Electronic transport close to boundaries of semi-infinite graphene and their interfaces

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.