We theoretically study the dc Josephson effect between two volumes of superfluid ${}^3\mathrm{He}-\mathrm{B}.$ We first discuss how the calculation of the current-phase relationships is divided into mesoscopic and macroscopic problems. We then analyze mass and spin currents and the symmetry of weak links. In quantitative calculations the weak link is assumed to be a pinhole, whose size is small in comparison to the coherence length. We derive a quasiclassical expression for the coupling energy of a pinhole, also allowing for scattering in the hole. Using a self-consistent order parameter near a wall, we calculate the current-phase relationships in several cases. In the isotextural case, the current-phase relations are plotted assuming a constant spin-orbit texture. In the opposite anisotextural case the texture changes as a function of the phase difference. For this we have to consider the stiffness of the macroscopic texture, and we also calculate some surface interaction parameters. We analyze the experiments by Marchenkov et al., [Phys. Rev. Lett. $83,$ 3860 (1999)], although the assumptions of the pinhole model were not quite satisfied there. We find that the observed $\pi$ states and bistability can hardly be explained with the isotextural pinhole model, but a quantitative agreement is achieved with the anisotextural model.

• Received 20 August 2001

DOI:https://doi.org/10.1103/PhysRevB.65.064530