A theorem for the existence of Majorana fermion modes in spin-orbit-coupled semiconductors

We prove a theorem for the existence of Majorana zero modes in a semiconducting thin film with a sizable spin-orbit coupling when it is adjacent to a $s$-wave superconductor. The theorem, which is analogous to the Jackiw-Rebbi index theorem for the zero modes in mass domain walls in one-dimensional Dirac theory, applies to vortices with odd flux quantum in a semiconducting film in which $s$-wave superconductivity and a Zeeman splitting are induced by proximity effect. Thus the theorem proves the existence of non-degenerate zero-energy Majorana excitations and the resultant non-Abelian topological order in the proposed semiconductor heterostructure.