13 - Electrons: band theory

Summary

Introduction

In the last chapter, we took a brash and somewhat unrealistic approach to treating the motion of electrons in a crystal. Although we know that the electron travels through a periodic potential caused by the regular arrangement of ion cores, we disregarded this “bumpy terrain” and considered instead only the barest consequences of the electron being trapped in the crystal “box” as a whole. In spite of its simplicity, this free electron model provided insightful explanations, not only for the origin of the small electronic contribution to specific heat and the temperature dependence of the electrical resistivity, but also for a host of emission phenomena, including the photoelectric effect.

However, the free electron model fails to provide any insight into additional questions regarding electrical conduction, such as (1) the anomaly of positive Hall coefficients that would imply positive charge carriers, and (2) the peculiar pattern of conductors, insulators and semiconductors that is found in the periodic table. In this chapter, we examine the nearly free electron model as a natural extension in which a weak, periodic potential is introduced. As a direct consequence of this addition, the continuum of electron energies in our free electron model now becomes separated into bands of allowed electron energy, separated by disallowed energy gaps. This separation of the electron energy into bands and gaps is key to understanding the division of materials into conductors, insulators and semiconductors, as well as providing a natural interpretation for the positive Hall coefficients.