Introduction to Quantum Hall Effect
In 1879, Edwin Hall, observed for the first time an effect on some materials that is called Hall effect. To describe the quantum case it is important first to define some fundamental quantities such as longitudinal resistivity and Hall conductivity. The last term indicates a viscous force proportional to the velocity that takes into account the impurities of the material and the interactions between the electrons. A very important and interesting property concerns the relationship between resistivity and transverse resistance or Hall resistance. The classical Hall effect is observed in the presence of weak magnetic fields. In the case of intense magnetic fields and at low temperatures it is necessary to use quantum mechanics. To determine the wave function of the system it is necessary to proceed defining the gauge used. For a system under the effect of a magnetic field, a symmetric gauge is chosen.