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


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 that the junction length is finite and magnetic impurity exists. The similarity and difference with conventional Josephson junction is discussed. The current is calculated in the semiconductor picture. Both the 2Ο€2\pi- and 4Ο€4\pi-period current-phase relations (I2Ο€(Ο•),I4Ο€(Ο•)I_{2\pi}(\phi), I_{4\pi}(\phi)) are studied. There is a sharp jump at Ο•=Ο€\phi=\pi and Ο•=2Ο€\phi=2\pi for I2Ο€I_{2\pi} and I4Ο€I_{4\pi} respectively in the clean junction. For I2Ο€I_{2\pi}, the sharp jump is robust against impurity strength and distribution. However for I4Ο€I_{4\pi}, the impurity makes the jump at Ο•=2Ο€\phi=2\pi smooth. The critical (maximum) current of I2Ο€I_{2\pi} is given and we find it will be increased by asymmetrical distribution of impurity.Comment: 7 pages, 5 figure

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