Semi-fluxons in long Josephson 0-pi-junctions

We investigate analytically long Josephson junctions with phase pi-discontinuity points. Such junctions are usually fabricated as a ramp between a superconductor like YBCO with d-wave symmetry of the order parameter and an s-wave superconductor like Nb. From the top, they look like zigzags with pi-jumps of the Josephson phase at the corners. These pi-jumps, at certain conditions, lead to the formation of half-integer flux quanta, which we call semi-fluxons, pinned at the corners. We derive a version of sine-Gordon equation which describes the dynamics of the Josephson phase in such structures, and obtain an explicit formula which describes the shape of a semi-fluxon. Some properties of semi-fluxons are discussed. We propose a way to construct artificial pi-junctions using only s-wave superconductors.