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We present a full microscopic theory based on the SU(2) covariant formulationof the quasiclassical formalism to describe the Josephson current through anextended superconductor-normal metal- superconductor (SNS) diffusive junctionwith an intrinsic spin-orbit coupling (SOC) in the presence of a spin-splittingfield h. We demonstrate that the ground state of the junction corresponds to afinite intrinsic phase difference 0 < {\phi}0 < 2{\pi} between thesuperconductor electrodes provided that both, h and the SOC-induced SU(2)Lorentz force are finite. In the particular case of a Rashba SOC we presentanalytic and numerical results for {\phi}0 as a function of the strengths ofthe spin fields, the length of the junction, the temperature and the propertiesof SN interfaces.

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