  Transport properties of 2D materials especially close to their boundary has
received much attention after the successful fabrication of graphene and other
fascinating materials afterwards. While most previous work is devoted to the
conventional lead-device-lead setup with a finite size center area, this
project investigates real space transport properties of infinite and
semi-infinite 2D system under the framework of Non-equilibrium Green function.
The commonly used method of calculating the Green function by inverting a
matrix in the real space directly can be unstable in dealing with large systems
as sometimes it gives non-converging result. Not to mention that the
calculation error and time increase drastically with size of the system. By
transforming from the real space to momentum space, we managed to replace the
matrix inverting process by Brillouin Zone integral process which can be
greatly simplified by the application of contour integral. Combining this
methodology with Dyson equations, we are able to calculate transport properties
of semi-infinite graphene close to its zigzag boundary and its combination with
other material including s-wave superconductor. Interference pattern of
transmitted and reflected electrons, graphene lensing effects and difference
between Specular Andreev reflection and normal Andreev reflection are verified
through our calculation.
