Wave-packet scattering at a normal-superconductor interface in two-dimensional materials: A generalized theoretical approach

A wave-packet time evolution method, based on the split-operator technique, is developed to investigate scattering of quasiparticles at a normal-superconductor interface arbitrary profile and shape. As practical application, we consider system where low-energy electrons can be described as Dirac particles, which case for most two-dimensional materials, such graphene transition-metal dichalcogenides. However, method easily adapted other cases in few-layer black phosphorus or any Schr\"odinger within effective mass approximation semiconductors. We employ revisit Andreev reflection mono-, bi-, trilayer graphene, specular- retro-reflection are observed scattered by steplike superconducting region. The effect opening zero-gap channel across region electron hole also addressed, an example versatility technique proposed here.