Remember how I said I'd talk about 'majority carrier transport' and 'minority carrier transport' this time?

Turns out, I lied.

'Majority carriers' and 'minority carriers' are terms that only make sense in the context of doped semiconductors, and to understand doped semiconductors you need to understand regular semiconductors.

To understand that you need to know what 'conduction' means, and from there it's just turtles all the way down.

So now, in the first of a series of digressions, I give you:

# Semiconductors physics, as painlessly as I can

November 3, 2012 11:32:40 PM CDT

You probably already know that atoms are made of protons, neutrons, and electrons. The protons and neutrons live in the nucleus. They're heavy (accounting for about 99.95% of the mass in the atom), and small (occupying about a billionth of the volume). The electrons orbit the nucleus. They're light (1/1800th the mass of a proton), and spread out over a large space compared to the nucleus.

You might know that electrons arrange themselves in layers called 'shells', and that the outermost shell of electrons is all we ever really interact with here in the macroscopic world.

If you had high school chemistry, you know that electron shells are composed of 'orbitals', and that it takes a certain amount of energy to keep an electron in a given orbital. Larger orbitals take the electron farther away from the nucleus, and require more energy.

If you learned basic quantum physics, you know that orbitals are more like categories of shapes than actual locations.. 's' orbitals are spherical, 'p' orbitals look like two spheres being pushed together and trying hard to avoid touching each other. 'd' and 'f' orbitals look like someone got a physicist drunk and taught him to tie balloon animals.

There are infinitely many versions of any given orbital, each corresponding to the electron having a certain amount of energy -- a low energy s-orbital would be a small sphere, a higher energy one would be a larger sphere. As electrons absorb energy their orbitals get bigger, as they release energy their orbitals get smaller.

There's something weird about that though: no matter how much energy an electron gains or loses, it's always an integer multiple of the same amount.. a value called "Planck's constant" which describes the energy of a photon.

### The photon.. nature's way of saying "screw you" to physicists:

Photons are one of the biggest headaches in physics. They refuse to make sense. You can build one experiment to prove photons must be waves and can't be particles. You can build another to prove that photons must be particles and can't be waves.

Both will work.

Eventually physicists gave up trying to understand the results and just accepted the fact that both experiments do work, and went on with their math. They built new theories around the idea that this minimum unit of energy -- this 'quantum' -- exists and does interesting things.

One of the interesting things it does is control electron orbitals.

Electrons gain and lose energy by absorbing or emitting photons, so the amount of energy they do gain or lose will always be some integer multiple of Planck's constant. An electron's orbital depends on the amount of energy it has, so that means the orbitals themselves come in distinct bundles.. in the jargon, they're 'quantized'.

It's easy enough to understand the idea of electrons moving from one orbital to another, the mind-blowing part is the idea that they never cross through the space in between. An electron with energy 1 lives in orbital 1. If it gains a quantum of energy, it lives in orbital 2.. but since energy is quantized it's impossible for it to ever have passed through a position that would correspond to 'orbital 1-1/2'. It just ceases to exist in orbital 1 and starts existing in orbital 2 like a bad special effect in a movie.

That process is called a 'quantum leap'.

### The point of all this:

There are lots of interesting ideas that fall out of quantized energy levels, but two relate directly to diodes:

Fermi energy levels: the lowest energy level for any orbital is called the 'Fermi level'. It's where the electron will end up if it emits every photon it possibly can. If an isolated atom's electrons are all at their Fermi levels, you just can't get any more energy out of the atom without breaking it down into subatomic particles and turning it into something else. That's the 'absolute zero' state for a single atom.

Orbital overlap: technically you can add an unlimited amount of energy to an electron and still have it stay in the same family of orbitals.. it's just really unlikely. Sooner or later you'll pump enough energy into the electron in an 's' orbital that it passes the Fermi level for a 'p' orbital. At that point it's easier for the electron to exist in the 'p' orbital, so that's where it goes.

### Molecular orbitals:

Now, when you put a bunch of molecules close together, it becomes possible for the electrons to skip from one nucleus to another. The basic rules are pretty much the same as for the overlap between orbitals in a single atom, but this time the new orbital belongs to another atom. You have to move the electron pretty far away from the nucleus for that to happen though, and we know that larger orbitals require more energy.

