ZONE BOUNDARY EFFECT IN SIMPLE METALS

Abstract

The electrons near the Fermi energy play a signi-ficant role in determining the low temperature behaviour of metals primarily because the electrons which contribute to the transport properties lie within the range ± kBT of the Fermi energy EF. In order to calculate the low tempera-ture properties, it is essential to know the band structure near EF,i.e.,the Fermi surface. Since kBT is almost zero, phonon effects are not important. The various transport properties as density of states, optical mass, Hall coeffi-cient and thermoelectric power etc. are well known for a spherical Fermi surface and can be found in any text book. When the Fermi surface shows distortion from the spherical shape, things become more difficult and more interesting.. For example the transport properties show an anisotropy. The anisotropy of the Fermi surface is due to the anisotro- py of the pse ,ce / ntial seen by the conduction electrons. This, in t.--7,V depends on the crystal structure. :Most two decades ago Ziman reported extensive calculations of various electronic properties of noble metals and their alloys by using an eight-cone model where the Fermi surface is almost spherical except along the <111> directions. Huntington and his coworkers have also studied electromigration, point defect scattering etc.. . using the nearly-free-electron model with special reference to zinc. Apart from these workers a number of other attempts have been made to see the effect of Fermi surface distortions on various electronic properties and these have been beautifully summarized by Havinga and van Marren. _Our motivation here is to determine how these properties change before the Bragg planes set in and after they have set in. In this thesis we have attempted to determine the effect of Fermi surface distortion induced by a pair of Bragg planes in the nearly-free-electron model, on some electronic properties. We feel that this is the simplest way to include Fermi surface anisotropy. In the nearly-free-electron model there are three regions of interest depend- 2 ing on the energy E o (i) E < - V; where G is the reciprocal lattice vector and V is the pseudopOtential. In this region the Fermi surface is free:-electron like. Alkali G2 (12 metals come in this region, (ii) 1- - V < E<4 + V ; the Fermi surface opens up in the direction of the Bragg planes. 2 Noble metals show this kind of behaviour, (iii) E > 3- + V; there are two sheets of the Fermi surface (a) same open sheet as in (ii) (monster) and (b) a closed sheet (lens). The nearly-free-electron metals Zn, Cd, Mg come in this region. Our motivation in choosing this model is•to see the change that- manifest when we go from the region (1) to the region (iii). None of the earlier workers have done this. In fact this is the main motivation of our work. In Chapter I of our work, we introduce the subject and review some of the earlier work in this field. We also discuss the present status of the problem and the various efforts made to study the electronic properties. The Fermi surface model, which is used in this thesis, is introduced and discussed here. In Chapter II, we have studied the influence of Bragg planes on simple properties such as density of states, optical mass, and the Fermi surface average of velocity squared. Although density of states N(E) has been studied by other workers, our purpose here is to set the pace and also to check our calculations by comparison with their work. We have calculated N(E) for different values of the pseudopotential parameter V. We see that our results for N(E) are in agreement with the results obtained by other workers. Then we report calculations giving by the effect of Bragg planes on the optical mass mopt. There have been two earlier calculations of mopt by Ziman and by Ashcroft and Sturm but none of these give the effect in going from (i) to (iii). Here we obtain analyti-cal expressions of mopt for the entire energy range. We seethatmoptstarts from a large value at E = 0 and G2 decreases rapidly with increasing E till E = -V. Here there is a sharp drop in mopt. Further increasing E, mopt decreas-es slowly with E. A quantity that is very important in transport theory is the average of the square of the Fermi velocity < v2 > over the Fermi surface. We have calculated G2 this and find that < v2 > has a sharp drop at E = 4 - V G2 and a smaller drop at E = -4- + V. The changes in all these G2 quantities at E = + V are explained in terms of changes in the Fermi surface topology.