The nature of the electronic states in disordered one-dimensional systems

Abstract

We consider an electron moving in the field of a one-dimensional infinite chain of identical potentials separated by regions of zero potential, the lengths s of these regions being distributed according to a probability density function p(8) . If we define the reduced phase of a real solution of the wave equation as the principal value of arctan ( — ψ'/kψ ) and є i as the reduced phase at the point x i immediately to the left of the i th atomic potential, it is shown for all bounded p(s) and sufficiently high electron energies that the є i are distributed according to a probability density function which depends on the direction of integration from a specified homogeneous boundary condition. This result is shown to imply that the eigenfunctions for such systems are localized in the sense that the envelope of such a function decays on average in an exponential manner on either side of some region. An analytical calculation for a random chain of δ-functions gives the decay of the nodes explicitly for high energies, and numerical calculations of the decay for a liquid model are presented. Further support for the theory is provided by computer calculations of some of the eigenfunctions of a chain of 1000 randomly placed δ-functions.