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

We present a full microscopic theory based on the SU(2) covariant formulation of the quasiclassical formalism 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 0 < {\phi}0 < 2{\pi} between the superconductor electrodes provided that both, h and the SOC-induced SU(2) Lorentz force are finite. In the particular case of a Rashba SOC we present analytic and numerical results for {\phi}0 as a function of the strengths of the spin fields, the length of the junction, the temperature and the properties of SN interfaces.