Band Theory of Solids


Introduction
This is a special section of Understanding the Properties of Matter not included in the published text.

It is special because it is an almost entirely theoretical look at solids. As such, it has a distinct flavour from the rest of the book where I have focused on experimental properties, and then developed theories only to the extent that they are needed to describe the experimental data.

My motivation in making this additional section available is that the problem they address is central to understanding what “really happens” inside solids. This central problem is the description of the nature of electronic states in solids.

The challenge

The challenge can be highlighted by considering two elements: argon and potassium. As solids we know from §6.2 and §7.3, that argon is described as a molecular solid. However, when we describe potassium, we would naturally discount the theory for argon as irrelevant, and adopt the description of free-electron gas described in §6.5. This may seem obvious: after all, one is a metal and the other an insulator. Well the essential question is: “Why is it obvious?”

Argon and potassium atoms differ by only a single electron in their outer shells. In describing the internal electronic structure of their atoms, we used the same “language” to describe both argon and potassium. What we would like is a unified picture, which will describe the nature of electronic states in all solids. From this we should be able to deduce that argon is an insulator and potassium is a metal. This is the challenge.

The problem

The problem with this approach is that the electronic states in solids are not simple. Some states are like the corresponding states on the isolated atoms, and some appear to be like free-electron states i.e. plane waves. Most states, at least most of the valence states, in solids are in a state somewhere between these two extremes, and the challenge is to find a way to describe such states.

Now the solution of this problem has “been known” for nearly half a century. I use the term “known” in quotation marks because it is not at all clear to me who it is who has been doing the “knowing”. I say this, because physical descriptions of the nature of electronic states are extremely rare, but dense mathematical tomes are plentiful. In this exposition, It is my intention to keep the mathematical complexities to an absolute minimum. However, the level of mathematical ability required to follow the arguments with regard to the tight-binding theory is definitely higher than for the topics in the main text.