Josephson junction on one edge of a two dimensional topological insulator affected by magnetic impurity

The current-phase relation in a Josephson junction formed by putting two s-wave superconductors on the same edge of a two dimensional topological insulator is investigated. We consider the case in which the junction length is finite and magnetic impurity exists. The similarities and differences with respect to a conventional Josephson junction are discussed. Both the 2 pi- and 4 pi-period current-phase relations (I-2 pi(phi), I-4 pi(phi)) are studied. There is a sharp jump at phi = pi and phi = 2 pi for I-2 pi and I-4 pi, respectively, in the clean junction. For I-2 pi, the sharp jump is robust against the impurity strength and distribution. However, for I-4 pi, an impurity makes the jump at phi = 2 pi smooth. The critical (maximum) current I-c,I-2 pi of I-2 pi is given and we find it will be increased by an asymmetrical distribution of the impurity.