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We investigate Josephson junctions on the surface of a three-dimensionaltopological insulator in planar, step, and edge geometries. The ellipticalnature of the Dirac cone representing the side surface states of thetopological insulator results in a scaling factor in the Josephson current in astep junction as compared to the planar junction. In edge junctions, thecontribution of the Andreev bound states to the Josephson current vanishes dueto spin-momentum locking of the surface states. Furthermore, we consider ajunction with a ferromagnetic insulator between the superconducting regions. Inthese ferromagnetic junctions, we find an anomalous finite Josephson current atzero phase difference if the magnetization is pointing along the junction (andperpendicular to the Josephson current). An out-of-plane magnetization withrespect to the central region of the junction opens up an exchange gap andleads to a non-monotonic behavior of the critical Josephson current forsufficiently large magnetization as the chemical potential increases.

Josephson effect in normal and ferromagnetic topological insulator planar, step and edge junctions