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

• Received 27 September 2022
• Revised 13 January 2023
• Accepted 23 March 2023

DOI:https://doi.org/10.1103/PhysRevB.107.165306