Quantization of the Hall conductance in a two-dimensional electron gas

We present a microscopic theory of the Hall conductance in a two-dimensional electron gas. Our approach is based on a single-particle picture and explicitly accounts for the effects of a random impurity potential. Within the geometry introduced by Laughlin a general expression is derived from which it is possible to evaluate the Hall conductance in terms of the properties of the electronic spectrum at the Fermi energy for any value of the magnetic field. When the chemical potential lies between the bulk extended states of well-defined neighboring Landau bands, the Hall conductance is quantized in integral multiples of $\frac{e^2}{h}$, even in the presence of a large density of localized states. Within our model the exactness of this quantization depends on the shape of the confining potential, the thickness of the sample, and the magnetic field.