Some orbitals just don't enough energy for the electron to escape the nucleus. Electrons in those orbitals are said to be 'localized'.. they never leave their original atom.

Other orbitals do have enough energy for the electrons to leave. The electrons in those orbitals are called 'non-localized'.

Sometimes it takes less energy for a pair of electrons to hop back and forth between two nucleii than to spend all their time orbiting a single nucleus. Electrons which do that are called a 'covalent bond' between the two atoms.

Sometimes it takes less energy for electrons to keep hopping from one nucleus to the next than it does for them to stay close to any specific nucleus. Those electrons form what's called a 'metallic bond'. The shiny surface you see in metals is a cloud of permanently non-localized electrons.

Given that molecular orbitals give electrons new ways to exist, you can define a Fermi level for a material. The electrons in metal don't suddenly localize to a single nucleus at absolute zero.. they continue to skip from one atom to the next, but they do it in such a way that you can't remove any more energy from the material.

### Enter Wolfgang Pauli:

This is where things start to get weird in a heavily mathematical way.

Protons, neutrons, and electrons obey what's called the 'Fermi-Dirac statistical model', which describes each particle with four numbers that I won't even try to explain. In a way, those numbers are the particle, and the 'Pauli exclusion principle' says no two particles can have the same values in all four places.. that would be like having two particles worth of energy trying to exist in the space where a single particle should be.

Those four values describe the particle's energy and orbital, and when you apply the exclusion principle to molecular orbitals, something interesting falls out:

### Energy bands:

Most molecular orbitals involve two electrons trading places between atoms, each taking the place of the other. For that to work, their energy levels have to be similar. For some orbitals in some materials, the math is such that two electrons sharing that orbital would have exactly the same four descriptive values.

The Pauli exclusion principle says that can't happen, so electrons don't go into those orbitals. The range of energy levels where that happens is called a 'forbidden band', or (since it creates a gap in the energy levels the electrons can have) a 'bandgap'.

The upshot is that you get groups of orbitals where it's easy for electrons to hop back and forth between energy levels, and gaps between those groups where electrons can't exist at all. To cross a bandgap takes another (larger) quantum leap.

### Valence and conduction bands:

For my purposes, the most interesting bandgaps happen in the range where electrons start having enough energy to leave their nucleus.

Electrons below the bandgap don't have enough energy to escape the pull of the nucleus, so they stay localized. The energy levels they can occupy are called the 'valence band'. 'Valence' is an old word for the perceived strength of an atom, which we now know to be based on the number of electrons it has in its outer shell.

Electrons above the bandgap have enough energy to become non-localized and move from one atom to the next. The energy levels they occupy are called the 'conduction band'.

As you can probably guess, the conduction band is where the phenomenon we call 'electricity' happens.

### Types of materials:

The relative positions of the valence and conduction bands tell you how a given substance will behave electrically.

Insulators have a really wide gap between the valence and conduction bands. It's 'statistically unlikely' for electrons to hop that that high on their own (about the same probability as a glass of water spontaneously dividing itself into a chunk of ice and a cloud of steam), and it can take several hundred or thousand volts of external influence to pull them across the gap.

Semiconductors are technically insulators, but their bandgap is small.. only a handful of volts. If you leave them alone, their electrons stay localized, but you can pop their electrons up into the conduction band by doing something as simple as shining a light on them. That's the photoelectric effect, and Albert Einstein was the first person to figure out what was going on.

Conductors have a negative bandgap. If you pull a single atom off by itself, the Fermi levels of its outer electrons will be fairly high. If you put a bunch of atoms together, there are molecular orbitals that take less energy than the highest Fermi levels for the single atom. The material ends up being a sea of metallically bonded electrons with atoms floating among them. It's extremely easy for the electrons in metals to pass energy back and forth, so metals conduct both heat and electricity easily.

### Silicon:

Silicon is a semiconductor. The bandgap between its valence and conduction bands is about 1.2v at absolute zero, but silicon happens to be arranged so that a combination of thermal and electrical energy can bump electrons up to the conduction band. At room temperature you only need about 1.1v to push electrons up into the conduction band.

That's still more than we'd really like, and it turns out there are ways we can lower the gap even further.

I'll talk about those next time.