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
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.
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