Study of integer quantum Hall effect in systems with isotropic and non-isotropic impurities

In this paper, we study the integer quantum Hall effect (IQHE) in systems with different types of impurities, considering the IQHE as a consequence of gauge invariance and the existence of a mobility gap. We consider the response of the Landau energy levels to a change of flux, in the presence of impurities in delta and Gaussian forms (in which the impurities are distributed isotropically). Then, we add a specified form of a quantum dot potential to the system with isotropic impurities, and observe the Hall effect and study the behavior of the system. Finally, we add non-isotropic impurities in systems showing the IQHE. We observe that if the impurity distribution is in the same direction in which the electric field is applied, the localized levels move out from the origin with respect to the isotropic case, and the Hall step will be clearly observed.