Conduction in non-crystalline systems: II. The metal-insulator transition in a random array of centres

Abstract

The metal-insulator transition for donors in a semiconductor is examined in the light of the work of Kohn on the nature of the transition. Kohn's work predicts, for a rigid crystalline lattice, an infinite series of second-order transitions as the interatomic distance approaches the transition point. The original work of Mott on the transition suggested that it should be of the first order with a discontinuous change in the number of free electrons at T=0, unless the dielectric constant k could be shown to tend to infinity at the transition point. In this paper it is suggested that Kohn's theory may allow this to occur. The evidence from the behaviour of doped semiconductors is analysed; in this case the effect of the random (non-crystalline) distribution of the centres must be allowed for. It is suggested that Kohn's charge density waves must be replaced by random fluctuations of charge density, occurring near the transition point and due, as is the transition, to electron-electron interaction. The possibility that k should tend to infinity exists here as in Kohn's model. It appears therefore that the random distribution of the centres does not change the nature of the transition in any essential way, and that it is second order in either case. The experimental work of Davis and Compton (1965) on the effect of compensation on the activation energies for conduction near the transition point in antimony-doped germanium is examined in the light of the model proposed. The most striking result is that the concentration of centres at which the transition occurs is almost independent of compensation. An explanation of this is suggested. Another unexpected result is that, in the metallic region where the concentration of centres is high, a small activation energy remains if the compensation K is large. It is suggested that in this case the one-electron wave-functions of electrons near the Fermi energy are localized, due to strong scattering by the charged acceptors, and it is shown that this is bound to occur if K is large enough.