  We show that an ordinary semiconducting thin film with spin-orbit coupling
can, under ap- propriate circumstances, be in a quantum topologically ordered
state supporting exotic Majorana excitations which follow non-Abelian
statistics. The key to the quantum topological order is the coexistence of
spin-orbit coupling with proximity-induced s-wave superconductivity and an
externally-induced Zeeman coupling of the spins. For the Zeeman coupling below
a critical value, the system is a non-topological (proximity-induced) s-wave
superconductor. However, for a range of Zeeman coupling above the critical
value, the lowest energy excited state inside a vortex is a zero-energy
Majorana fermion state. The system, thus, has entered into a non-Abelian s-wave
superconducting state via a topological quantum phase transition (TQPT) tuned
by the Zeeman coupling. In the one-dimensional version of the same structure
and for the Zeeman coupling above the critical value, there are localized
Majorana zero-energy modes at the two ends of a semiconducting quantum
nanowire. In this case, the Zeeman coupling can be induced more easily by an
external magnetic field parallel to the wire, obviating the need for a magnetic
insulator. We show that, despite the fact that the superconducting pair
potential in the nanowire is explicitly s-wave, tunneling of electrons to the
ends of the wire reveals a pronounced zero-bias peak. Such a peak is absent
when the Zeeman coupling is below its critical value, i.e., the nanowire is in
the non-topological s-wave superconducting state. We argue that the observation
of this zero-bias tunneling peak in the semiconductor nanowire is possibly the
simplest and clearest experiment proposed so far to unambiguously detect a
Majorana fermion mode in a condensed matter system.
