We show that a film of a semiconductor in which s-wave superconductivity and Zeeman splitting are induced by the proximity effect, supports zero-energy Majorana fermion modes in the ordinary vortex excitations. Since time-reversal symmetry is explicitly broken, the edge of the film constitutes a chiral Majorana wire. The heterostructure we propose-a semiconducting thin film sandwiched between an s-wave superconductor and a magnetic insulator-is a generic system which can be used as the platform for topological quantum computation by virtue of the existence of non-Abelian Majorana fermions.

In 2D chiral p-wave superconductors, the zero-energy Majorana fermion excitations trapped at vortex cores are protected from the thermal effects by the mini-gap, ∆ 2 /ǫF (∆: bulk gap, ǫF : fermi energy), which is the excitation gap to the higher-energy bound states in the vortex cores. Robustness to thermal effects is guaranteed only when T ≪ ∆ 2 /ǫF ∼ 0.1 mK, which is a very severe experimental constraint. Here we show that when s-wave superconductivity is proximity-induced on the surface of a topological insulator or a spin-orbitcoupled semiconductor, as has been recently suggested, the mini-gaps of the resultant non-Abelian states can be orders of magnitude larger than in a chiral p-wave superconductor. Specifically, for interfaces with sufficient barrier transparencies, the mini-gap can be as high as ∼ ∆ ≫ ∆ 2 /ǫF , where ∆ is the bulk gap of the s-wave superconductor.

After a recent series of rapid and exciting developments, the long search for the Majorana fermion -the elusive quantum entity at the border between particles and antiparticles -has produced the first positive experimental results, but is not over yet. Originally proposed by E. Majorana in the context of particle physics, Majorana fermions have a condensed matter analog in the zero-energy bound states emerging in topological superconductors. A promising route to engineering topological superconductors capable of hosting Majorana zero modes consists of proximity coupling semiconductor thin films or nanowires with strong spin-orbit interaction to conventional s-wave superconductors in the presence of an external Zeeman field. The Majorana zero mode is predicted to emerge above a certain critical Zeeman field as a zero-energy state localized near the order parameter defects, viz., vortices for thin films and wire-ends for the nanowire. These Majorana bound states are expected to manifest non-Abelian quantum statistics, which makes them ideal building blocks for fault-tolerant topological quantum computation. This review provides an update on current status of the search for Majorana fermions in semiconductor nanowires by focusing on the recent developments, in particular the period following the first reports of experimental signatures consistent with the realization of Majorana bound states in semiconductor nanowire-superconductor hybrid structures. We start with a discussion of the fundamental aspects of the subject, followed by considerations on the realistic modeling which is a critical bridge between theoretical predictions based on idealized conditions and the real world, as probed experimentally. The last part is dedicated to a few intriguing issues that were brought to the fore by the recent encouraging experimental advances. CONTENTS