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 φ(E) is introduced which specifies the fraction of space satisfying $V<E$. Extended states appear above a critical density φc; this provides our working criterion φ(Ec)=φc for the location of the percolation threshold Ec. In one dimension φc=1, and in two dimensions we find that φc=1/2 for an important class of random potentials. In three dimensions we obtain an estimate of φc=0.15 from an empirical rule. The percolation criterion for the location of the mobility edge Ec 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 Ec 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.