Singular effects on the Hall conductivity.

Abstract

Condensed Matter physics is a branch of Physics which includes the laws of quantum mechanics, electromagnetism and statistical mechanics. In this area, we consider a two-dimensional electron gas (2DEG), which can be defined as a system of electrons that are confined by opposing forces to a thin planar region. Nowadays, it is more interesting to study the impact of the geometry or of some topological defects introduced on the sample, to get closer of reality. In the work that we have proposed to realize, we have considered a sample immersed on a magnetic field. Or more simply, we wanted to study the Hall effect for particular cases, on usual semiconductors. In fact, we have tried to understand how we can influence the behavior of the Hall conductivity, with a curved surface (rather than the flat case) and by introducing some topological defects. In a first work, we have considered the single cone immersed into a magnetic field. We realized that we can consider the conical tip in two different ways : with and without a singularity at the tip. We saw that we obtained different Landau Levels LL, which change the type of plateaus that we obtain and the shifting in the Hall conductivity. After discussing this, we changed some parameters (the opening angle α of the cone, the geometric potential) to see how this influenced the Hall conductivity when we varied the magnetic field. We compared two types of geometric models : the da Costa and the Dirac models. In a second work, we have studied the effect of a screw dislocation on the Hall conductivity. For that, we have changed the torsion parameter of the medium β. We did not observe any shift of the Hall conductivity, but we noticed a change of the size of the conduction plateaus. In order to appreciate the presence of both a screw dislocation and a disclination, we have considered a 2DEG with a dispiration. By changing α, which is the deficit/excess angle, we obtained, this time, a shift of the Hall conductivity. In summary, we tried to explain the behaviour of the Hall conductivity in the presence of singularities and how taking into account the geometry is important.