Conduction in non-Crystalline systems

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Abstract
The present author (1967, 1968) has used the work of Anderson (1958) to deduce that under certain conditions the conductivity <σ> due to a degenerate gas of electrons in a disordered lattice tends to zero with temperature, even though the density of states N(EF ) at the Fermi energy EF is finite. Neither Anderson's conclusions nor those of the present author have been universally accepted, and this paper examines in detail the method of Mott (1968) which shows that, for a very disordered Anderson lattice, the conductivity <σ(ω)> at frequency ω behaves (apart from a logarithmic term) like ω2 for small values of ω, and therefore tends to zero with ω. The method is the laborious one of examining all configurations of the ensemble and showing that any non-zero contributions to <σ(0)> tend to zero exponentially as the volume ω of the specimen considered tends to infinity. Some discussion is given of the energy Ec which separates localized from non-localized states and of the behaviour of <σ(0)> for values of EF near to this value; a numerical value is deduced and compared with experiments. Applications to semiconductors are discussed.