spin projection parallel to the exchange field direction. In the particular case of a Rashba SOC we present analytic and numerical results for as a function of the strengths of the spin fields, the length of the junction, the temperature and the properties of SN interfaces.