Effects of Imperfections and Disorder

Abstract

As was remarked in the first chapter, in order to have the QHE it is essential to break translational invariance. In practice this happens automatically because real material samples have some degree of imperfection. In this lecture, we deal with the effects of imperfections in some detail, although still pretty much at an elementary level. This will be done by introducing various models for impurities. A combination of these models is probably quite realistic. The aim is twofold: First, we want to appreciate better what kind of imperfection systems will give rise to a mobility gap at the Fermi level. This, we recall, is a situation in which there are no extended, current carrying states at the Fermi level. There may be, and in practice are, localized states there. A mobility gap is the essential condition for the QHE, according to the gauge argument of Laughlin given in Chap. 1.7. Secondly, we want to develop an understanding of what localized states are really like, and what extended states are like. We would also like some explicit, semirealistic models which exhibit the QHE.