'Energy' is a great name for a band*

^*Actually not really. It sounds cool but it would be impossible to look up on google, and  anyway I would guess it's already being used. But I don't know cause it's hard to look up on google. 

Where was I? Ah, yes: there's a fascinating relationship between the microchips that allow you to read these words and the colorful patterns on the wings of a butterfly.

The original idea of this post was to explain what I'm working on, at least qualitatively. I thought a good introduction to that could be the wings of a butterfly, but then I realized an even better starting point would be the unlikely relationship that they share with microprocessors, so I thought I'd start off there... and then I finished the post and never got to talking about what I'm working on, and neither did I get to the butterfly. But I will get to both, soon.

Anyhow, hopefully I got you interested. Good. Now I'll go into the boring stuff. Semiconductors seem to be boring. Honestly, even I was bored when I first had to learn the details. Think about it - do you even have an idea what a semiconductor is? My observations are that the majority of scientifically interested people (not physicists, though), who have an idea about quantum mechanics and relativity and superconductivity and dark matter and the Higgs boson, have a hard time defining what a semiconductor is, and, consequently, why they are the backbone of practically any gadget around us.  Occasionally someone would know that a semiconductor is a material with conductivity between that of a conductor and that of an insulator, but a fairly intelligent person can practically read that off their name. Essentially, they're not quantum-black-hole-God-particle-wormhole material, they’re boring stuff... which has meanwhile had more impact on our lives than literally anything else in literally the whole history of humanity. So I'll dare try to lay out the basics here - please bear with me if you start yawning.

Let's start all the way back: throwing a ball. Or a potato. When you throw one (or the other), there is a fixed relationship between its energy and its momentum. The harder you throw it, both of those increase, but the energy increases with the velocity v squared (E = mv2/2), while the momentum scales just with v (p = mv). So, if we were to draw a graph of energy versus momentum, it would look something like that. 

This holds true for any 'classical' (= large enough) moving object, but it turns out that even with quantum mechanics taken into account, the energy-momentum graph for massive elementary particles still looks like the one above! If you're ever asked to name one similarity between an electron and a potato, you're welcome. No, really, if you do get that question and say that energy is proportional to the momentum squared, you'll make an impression.

By the way in line with my previous posts, do notice that this does not pre-suppose any particle nature of the electron - energy and momentum can be defined independently of that. However, in both cases the graph above only holds true when we consider free propagation. If there is interaction with the surrounding world, the graph could change significantly. In most situations of practical interest, this is straightforward to take into account for potatoes and other objects from the classical lot, but elementary particles become much more tricky when interactions are turned on. To make matters worse, electrons love to interact with their environment, because they are charged particles, and there are usually other charged particles around them (including, but not limited to, other electrons). This makes characterizing the way electrons move in a given material an extremely difficult task. On the other hand, it is arguably the most important characterization we could do, since a large number of a material’s physical, chemical, optical, and electrical properties can at least in principle be extracted on this basis. And so, this study, usually referred to as solid-state physics (the term is a bit broader than just the motion of electrons, but that's the gist of it), has produced an astonishing number of PhD theses.

Now, in all crystals, which is to say materials with some structure in them, you have a lattice of ions through which the electrons propagate. This of course modifies the Energy-momentum graph, and it starts looking somewhat like this.

The jump discontinuities occur at exactly the same intervals, and so the graph above is usually drawn in the "folded" version, like this

It is easy (well, for a physicist, at least) to know where to 'fold' the graph - the fact that the crystal lattice is periodic ensures the periodicity of the 'special points' on the momentum axis, and you fold at the first one of those. This often has some deep physical significance, but anyway, you could as well just think of it as a more compact representation of the full plot.

Hope you're still with me here – we're getting to the point where you'll understand the title of the post. The energy intervals within which an energy-momentum graph exists are called 'energy bands', cause, well, they kinda do look like bands when you highlight them:

How much of the energy spectrum is covered by these bands, and in what way, determines a lot of the properties of the material. In fact, perhaps even more importantly – although technically carrying the same information – are the regions outside the energy bands, which are called band gaps:

The significance of those is a fairly bizarre one: electrons of that energy simply cannot exist in the material! That's now quite different from throwing a potato – you can throw it with any speed you like, and it will propagate with the corresponding kinetic energy – there are no forbidden energies. That's obviously not always the case for an electron in a crystal. Now, remember that electrica current is nothing else but motion of electrons. Because of this, the energy bands - and the presence of band gaps - fully determine whether a material is a conductor, a semiconductor, or an isolator. To see this, we need to add the last piece to the story, which is the fact that in every material there are a number of electrons that could move around, if 'pushed' (which in physical terms is done by applying voltage). If they're not pushed, they occupy all the lowest energy 'states' – this is because nature in general is quite lazy, and does everything with the lowest possible amount of energy, unless it has a very good reason to do otherwise. Now, every material can be characterized by a certain number called the Fermi energy, whose significance is that there are electrons with all energies below that level (and none above, at least not at 0 temperature). This together with the 'band structure' of the material determine its electrical properties in the following way. 

In brief, if the Fermi level crosses a band, then arbitrarily small voltage makes the electrons move, i.e. creates current, which the material 'conducts'. If the Fermi level is in a band gap, the applied voltage needs to be large enough so that the difference in energy between the Fermi level and the closest higher-energy band can be overcome. Thus, technically, there is no difference between a semiconductor and an isolator: it is in principle possible for both of them to conduct electricity, but only if enough voltage is applied. What makes a material an insulator is then only the practical consideration that an impossibly high amount of voltage is needed to make its electrons move in a certain direction, while for a semiconductor that value is achievable. 

Now, another important aspect of semiconductors which ultimately renders them useful for CPUs is that they can be doped

The red dots in the image above represent 'impurities' which are artificially introduced to make the semiconductor either more or less conductive (we can do both). I'll skip the rest of the details: the important point is that by making contacts ('junctions') between differently doped semiconductor pieces, you have a lot of non-trivial control on the way current flows through a device. Two examples are diodes (the D in LED) and transistors. The latter, very schematically, work in the following way

Essentially, the transistor is a 'switch': if there is no current at the 'gate', the switch is off, and no current can pass from the source to the drain. The switch can be turned on by current flowing at the gate. And that's all you need to make a logic circuit! No current is your 0, yes current is your 1. You connect the drain of one transistor to the gate of another, and your 0 or 1 influences the next 0 or 1. Depending on how you make the connections, there's no end to the possibilities of the allowed operations that you can hard-wire in a chip. Of course you only go for some basic ones, and then leave it to a programer to play around in arranging those in ever-more-complicated patterns, so that you can all enjoy your cat pictures and your instant connectivity with anyone anywhere around the world and your daily dose of videos of naked people doing dirty stuff and reading this post and God knows what else you're into that's only made possible because of what I explained here.

Hope it wasn't so boring after all.