Snyder, Rodney AABSTRACTStudying topological states of matter offers potential routes to novel excitations withproperties beyond those of simple electrons. While the range of materials and tech-niques for studying topological systems is still expanding, only a handful of systemshave been thoroughly studied. Topological crystalline insulators (TCI) and the richstates present in ZrTe5offer alternative routes to studying topological states of matterwith surface states that have a distinct character compared to those in more common3D topological insulators.First, I report on the fabrication of Josephson junctions using a MBE-grown can-didate TCI material — Pb-doped SnTe — as weak links. We then perform transportcharacterization measurements on these devices including the effects of DC, RF, andmagnetic field biases. I compare my results to a model that uses a skewed current-phase relationship and the resistively shunted Josephson junction model. I find goodqualitative agreement between experimental data and numerically calculated data.In the second section, I describe the fabrication of devices made from mechanicallyexfoliated ZrTe5nanowires. I then present results from magnetoconductance measure-ments from 50 mK to 1 K on these devices. Finally I apply existing models for weakantilocalization and universal conductance fluctuations for a quasi-1D system in orderto extract a coherence length up to 1.5μm at 50 mK and 700 nm at 1 KenDissertationPhysics