Vortex state in unconventional junctions of superconductors with symmetry

Based on the Ginzburg-Landau theory, we derive a general current expression for a Josephson junction made of two superconductors with order parameters of different mixing of d-wave and s-wave symmetry. It is found that the characteristic phase (Formula presented) occurring in the current-phase relation is crucially determined by the two-component order-parameter configuration. By solving the sine-Gordon equation constructed from this expression, we study the static properties of a frustrated Josephson junction consisting of two junctions with characteristic phases (Formula presented) and (Formula presented). In the short junction limit, the order-parameter symmetry can manifest in the diffraction pattern of the critical current passing through the junction. In the long junction limit, we show that a spontaneous vortex with the flux (Formula presented) is induced near the center of the junction. As a result, the interference pattern of the critical currents versus an applied magnetic field is insensitive to the order-parameter profile. These analyses could be used to describe both (Formula presented)-wave/s-wave superconducting corner junctions and (Formula presented)-wave tricrystal junctions.