Электроны Глава 4

Kaganov M I Chapter 4

М. И. Каганов


Electron energy spectrum of crystals

The influence of quantum mechanics is already felt in the combination of words “energy spectrum”. The word “spectrum” comes from the wave side of things and the word “energy” comes from the corpuscular side of things. The discussion to follow of the electron properties of crystals has several goals:

(a)    to understand why some crystals, the so-called conductors, conduct electric current well, while other crystals, insulators, conduct electric current poorly;

(b)    to explain the existence of materials with intermediate properties: semiconductors, semimetals;

(c)    to analyze, using a number of examples, how quantum mechanics and quantum statistics explain the behaviour observed in experiment.

Our exposition of the electron properties of solids will not follow the historical evolution of the relevant concepts (just as we did not follow it in the preceding chapters). We shall try to describe the electron properties of solids in terms

of today’s concepts. As mathematical techniques are taboo in this book, we shall resort to analogy, intuition’s assistant. We must confess that this chapter will require more effort on the part of the reader than the preceding chapters.

An electron in a crystal is in a force field created by ions which are arranged in space periodically. This phrase is already a “quantum” phrase. A classical electron is found either in the field of one ion or in the field of another ion. A quantum electron, however, does not fear potential barriers and can move from ion to ion with relative freedom. As far as this property is concerned (in scientific language, delocalization) an electron in a crystal resembles a free electron. The energy of a free electron, e, is a function of momentum 1 p. In order to avoid the mathematical techniques of quantum mechanics, we will use the de Broglie relation in its initial form: Hk — = p, where k is, as before, the wave vector. When we were speaking about the structure of crystals at the beginning of the preceding chapter, we emphasized that a periodic crystal “creates” reciprocal space (k-space) that is also periodic. Then, in the chapter about phonons (p. 133-4), we transformed it into the periodic p-space. All quantities dependent on momentum are periodic functions of p. Therefore, the energy of an electron in a crystal, e, is a periodic function of p.