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Up to know all the experimental results concerning the integer and fractionalquantum Hall effect are related to semiconductor heterostructures (and morerecently with graphene). The common characteristic of all these systems is thepresence of a reservoir of electrons, which, in fact, in the initial stage isthe source of the electrons, providing the two-dimensional electron gas (2DES).Then, any physical realization of a 2DES is necessarily embedded in a 3Dstructure, which establishes the Fermi level. Hence, the 2DES appears to be anopen system. In this paper we present an analytical approach to the integerquantum Hall effect (IQHE) and the Shubnikov-de Haas (SdH) phenomena in the2DES, basing us in fundamental principles and showing the secondary role of thelocalized electron states in both phenomena. In fact, we show that the IQHE isa consequence of the fluctuations of electrons in the 2DES. Once we obtain thedensity of states of the 2DES under the application of a magnetic field wecalculate both magnetoconductivities (diagonal and Hall) deducing them from theBoltzman semiclassical equation. The model proposed reproduces both phenomena,the width of the Hall plateaus (with the precision reached in the experimentalmeasurements, of the order of 10-8-10-9) and the corresponding minima of thediagonal magnetoresistivity, and also the dependence with temperature of theIQHE and SdH.

A model for the study of the Shubnikov-de Haas and the integer quantum Hall effects in a two dimensional electronic system