Conduction in non-crystalline systems. X. Mobility and percolation edges

Abstract

A distinction is made between a mobility edge, which is a quantum-mechanical phenomenon predicted for non-crystalline conductors, and a percolation edge which occurs in classical theory as a consequence of long-range fluctuations in the degree of disorder. The mobility edge is an energy, denoted by E _c, at which the conductivity σ(E) at zero temperature of a degenerate electron gas with Fermi energy E jumps discontinuously from zero to a value σ_min (the minimum metallic conductivity); for energies above a percolation edge E _p there is no discontinuity and σ varies as (E-E _p)1·6 For wells of random depth (the Anderson model) σ_min is given by the author's formula (∼ 0·025 e2/ka), the constant however depending strongly on the Anderson localization parameter. When there are long-range fluctuations, as for instance in an inhomogeneous glass, then as soon as tunnelling is allowed there will still be a discontinuity in σ it will however be much smaller than the value given above and will occur at a value of E near to E _p. These conclusions differ from those put forward by Cohen and more recently by Cohen and Jortner, who state that a minimum metallic conductivity does not exist in any non-crystalline substance. Cohen and Jortner have proposed also that in certain liquids fluctuations in density or composition, unrelated to the proximity of critical points, extend over large enough distances to allow the use of classical percolation theory. We find that this hypothesis does not depend on the truth or otherwise of their views on a minimum metallic conductivity, though we give reasons for doubting its validity for some of the liquids discussed by them. We discuss also liquids near critical points where long-range fluctuations certainly exist. A distinction is made between the critical poiAnt resulting from the liquid-vapour transition, as in mercury, and that resulting from metal-insulator transitions of Mott-Hubbard type, as in caesium, liquid salts such as KI-K, and metal-ammonia solutions.