  We prove an index theorem for the existence of Majorana zero modes in a
semiconducting thin film with a sizable spin-orbit coupling when it is adjacent
to an 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. The momentum-space construction of the
zero-mode solution presented here is complementary to the approximate
real-space solution of the Bogoliubov-de Gennes equations at a vortex core [J.
D. Sau et al., arXiv:0907.2239], proving the existence of non-degenerate
zero-energy Majorana excitations and the resultant non-Abelian topological
order in the semiconductor heterostructure. With increasing magnitude of the
proximity-induced pairing potential, the non-Abelian superconducting state
makes a topological quantum phase transition to an ordinary s-wave
superconducting state which no topological order.
