Localization in thin films: cross-over from two dimensions to three dimensions

Abstract

We have examined the localization of electronic states in disordered thin films as a function of film thickness. The study is motivated by the following consideration. According to the present understanding, in two-dimensions all electronic states are localized for any strength of disorder, however weak. Whereas in three-dimensions, there is a threshold disorder W_3c, only above which all the band states are localized. We consider a film of thickness b, with a disorder strength smaller than W_3c. With increasing thickness, one might expect a dimensional cross-over, so that film becomes conducting. We have examined questions that arise in this context by two techniques. The first is a finite size scaling technique due to Pichard and Sarma, in which the localization length is calculated numerically for strips and bars as a function of their lateral dimensions. Using this technique we study the delocalization of states at the band centre, as the thickness of the film is increased for moderate to strong disorders. The second technique involves the incorporation of quantum corrections to conductivity by extending the weak scattering methods to films of finite thickness. The two techniques complement each other, as the first one is more suitable for strong disorder, while the latter for weak disorder.