Multifractal analysis of extended states and universality in the IQHE with gate-controlled spin splitting in the LLL approximation

The discovery of the Integer Quantum Hall Effect gave rise to a great deal of research to explain the experimental observations. Although a little progress could be made from analytical approaches the success of numerical solutions to the problem started to take off since most of the experimental data could be understood in terms of the numerical models. With these models it became possible to explain the IQHE as a transition between localized and extended states. Several authors were able to describe the transition from the properties of extended states which show a multifractal nature.;In this work we will present a numerical model that shows the existence of localized and extended states; and we will be able to characterize the transition in terms of the properties of these states. With the numerical solutions it is possible to obtain the multifractal spectra of extended states and show that the scaling exponents exhibit universality relative to disorder strength and system size as was determined several years ago. Although the subject has been abandoned for some time since all the experimental data has be explained through numerical models, with the recent interest on spin FET transistors we considered that it would be interesting to investigate the transition by taking into account the spin of the electron.;In this dissertation we present results based on the multifractal analysis of extended states in the lowest Landau level approximation when the interaction between the spin magnetic moment of the electron and its orbital motion are taken into account. We obtain results that predict the universality of multifractal spectra in the IQHE when the spin-orbit coupling term is introduced in the Hamiltonian. The universal behavior of the scaling exponents relative to disorder strength and system size was found earlier ([27], [31] and [34]); but in this case we also predict universality for different values of the spin-orbit coupling parameter which, in principle, can be tuned through the gate voltage.