Carriers in a two-dimensional lattice under magnetic and electric fields

Abstract

We have studied the propagation of carriers in a two-dimensional system under the influence of magnetic and electric fields. To take into account the magnetic field we have used the Peierls substitution, and have considered the so-called Landau gauge for the vector potential. The nature of the propagation of the wave packet under only a magnetic field is controlled by the ratio between the magnetic flux through the unit cell to the flux quantum: $\ensuremath{\alpha}=\ensuremath{\Phi}/{\ensuremath{\Phi}}_{0}.$ For rational values of \ensuremath{\alpha} we have obtained ballistic propagation for sufficient long times. But for irrational \ensuremath{\alpha} the wave remains localized in a definite region due to incommensurability. The inclusion of the electric field changes this picture. In fact, when the electric field is included, the degeneracy between the on-site energies is broken along the field direction, thus inhibiting hopping along it, while in the perpendicular direction to the applied field propagation is favored. This behavior is common to both rational and irrational \ensuremath{\alpha}. Finally, for certain values of the electric and magnetic fields the wave packet performs an oscillatory movement; this is the phenomenon of dynamic localization.