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 єi as the reduced phase at the point xi immediately to the left of the ith atomic potential, it is shown for all bounded p(s) and sufficiently high electron energies that the єi 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.