Lattice models for Josephson junctions and graphene superlattices

Abstract

In this thesis we study quantum transport phenomena on the

nanometer scale, in two classes of materials: topological

insulators with induced superconductivity and graphene superlattices. Both topics are motivated by recent experimental developments: the first topic arose from the search for Majorana fermions in a quantum spin Hall insulator, the second topic arose from the search for massive Dirac fermions in the Kekulé band structure of graphene on a copper substrate. We focus on lattice models, solving them both numerically and analytically.