The localization of electrons in ordered and disordered systems III. Quantum-mechanical particles in `free' bands

Abstract

At energies above the lattice minimum electrons can only be localized by coherent wave-mechanical diffraction from the assembly of atoms. In crystalline semiconductors, for example, this gives rise to band gaps containing impurity levels. What features of the arrangement of atoms are necessary to produce such effects? Can band gaps be produced by diffraction from a structure without long-range order? The linear chain model is useless, because it cannot be topologically disordered. We suppose only that we know the successive probability distributions for configurations of 2, 3, 4, etc., atoms. As shown by Edwards, these give rise to successive terms, Σ₂, Σ₃, Σ₄, etc., in the perturbation series for the self-energy correction Σ(, k) in the average Green function. These terms are rederived and expressed rather compactly by the method of cumulants. Assuming directional averaging over many crystallites, we find that we must go to Σ₄ to produce the known band gaps even in a coarsely polycrystalline specimen. In any system without long-range order, the four-body distribution must have short-range orientational order - a tendency for the atoms to lie in chains, squares, tetrahedra, etc. - to make similar contributions to Σ. The Bernal liquid does not have this property; in other close-packed structures, local orientational correlations merely imply polycrystalline order. But the `tetrahedral glass' model of amorphous germanium, which is topologically disordered, nearly preserves relative bond angles, and hence may still give rise to gaps in the electron spectrum.