Dynamics of wave packets in two-dimensional crystals under external magnetic and electric fields: Formation of vortices

Abstract

In the present work, we deal with the dynamics of wave packets in a two-dimensional crystal under the action of magnetic and electric fields. The magnetic field is perpendicular to the plane and the electric field is on the plane. In the simulations, we considered a symmetric gauge for the vector potential while the initial wave packet was assumed to have a Gaussian structure with given velocities. The parameters that control the kind of time evolution of the packets are the width of the Gaussian, its velocity, and the intensity and direction of the electric field as well as the magnitude of the magnetic field. In order to characterize the kind of propagation, we evaluated the mean-square displacement and the participation function, and, more importantly, we were able to follow the wave at different times, which allowed us to see the time evolution of the centroid of the wave packets. We observed that the dynamics is such that the wave function splits into two or more components and reconstructs successively as time goes; vortices form. As for the inclusion of the electric field, we observed a complex behavior of the wave packet as well as noted that the vortices propagate in a direction perpendicular to the applied electric field, a similar behavior presented by the classical treatment. In our case, we give a quantum mechanics explanation for that.