# eli5: Band-gaps, carrier electrons, electron holes

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Hello!
Could someone explain electron holes to me like I’m five? Are they a concrete positive charge, or a mere absence of an electron? I have a lot of question marks in my head regarding this concept. I would appreciate resources that help to understand it too.
Thank you!

Tldr: what’s an electron hole?

In: Chemistry

An electron hole is not a real particle, it represents an empty state where an electron is supposed to be. Since you are removing a negative charge you can think of the empty state as having a positive charge.

An electron hole is conceptually, an anti-electron (without being a proton). Or, a space an electron could occupy, that when under an electron flow, moves in the opposite direction.

Imagine if you will, a line of people. The people can only move forward if there is a gap in the line in front of them. So, person 1 steps into that gap, leaving a gap behind them. The person behind them steps forward to fill the recently open gap, and another new gap opens up behind *them*. From an outside observer, this would look like the “gap” in the line is moving backwards, while the line of people moves forwards.

In a lattice you can describe electron states with the Bloch equation. Typically in quantum mechanics its difficult to solve the full Schrödinger equation for a given porblem. The equation is made of two main parts a kinetic and potential energy term and what you can do is to solve for just the kinetic part adhering to the periodic nature of the lattice structure. (You introduce some constraint like the wavefunction needs to be periodic for lattice elements.) This is the free solution and if you plot energy states with respect to k (which is the reciprocal lattice constant, its a useful quality whilst using the regular lattice constant can be problematic, similar to the difference between wavelength and wavenumber vector the latter is often more convenient) you get parabolas. It looks something like [this](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRYnRwBgtxnexj2YLit0V-SKb_hVIyjhpI8EJ04DmvAiA&s) except they are actually parabolas and touch.

Once you have the free solution you can pertube it with a small potential and what happens as you can see it on the figure, these bands split and you’ll end up with gaps between energy states. Given that we have electrons you can only put 2 of them into a given energy state due to the Pauli exclusion principle. (The lines on the figure aren’t continuous, you can fit a certain amount of electrons in each band.) In order to knock electrons to higher bands you need energy. Lets look at a material where the lower band is filled, its an insulator, as you can see you need a lot of energy to get things to a higher band. Its slightly more complete than this but basically there is a band up to which the electrons sit in peace called the valence band and above that electrons can move freely hence the name conductance band. (Just cause electrons jump to a higher band doesn’t automatically mean they can be moved but lets not complicate things too much.)

So when a valence band is filled and next up is a huge gap you have an insulator. Say the third or fourth band is filled half way. So you can push electrons to different energy states with minimal energy and so you have a good conductor. But say we have things filled to the second band, the gap to the third is small and if so you call that a semiconductor. On 0 K they wouldn’t conduct but a bit of heat is enough to push electrons into the conductance band above.

Without going to much into the details you can introduce contamination into a semiconductor like silicone. A silicone atom has 4 valence electrons with which it can form a lattice with other atoms. These bonds aren’t to hefty to break hence the small gap. But say you want the silicone to conduct, you can dope it with an element that has 5 valence electrons, thats +1 electron with no pair so isn’t not bound in the lattice thats much, it need even less energy to jump to the conductance band. You have enough of these extra electrons from you contaminants, you connect some voltage across the silicone and now it will conduct. But then you can also contaminate with something that has 3 valence electrons and as you can imagine this can compensate the other type of contamination so its now basically like pure silicone. Of course the funny part is that you can play with local contamination making different regions and build circus with that.

If you want to describe how silicone conducts wih contaminants in mind you don’t need two very different equation to describe the electrons as the negative free charge carries and holes to which electrons can jump. Electrons jumping into holes is like the hole is moving in the opposite direction. It acts as a positive charge. Say the electrons flow in the positive direction than the holes move in the negative direction and we know how negative charge flowing in one direction is equivalent to positive charge flowing in the other direction. So all you need is to set the charge of the hole to +e and you can use the same expressions for the number of charge carries as with electrons.

Holes are an absence of an electron.

I’m not sure how simple I can make it, but here’s a try.

To conduct electricity, electrons have to be able to move freely.

You might be familiar with the concept that electrons sit on specific energy levels and cannot share them. This means every electron has a place. The only electrons that can participate in anything, chemistry or conduction, are only the very outermost ones, called Valence electrons.

When you have enough atoms in a material, you can start sort of mushing everything together and stop thinking about individual atoms. So now imagine that all these valence electrons more like balls on a shelf.

Now, in conductors like most metals, these electrons are already free to move. That’s just how metals work. So they just slide around their shelf freely. Call that the conduction band

But in other materials, they’re still sitting in their designated spaces. Think of it as having a cup they go into on their shelf. Call that the valence band. They can move from cup to cup, but only if there’s an empty one next to them.

Now, above the valence band you’ll still have the shelf of the condition band. If given a kick, an electron can moved from its cup onto it and go ahead and slide around freely, conduct. Semi-conductors are materials where that gap between shelves is small enough to actually be practical to use.

Now, you moved the electron from the cup. You have an empty cup. It’s an absence of an electron, not a real thing. If you zoom in, you see that it’s just electrons moving around.

But when you zoom out, something interesting happens. It starts to seem like the hole is moving about the shelf, sort of like the electrons on the shelf above it. Like some sort of a actual particle. And you can even describe that behaviour with exactly the same math as you do for the electrons. When you do that, you have to give it positive charge, because compared to it’s surrounding it’s more positive by exactly one electron charge.

So while it doesn’t really exist, the way a hole left behind by an electron behaves and the properties it has make it very useful to pretend it’s a “thing”.