Enhancement of topological regime in elongated Josephson junctions

We theoretically study topological superconductivity in elongated planar Josephson junctions. Josephson junctions host a topological phase accompanied by the presence of zero-energy Majorana-bound states in a certain region of the superconducting phase difference with a span determined by the strength of an external magnetic field. We demonstrate that the topological superconducting phase range can be greatly increased by elongation of the junction, which causes an amplification of the Zeeman-induced phase shift of Andreev bound states. We show that the appearance of trivial in-gap states that occurs in elongated junctions can prohibit the creation of Majorana modes, but it can be mitigated by further proximitization of the junction with additional superconducting contacts. The topological transition in this system can be probed by measurements of the critical current and we show that the elongation of the junction leads to a linear decrease of the transition critical magnetic field beneficial for experimental studies.