Majorana bound states in hybrid two-dimensional Josephson junctions with ferromagnetic insulators

We consider a Josephson junction consisting of superconductor/ferromagnetic insulator (S/FI) bilayers as electrodes which proximizes a nearby two-dimensional (2D) electron gas. By starting from a generic Josephson hybrid planar setup, we present an exhaustive analysis of the interplay between the superconducting and magnetic proximity effects and the conditions under which the structure undergoes transitions to a nontrivial topological phase. We address the 2D bound-state problem by using a general transfer-matrix approach that reduces the problem to an effective 1D Hamiltonian. This allows for a straightforward study of the topological properties in different symmetry classes. As an example we consider a narrow channel coupled with multiple ferromagnetic superconducting fingers, and discuss how the Majorana bound states can be spatially controlled by tuning the superconducting phases. Following our approach, we also show the energy spectrum, the free energy, and finally the multiterminal Josephson current of the setup.