Kohn's Theory Research Articles

Conduction in non-crystalline systems

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.

A Theory of the Magneto-Optical Absorption

A theory of the magneto-optical absorption is developed by means of the Luttinger and Kohn theory of the effect of magnetic field on the energy bands in the case of a simple band model. The main purpose is to study the absorption line shape. As the width of the ab­ sorption line we get a value which agrees with experiment. It does not exhibit appreciable dependence on the magnetic field and temperature. The absorption coefficient in the absence of a magnetic field is also discussed. Recently, optical absorption experiments in semi-conductors have been performed in a strong magnetic field. These results show that absorption edges shift to shorter wave-length side in proportion to the magnetic field and the absorption spectrum shows an oscillation which is not observed in the absence of a magnetic field. These facts can be interpreted as the optical absorption which is accompanied with transitions of electrons between Landau levels of the valence band and of the conduction band. This is the oscillatory magneto-optical absorption effect_ll In this effect we are interested in i) positions of absorption peaks, ii) selec­ tion rules and iii) absorption line shapes, which are determined by level scheme, symmetry of band structure and scattering mechanism of carriers. Elliott et al,2l discussed the problems i) and ii). In this paper the electron-phonon interaction is taken into account and the absorption line shape is discussed as the main prob­ lem. We aim to gain some information about the relation between absorption line shapes and scattering mechanisms in the case of a strong magnetic field. In order to avoid the complexity, treatment is done for a simple spherical band model. In § 2, we shall make general considerations. In § 3, we shall discuss the absorp­ tion coefficient by. taking account of phonon effects. We will give numerical ex­ amples and discussions in § 4.