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The search for a Majorana Fermion has been an area of intense interest incondensed matter research of late. This elusive particle, predicted to exist in1937, has been sought after for both fundamental and practical reasons. On thefundamental level, no particle to date has been observed to be a Majoranafermion, meanwhile on the practical level a Majorana fermion, if found, wouldrepresent a non-abelian anyon and could thus be used to build a quantumcomputer. The search for a Majorana Fermion has recently shifted to topologicalsuperconductivity. Topological superconductors are categorized by thenontrivial wind- ing of their order parameter phase and for this reason areexpected to support Majorana Fermions in their vortex cores. Owing to this, thestudy of topological superconductors has intensified in recent years. Currentproposals for a device that may behave as a topological superconductor arebased on semiconductor heterostructures, where the spin-orbit coupled bands ofa semiconductor are split by a band gap or Zeeman field and superconductivityis induced by proximity to a conventional superconductor. In this setup,topological superconductivity is obtained in the semiconductor layer and theproposed heterostructures typically include two or three layers of differentmaterials. In this thesis we propose a simplification to these types ofdevices, suggesting a way in which the superconducting layer can be replaced.Part of our proposal includes a model Hamiltonian for these types of systems.This thesis will also develop several different methods to analyze this modelHamiltonian in various different parameter regimes with the ultimate goal ofclassifying its topology.

McGill University M. Sc. Thesis: Topological Superconductivity without Proximity Effect