Spontaneous flux and magnetic-interference patterns in 0-$\pi$ Josephson junctions

The spontaneous flux generation and magnetic field modulation of the critical current in a 0-$\pi$ Josephson junction are calculated for different ratios of the junction length $L$ to the Josephsen penetration depth ${\lambda }_J,$ and different ratios of the $0$-junction length to the $\pi$-junction length. These calculations apply to a Pb-YBa${}_2$Cu${}_3$O${}_{7-\delta }$ (YBCO) $c$-axis-oriented junction with one YBCO twin boundary, as well as other experimental systems. Measurements of such a junction can provide information on the nature of the $c$-axis Josephson coupling and the symmetry of the order parameter in YBCO. We find spontaneous flux even for very short symmetric 0-$\pi$ junctions, but asymmetric junctions have qualitatively different behavior.