Percolation on a Continuum and the Localization-Delocalization Transition in Amorphous Semiconductors

The concept of percolation has been set in a form which is more directly germane than the existing theory of percolation on lattices to the question of the localization-delocalization transition (mobility edge) in amorphous semiconductors. The problem of percolation on a continuum has been formulated in the context of motion in a random potential V (r). An energy-dependent dimensionless density (E) is introduced which specifies the fraction of space satisfying V<E. Extended states appear above a critical density ₂; this provides our working criterion (E₂) =₂ for the location of the percolation threshold E₂. In one dimension ₂=1, and in two dimensions we find that ₂=12 for an important class of random potentials. In three dimensions we obtain an estimate of ₂=0. 15 from an empirical rule. The percolation criterion for the location of the mobility edge E₂ has been applied to several types of disordered potentials. The oft-invoked Gaussian potential distribution has been treated and the results compared with those of several recent calculations. Random-walk techniques can be used to attack more general random potentials; we have used this approach to explicity calculate E₂ for the potential of an array of random dipoles, which is an initial model for an amorphous molecular solid. A way of including short-range order is briefly discussed.