Propagação de pacotes de onda em fosforeno na presença de fronteiras de grão

Abstract

Experimentally obtained samples of materials inevitably present impurities and/or structural defects as a result of their synthesis and post-synthesis processes, which may drastically affect their physical properties. In particular, they were reported many times in the recently discovered two-dimensional (2D) materials, a new class of materials consisting of a single layer of atoms that are stable at finite temperatures, such as graphene and phosphorene. One common example of structural defect present in samples of 2D materials is the grain boundary (GB). GBs are the interface of domains with different lattice orientations forming the so-called polycrystalline materials. In this work, we consider the monolayer phosphorene due to its higly anisotropic dispersion relation, which allows us to investigate the role of anisotropy on the propagation of electrons. A polycrystalline phosphorene is modeled by a parabolic dispersion with spatially-dependent anisotropic effective masses that varies abruptly from one domain to another. This approach can also be used as a platform for studies of quantum-optical analogies, such as amphoteric refraction (positive or negative refraction depending on the incidence angle) of electrons in heterostructures. In particular, it is observed a modification of the reflection law and a collimation effect of an electrom beam as the orientation of the incidence region varies. Simulations of gaussian wave packets propagating through a GB were performed by the split-operator technique in order to illustrate the effect of the GB and compare its trajectories with the ones for ballistic electrons, which are basically the plane-wave solutions of a tunneling-like problem obtained by the 2D Schro¨dinger equation with anisotropic effective masses. The energy and momentum conservation of the component parallel to the interface were used to predict the direction of propagation of the reflected and transmitted wave packets. It was shown that, the curvature of the constant-energy surface can influence the direction of gaussian wave packets for a given value of the average wave vector. Total internal reflection can also be observed even in the absence of external potentials