Localized Electron Eigenstates in One-dimensional Liquids

The electronic structure of liquid insulators

The electronic structure of disordered systems is analyzed in the case of non-interacting electrons tightly bound to their ions. Particular attention is paid to the behaviour of the band gap in the presence of disorder. The Kronig-Penney and tight-binding solutions are derived for perfect lattices. With the physically reasonable restriction that the ions do not approach closer to each other than a certain distance, the band gap is discussed in successively more difficult cases. The closure of the gap between neighbouring levels in one dimension depends on the relative parity of the two levels: if the parities are the same the gap cannot close. The hardest case discussed, that of neighbouring -levels in three dimensions, gives the same result. A detailed discussion of the approximations is given.

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

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.